On the Two Point Boundary Value Problem for Systems of Linear ...

On the Two Point Boundary Value Problem for Systems ofLinear Generalized Differential Equations with SingularitiesMalkhaz AshordiaA. Razmadze Mathematical Institute of I. Javakhishvili Tbilisi State UniversitySukhumi State University, Tbilisi, GeorgiaE-mail: ashord@rmi.geMurman KvekveskiriSukhumi State University, Tbilisi, GeorgiaFor the system of linear generalized ordinary differential equations with singularitiesthe two point boundary value problemdx(t) = dA(t) · x(t) + df(t) for t ∈ ]a, b[ , (1)x i (a+) = 0 (i = 1, . . . , n 0 ), x i (b−) = 0 (i = n 0 + 1, . . . , n), (2)is considered, where −∞ < a < b < +∞, x(t) = (x i (t)) n i=1 , n 0 ∈ {1, . . . , n}, f = (f l ) n l=1 :[a, b] → R n is a vector-function whose components have bounded variations, and A = (a il ) n i,l=1 :[a, b] → R n×n is a matrix-function such that the functions a i1 , . . . , a il have bounded variationson every closed interval from ]a, b] for i ∈ {1, . . . , n 0 } and on every closed interval from [a, b[ fori ∈ {n 0 + 1, . . . , n}.There are established sufficient conditions for the unique solvability of this problem in the casewhen considered system is singular, i. e., the components of the matrix-function A may haveunbounded variation on the interval [a, b].By BV loc (]a, b[ , R n×m ) we denote the set of all matrix-functions X : ]a, b[ → R n×m with boundedvariation on every closed interval from ]a, b[ .By a solution of the problem (1), (2) we mean a vector-function x = (x i ) n i=1 ∈ BV loc(]a, b[ , R n )satisfying the condition (2) and the system (1), i.e., such that x(t) = x(s)+∫ tsdA(τ)·(τ)+f(t)−f(s)for a < s ≤ t < b, where integral is considered in the Lebesgue–Stieltjes sense.Let det(I n + (−1) j d j A(t)) ≠ 0 for t ∈ ]a, b[ (j = 1, 2) and let γ α (· , s) be a unique solution of theCauchy problem dγ(t) = γ(t) dα(t), γ(s) = 1.Definition. Let n 0 ∈{1, . . . , n}. We say that a matrix-function C =(c il ) n i,l=1 ∈BV([a, b], Rn×n )belongs to the set U(a+, b−; n 0 ) if the functions c il (i ≠ l; i, l = 1, . . . , n) are nondecreasing on [a, b]and the systemsgn(n 0 + 1 )n∑2 − i · dx i (t) ≤ x l (t) dc il (t) for t ∈ ]a, b[ (i = 1, . . . , n)has no nontrivial, nonnegative solution satisfying the condition (2).l=1Theorem. Let the vector-function f have bounded variation, and let the matrix-function A =(a il ) n i,l=1 ∈ BV loc(]a, b[ , R n×n ) be such that the conditions(s0 (a ii )(t) − s 0 (a ii )(s) ) sgn(n 0 + 1 )2 − i ≤ s 0 (c ii − α i )(t) − s 0 (c ii − α i )(s),(−1) j(∣ ∣ 1 + (−1) j d j a ii (t) ∣ ) (− 1 sgn n 0 + 1 )2 − i (≤ d j cii (t) − α i (t) ) ;∣ s0 (a il )(t) − s 0 (a il )(s) ∣ ≤ s0 (c il )(t) − s 0 (c il )(s) (i ≠ l),|d j a il (t)| ≤ d j c il (t) (i ≠ l)3

hold for a < s < t < b (i, l = 1, . . . , n), where C = (c il ) n i,l=1 ∈ U(a+, b−; n 0), α i : ]a, b[ → R(i = 1, . . . , n) are nondecreasing functions andlim d 1α i (t) < 1 (i = 1, . . . , n 0 ),t→a+lim d 2α i (t) < 1 (i = n 0 + 1, . . . , n 0 ); (3)t→b−lim sup { γ αi (t, a + 1/k) : k = 1, 2, . . . } = 0 (i = 1, . . . , n 0 ),t→a+lim sup { γ αi (t, b − 1/k) : k = 1, 2, . . . } = 0 (i = n 0 + 1, . . . , n 0 ).t→b−Then the problem (1), (2) has one and only one solution.Corollary. Let the vector-function f have bounded variation, and let the matrix-function A =(a il ) n i,l=1 ∈ BV loc(]a, b[, R n×n ) be such that the conditions(s0 (a ii )(t) − s 0 (a ii )(s) ) sgn(n 0 + 1 2 − i )≤ h ii(s0 (β)(t) − s 0 (β)(s) )− ( s 0 (α)(t) − s 0 (α)(s) ) ,(−1) j(∣ ∣ 1 + (−1) j d j a ii (t) ∣ ) (− 1 sgn n 0 + 1 )2 − i ≤ h ii d j β(t) − d j α(t) ) ,∣ s0 (a il )(t) − s 0 (a il )(s) ∣ ( ≤ hil s0 (β)(t) − s 0 (β)(s) (i ≠ l),|d j a il (t)| ≤ h il d j β(t) (i ≠ l)hold for a < s < t < b (i, l = 1, . . . , n), where α is a nondecreasing on ]a, b[ function satisfying theconditions (3) and (4); β is a nondecreasing on [a, b] function having no more than a finite numberof discontinuity points, h ii ∈ R, h il ∈ R + (i ≠ l; i, l = 1, . . . , n). Let, moreover, ρ 0 r(H) < 1, where(4)H = (h ik ) n i,k=1 , { 2∑ }ρ 0 = max λ mj : m = 0, 1, 2 , λ 00 = 2 (s0 (β)(b) − s 0 (β)(a) ) ,πj=0λ 0j = λ j0 = ( s 0 (β)(b) − s 0 (α)(a) ) 1 ( 2s j (β)(b) − s j (β)(a) ) 1 2(j = 1, 2),λ mj = 1 ( ) 1µαm µ 2 αj sin −1 π(m, j = 1, 2),24n αm + 2µ αm = max { d m α(t) : t ∈ [a, b] } (m = 1, 2),and n αm is a number of points from [a, b] for which d m α(t) ≠ 0 for every m ∈ {1, 2}. Then theproblem (1), (2) has one and only one solution.4

Uniform Estimates and Existence of Solution withPrescribed Domain to a Nonlinear Third-OrderDifferential EquationI. AstashovaLomonosov Moscow State University, Moscow State University of Economics, Statistics andInformatics, Moscow, RussiaE-mail: ast@diffiety.ac.ru1 IntroductionIn 1990, after my lecture at Enlarged Sessions of the Seminar of I.Vekua Institute of AppliedMathematics, professor T. A. Chanturia posed me a question about existence for any finite x ∗ , x ∗ ,x ∗ < x ∗ of a non-extensible solution y(x) with domain (x ∗ , x ∗ ) to the equation y ′′′ +p(x)|y| k−1 y = 0.For such equation of the second order some related result was obtained in [1]. I’ve got an answerfor the third-order equation in 1992 ([3]), but unfortunately T. A. Chanturia could not see it. . .The proof of this result was very complicated, but with the help of uniform estimates of solutions([8]) it became much better.2 Uniform Estimates of SolutionsConsider the differential equationy ′′′ + p(x, y, y ′ , y ′′ )|y| k−1 y = 0, k > 1, (1)the function p(x, y 0 , y 1 , y 2 ) is continuous in x and Lipschitz continuous in y 0 , y 1 , y 2 with0 < p ∗ ≤ p(x, y 0 , y 1 , y 2 ) ≤ p ∗ , (2)where p ∗ , p ∗ are positive constants.Put β = k−13> 0.Theorem 1. For any k > 1, p ∗ > 0, p ∗ > p ∗ , h > 0 there exists a constant C > 0 such that forany p(x, y, y ′ , y ′′ ) satisfying (2), any solution y(x) to (1) satisfying the condition |y(x 0 )| = h > 0in some point x 0 ∈ R cannot be extended to the interval (x 0 − C h −β , x 0 + C h −β ).Theorem 2. For any k > 1, p ∗ > 0, p ∗ > p ∗ there exists a constant C > 0 such that for anyp(x, y, y ′ , y ′′ ) satisfying (2) and any solution y(x) to (1) defined on [−a, a] it holds |y(0)| ≤ ( Ca ¯g)1/β i .Theorem 3. For any k > 1, p ∗ > 0, p ∗ > p ∗ there exists a constant C > 0 such that for anyp(x, y, y ′ , y ′′ ) satisfying (2) and any solution y(x) to (1) defined on [a, b] it holds|y(x)| ≤ C min(x − a, b − x) −1/β . (3)Remark 1. In [5] uniform estimates for positive solutions with the same domain to the equationn−1∑y (n) + a j (x)y (i) + p(x)|y| k−1 y = 0with continuous functions p(x) and a j (x), n ≥ 1, k > 1 were obtained.estimates for absolute values of all solutions to the equationwere proved.j=0n−1∑y (n) + a j (x)y (i) + p(x)|y| k = 0j=05In [6] similar uniform

3 Existence of Solution with Prescribed DomainConsider the differential equation (1) with the same propositions about the function p(x, y 0 , y 1 , y 2 ).Definition. A solution y(x) has a resonance asymptote x = x ∗ iflim y(x) = +∞, limx→x∗ y(x) = −∞.x→x ∗Theorem 4. Suppose that condition (2) holds. Let y(x) be a solution to (1) defined on [x 0 , x ∗ )with the resonance asymptote x = x ∗ . Then the position of the asymptote x = x ∗ depends continuouslyon y(x 0 ), y ′ (x 0 ), y ′′ (x 0 ).Theorem 5. Suppose that condition (2) holds. Then for any finite x ∗ < x ∗ there exists anon-extensible solution y(x) to (1) defined on (x ∗ , x ∗ ) with the vertical asymptote x = x ∗ and theresonance asymptote x = x ∗ .Corollary 1. Suppose that condition (2) holds. Then for any x ∗ ∈ R there exists a Knesersolution of (1) with the vertical asymptote x = x ∗ defined on the interval (x ∗ , +∞) and tending to0 as x → +∞.Corollary 2. Suppose that condition (2) holds. Then for any x ∗ ∈ R there exists a nonextensiblesolution y(x) of (1) with the resonance asymptote x = x ∗ defined on the interval (−∞, x ∗ )and tending to 0 as x → −∞.Theorem 6. Suppose that condition (2) holds. Then for any finite or infinite x ∗ < x ∗ thereexists a non-extensible solution y(x) of (1) with domain (x ∗ , x ∗ ).Remark 2. In [4], [7] asymptotic behavior of all possible solutions to (1) is described.AcknowledgementsThe work was partially supported by the Russian Foundation for Basic Researches (Grant 11-01-00989) and by Special Program of the Ministery of Education and Science of the Russian Federation(Project 2.1.1/13250).References[1] V. A. Kondratiev and V. A. Nikishkin, On positive solutions of y ′′ = p(x)y k . (Russian) SomeProblems of Qualitative Theory of Differential Equations and Motion Control, Saransk, 1980,134–141.[2] I. T. Kiguradze and T. A. Chanturia, Asymptotic properties of solutions of nonautonomousordinary differential equations. Mathematics and its Applications (Soviet Series), 89. KluwerAcademic Publishers Group, Dordrecht, 1993.[3] I. V. Astashova, On the existence of solution with prescribed domain for an equation of thethird order. (Russian) Dokl. Sem. IPM Imeni I. N. Vekua, Tbilisi, TSU 7 (1992), No. 3, 16–19.[4] I. V. Astashova, Application of dynamical systems to the investigation of the asymptoticproperties of solutions of higher-order nonlinear differential equations. (Russian) Sovrem. Mat.Prilozh. No. 8 (2003), 3–33; translation in J. Math. Sci. (N. Y.) 126 (2005), No. 5, 1361–1391.[5] I. V. Astashova, Uniform estimates for positive solutions of quasilinear ordinary differentialequations. (Russian) Izv. Ross. Akad. Nauk Ser. Mat. 72 (2008), No. 6, 85–104; translationin Izv. Math. 72 (2008), No. 6, 1141–1160.6

[6] I. V. Astashova, Uniform estimates of positive solutions to quasi-linear differential inequalities.(Russian) Trudy Seminara Imeni I. G. Petrovskogo 143 (2007), No. 26, 27–36; translation inJ. Math. Sci. 143 (2007), No. 4, 3198–3204.[7] I. V. Astashova, Qualitative properties of solutions to quasilinear ordinary differential equations.(Russian) MESI, Moscow, 2010 (ISBN 978-5-7764-0647-8).[8] I. V. Astashova, On uniform estimates of solutions to nonlinear third-order differential equations.Trudy Seminara Imeni I. G. Petrovskogo, 2011, No. 29 (to appear).7

On the Convergence of Difference Schemes for GeneralizedBBM EquationGivi BerikelashviliA. Razmadze Mathematical Institute of I. Javakhishvili Tbilisi State University,Georgian Technical University, Tbilisi, GeorgiaE-mail: bergi@rmi.geWe consider the initial boundary value problem for generalized Benjamin–Bona–Mahony (BBM)equation∂u∂t + ∂u∂x + γ ∂(u)m∂x− µ ∂3 u∂x 2 ∂t = 0, (x, t) ∈ Q T , (1)u(a, t) = u(b, t) = 0, t ∈ [0, T ), u(x, 0) = φ(x), x ∈ [a, b], (2)where γ and µ are positive constants, m ≥ 2 is a positive integer, and Q T := (a, b) × (0, T ). Inthe cases m=2,3 (1) represents the BBM (or regularized long-wave) and modified BBM equations,respectively.For convenience we introduce the notationx i = a + ih, t j = jτ, i = 0, 1, 2, . . . , n, j = 0, 1, 2, . . . , J,where h = (b − a)/n and τ = T/J denote the spatial and the temporal mesh size, respectively. Letu j i := u(x i, t j ), U j i ∼ u(x i, t j ),(U j i ) x := U j i+1 − U j ih(U j i ) t := U j+1i− U j iτn−1∑(U j , V j ) :=, (U j i )¯x := U j i − U j i−1, (U j ih) x ◦ := 1 2 ((U j i ) x + (U j i )¯x),, (U j i )¯t := U j i − U j−1iτi=1hU j i V ji , (U j , V j ] :=∥U j ∥ 2 := (U j , U j ), ∥U j ]| 2 := (U j , U j ], ∥U j ∥ 2 W 1 2, (U j i ) ◦ t := 1 2 ((U j i ) t + (U j i )¯t),n∑i=1hU j i V ji ,:= ∥U j¯x]| 2 + ∥U j ∥ 2 .We approximate the problem (1), (2) with the help of the difference scheme:whereLU j i= 0, i = 1, 2, . . . , n − 1, j = 0, 1, . . . , J − 1, (3)U j 0 = U j n = 0, j = 0, 1, . . . , J, U 0 i = φ(x i ), i = 0, 1, . . . , n, (4)LU j i := (U j i ) t ◦ + 1 j+1(Ui+ U j−1i) ◦2 x++ γm2(m + 1) ΛU j i − µ(U j i )¯xx t ◦ , i = 1, n − 1, j = 1, 2, . . . , J − 1,LU 0 i := (U 0 i ) t + 1 2 (U 1 i + U 0 i ) ◦ x++ γm2(m + 1) ΛU 0 i − µ(U 0 i )¯xxt , i = 1, n − 1, j = 0,ΛU j i := (U j i )m−1 (U j+1i+ U j−1i) ◦ x+ ( (U j i )m−1 (U j+1i+ U j−1i) ) ◦ x, j = 1, 2, . . . , J − 1,ΛU 0 i := (U 0 i ) m−1 (U 1 i + U 0 i ) ◦ x+ ( (U 0 i ) m−1 (U 1 i + U 0 i ) ) ◦ x.8

The obtained algebraic equations are linear with respect to the values of desired function for eachnew level.Theorem 1. The finite difference scheme (3), (4) is uniquely solvable and possesses the followinginvariantE j := ∥U j ∥ 2 + µ∥U j¯x]| 2 = ∥φ∥ 2 + µ∥φ¯x ]| 2 := E 0 , j = 1, 2, . . . .Definition. Let U, V are solutions of a difference scheme respectively with any initial dateU 0 , V 0 . If there exists a constant c(T ) > 0, independent of mesh sizes τ and h, such that∥U j − V j ∥ 1h ≤ c(T )∥U 0 − V 0 ∥ 2h , j ≥ 1,where ∥ · ∥ 1h and ∥ · ∥ 2h are suitable norms on the set of qrid functions, then we say that differencescheme is stable with respect to initial data. A difference scheme is said to be absolutely stable ifit is stable for any τ and h.Theorem 2. Difference scheme (3), (4) is absolutely stable with respect to initial data.Using procedure proposed by R. D. Lazarov, V. L. Makarov and A. A. Samarskii [1] anddeveloped in [2], we obtain convergence rate estimates that are compatible with the smoothness ofthe desired solution.Theorem 3. Let the exact solution of the initial-boundary value problem (1), (2) belong toW2 k(Q T ), k > 1. Then the convergence rate of the finite difference scheme (3), (4) is determined bythe estimate∥U j − u j ∥ W 12≤ c(τ k−1 + h k−1 )∥u∥ W k2 (Q T ) , 1 < k ≤ 3,where c = c(u) denotes positive constant, independent of h and τ.References[1] R. D. Lazarov, V. L. Makarov, and A. A. Samarskii, Application of exact difference schemesfor constructing and investigating difference schemes on generalized solutions. (Russian) Mat.Sb. (N.S.) 117(159) (1982), No. 4, 469–480, 559.[2] A. A. Samarskii, R. D. Lazarov, and V. L. Makarov, Difference schemes for differential equationswith generalized solutions. (Russian) Visshaja Shkola, Moscow, 1987.9

Asymptotic Behavior of Solutions of Differential Equations of theType y (n) = α 0 p(t) n−1 ∏φ i (y (i) )i=0M. O. BilozerovaI. I. Mechnikov Odessa National University, Odessa, UkraineE-mail: Marbel@ukr.netThe differential equationn−1∏y (n) = α 0 p(t) φ i (y (i) ), (1)where α 0 ∈ {−1, 1}, p : [a, ω[ 1 → ]0, +∞[ (−∞ < a < ω ≤ +∞), φ i : ∆ Yi → ]0, +∞[ (i = 0, . . . , n)are the continuous functions, Y i ∈ {0, ±∞}, ∆ Yi is either the interval [yi 0, Y i[ 2 or the interval ]Y i , yi 0],is considered. We suppose also that every φ i (z) is regularly varying as z → Y i (z ∈ ∆ Yi ) of indexσ i and n−1 ∑σ i ≠ 1.i=0According to the type of the functions φ 0 , . . . , φ n−1 it is clear that the equation (1) is in somesense similar to the well known differential equationi=0n−1∏y (n) = α 0 p(t) |y (i) | σ i. (2)We call the solution y of the equation (1) the P ω (λ 0 n−1 )-solution, where −∞ ≤ λ0 n−1 ≤ +∞, ifthe following conditions take placei=0y (i) : [t 0 , ω[ → ∆ Yi ,lim y (i) (y (n−1) (t)) 2(t) = Y i (i = 0, . . . , n − 1), limt↑ω t↑ω y (n) (t)y (n−2) (t) = λ0 n−1.All P ω (λ 0 n−1 )-solutions of the equation (2) were investigated in [2], [3]. In case n = 2 for allP ω (λ 0 n−1 )-solutions of the equation (1) the necessary and sufficient conditions of existence andasymptotic representations as t ↑ ω were found later [4]–[7]. The methods of this investigations areused in this work for the equation (1), where n ≥ 2.The cases λ 0 ∈ { 0, 1 2 , 2 3 , . . . , n−1} n−2 are singular by the studying of Pω (λ 0 n−1 )-solutions of theequation (1). P ω (λ 0 n−1 )-solutions, where λ 0 ∈ { 0, 1 2 , 2 3 , . . . , n−2n−1}are regularly varying functions ast ↑ ω of indexes {0, 1, . . . , n−1}. To investigate such solutions we have to put additional conditionson the functions φ 0 , . . . , φ n−1 and the function p. The necessary and sufficient conditions of theexistence of P ω (λ 0 n−1 )-solutions of the equation (1) for λ0 n−1 ∈ { 0, 1 2 , 2 3 , . . . , n−2n−1}are found in thiswork. The asymptotic representations as t ↑ ω for such solutions and their derivatives from thefirst to (n − 1)-th order are found too. We will illustrate this results for the case λ 0 = 0, that isone of the most difficult for investigation.We call the slowly varying as z → Y (z ∈ ∆) function θ satisfies the condition S if for every continuouslydifferentiable function L : ∆ → ]0; +∞[ such that lim′ (z) zLz→Y L(z)= 0, the next representationz∈∆takes placeθ(zL(z)) = θ(z)[1 + o(1)] as z → Y (z ∈ ∆).1 If ω > 0 we will take a > 0.2 If Y i = +∞ (Y i = −∞) we take y 0 i > 0 (y 0 i < 0) correspondingly.10

Let us introduce subsidiary notations.⎧a, if ⎪⎨A 0 ω =ω, if⎪⎩C ={tπ ω (t) =t − ωI 0 (t) =∫ ωa∫ ωa∫ tA 0 ωn−31∑, µ = (i + 2 − n)σ i ,1 − σ n−1for ω = +∞,for ω < +∞,p(τ) dτ, I 1 (t) =p(τ) dτ = +∞,p(τ) dτ < +∞,⎧∫ tA 1 ω⎪⎨ a, ifA 1 ω =⎪⎩ ω, ifi=0θ n−1 (z) = φ n−1 (z)|z| −σ n−1,I 0 (τ)(∣C|π ω (τ)| µ θ |I0 (τ)| C ) ∣ Cn−1 ∣∣ dτ,∫ ωa∫ ωThe following conclusion takes place for the equation (1).ay 0 n−1I 0 (τ)(∣|π ω (τ)| µ θ |I0 (τ)| C ) ∣ Cn−1 ∣∣yn−10 dτ = +∞,I 0 (τ)(∣|π ω (τ)| µ θ |I0 (τ)| C ) ∣ Cn−1 ∣∣ dτ < +∞.Theorem. Let the function θ n−1 satisfy the condition S and σ n−1 ≠ 1. Then the followingconditions are necessary for the existence of P ω (0)-solutions of the equation (1):I 1 ′ lim(t)I 0(t)t↑ω p(t)I 1 (t) = 0,lim y 0t↑ωn−2|I 1 (t)| (1−σ n−1)/(1− n−1 ∑)σ jj=0y 0 n−1limt↑ω y0 n−1|I 0 (t)| C = Y n−1 , (3)= Y n−2 , limt↑ωy 0 i |π ω (t)| n−i−2 = Y i , (4)α 0 y 0 n−1CI 0 (t) > 0, y 0 n−2y 0 n−1I 1 (t) > 0, y 0 i y 0 i+1(n − i − 1)(n − i − 2) > 0 if t ∈ [a, ω[ , (5)πwhere i = 0, . . . , n − 3. If there exists a finite or an infinite limit lim ω (t)p(t)t↑ωI 0 (t), then the conditions(3)–(5) are sufficient for the existence of such solutions of the equation (1). Moreover, for anyP ω (0)-solution of (1) the following asymptotic representationsy (n−1) (t)= α 0 (1 − σ n−1 )I 0 (t)[1 + o(1)],∏φ j (y (j) (t))n−1j=0y (n−1) (t)y (n−2) (t) = I′ 1 (t)y[1 + o(1)],I 1 (t)hold as t ↑ ω, where i = 0, . . . , n − 3.(i) (t)y (n−2) (t) = [π ω(t)] n−i−2[1 + o(1)](n − i − 2)!References[1] E. Seneta, Regularly varying functions. Lecture Notes in Mathematics, Vol. 508. Springer-Verlag, Berlin–New York, 1976.[2] V. M. Evtukhov, A class of monotone solutions of an nth-order nonlinear differential equationof Emden–Fowler type. (Russian) Soobshch. Akad. Nauk Gruzii 145 (1992), No. 2, 269–273.[3] V. M. Evtukhov, Asymptotic representations of monotone solutions of an nth-order nonlineardifferential equation of Emden–Fowler type. (Russian) Dokl. Akad. Nauk 324 (1992), No. 2,258–260; English transl.: Russian Acad. Sci. Dokl. Math. 45 (1992), No. 3, 560–562 (1993).11

[4] V. M. Evtukhov and M. A. Belozerova, Asymptotic representations of solutions of secondorderessentially nonlinear nonautonomous differential equations. (Russian) Ukrain. Mat. Zh.60 (2008), No. 3, 310–331; English transl.: Ukrainian Math. J. 60 (2008), No. 3, 357–383.[5] M. A. Belozerova, Asymptotic properties of a class of solutions of essentially nonlinear secondorderdifferential equations. (Russian) Mat. Stud. 29 (2008), No. 1, 52–62.[6] M. A. Belozerova, Asymptotic representations of solutions of differential equations of thesecond order with nonlinearities, that are in some sense near to the power type. (Ukrainian)Nauk. Visn. Thernivetskogo univ, Thernivtsi: Ruta 374 (2008), 34–43.[7] M. A. Belozerova, Asymptotic representations of solutions of second-order nonautonomousdifferential equations with nonlinearities close to power type. (Russian) Nelīnīīnī Koliv. 12(2009), No. 1, 3–15; English transl.: Nonlinear Oscil. (N. Y.) 12 (2009), No. 1, 1–14.12

On Stability of Delay SystemsAlexander DomoshnitskyDepartment of Mathematics and Computer Sciences, Ariel University Center, Ariel, IsraelE-mail: adom@ariel.ac.ilConsider the systemn∑x ′ i(t) + p ij (t)x j (t − τ ij (t)) = f i (t), i = 1, . . . , n, t ∈ [0, +∞),j=1x i (θ) = 0 for θ < 0.(1)Its general solution has the following representationx(t) =∫ t0C(t, s)f(s) ds + X(t)α,where the n × n matrix C(t, s) is called the Cauchy matrix of equation (1), X(t) is a n × nfundamental matrix of the system(M i x)(t) ≡ x ′ i(t) +n∑p ij (t)x j (t − τ ij (t)) = 0, i = 1, . . . , n, t ∈ [0, +∞),j=1x i (θ) = 0 for θ < 0,such that X(0) = I (I is the unit n × n matrix), f = col(f 1 , . . . , f n ), α = col(α 1 , . . . , α n )(1 0 )Definition 1. Let us say that system (1) is exponentially stable if the Cauchy and fundamentalmatrices X(t) = {X ij (t, s)} i,j=1,...,n and C(t, s) = {C ij (t, s)} i,j=1,...,n satisfy the exponentialestimate, i.e. there exist N and γ such thatfor 0 ≤ s ≤ t < +∞, i, j = 1, . . . , n.|C ij (t, s)| ≤ Ne −γ(t−s) , |X ij (t)| ≤ Ne −γt ,Theorem 1. Let the following conditions be fulfilled:1) p ij ≤ 0 for i ≠ j, i, j = 1, . . . , n − 1;2)∫ tt−τ ii (t)p ii (s)ds ≤ 1 efor i = 1, . . . , n − 1;3) p jn ≥ 0, p nj ≤ 0 for j = 1, . . . , n − 1, p nn ≥ 0;4) there exist positive α and β i such thatn−1∑p nn (t)e ατnn(t) − p nj (t)β j e ατnj(t) ≤ α ≤j=1≤{min − p in (t) 1 n−1∑e ατin(t) + p ij (t) β }je ατ jn(t), t ∈ [0, +∞).1≤i≤n−1 β i β i13j=1

Then the elements of the n-th row of the Cauchy matrix of system (1) satisfy the inequalities:C nn (t, s) > 0, C nj (t, s) ≥ 0 for j = 1, . . . , n − 1, 0 ≤ s ≤ t < +∞. If in addition there exist positiveε and z 1 , . . . , z n−1 such thatn−1∑p ij (t) ≥ ε > 0,j=1n−1∑p in (t) − p ij (t)z j ≥ ε > 0, i = 1, . . . , n − 1,j=1n−1∑p nn (t) + |p nj (t)|z j ≥ ε > 0,j=1t ∈ [0, +∞), (2)then system (1) is exponentially stable.Remark 1. It was assumed in condition 3) that p jn ≥ 0, for j = 1, . . . , n − 1, and Wazewskii’scondition, generally speaking, is not fulfilled.Remark 2. A possible case is p nn = 0 and the principle of main diagonal dominance (evenin its generalized form, assuming, for example, that the matrix {p ij } i,j=1,...,n is M-matrix), is notfulfilled.Remark 3. The inequalitycannot be set instead of the inequalityn−1∑p nn (t) + |p nj (t)|β j > 0, t ∈ [0, +∞)j=1n−1∑p nn (t) − p nj (t)β j ≥ ε > 0, t ∈ [0, +∞)j=1in the condition (2) of Theorem 1. Actually in the case p nj (t) ≡ 0 for j = 1, . . . , n − 1 andp nn (t) = 1 , the component xt 2 n (t) of the solution vector of the homogeneous system (1 0 ) does nottend to zero when t → +∞.Remark 4. Conditions 1) and 2) imply that all elements of the Cauchy matrix K(t, s) ={K ij (t, s)} i,j=1,...,n−1 of auxiliary systemare nonnegative.n−1∑x ′ i(t) + p ij (t)x j (t − τ ij (t)) = f i (t), i = 1, . . . , n − 1, t ∈ [0, +∞), (3)j=1Remark 5. The inequalityn−1∑p ij (t) ≥ ε > 0, t ∈ [0, +∞)j=1implies that the Cauchy matrix of the auxiliary system (3) satisfies the exponential estimate andthis is essential in the case of separated n-th equation, i.e. when p nj (t) ≡ 0, p jn (t) ≡ 0 forj = 1, . . . , n − 1.14

On Well-Posedness with Respect to Functional for anOptimal Control Problem with Distributed DelaysPhridon DvalishviliI. Javakhishvili Tbilisi State University, Tbilisi, GeorgiaE-mail: pridon.dvalishvili@tsu.geLet Rx n be an n-dimensional vector space of points x = (x 1 , . . . , x n ) T , where T means transpose;O ⊂ Rx n and V ⊂ Ru r be open sets; the (n + 1)-dimensional function F (t, x, u) = (f 0 , f) T becontinuous on the set I × O × V and continuously differentiable with respect to x, where I =[t 0 , t 1 ]; next, τ > 0, θ > 0 given numbers, let Φ and Ω be sets of continuous initial functionsφ(t) ∈ O, t ∈ [t 0 − τ, t 0 ] and measurable control functions u(t) ∈ U, t ∈ [t 0 − θ, t 1 ], respectively,where U ⊂ V is a compact set.Let x 0 ∈ O and φ ∈ Φ be fixed initial vector and function. To each control function u ∈ Ω weassign the differential equation with distributed delayswith the initial conditionẋ(t) =∫ 0 ∫ 0−τ −θf(t, x(t + s), u(t + ξ)) ds dξ, t ∈ I (1)x(t) = φ(t), t ∈ [t 0 − τ, t 0 ), x(t 0 ) = x 0 . (2)Definition 1. Let u ∈ Ω be a given control. A function x(t) = x(t; u) ∈ O, t ∈ [t 0 − τ, t 1 ],is called a solution of Eq. (1) with the initial condition (2) or a solution corresponding to u anddefined on [t 0 − τ, t 1 ], if it satisfies condition (2) and is absolutely continuous on the interval [t 0 , t 1 ]and satisfies Eq. (1) almost everywhere on [t 0 , t 1 ].Definition 2. A control u ∈ Ω is said to be admissible if the corresponding solution x(t) =x(t; u) is defined on the interval [t 0 − τ, t 1 ].whereWe denote the set of admissible controls by Ω 0 .Definition 3. A control u 0 ∈ Ω 0 is said to be optimal if for any u ∈ Ω 0 we haveJ(u) =∫ t 1 { ∫0∫ 0t 0 −τ −θJ(u 0 ) ≤ J(u),The optimal control u 0 is called solution of the problem (1)–(3).}f 0 (t, x(t + s), u(t + ξ)) ds dξ dt. (3)Introduce the following notations: By Y we denote the set of continuous functions y(t) ∈ O,t ∈ I 1 = [t 0 − τ, t 0 ) ∪ (t 0 , t 1 ], with cly(I 1 ) ⊂ O is the compact set;F (t, y(·), u) =∫ 0−τF (t, y(t + s), u) ds, y ∈ Y, P (t, y(·)) = { F (t, y(·), u) : u ∈ U } .Theorem 1. Let the following conditions hold: Ω 0 ≠ Ø; there exists a compact set K 0 ⊂ Osuch that x(t; u) ∈ K 0 , t ∈ [t 0 − τ, t 1 ], ∀ u ∈ Ω 0 ; for any (t, y) ∈ I × Y the set P (t, y(·)) is convex.Then the problem (1)–(3) has a solution u 0 .15

Let K 1 ⊂ O be a compact set containing a certain neighborhood of the set K 0 . By E we denotethe set of functions G(t, x, y) = (g 0 , g) T ∈ Rxn+1 continuous on the set I × O and continuouslydifferentiable with respect to x and satisfying the condition∫|G δ (t, x)| dt ≤ const, ∀ x ∈ K 1 .ITheorem 2. Let the conditions of Theorem 1 hold. Then for any ε > 0 there exists a numberδ > 0 such that for an arbitrary vector x 0δ ∈ O and functions φ δ ∈ Φ, G δ ∈ E satisfying theconditions|x 0 − x 0δ | + ∥φ − φ δ ∥ + ∥G δ ∥ ≤ δ,the perturbed optimal problemẋ(t) =J(u, δ) =∫ 0 ∫ 0−τ −θ∫ t 1 { ∫0∫ 0t 0 −τ −θhas a solution u 0δ and the following inequalityis fulfilled. Here{}f(t, x(t + s), u(t + ξ)) + g(t, x(t + s)) ds dξ, t ∈ I 1 ,x(t) = φ δ (t), t ∈ [t 0 − τ, t 0 ), x(t 0 ) = x 0δ ,[] }f 0 (t, x(t + s), u(t + ξ)) + g 0 (t, x(t + s)) ds dξ dt|J(u 0 ) − J(u 0δ )| ≤ ε∥ φ − φ δ ∥= sup { |φ(t) − φ δ (t)| : t ∈ I 1},{ ∣∣∣ ∫t ′′∥G δ ∥ = sup g(t, x) dt∣ : t ′ , t ′′ ∈ I, x ∈ K 1}.Theorems 1 and 2 are proved by scheme given in [1], [2].t ′References[1] T. A. Tadumadze, Some problems in the qualitative theory of optimal control. (Russian) Tbilis.Gos. Univ., Tbilisi, 1983.[2] P. A. Dvalishvili, Some problems in the qualitative theory of optimal control with distributeddelay. (Russian) Tbiliss. Gos. Univ. Inst. Prikl. Mat. Trudy 41 (1991), 83–113.16

Asymptotic Representations of Solutions of Ordinary DifferentialEquations of N-th Order with Regularly Varying NonlinearsV. M. Evtukhov and A. M. KlopotI. I. Mechnikov Odessa National University, Odessa, UkraineE-mail: emden@farlep.net, mrtark@gmail.comWe consider the differential equationy (n) =m∑k=1n−1∏α k p k (t) φ kj (y (j) ), (1)where α k ∈ {−1; 1} (k = 1, m), p k : [a, ω[ → ]0, +∞[ (k = 1, m) are continuous functions, φ kj :△ Yj → ]0, +∞[ (k = 1, m; j = 0, n − 1) are continuous and regularly varying at y (j) −→ Y jfunctions of orders σ kj , −∞ < a < ω ≤ +∞, △ Yj one-sided neighbourhood Y j , Y j is either 0, or±∞.Continuous function φ : ∆ Y → ]0, +∞[ , where Y is either 0, or ±∞ and ∆ Y one-sidedneighbourhood Y , is regularly variyng at y → Y , if there exists a number σ ∈ R such that= λ σ for any λ > 0. In this case the number σ is called the order of regularly varyinglim y→Yy∈∆Yφ(λy)φ(y)function. Regularly varying at y → Y zero-order function is called slowly changing function.Since each regularly varying at y → Y function φ of σ order is represented as φ(y) = |y| σ L(y),where L- slowly changing function at y → Y , then the differential equation (1) is asymptoticallyclose at y (j) → Y j (j = 0, n − 1) to the equationy (n) =m∑k=1j=0n−1∏α k p k (t) |y (j) | σ kj.Asymptotic behavior of certain classes of solutions of those equation is investigated in works [1], [2].A solution y of the equation (1) is called P ω (Y 1 , . . . , Y n−1 , λ 0 )- solution, where −∞ ≤ λ 0 ≤ +∞,if it is defined on an interval [t 0 , ω[ ⊂ [a, ω[ and satisfies the following conditionsy (j) (t) ∈ ∆ Yj at t ∈ [t 0 , ω[ , limt↑ωy (j) (t) = Y j (j = 0, n − 1), limt↑ω[y (n−1) (t)] 2y (n) (t)y (n−2) (t) = λ 0.Choose the numbers b j ∈ ∆ Yj (j = 0, n − 1) such that |b j | < 1 at Y j = 0, b j > 1 (b j < −1) atY j = +∞ (Y j = −∞), and introduce the numbers{1, if ∆ν 0j = sign b j , ν 1j =Yj − left neighborhood Y j ,(j = 0, n − 1).−1, if ∆ Yj − right neighborhood Y j ,Note that P ω (Y 0 , . . . , Y n−1 , λ 0 ) solution of equation (1) satisfies the following conditionsNext, letj=0ν 0j ν 1j < 0, if Y j = 0, ν 0j ν 1j > 0, if Y j = ±∞ (j = 0, n − 1). (2)a 0i = (n − i)λ 0 − (n − i − 1) (i = 1, . . . , n) at λ 0 ∈ R,{{t, if ω = +∞,π ω (t) =t − ω, if ω < +∞, , β = 1, if ω = +∞,−1, if ω < +∞, ,17

n−1∑γ k = 1 − σ kj , µ kn =j=0n−2∑σ kj (n − j − 1), C k =j=0∫ tJ kn (t) = p k (τ) ∣ πω (τ) ∣ a, if ⎪⎨µ kndτ, A kn =A kn⎧ω, if⎪⎩∫ ωa∫ ωan−2∏j=0(λ 0 − 1) n−j−1 ∣ ∣∣∣σ kj∣(k = 1, m),n−1 ∏a 0ii=j+1p k (t) ∣ ∣π ω (t) ∣ ∣ µ kndt = +∞,p k (t) ∣ ∣ πω (t) ∣ ∣ µ kndt < +∞(k = 1, m).Theorem. Let λ 0 ∈ R \ { 0, 1 2 , . . . , n−2n−1 , 1} and for some s ∈ {1, . . . , m} the inequality γ s ≠ 0and the conditionslim supt↑ωln p k (t) − ln p s (t)∣ ln |πω (t)| ∣ ∣< β n−1∑(σ sj − σ kj )a 0j+1 for all k ∈ {1, . . . , m} \ {s}λ 0 − 1j=0be fulfilled. Then for P ω (Y 0 , . . . , Y n−1 , λ 0 ) solutions of equation (1) to be exist it is necessary andif algebraic relatively ρ equationn−1∑n−1∏σ sj a 0ij=0 i=j+1 i=1j∏n−1∏(a 0i + ρ) = (1 + ρ) (a 0i + ρ)does not have roots with zero real part, it is also sufficient that inequality (2), the inequalitiesν 0j ν 0j+1 a 0j+1 (λ 0 − 1)π ω (t) > 0 (j = 0, n − 2), α s ν on−1 γ s J sn (t) > 0 at t ∈ ]a, ω[and the conditionπ ω (t)Jlimsn(t)′ = γ st↑ω J sn (t) λ 0 − 1are satisfied. Moreover, for each such a solution at t ↑ ω the asymptotic representationsy (j) (t) = [(λ 0 − 1)π ω (t)] n−j−1y (n−1) (t)[1 + o(1)] (j = 0, 1, . . . , n − 2),n−1 ∏a 0in−1 ∏j=0i=j+1|y (n−1) (t)| γ sL sj([(λ0 −1)π ω (t)] n−j−1n−1 ∏a 0ii=j+1hold, where L sj (y (j) ) = |y (j) | −σ sjφ sj (y (j) ) (j = 0, n − 1).i=1) = α sν 0n−1 γ s C s J sn (t)[1 + o(1)]y (n−1) (t)References[1] A. V. Kostin, Asymptotics of the regular solutions of nonlinear ordinary differential equations.(Russian) Differentsial’nye Uravneniya 23 (1987), No. 3, 524–526.[2] V. M. Evtukhov and E. V. Shebanina, Asymptotic behaviour of solutions of nth order differentialequations. Mem. Differential Equations Math. Phys. 13 (1998), 150–153.18

On Mixed Type Quasi-Linear Equations with General Integrals,Represented by Superposition of Arbitrary FunctionsJondo GvazavaA. Razmadze Mathematical Institute of I. Javakhishvili Tbilisi State University, Tbilisi, GeorgiaE-mail: jgvazava@rmi.geWe discuss second-order quasi-linear equations with two independent variables x, t. The coefficientsof the principal part of the equation are assumed to be polynomials with respect to unknownsolutions and to their derivatives of the first order, while the characteristic roots expressed bythese values are always real. Depending on the solutions, on some sets of points these roots maycoincide. Therefore the equations under consideration should be referred to a class of equationsof mixed type, in particular, to hyperbolic equations with admissible parabolic degeneration. Forevery particular equation of such a type, a structure of points of parabolic degeneration dependscompletely on the data of the initial, or of some other formulated problem. In the present reportwe will consider equations which admit explicit representation of general integrals in the form ofsuperposition of arbitrary functions. A number of these arbitrary elements is equal to two, andtheir arguments, except independent variables, contain an unknown solution as well. Such is, forexample, the equation∂u ∂ 2 u[∂t ∂t 2 + u ∂u∂t − ∂u ] ∂u2∂x ∂x∂t − u ∂u ∂ 2 u= 0, (1)∂x ∂t2 having certain practical application. A class of its hyperbolic solutions is defined by the conditionH ≡ ∂u∂x + u ∂u ≠ 0. (2)∂tIf H is identical zero for some solution, then the solution itself is parabolic. If, however, thecondition (2) is fulfilled not everywhere, then the corresponding solution is called mixed one. Ageneral integral of the equation has the formuf ′ (u) − f(u) + g[x − f(u)] = t, (3)where f, g ∈ C 2 (R 1 ) are arbitrary functions. Consider now the Cauchy problem with the initialconditionsu ∣ ∂ut=0= τ(x), ∣ = ν(x), (x, t) ∈ H, τ, ν ∈ C 2 (J) (4)∂t t=0on the data supportJ := { (x, t) : t = 0, α ≤ x ≤ β } .The functional equationsτ(x) = ζ,∫ xατ(z)ν(z)τ ′ dz = ξ (5)(z) + τ(z)ν(z)with respect to x are assumed to be uniquely solvable and we denote their solutions by, respectively,x = T (ζ) and x = G(ξ). In this case, subjecting the general integral (3) to the conditions (4), onewill be able to define arbitrary functions in terms of the initial perturbationsf[ζ] = f[τ(α)] + f ′ [τ(α)] [ τ(T (ζ)) − τ(α) ] +g(ξ) = g[α − f ′ (α)] +G(ξ)∫α19T∫(ζ)α[τ(T ζ)) − τ(ζ)]τ ′ (z)τ ′ (z) + τ(z)ν(z)τ(z)τ ′ (z)τ ′ (z) + τ(z)ν(z) dz.dz,

Having defined in such a way arbitrary functions and substituting them into the representation(3), we construct the integral of the problem (1), (4). To describe a structure of the domainof definition of the integral of the problem, we use the properties of characteristic invariants,represented in the given case by the right-hand sides of the functional equations (5), in particular,the fact that along any characteristic of the corresponding family they are constant. A globalcharacter of that property allows one to construct all characteristics emanated from the points ofthe data support of the problem (1), (4) and, consequently, the domain of definition of the integralby means of a set of points of intersection of these characteristics.20

Fredholm Type Theorem for Systems of Functional-DifferentialEquations with Positively Homogeneous OperatorsRobert HaklInstitute of Mathematics, Academy of Sciences of the Czech Republic, branch in Brno, Brno,Czech RepublicE-mail: hakl@ipm.czConsider the system of functional-differential equationsu ′ i(t) = p i (u 1 , . . . , u n )(t) + f i (u 1 , . . . , u n )(t) for a. e. t ∈ [a, b] (i = 1, . . . , n) (1)together with boundary conditionsl i (u 1 , . . . , u n ) = h i (u 1 , . . . , u n ) (i = 1, . . . , n). (2)Here, p i , f i : [C([a, b]; R)] n → L ( [a, b]; R ) are continuous operators satisfying Carathéodory condition,i.e., they are bounded on every ball by an integrable function, and l i , h i : [C([a, b]; R)] n → Rare continuous functionals which are bounded on every ball by a constant. Furthermore, we assumethat p i and l i satisfy the following condition: there exist positive real numbers λ ij and µ i such thatλ ij λ jm = λ im whenever i, j, m ∈ {1, . . . , n}, and for every c > 0 and u k ∈ C([a, b]; R) (k = 1, . . . , n)we havecp i (u 1 , . . . , u n )(t) = p i (c λ i1u 1 , . . . , c λ inu n )(t) for a. e. t ∈ [a, b],c µ il i (u 1 , . . . , u n ) = l i (c λ i1u 1 , . . . , c λ inu n ).By a solution to (1), (2) we understand an absolutely continuous vector-valued function (u i ) n i=1 :[a, b] → R n satisfying (1) almost everywhere in [a, b] and (2).Remark 1. From the above-stated assumptions it follows that λ ii = 1, λ ij = 1/λ ji for everyi, j ∈ {1, . . . , n}.Notation 1. Define, for every i ∈ {1, . . . , n}, the following functions{q i (t, ρ) def ∣∣fi= sup (u 1 , . . . , u n )(t) ∣ }: ∥uk ∥ C ≤ ρ λ ik, k = 1, . . . , n for a. e. t ∈ [a, b],{η i (ρ) def ∣∣hi= sup (u 1 , . . . , u n ) ∣ }: ∥uk ∥ C ≤ ρ λ ikµ i , k = 1, . . . , n .Theorem 1. Let∫ bq i (s, ρ)limds = 0,ρ→+∞ ρaη i (ρ)lim = 0 (i = 1, . . . , n). (3)ρ→+∞ ρIf the problemu ′ i(t) = (1 − δ)p i (u 1 , . . . , u n )(t) − δp i (−u 1 , . . . , −u n )(t) for a. e. t ∈ [a, b] (i = 1, . . . , n),(1 − δ)l i (u 1 , . . . , u n ) − δl i (−u 1 , . . . , −u n ) = 0 (i = 1, . . . , n)has only the trivial solution for every δ ∈ [0, 1/2], then the problem (1), (2) has at least one solution.21

Sketch of proof. There exists r > 0 such that for any absolutely continuous vector-valued fucntion(u i ) n i=1 defined on [a, b] and every δ ∈ [0, 1/2], the a priori estimaten∑k=1∥u k ∥ λ k1C≤ r n∑i=1(∥ ˜f)i ∥ λ i1L+ |˜h i | λ i1µ iholds, where˜f i (t) def= u ′ i(t) − (1 − δ)p i (u 1 , . . . , u n )(t) + δp i (−u 1 , . . . , −u n )(t) for a. e. t ∈ [a, b] (i = 1, . . . , n),Put˜hidef= (1 − δ)l i (u 1 , . . . , u n ) − δl i (−u 1 , . . . , −u n ) (i = 1, . . . , n).x = ( (u i ) n i=1, (α i ) n i=1)∈ X = [C([a, b]; R)] n × R n with the norm ∥x∥ =A(x) def=((∫ tu i (a) + α i +Ω =a{x ∈ X :) np i (u 1 , . . . , u n )(s) + f i (u 1 , . . . , u n )(s) ds ,i=1n∑k=1n∑ (∥uk ∥ C + |α k | ) ,k=1(αi + l i (u 1 , . . . , u n ) − h i (u 1 , . . . , u n ) ) )n,i=1(∥uk ∥ λ k1C + |α k| ) < ρ 0}for sufficiently large ρ 0 .Then using Krasnosel’skii theorem (see [1, Theorem 41.3, p. 325]):A(x) − x∥A(x) − x∥ ≠A(−x) + x∥A(−x) + x∥ (x ∈ ∂Ω) =⇒ ∃ x 0 ∈ Ω such that A(x 0 ) = x 0 ,we prove the assertion of this theorem. For more detailed idea of the proof one can see [2].If the operators p i and l i are homogeneous, i.e., ifp i (−u 1 , . . . , −u n )(t)=−p i (u 1 , . . . , u n )(t) for a. e. t ∈ [a, b], u j ∈C([a, b]; R) (i, j = 1, . . . , n), (4)l i (−u 1 , . . . , −u n ) = −l i (u 1 , . . . , u n ) u j ∈ C([a, b]; R) (i, j = 1, . . . , n) (5)hold, then from Theorem 1 we obtain the following assertion.Corollary 1. Let (3), (4), and (5) be fulfilled. If the problemu ′ i(t) = p i (u 1 , . . . , u n )(t) for a. e. t ∈ [a, b] (i = 1, . . . , n), (6)l i (u 1 , . . . , u n ) = 0 (i = 1, . . . , n) (7)has only the trivial solution, then the problem (1), (2) has at least one solution.For a particular case when p i are defined byp i (u 1 , . . . , u n )(t) def= ˜p i (t)|u i+1 (t)| λ isgn u i+1 (t) for a. e. t ∈ [a, b] (i = 1, . . . , n − 1), (8)p n (u 1 , . . . , u n )(t) def= ˜p n (t)|u 1 (t)| λ nsgn u 1 (t) for a. e. t ∈ [a, b], (9)where ˜p i ∈ L ( [a, b]; R ) , we have the following assertion.22

Corollary 2. Let (3), (5), (8), and (9) be fulfilled. Let, moreover,n∏λ i = 1,i=1and let the problem (6), (7) have only the trivial solution. Then the problem (1), (2) has at leastone solution.resp.In [3], the problem (1), (2) is studied with n = 2, p 1 , p 2 defined by (8), (9), andl 1 (u 1 , . . . , u n ) def=l 1 (u 1 , . . . , u n ) def=∫ a 0a∫ a 0au 1 (s) dα 1 (s),u 1 (s) dα 1 (s),l 2 (u 1 , . . . , u n ) def=l 2 (u 1 , . . . , u n ) def=∫ bb 0u 1 (s) dα 2 (s), (10)∫ bb 0u 2 (s) dα 2 (s), (11)where a < a 0 ≤ b, a ≤ b 0 < b, α 1 : [a, a 0 ] → R and α 2 : [b 0 , b] → R are functions of boundedvariation. For this particular case, Theorem 1 yieldsCorollary 3. Let n = 2, (3) be fulfilled, λ 1 λ 2 = 1, k ∈ {1, 2}, and letu ′ 1 = ˜p 1 (t)|u 2 | λ 1sgn u 2 , u ′ 2 = ˜p 2 (t)|u 1 | λ 2sgn u 1 ,∫ a 0au 1 (s) dα 1 (s) = 0,∫ bb 0u k (s) dα 1 (s) = 0have only the trivial solution. Then the problem (1), (2) with p i and l i defined by (8), (9), and(10), resp. (11) has at least one solution.References[1] M. A. Krasnosel’skiǐ and P. P. Zabreǐko, Geometric methods of nonlinear analysis. (Russian)Monographs in Nonlinear Analysis and Applications. Izdat. “Nauka”, Moscow, 1975.[2] A. Lasota, Une généralisation du premier théorème de Fredholm et ses applications à la théoriedes équations différentielles ordinaires. Ann. Polon. Math. 18 (1966), 65–77.[3] I. Kiguradze and J. Šremr, Solvability conditions for non-local boundary value problems fortwo-dimensional half-linear differential systems. Nonlin. Anal., TMA 74 (2011), No. 17, 6537–6552.23

Periodic Solution to Two-Dimensional Half-Linear SystemRobert HaklInstitute of Mathematics, Academy of Sciences of the Czech Republic, branch in Brno, Brno,Czech RepublicE-mail: hakl@ipm.czManuel ZamoraDepartamento de Matemática Aplicada, Facultad de Ciencias, Granada, SpainOn the interval [0, ω] we consider the system of differential equationsu ′ 1 = p 1 (t)|u 2 | λ 1sgn u 2 + f 1 (t, u 1 , u 2 ),u ′ 2 = p 2 (t)|u 1 | λ 2sgn u 1 + f 2 (t, u 1 , u 2 ),subjected to the periodic-type boundary conditionsu 1 (0) − u 1 (ω) = h 1 (u 1 , u 2 ), u 2 (0) − u 2 (ω) = h 2 (u 1 , u 2 ). (2)Here, p i ∈ L([0, ω]; R), f i ∈ Car([0, ω] × R 2 ; R), h i : C([0, ω]; R) × C([0, ω]; R) → R are continuousfunctionals bounded on every ball, and λ i > 0 such that λ 1 λ 2 = 1.Notation 1. Define the following functions{q i (t, ρ) def ∣∣= sup f i (t, x 1 , x 2 ) ∣ }: |x i | ≤ ρ, |x 3−i | ≤ ρ λ 3−ifor a. e. t ∈ [0, ω] (i = 1, 2),{}η i (ρ) def= sup |h i (u 1 , u 2 )| : ∥u i ∥ C ≤ ρ, ∥u 3−i ∥ C ≤ ρ λ 3−i(i = 1, 2).Theorem 1. Let∫ ωq i (s, ρ)limds = 0,ρ→+∞ ρ0Let, moreover, σ ∈ {1, −1} be such thatand let there exist α i ∈ AC([0, ω]; R) (i = 1, 2) such that(1)η i (ρ)lim = 0 (i = 1, 2). (3)ρ→+∞ ρσp 1 (t) ≥ 0 for a. e. t ∈ [0, ω], p 1 ≢ 0, (4)α ′ 1(t) = p 1 (t)|α 2 (t)| λ 1sgn α 2 (t) for a. e. t ∈ [0, ω],α 1 (0) = α 1 (ω),α 2(t) ′ ≤ p 2 (t)|α 1 (t)| λ 2sgn α 1 (t) for a. e. t ∈ [0, ω], α 2 (0) ≤ α 2 (ω),σα 1 (t) > 0 for t ∈ [0, ω],{}meas t ∈ [0, ω] : α 2(t) ′ < p 2 (t)|α 1 (t)| λ 2sgn α 1 (t) + α 2 (ω) − α 2 (0) > 0.Then the problem (1), (2) has at least one solution.Corollary 1. Let (3) and (4) be fulfilled with σ ∈ {1, −1}. Let, moreover,∫ ω0[σp 2 (s)] − ds

ω∫Remark 1. Theorem 1 and Corollary 1 are applicable in the case when σp 2 (s) ds > 0. For theω∫case when σp 2 (s) ds < 0 one can use the following assertion.0Theorem 2. Let (3) and (4) be fulfilled with σ ∈ {1, −1}. Let, moreover,0∫ ωσp 2 (s) ds < 0,∫ ω( ∫ωλ1σp 1 (s) ds [σp 2 (s)] − ds)< 4 1+λ 1.000Then the problem (1), (2) has at least one solution.Sketch of the proofs. According to the general result established in [1] one can see that the followingassertion holds:Proposition 1. Let (3) be fulfilled. If the problemu ′ 1 = p 1 (t)|u 2 | λ 1sgn u 2 ,u ′ 2 = p 2 (t)|u 1 | λ 2sgn u 1 ,u 1 (0) − u 1 (ω) = 0, u 2 (0) − u 2 (ω) = 0 (6)has only the trivial solution, then the problem (1), (2) has at least one solution.Then the conditions of Theorems 1 and 2 and Corollary 1 are obtained by direct analysing thenon-trivial solutions of the problem (5), (6).Remark 2. Results obtained are unimprovable in that sense that neither of the strict inequalitiesestablished in Corollary 1 and Theorem 2 can be weakened.Remark 3. When λ i = 1, p 1 ≡ 1, h i ≡ 0, f 1 ≡ 0, f 2 (t, x, y) = f(t) for a. e. t ∈ [0, ω], x, y ∈ Rwith f ∈ L([0, ω]; R), then the problem (1), (2) becomes a periodic problem for the second-orderlinear equationu ′′ = p 2 (t)u + f(t), u(0) = u(ω), u ′ (0) = u ′ (ω).In this case, Theorems 1 and 2, and Corollary 1 coincide with the results obtained in [2].References[1] R. Hakl, Fredholm type theorem for functional-differential equations with positively homogeneousoperators. (submitted)[2] R. Hakl and P. J. Torres, Maximum and antimaximum principles for a second order differentialoperator with variable coefficients of indefinite sign. Appl. Math. Comput. 217 (2011), No. 19,7599–7611.(5)25

Necessary Conditions of Optimality for Optimal Problems withGeneral Variable Delays and the Mixed Initial ConditionMedea IordanishviliI. Javakhishvili Tbilisi State University, Tbilisi, GeorgiaE-mail: imedea@yahoo.comLet Rx n be an n-dimensional vector space of points x = (x 1 , . . . , x n ) T , where T means transpose;P ⊂ Rp, k Z ⊂ Rz e and V ⊂ Ru r be open sets and O = { x = (p, z) T ∈ Rx n : p ∈ Rp, k z ∈ Rz} e , withk +e = n; the n-dimensional function f(t, p 1 , . . . , p s , z 1 , . . . , z m , u 1 , . . . , u ν ) be continuous on the set[a, b]×P s ×Z m ×V ν and continuously differentiable with respect to p i , i = 1, s and z j , j = 1, m; thefunctions q i (t 0 , t 1 , p, z, x), i = 0, l be continuously differentiable on the set [a, b] × [a, b] × P × Z × O.Let us consider the optimal control problem:ẋ(t) = f(t, p(τ 1 (t)), . . . , p(τ s (t)), z(σ 1 (t)), . . . , z(σ m (t)), u(θ 1 (t)), . . . , u(θ ν (t))), u(·) ∈ Ω,x(t) = (p(t), z(t)) T = (φ(t), g(t)) T , t ∈ [τ 0 , t 0 ),x(t 0 ) = (p 0 , g(t 0 )) T , p 0 ∈ P, φ(·) ∈ Φ, g(·) ∈ G,q i( t 0 , t 1 , p 0 , g(t 0 ), x(t 1 ) ) = 0, i = 1, l,q 0 (t 0 , t 1 , p 0 , g(t 0 ), x(t 1 )) −→ min,where the functions τ i (t), i = 1, s are continuously differentiable and satisfying the conditionsτ i (t) ≤ t, ˙τ i (t) > 0; the functions σ i (t), i = 1, m, θ j (t), j = 1, ν satisfy the similar conditions; Φand G are sets of continuous initial functions φ : [τ 0 , b] → P 1 , and g : [τ 0 , b] → Z 1 , where P 1 ⊂ Pand Z 1 ⊂ Z are compact convex sets, τ 0 = min{τ 1 (a), . . . , τ s (a), σ 1 (a), . . . , σ m (a)}; Ω is the set ofpiecewise continuous control functions, u : [θ, b] → U, with finite number of points of discontinuity,θ = min{θ 1 (a), . . . , θ ν (a)}, U ⊂ V is a compact convex set.In the paper, on the basis of variation formulas of solution [1] and by a scheme given in [2],necessary conditions of optimality are obtained: in the form of linearized maximum principle forcontrol and initial function, in the form of equalities and inequalities for initial and final moments.References[1] L. Alkhazishvili and M. Iordanishvili, Local variation formulas for solution of delay controlleddifferential equation with mixed initial condition. Mem. Differential Equations Math. Phys.51 (2010), 17–41.[2] G. L. Kharatishvili and T. A. Tadumadze, Formulas for the variation of a solution and optimalcontrol problems for differential equations with retarded arguments. (Russian) Sovrem. Mat.Prilozh. No. 25, Optimal. Upr. (2005), 3–166; translation in J. Math. Sci. (N. Y.) 140 (2007),No. 1, 1–175.26

Asymptotic Equivalence of Linear Systems for SmallLinear PerturbationsN. A. IzobovInstitute of Mathematics of the National Academy of Sciences of Belarus, Minsk, BelarusE-mail: izobov@im.bas-net.byS. A. MazanikBelarusian State University, Minsk, BelarusE-mail: smazanik@bsu.by1 Tests of Lyapunov’s Reducibility of Linear SystemsConsider the linear systemsẋ = A(t)x, x ∈ R n , t ∈ I = [0, +∞), (1)with piecewise continuous and bounded (by the constant a ≥ ∥A(t)∥ for t ∈ I) coefficients. Alongwith systems (1), we will consider the systemsẏ = (A(t) + Q(t))y, y ∈ R n , t ∈ I, (2)likewise with piecewise continuous and bounded on I coefficients.Systems (1) and (2) are asymptotically equivalent (Lyapunov-equivalent, reducible) if thereexists a linear transformation x = L(t)y, transferring one of the systems into another, where thematrix L(t) is the Lyapunov one, i.e., satisfying the condition{sup |L(t)∥ + ∥L −1 (t)∥ + ∥ ˙L(t)∥ } < +∞.t∈IOne of the tests of asymptotical equivalence of systems (1) and (2) is reflected [1] in the followingassertion.Theorem 1. If ∥ +∞ ∫Q(u) du ∥ ≤ Ce −σt , t ∈ I, σ > 2a, where C is some constant, then thetsystems (1) and (2) are asymptotically equivalent.The following statement [1] establishes that the estimate σ > 2a is unimprovable in a whole setof linear systems (1) with piecewise continuous matrices of coefficients.Theorem 2. For any number a > 0 there exist system (1) with piecewise continuous matrix ofcoefficients with the norm ∥A(t)∥ ≤ a for t ∈ I and the piecewise continuous matrix Q(t), satisfyingthe condition ∥ +∞ ∫Q(u) du ∥ ≤ Ce −2at , t ∈ I, such that systems (1) and (2) are not asymptoticallyequivalent.tThe following assertion establishes [2] the integral test of asymptotic equivalence of systems (1)and (2).limt→+∞Theorem 3. If the matrix of perturbations Q(t) of system (2) satisfies the condition+∞ ∫ ∥∥X A (t, τ)Q(τ)X A (τ, t) ∥ dτ < 1, where X A (t, τ) is the Cauchy matrix of system (1), thentsystem (2) is equivalent to system (1).27

2 Coefficients and Exponents of Reducibility of Linear SystemsLet the perturbation Q(t) satisfy the conditionor the more general condition∥Q(t)∥ ≤ C(Q)e −σt , σ ≥ 0, t ≥ 0, (3)λ[Q] ≡limt→+∞ t−1 ln ∥Q(t)∥ ≤ −σ ≤ 0. (4)These perturbations for σ = 0 in both cases (3) and (4) we assume additionally to be vanishingat infinity: Q(t) → 0 as t → +∞. To every system (1) we put into correspondence the sets R(A)and R λ (A) of those values of the parameter σ in (3) and (4) for which perturbed system (2) for anyperturbation Q(t) satisfying (3) or, respectively, (4), is asymptotically equivalent to non-perturbedsystem (1).Definition. An exact lower bound r(A) of the set R(A) (an exact lower bound ρ(A) of the setR λ (A)) will be called a coefficient of reducibility (an exponent of reducibility) of system (1).Theorem 4 ([2]). The coefficient of reducibility r(A) and the exponent of reducibility ρ(A) ofevery system (1) with piecewise continuous bounded coefficients coincide.This fact allows one to define a new asymptotic invariant of linear systems, i.e., the coefficientof reducibility of the system r A , as a general value of its coefficient and exponent of reducibility.However, despite the fact that the above-mentioned values coincide for every system (1), the behaviorof the coefficient of reducibility r A is distinct with respect to perturbations (3) and (4). Thefollowing theorem [2] establishes this difference.Theorem 5. For any number a > 0, there exists system (1) with the coefficient of reducibilityr A = 2a such that system (2) with any piecewise continuous perturbation Q satisfying condition (3)with Q is reducible to the initial system (1) and non-reducible to that system for some perturbationQ satisfying the condition (4) with σ = r A .Thus it follows from the above results (see also [3, 4]) that both the sets R(A) and R λ (A)are the intervals, and (2a, +∞) ⊂ R λ (A) ⊂ R(A). In addition, despite the fact that the valuesr(A) and ρ(A) coincide for any system (1), their properties are distinct: there exist systems (1)in which R(A) = R λ (A) = (r A , +∞) and, at the same time, there exist systems (1) for whichR(A) = [r A , +∞) and R λ (A) = (r A , +∞). Moreover [5], unlike the coefficient r(A) which can orcannot belong to the set R(A), the exponent of reducibility ρ(A) of system (1) never belongs tothe set R λ (A).Establishment of the above-mentioned properties of the coefficient of reducibility of the lineardifferential system allows one to investigate certain parametric properties of the so-called sets ofnon-reducibility N r (a, σ) and N ρ (a, σ) σ ∈ (0, 2a] of all those systems (1) for every of which thereexists a non-reducible to it system (2) with the matrix Q(t) satisfying, respectively, either condition(3), or the more general condition (4).These sets are non-empty for σ ∈ (0, 2a] and empty for σ > 2a. Moreover, these sets get strictlynarrow as parameter σ ∈ (0, 2a] increases and for σ ∈ (0, 2a] they do not coincide with each other.References[1] N. A. Izobov and S. A. Mazanik, On asymptotically equivalent linear systems under exponentiallydecaying perturbations. (Russian) Differ. Uravn. 42 (2006), No. 2, 168–173; Englishtransl.: Differ. Equ. 42 (2006), No. 2, 182–187.[2] N. A. Izobov and S. A. Mazanik, A general test for the reducibility of linear differential systems,and the properties of the reducibility coefficient. (Russian) Differ. Uravn. 43 (2007), No. 2,191–202; English transl.: Differ. Equ. 43 (2007), No. 2, 196–207.28

[3] N. A. Izobov and S. A. Mazanik, The coefficient of reducibility of linear differential systems.Mem. Differential Equations Math. Phys. 39 (2006), 154–157.[4] N. A. Izobov and S. A. Mazanik, Asymptotic equivalence of linear differential systems underexponentially decaying perturbations. Analytic methods of analysis and differential equations:AMADE 2006, 69–80, Camb. Sci. Publ., Cambridge, 2008.[5] N. A. Izobov and S. A. Mazanik, On a property of the reducibility exponent of linear differentialsystems. (Russian) Differ. Uravn. 44 (2008), No. 3, 323–328; English transl.: Differ. Equ. 44(2008), No. 3, 334–340.29

The Initial-Characteristic Problems for Wave Equations withNonlinear Damping TermOtar JokhadzeA. Razmadze Mathematical Institute of I. Javakhishvili Tbilisi State University, Tbilisi, GeorgiaE-mail: ojokhadze@yahoo.comFor one-dimensional wave equations with nonlinear damping term [1]–[4]u tt − u xx + h(u t ) = f(x, t), (1)in the half-plane Ω := {(x, t) : x ∈ R, t > 0} let us consider the Initial-Cauchy problem with thefollowing conditionsu(x, 0) = φ(x), u t (x, 0) = ψ(x), x ∈ R, (2)where f, h, φ, ψ are given, and u unknown real functions.Conditions, imposed on nonlinear function h, are obtained guaranteeing the existence of aglobal classical solution of (1), (2). The violation of those conditions may cause the blow-up of thesolution.Theorem 1. Let the conditionsbe fulfilled andh ∈ C 2 (R), f ∈ C 1 (Ω), φ ∈ C 2 (R), ψ ∈ C 1 (R)h ′ (s) ≥ −M, s ∈ R, M := const > 0. (3)Then there exists a unique global classical solution u ∈ C 2 (Ω) of the problem (1), (2).Violation of the condition (3) may, generally speaking, cause an absence [5] of the classicalsolution of the problem (1), (2).Let h(s) = −|s| α s, s ∈ R, α > 1 and the function ψ ≥ 0 has the compact supports, for example,the segment [x 1 , x 2 ] ⊂ R; T ∞ := (αc α 1 c 2) −1 > 0, c 2 := (x 2 − x 1 ) −α andTheorem 2. If,∫ x 2c 1 := 1 ψ(x)dx > 0. (4)2x 1φ ′′ (x) ≥ 0, ψ(x) ≥ 0, x ∈ R, f(x, t) ≥ 0, (x, t) ∈ Ω, (5)then for T ∞ > 0 the problem (1), (2) has no classical solution.Remark. Naturally arise a question. What is happening, when some of the conditions (4), (5)are violated.Let the condition (4) be violated i.e. the function ψ ≥ 0 has not compact supportφ(x) = αx + β, α, β := const, x ∈ R, ψ ≡ 0, f ≡ 0.Then the problem (1), (2) has the global classical solution u = αx + β.Let the first condition of (5) be violated:φ = −x 2 , x ∈ R, ψ ≡ 1, f ≡ 1.30

Then the problem (1), (2) has the global classical solution u = −x 2 + t.Let now the second condition of (5) be violated:φ ≡ 0, ψ ≡ −1, f ≡ 1.Then the problem (1), (2) has the global classical solution u = −t.Let at last the third condition of (5) be violated:φ ≡ 1, ψ ≡ 1, f ≡ −1.Then the problem (1), (2) has the global classical solution u = 1 + t.Similarly can be considered the characteristic (Goursat) and initial characteristic (First Darbouxand Chachy–Goursat) problems and obtained corresponding results.References[1] J.-L. Lions and W. A. Strauss, Some non-linear evolution equations. Bull. Soc. Math. France93 (1965), 43–96.[2] J. L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod;Gauthier-Villars, Paris, 1969.[3] L. Hörmander, Lectures on nonlinear hyperbolic differential equations. Mathematiques & Applications(Berlin) [Mathematics & Applications], 26. Springer-Verlag, Berlin, 1997.[4] O. Jokhadze, The Cauchy problem for one-dimensional wave equations with nonlinear dissipativeand damping terms. Proc. A. Razmadze Math. Inst. 155 (2011), 126–130.[5] E. Mitidieri and S. I. Pohozaev, A priori estimates and the absence of solutions of nonlinearpartial differential equations and inequalities. (Russian) Tr. Mat. Inst. Steklova 234 (2001),1–384; English transl.: Proc. Steklov Inst. Math. 2001, No. 3(234), 1–362.31

The Multidimensional Darboux Problem for a Class ofNonlinear Wave EquationsSergo KharibegashviliA. Razmadze Mathematical Institute of I. Javakhishvili Tbilisi State University, Tbilisi, GeorgiaE-mail: khar@rmi.geConsider the nonlinear wave equation of the typeLu := ∂2 un∑∂t 2 − ∂ 2 u∂x 2 ii=1+ f(u) = F, n > 1, (1)where f and F are given real functions, f is a nonlinear function, and u is an unknown real function.Denote by D : t > |x|, x n > 0, the half of the light cone of the future which is boundedby the part S 0 : D ∩ {x n = 0} of the hyperplane x n = 0 and by the half S : t = |x|, x n ≥ 0,of the characteristic conoid C : t = |x| of equation (1). Assume D T := {(x, t) ∈ D : t < T },S 0 T := {(x, t) ∈ S0 : t ≤ T }, S T := {(x, t) ∈ S : t ≤ T }, T > 0. When T = ∞, it is obvious thatD ∞ = D, S 0 ∞ = S 0 and S ∞ = S.For equation (1) we consider the multidimensional version of the Darboux problem: find in thedomain D T a solution u(x, t) of that equation with the boundary conditionsu ∣ ∣S 0T= 0, u ∣ ∣ST= 0. (2)Below we consider the following conditions imposed on the function f:f ∈ C(R), |f(u)| ≤ M 1 + M 2 |u| α , α = const ≥ 0, (3)∫ u0f(s)ds ≥ −M 3 − M 4 u 2 , (4)where M i = const ≥ 0, i = 1, 2, 3, 4.Note that in case α ≤ 1 the inequality (3) results in the equality (4).0Let W 1 2 (D T , ST 0 ∪ S T ) := { u ∈ W2 1(D T ) : u ∣ S 0 = 0 } , where W kT ∪S T 2 (D T ) is the well-knownSobolev space consisting of the functions u ∈ L 2 (D T ) whose all generalized derivatives up to the∣k-th order, inclusive, also belong to the space L 2 (D T ), while the equality u = 0 is understoodin the sense of the trace theory.∣S 0T ∪S TDefinition 1. Let F ∈ L 2 (D T ). A function u ∈ 0 W 1 2 (D T , S 0 T ∪ S T ) is said to be a stronggeneralized solution of the problem (1), (2) of the class W 1 2 in the domain D T if there exists asequence of functions u m ∈ 0 C 2 (D T , S 0 T ∪ S T ) := { u ∈ C 2 (D T ) : u ∣ ∣S 0T ∪S Tin the space 0 W 1 2 (D T , S 0 T ∪ S T ) and Lu m → F in the space L 2 (D T ).= 0 } such that u m → uTheorem 1. Let F ∈ L 2,loc (D ∞ ) and F ∈ L 2 (D T ) for any T > 0. Let 0 ≤ α < n+1n−1and thefunction f satisfy the inequality (3). Moreover, in case α > 1, let the function f satisfy also thecondition (4). Then the problem (1), (2) is globally solvable in the class W2 1 , i.e. for any T > 0 thisproblem has at least one strong generalized solution of the class W2 1 in the domain D T in the senseof Definition 1.Theorem 2. Let F ∈ L 2,loc (D ∞ ) and F ∈ L 2 (D T ) for any T > 0. Let 1 < α < n+1n−1. For thefunction f let the condition (3) be fulfilled but the condition (4) may be violated. Then the problem32

(1), (2) is locally solvable in the class W 1 2 , i.e. there exists a number T 0 = T 0 (F ) > 0 such that forT ≤ T 0 this problem has at least one strong generalized solution of the class W 1 2 in the domain D Tin the sense of Definition 1.Note that in case f(u) = −|u| α , 1 < α < n+1n−1, the condition (4) is violated.Theorem 3. Let f(u) = −|u| α , 1 < α < n+1n−1 . If F ∈ L 2,loc(D ∞ ), F ∈ L 2 (D T ) for any T > 0,and F ≥ 0, F (x, t) ≥ ct −k for t ≥ 1, where c = const > 0, 0 < k = const ≤ n + 1, then there existsa positive number T 1 = T 1 (F ) such that, for T > T 1 , the problem (1), (2) cannot have a stronggeneralized solution of the class W 1 2 in the domain D T in the sense of Definition 1.33

On Two-Point Boundary Value Problems for Higher OrderQuasi-Halflinear Differential Equations with Strong SingularitiesIvan KiguradzeA. Razmadze Mathematical Institute of I. Javakhishvili Tbilisi State University, Tbilisi, GeorgiaE-mail: kig@rmi.geIn the open interval ]a, b[ we consider the differential equation( ∏m )u (2m) = p(t) |u (i−1) | α isgn u + q(t, u) (1)i=1with the Dirichlet or the focal boundary conditionslimt→a u(i−1) (t) = 0,limt→a u(i−1) (t) = 0,lim u (i−1) (t) = 0 (i = 1, . . . , m); (2)t→blim u (i−1) (t) = 0 (i = 1, . . . , m). (3)t→bHere m is a natural number, p : ]a, b[ → R and q : ]a, b[ ×R → R are continuous functions and α i(i = 1, . . . , m) are nonnegative numbers such thatwhereα 1 > 0,m∑α i = 1.We say that the equation (1) has a strong singularity at the point a (at the point b) if∫ ta(s − a) α−1 |p(s)| ds = +∞( ∫bti=1)(b − s) α−1 |p(s)| ds = +∞α = 2m −m∑iα i .i=1for a < t < b,The obtained by us sufficient conditions of solvability of the problem (1), (2) (of the problem(1), (3)) cover the case, where the equation (1) has strong singularities at the points a and b (hasa strong singulary at the point a).By C 2m,m (]a, b[) we denote the space of 2m-times continuously differentiable functions u :]a, b[ → R such that∫ ba|u (m) (s)| 2 ds < +∞. Put1m∏ (2 2m−i+1 ) αi, 1m∏γ 1 =γ2 =(2m−1)!! (2m−2i+1)!! (m−1)! √ 2m−1i=1i=1φ 1 (t) = ( (t − a) −2m + (b − t) −2m) m∏ 12φ 2 (t) = ( (t − a) 1−2m + (b − t) 1−2m) 1 2i=1m∏i=1((m−i)! √ 2m−2i+1) −αi,((t − a) 2i−2m−2 + (b − t) 2i−2m−2) α i2,((t − a) 2i−2m−1 + (b − t) 2i−2m−1) α i2,q ∗ (t, y) = max { |q(t, x)| : |x| ≤ y } , q ∗ (t, y) = inf { |q(t, x)| : |x| ≥ y } .34

Along with (1), we consider the differential equationdepending on the parameter λ ∈ [0, 1].Theorem 1. Let( ∏m )u (n) = λp(t) |u (i−1) | α isgn u, (4)i=1(−1) m p(t) ≤ lφ 1 (t) + p 0 (t)φ 2 (t) for a < t < b, (5)where l and p 0 : ]a, b[ → [0, +∞[ are, respectively, a nonnegative number and a continuous functionsuch thatIf, moreover,γ 1 l < 1,∫ bap 0 (t) dt < +∞.∫ b[ ] m−1limq(t, [(t − a)(b − t)] m− 1 2ρ)(t − a)(b − t)2dt = 0 (6)ρ→+∞ρaand for an arbitrary λ ∈ [0, 1] the problem (4), (2) has only the trivial solution in the spaceC 2m,n (]a, b[), then the problem (1), (2) has at least one solution in the same space.Theorem 2. Let the conditions (5) and (6) hold, where l and p 0 : ]a, b[ → [0, +∞[ are, respectively,a nonnegative number and a continuous function such thatγ 1 l + γ 2∫bap 0 (t) dt < 1. (7)Then the problem (1), (2) in the space C 2m,m (]a, n[) has at least one solution.Theorem 3. Let along with (5) and (6) the conditions(−1) m p(t) ≥ 0, (−1) m q(t, x) ≥ 0 for a < t < b, x ∈ R,limρ→0∫ ba(t − a) m (b − t) m q ∗(t, (t − a) m (b − t) m ρ)ρdt = +∞hold, where l and p 0 : ]a, b[ → [0, +∞[ are, respectively, a nonnegative number and a continuousfunction satisfying the inequality (7). Then the problem (1), (2) in the space C 2m,m (]a, b[) alongwith the trivial solution has a positive and a negative on ]a, b[ solutions.Analogous results have been established for the problem (1), (3).Remark. In Theorem 1 (in Theorems 2 and 3) the condition γ 1 l < 1 (the condition (7)) isunimprovable and it cannot be replaced by the conditionAcknowledgementsγ 1 l ≤ 1( ∫bγ 1 l + γ 2a)p 0 (t) dt ≤ 1 .Supported by the Shota Rustaveli National Science Foundation (Project # GNSF/ST09 175 3-101).35

On Conditional Well-Posedness of Nonlocal Problems forFourth Order Singular Linear Hyperbolic EquationsTariel KiguradzeFlorida Institute of Technology, Melbourne, USAE-mail: tkigurad@fit.eduIn the rectangle Ω = [0, a] × [0, b] consider the linear hyperbolic equationu (2,2) =with the nonlocal boundary conditionsHere∫ a02∑i=1 k=1u(s, y) dα i (s) = 0 for 0 ≤ y ≤ b,2∑h ik (x, y)u (i−1,k−1) + h(x, y) (1)∫ b0u(x, t) dβ k (t) = 0 for 0 ≤ x ≤ a (i, k = 1, 2). (2)u (i,k) (x, y) = ∂i+k u(x, y)∂x i ∂y k (i, k = 0, 1, 2),h ik : Ω → R (i, k = 1, 2) are measurable functions, h ∈ L(Ω), and α i : [0, a] → R and β i : [0, b] → R(i = 1, 2) are functions of bounded variation. Moreover,where∫ a∆ 1 (x) = α 2 (a)xα i (0) = 0, β i (0) = 0, ∆ i (0) = 1 (i = 1, 2), (3)∫ aα 1 (s) ds − α 1 (a)xα 2 (s) ds,∫ b∆ 2 (y) = β 2 (b)y∫ bβ 1 (t) dt − β 1 (b)yβ 2 (t) dt.We employ the concepts of well-posedness and conditional well-posedness for problem (1), (2)that were introduced in [1].Along with the equation (1) consider the corresponding homogeneous equationu (2,2) =and introduce the functions:{1 for s ≥ t,χ(s, t) =0 for s < t,g 1 (x, s) =g 2 (y, t) =∫ a0∫ b0α 1 (τ) dτ∫ as2∑i=1 k=1∫ aα 2 (τ) dτ −2∑h ik (x, y)u (i−1,k−1) , (1 0 )sα 1 (τ) dτ∫ a0α 2 (τ) dτ+ (s − a)∆ 1 (0) + (a − x)∆ 1 (s) + χ(x, s)(x − s) for 0 ≤ x, s ≤ a,β 1 (τ) dτ∫ bt∫ bβ 2 (τ) dτ −tβ 1 (τ) dτ∫ b0β 2 (τ) dτ+ (t − b)∆ 2 (0) + (b − y)∆ 2 (t) + χ(y, t)(y − t) for 0 ≤ y, t ≤ b,36

φ 11 (x) = max { |g 1 (x, s)| : 0 ≤ s ≤ a } ,φ 12 (x) = sup {∣ ∣∆ 1 (s) − χ(x, s) ∣ }: 0 ≤ s ≤ a, s ≠ x ,φ 21 (y) = max { |g 2 (y, t)| : 0 ≤ t ≤ b } , φ 22 (y) = sup{ ∣∣∆ 2 (t) − χ(y, t) ∣ ∣ : 0 ≤ t ≤ b, t ≠ y } .Theorem 1. If along with (3) the condition∫ b ∫ a00φ 1i (x)φ 2k (y)|h ik (x, y)| dx dy < +∞ (i, k = 1, 2)holds, then problem (1), (2) is conditionally well-posed if and only if the corresponding homogeneousproblem (1 0 ), (2) has only the trivial solution.Theorem 2. If along with (3) the inequality2∑2∑i=1 k=1 0∫ b ∫ a0φ i (x)ψ k (y)|h ik (x, y)| dx dy < 1 (4)holds, then problem (1), (2) is conditionally well-posed. Moreover, if h ik ∈ L(Ω) (i, k = 1, 2), thenproblem (1), (2) is well-posed.Theorem 3. If conditions (3) and (4) hold, and∫ b ∫ a00|h 11 (x, y)| dx dy = +∞,then problem (1), (2) is conditionally well-posed but not well-posed.References[1] I. Kiguradze and T. Kiguradze, Conditions for Well-Posedness of Nonlocal Problems for HigherOrder Linear Differential Equations with Singularities. Georgian Math. J. 18 (2011), No. 4.37

Relaxational Oscillations and Diffusive Chaos inBelousov’s ReactionA. Yu. KolesovP. G. Demidov Yaroslavl State University, Yaroslavl’, RussiaE-mail: kolesov@uniyar.ac.ruN. Kh. RozovLomonosov Moscow State University, Moscow, RussiaE-mail: fpo.mgu@mail.ruThe surprising phenomenon of chemistry, the reaction discovered by B. P. Belousov in 1951(frequently called also as Belousov–Zhabotinsky’s reaction), is an instructive episode in the historyof home natural science deserving separate narration. The mathematical model of the reactionearns the attention and it should be included in the compulsory course of ordinary differentialequations, since its discussion may, on the one hand, be anticipated by a rather simply organizablevisual experiment and, on the other hand, it shows an exclusive value and might of mathematizationfor penetration into the essence of natural phenomena.We will consider modification of the mathematical model of Belousov’s reaction, namely, asystem of three ordinary differential equationsẋ = r 1[1 + a(1 − z) − x]x, ẏ = r2 [x − y]y, ż = r 3[αx + (1 − α)y − z]z, (1)where x, y, z are the analogues of concentration densities of chemical substances; the parametersr 1 , r 2 , r 3 and a are positive ones, and the parameter α ∈ (0, 1). The most natural from the chemicalpoint of view is the assumption that the parameter a is “very large” and the rest parameters areof order 1.Belousov’s reaction has, for the first time, shown experimentally the possibility for the chemicalreaction to run periodically, and moreover, the stages running “very fast” in the course of the reactionalternate with those running “rather slowly”. Such periodical processes are called relaxationaloscillations. Therefore of interest is the study of relaxation regime in system (1) in which we passfrom the parameter a to the small parameter ε = 1/a.We fix an arbitrary compact set Ω 0 of the semi-strip {(u 0 , v 0 ) : u 0 > 0, 0 < v 0 < 1} and denoteby()L ε (u 0 , v 0 ) = x(t, u 0 , v 0 , ε), y(t, u 0 , v 0 , ε), z(t, u 0 , v 0 , ε) : t ≥ 0, (u 0 , v 0 ) ∈ Ω 0 (2)the trajectory of the systemεẋ = r 1[1 − z + ε(1 − x)]x, ẏ = r2 [x − y]y, ż = r 3[αx + (1 − α)y − z]z, (3)emanating for t = 0 from the point (x, y, z) = (1, u 0 , v 0 ). We introduce into consideration thesecond positive root t = T (u 0 , v 0 , ε) (if it exists) of the equation x(t, u 0 , v 0 , ε) = 1 and on theintersecting plane {(x, y, z) : x = 1} we define the Poincaré successor operator Π ε (u 0 , v 0 ):Π ε (u 0 , v 0 ) = ( y(t, u 0 , v 0 , ε), z(t, u 0 , v 0 , ε) )∣ ∣∣t=T(u0 ,v 0 ,ε) . (4)Theorem 1. On the set Ω 0 , in the metric of the space C 1 (Ω 0 ; R 2 ) there exists the limitingoperatorlim Π ε(u 0 , v 0 ) = Π 0 (u 0 , v 0 ) (5)ε→0describing constructively.38

Theorem 2. Under the conditionα 0, 0 < v 0 < 1} there exists atleast one stable fixed point, and for all sufficiently small ε > 0 the initial operator Π ε has a stablefixed point (u ε , v ε ) with asymptotically close to (u 0 , v 0 ) components. In system (3), this point isassociated with the stable relaxational cycle L ε .Theorem 3. The time of motion of the phase point of system (3) along the “rapid segments”of the trajectory L ε is of order ε ln(1/ε), while the time of motion along its “slow segments” admitsas ε → 0 a finite positive limit.The distributed model, associated with system (1), i.e., the parabolic boundary value problem∂x∂t = ∂ 2 xdD0 1∂s 2 + r [ ] ∂x1 1 + a(1 − z) − x x, ∣ = ∂x∣ = 0,∂s s=0 ∂s s=1∂y ∂ 2 y∂y∣ = ∂y∣ = 0,s=0 s=1∂z∂t = dD0 3∂t = dD0 2∂s 2 + r 2[x − y]y,∂s ∂s∂ 2 z∂s 2 + r [ ] ∂z3 αx + (1 − α)y − z z, ∣ = ∂z∣ = 0∂s s=0 ∂s s=1on the segment 0 ≤ s ≤ 1, is also considered. Here d, D j , j = 1, 2, 3 are positive parameters. Ofinterest is the investigation of attractors arising in its phase space (x, y, , z) ∈ C([0, 1]; R 3 ) as ddecreases.For the distributed model (7), by means of numerical experiments we have managed to establisha phenomenon of diffusion chaos, an unrestricted growth of dimensions of chaotic attractors as d →0, i.e., for proportional decrease of coefficients of diffusion. Two types of chaotic autooscillations,the relaxation chaos and that of the type of self-organization, are discovered.(7)References[1] B. P. Belousov, Periodically acting reaction and its mechanisms. Autowave processes in systemswith diffusion. Gor’kii. Institute of Applied Physics Acad. Sci. SSSR, 1981, 176–186.[2] A. M. Zhabotinskii, Concentrational autooscillations. Nauka, Moscow, 1974.39

On Three Problems of the Qualitative Theory ofDifferential EquationsA. V. Kostin, L. L. Kol’tsova, T. Yu. Kosetskaya, and T. V. KondratenkoI. I. Mechnikov Odessa National University, Odessa, UkraineE-mail: a − kostin@ukr.net; koltsova.liliya@gmail.comI. The asymptotics as t → +∞ of solutions of monotone type for a real nonlinear first orderdifferential equation (ODE-1)s∑k=1n∑p k (t)y α k(y ′ ) β k+ p k (t)y α k(y ′ ) β k= 0 (1)k=s+1is considered.To find formal asymptotic representations for solutions of monotone type, it is assumed thatthere exists at least one such a solution y(t) of equation (1), and for that solution asymptoticallybasic are the summands appearing in the sums ∑k=1. Under that assumption, we have obtained fory(t) possible formal asymptotic representations (exact, or requiring more precise determination).An asymptotic character of the obtained formal asymptotic representations is investigated. ( L.L. Kol’tsova and A. V. Kostin, The results of the work are submitted for publication in Mem.Differential Equations Math. Phys. (Tbilisi).II. We investigate a classical problem dealing with asymptotic stability (AS) of the real linearhomogeneous differential equation ODE-n, n ′ geq2 (n is order of ODE),y (n) + p 1 (t)y (n−1) + · · · + p n (t)y = 0, t ∈ I = [t 0 , +∞[ (2)under the condition that the roots λ i (t) (i = 1, n) of the corresponding characteristic equation aresuch that λ i (t) ∈ C 1 +∞ ∫(I), Reλ i (t) < 0, Re λ i (t)dt = −∞, ∃ Re λ i (+∞), −∞ ≤ Re λ i (+∞) ≤ 0t 0(i = 1, n).The case n = 2 is considered rather thoroughly. We have managed to prove the property ofasymptotic stability in the case of real λ i (t) (i = 1, 2) in the following subcases:(1) λ 1 (+∞) ∈ R − =] − ∞, 0[ , λ 2 (+∞) = 0, λ ′ 1 (t) = o(λ 2(t));(2) λ i (+∞) = 0 (i = 1, 2), λ ′ 1 (t) = o(λ 1(t)λ 2 (t));(3) λ 1 (+∞) = 0, λ 2 (+∞) = −∞, λ ′ 1 (t) = o(λ2 1 (t));(4) λ i (+∞) = −∞ (i = 1, 2), λ ′ 1 (t) = o(λ2 1 (t)), λ′ 1 (t) = o(λ 1(t)λ 2 (t)).The subcase λ i (+∞) ∈ R − (i = 1, 2) is known.The case of complex-conjugate roots λ i (t) (i = 1, 2) is considered analogously. (T. Yu. Kosetskaya,A. V. Kostin).III. We investigate the problem on the existence of a particular solution y(t) from some classof real functions{K f(t) : f(t) ∈ C 2 (R), sup |f(t)| < +∞ }Rin ODE-2,y ′′ + ay ′ + by = f(t) + µF (t, y, y ′ ), (3)40

where (t, y, y ′ ) ∈ G { t ∈ R, |y −u(t)| ≤ h 1 , |y ′ −u ′ }(t)| ≤ h 2 , hi < +∞ (i = 1, 2), a, b ∈ R, f(t) ∈ K,the roots λ i (i = 1, 2) of the equation λ 2 + aλ + b = 0 are such that Re λ i ≠ 0 (i = 1, 2), and thecondition∫ t{+∞ (R λ i > 0)f(τ) exp Re λ i (t − τ) dτ ∈ K, A i =−∞ (R λ i < 0) , f(t) ∈ KA iis fulfilled, u(t) ∈ K is a unique solution of the class K of the equationy ′′ + ay ′ + by = f(t),F, F y, ′ Fy ′ (∈ C(G), sup |F | + |F′ ′ y | + |Fy ′ | ) < +∞ (F (t, f(t), f ′ (t)) ∈ K, if f(t) ∈ K).′GUsing the Perron transformation, we have obtained the estimate for a small parameter µ whichguarantees the solvability of the problem. In the capacity of the class K one can consider a classof almost-periodic, slowly varying and another functions. (A. V. Kostin, T. V. Kondratenko)41

Investigation of Stability of Stochastic Differential Equations byUsing Lyapunov Functions of Constant SignsA. A. LevakovBelarusian State University, Minsk, BelarusE-mail: alevakov@bsu.byThe method of Lyapunov functions is one of the most effective one for the investigation ofstability of differential systems, in particular, of stochastic differential systems. The main purposeof the report is the theorem on the stability of stochastic differential equations by using Lyapunovfunctions of constant signs.Consider the stochastic differential equationdx(t) = f(x(t))dt + g(x(t))dW (t), (1)with the Borel-measurable functions f : R d → R d , g : R d → R d×d .Definition 1. If there exists a process x(t) given on some probability space (Ω, F, P ) with aflow of σ-algebras F t , satisfying the following conditions:(1) there exists a (F t )-moment of the stop e such that the process x(t)1 [0,e) (t) is (F t )-coordinated,has continuous trajectories for t < e a.s. and lim sup ∥x(t)∥ = ∞ if e < ∞;t↑e(2) there exists the (F t )-Brownian motion W (t), W (0) = 0 a.s.;(3) the processes f(x(t)) and g(x(t)) belong, respectively, to the spaces L loc1 and L loc2 , where Lloc iis a set of all measurable (F t )-coordinated processes ψ such that for every moment of theσ∫stop σ, 0 ≤ σ < e the condition ∥ψ(s, ω)∥ i ds < ∞ a.s. i ∈ {1, 2} is fulfilled;(4) with probability I for all t ∈ [0, e), the equality0∫ tx(t) = x(0) +0∫ tf(x(τ)) dτ +0g(x(τ)) dW (τ)holds, and the set (x(t), Ω, F, P, F t , W (t), e) (or briefly, x(t)) is called a weak solution ofequation (1).We choose rows of the matrix g with numbers β 1 , . . . , β l , β 1 < · · · < β l , and let β l+1 < · · · < β dbe numbers of the rest rows. We construct the matrix⎛g β1 gβ ⊤ 1. . . g β1 g ⊤ ⎞β lσ β1 ,...,β l(x 1 , . . . , x d ) = ⎝ . . . . . . . . . ⎠ ,g βl gβ ⊤ 1. . . g βl gβ ⊤ lwhere g βj is the row with the number β j of the matrix g, and also we construct the sets H 1 , H 2 ;H 1 (β 1 , . . . , β l ) = { (x β1 , . . . , x βl ) | for any open neighborhood U (xβ1 ,...,x βl ) of the point (x β1 , . . . , x βl )there exists a number a > 0 such that the integral∫sup(det σβ1 ,...,β l(x 1 , . . . , x d ) ) −1 dxβ1 . . . dx βl(x βl+1 ,...,x βd )∈D 2 (0,a)U (xβ1 ,...,x βl )42

is either indeterminate, or is equal to ∞}, where D 2 (0, a) = { (x βl+1 , . . . , x βd ) | (x 2 β l+1+ · · · +x 2 β d) 1/2 ≤ a } ; H 2 (β 1 , . . . , β l ) = { (x β1 , . . . , x βl ) ∈ H c 1 (β 1, . . . , β l ) | for any open neighborhoodU (xβ1 ,...,x βl ) of the point (x β1 , . . . , x βl ) there exists a number a > 0 such that the function(sup det σβ1 ,...,β l(x 1 , . . . , x d ) ) −1 : U → [0, ∞](x βl+1 ,...,x βd )∈D 2 (0,a)is not Borel-measurable } (under the complement H c 1 of the set H 1 is understood the complementin the space of variables (x β1 , . . . , x βl ), and under the open neighborhood is understood the neighborhoodwhich is open in the space of the same variables (x β1 , . . . , x βl )).Let Ĥ(β 1, . . . , β l ) = H 1 (β 1 , . . . , β l ) ∪ H 2 (β 1 , . . . , β l ).We will say that the real function h(x) = h(x 1 , . . . , x d ) satisfies Condition C) if there existindices β 1 , . . . , β l , β 1 < · · · < β l ≤ d such that:1) the function h for every fixed (x β1 , . . . , x βl ) is continuous with respect to the rest components(x βl+1 , . . . , x βd ) of the vector x;2) in the space of variables (x β1 , . . . , x βl ) there is a closed set H with the properties:(2a) H ⊃ Ĥ(β 1, . . . , β l );(2b) the set {(x 1 , . . . , x d )| (x β1 , . . . , x βl ) ∈ H} belongs to the set of points of continuity of themapping h;(2c) the function σ β1 ,...,β l(x 1 , . . . , x d ) for every fixed (x β1 , . . . , x βl ) ∈ H c is continuous withrespect to the variables (x βl+1 , . . . , x βd ).Let the functions f(x) and g(x) be Borel-measurable and locally bounded, the components ofthe functions f(x), σ(x) = g(x)g ⊤ (x) satisfy Condition C). Then for any given probability ν on(R d , β(R d )) equation (1) has a weak solution with an initial distribution ν [1].Definition 2. A zero solution is said to be ϖ-stable, if for any ε > 0 there exists δ > 0 suchthat for a weak solution x(t) of equation (1), for which ∥x(0)∥ ≤ δ a.s., we have E(∥x(t)∥ ϖ ) ≤ ε∀ t ≥ 0 (E is a mathematical expectation).Definition 3. A zero solution is said to be asymptotically ϖ-stable, if it is ϖ-stable andthere exists M > 0 such that, for any weak solution x(t) for which ∥x(0)∥ ≤ M a.s., the relationlimt→+∞ E(∥x(t)∥ϖ ) = 0 is fulfilled.Condition L). There exists a twice continuously differentiable function V : R d → R + suchthat ∀x ∈ R d ,∂V (x)BV (x) =∂xf(x) + 1 ( ∂ 22 tr V (x))∂x 2 g(x)g ⊤ (x) ≤ 0.Assume M V = {x ∈ R d | BV (x) = 0}. We say that a weak solution (x(t), W (t), Ω, F, P, F t )belongs to the set M V if)g(x(t))g ⊤ (x(t)) = 0∂V (x(t))∂xfor (µ × P )- almost all (t, ω) ∈ R + × Ω.f(x(t)) + 1 2 tr ( ∂ 2 V (x(t))∂x 2Condition A). There exist the constants r > 1, σ > 0, M > 0 such that for any weak solutionx(t) of the system (1), satisfying the condition ∥x(0)∥ ≤ σ, the inequality E(∥x(t)∥ r ) ≤ M ∀ t ≥ 0is fulfilled.Condition B). The system (1) has no nonzero weak solutions x(t) such that x(0) = 0 a.s.Definition 4. The process x(t), t ∈ ] − ∞, 0] given on some probability space (Ω, F, P ) is saidto be a weak solution of equation (1) on the interval ] − ∞, 0], if:43

1) for every t 0 ∈ ] − ∞, 0[ , there exists the extension (˜Ω, ˜F, ˜P ) of the space (Ω, F, P ) and onthat extension there exists the flow (˜F t ), t ∈ [t 0 , 0], such that on (˜Ω, ˜F, ˜P ) with (˜F t ) one candefine the (˜F t )-Brownian motion W (t), W (t 0 ) = 0 a.s.;2) the processes f(x(t)) and g(x(t)) belong, respectively, to the spaces L 1 and L 2 , where L i is theset of all measurable (˜F t )-coordinated processes ψ(t), t ∈ [t 0 , 0] such that ∥ψ(s, ω)∥ i ds < ∞t 0a.s., i ∈ {1, 2};∫ t∫ t3) with probability 1 for all t ∈ [t 0 , 0] the equality x(t) = x(t 0 ) + f(x(τ)) dτ + g(x(τ)) dW (τ)t 0 t 0holds.The other necessary definitions and notation can be found in [1].Theorem. Let the functions f(x) and g(x) be Borel-measurable and locally bounded, the componentsof the functions f(x), σ(x) = g(x)g ⊤ (x) satisfy Condition C), the system (1) satisfyConditions A), B) and L), and let 0 < s < r. If there exists a constant a > 0 such that thesystem (1) has no nonzero weak solutions x(t) on the interval ] − ∞, 0] possessing the properties:x(t) ∈ m V = {x ∈ R d | V (x) =} ∀ t ∈ ] − ∞, 0] a.s.; E(∥x(t)∥ s ) ≤ a ∀ t ∈ ] − ∞, 0], then azero solution of the system (1) is s-stable. If, moreover, there exists a constant b > 0 such thatthe system has no nonzero weak solutions x(t), t ∈ [0, ∞[ , satisfying the conditions x(t) ∈ M V ,E(∥x(t)∥ s ) ≤ b ∀ t ∈ [0, ∞[ , then a zero solution is asymptotically s-stable.∫ 0References[1] A. A. Levakov, Stochastic differential equations. Belorusian State University, Minsk, 2009.44

On Hybrid StabilizationElena LitsynThe Weizmann Institute of Science, Rehovot, IsraelThe Research Institute, The Colledge of Judea and Samaria, Ariel, IsraelE-mail: elena@wisdom.weizmann.ac.ilYurii V. NepomnyashchikhPerm State University, Perm, RussiaE-mail: yura@prognoz.psu.ruArcady PonosovInstitutt for matematiske fag, NLH, NorwayE-mail: matap@imf.nlh.no1 IntroductionHybrid systems are those combining both discrete and continuous dynamics. Many examples ofhybrid systems can be found in manufacturing systems, intelligent vehicle highway systems, veriouschemical plants. Hybrid systems also arise when there is a necessity of combining logical decisionwith the generation of continuous control laws.An important question is how to stabilize a continuous control plant through an interactionwith a discrete time controller. Such a ”hybrid” feedback may help when the ordinary feedbackfails to stabilize the system.An example of a linear system which cannot be stabilized by the ordinary output feedback isthe harmonic oscillator:dξdt = η,dηdt= −ξ + u, u = u(y), y = ξ. (1.1)The only measured quantity (output), which is allowed to control, is the position variable ξ. Althoughthis last system is both controllable and observable, it cannot be stabilized by (even discontinuous)output feedbacks.It was however shown by Z. Artstein (1995), there exists a special hybrid feedback control, underwhich system (1.1) becomes asymptotically stable. Z. Artstein conjectured also that hybrid controlscan stabilize general linear systems of ordinary differential equations.2 The Main ResultWe give here the following affirmative answer to Artstein’s conjecture on the existence of a hybridstabilizer.Theorem 2.1. Under assumptions of controllability of (A, B) and observability of (A, C) thesystemẋ = Ax + Bu,u = u(y), y = Cxis stabilizable by a hybrid feedback control designed with the help of a discrete automaton which hasat most countable number of locations.Proof is based on the classical stabilization technique as well as on some recent results in thetheory of functional-differential equations in an essential way.45

3 Applications to Theorem 2.1We give here two examples.1. Predator-Prey InteractionsAn example of a situation, where hybrid feedback controls may be of use, is given by a populationmodel with an arbitrary number of species, some of them being observable and the others not.Although such a model is nonlinear, but linearization about the equilibrium state provides a linearsystem with a control depending on a part of variables while the rest variables may be not observableat all, so that these variables cannot be used for setting up a control function. A stabilization ofthe unstable equilibrium state may become then problematical if we use ordinary feedback, only.What does help is hybrid feedback controls.2. A Contribution to the Theory of Love AffairsHere is another example which illustrates the power of stabilization by hybrid feedback controls.The mathematical model for the dynamics of love affairs is given by a 2×2-system of linear equations(Strogatz, 1994). We consider the following particular case, which is called the star-crossed romancebetween Romeo and Juliet.Ṙ = aJ, J ˙ = −bR, (3.1)R(t) = Romeo’s love/hate for Juliet at time t,J(t) = Juliet’s love/hate for Romeo at time t(love gives positive sign to variables, while hate makes variables negative).From the system (3.1) we obtain the following tragic picture.The more Romeo loves Juliet, the more Juliet wants to run away and hide. But when Romeogets discouraged and backs off, Juliet begins to find him strangely attractive. Romeo, on the otherhand, tends to echo her; he warms up when she loves him, and grows cold when she hates him. . . Thesad outcome of their affair is, of course, a never-ending cycle of love and hate, because solutionsof the system (3.1) are ellipses with a center at (0, 0).We observe that ordinary feedback controls do not help Romeo and Juliet in this unpleasantsituation. If, for instance, we insert a feedback control like u = αR in the first equation suddenlyturns Romeo either into an “eager beaver” (α > 0), or into a “cautious lover” (α < 0) which seemsto be quite unrealistic as soon as one particular Juliet is concerned. The same applies to Juliet.The only possibility is therefore to try making influence on the constants a and b in the system(3.1). But any feedback control like u = αJ or/and v = βR will never change the sad and tragicpicture of never-ending ellipses in the phase-plane, because the corresponding coefficient matrixwill always have imaginary eigenvalues!We propose a hybrid feedback scenario, which according to Theorem 2.1 does make solutionsof the system (3.1) asymptotically stable. This scenario can be described explicitly and does leadto a kind of “happy end”. Naturally, the relationship between Romeo and Juliet will fizzle out tomutual indifference and the disaster will be prevented.Unfortunately, it is impossible to use a similar procedure to help Romeo and Juliet becomingeventually daring to each other, because hybrid feedback controls can stabilize solutions, but theycannot exclude oscillations.46

On Oscillation and Nonoscillation of Two-DimensionalLinear Differential SystemsAlexander Lomtatidze and Jiří ŠremrInstitute of Mathematics, Academy of Sciences of the Czech Republic, branch in Brno, Brno,Czech RepublicE-mail: bacho@math.muni.cz; sremr@ipm.czConsider the systemu ′ = q(t)v,v ′ = −p(t)u,where p, q : [0, +∞[ → R are locally Lebesgue integrable functions such that(1)q(t) ≥ 0 for a. e. t ≥ 0,∫+∞q(s) ds < +∞,and q ≢ 0 in any neighbourhood of +∞. Under a solution of system (1) we understand a vectorfunction(u, v): [0, +∞[ → R 2 with locally absolutely continuous components satisfying equalities(1) almost everywhere in [0, +∞[ . A solution (u, v) of system (1) is said to be nontrivial if u ≢ 0in any neighbourhood of +∞. A nontrivial solution (u, v) of system (1) is called oscillatory if thefunction u has a sequence of zeros tending to infinity.Definition 1. System (1) is said to be oscillatory if every nontrivial solution of this system isoscillatory, and nonoscillatory otherwise.whereFor any λ > 1, we put∫ tc(t; λ) := (λ − 1)f λ−1 (t)0f(t) :=∫0( ∫s)q(s)f λ f λ (ξ)p(ξ) dξ ds for t ≥ 0,(s)0+∞tq(s)ds for t ≥ 0.The following theorem is an analogue of the well-known Hartman–Wintner theorem.orTheorem 1. Let there exist λ > 1 such that eitherThen system (1) is oscillatory.−∞ < lim inft→+∞lim c(t; λ) = +∞t→+∞c(t; λ) < lim sup c(t; λ).t→+∞If we take this theorem into account it is obvious that, for given λ > 1, the following two casesremain uncovered. The first case, wherethere exists a finite limitlim c(t; λ), (2)t→+∞47

and the second case, wherelim inf c(t; λ) = −∞.t→+∞Below, we establish new oscillation and nonoscillation criteria assuming that (2) holds for someλ > 1. Having such λ, we denotewhereQ(t; λ) :=Moreover, for any µ < 1, we put( ∫1t)f λ−1 c 0 (λ) − f λ (s)p(s) ds(t)0c 0 (λ) =for t ≥ 0,lim c(t; λ). (3)t→+∞Finally, let∫ tH(t; µ) := f 1−µ (t)0f µ (s)p(s) ds for t ≥ 0.Q ∗ (λ) = lim inf Q(t; λ),t→+∞ Q∗H ∗ (µ) = lim inf H(t; µ),t→+∞ H∗Now we formulate our main results.(λ) = lim sup Q(t; λ),t→+∞(µ) = lim sup H(t; µ).t→+∞Theorem 2. Let there exist λ > 1 such that condition (2) holds andlim supt→+∞−1 (c0f λ−1 (λ) − c(t; λ) ) > 1 (t) ln f(t)4 ,where the number c 0 (λ) is defined by formula (3). Then system (1) is oscillatory.Corollary 1. Let there exist λ > 1 and µ < 1 such that condition (2) holds andThen system (1) is oscillatory.or( ) 1lim inf Q(t; λ) + H(t; µ) >t→+∞4(λ − 1) + 14(1 − µ) . (4)Corollary 2. Let there exist λ > 1 such that condition (2) holds and eitherQ ∗ (λ) >H ∗ (µ) >for some µ < 1. Then system (1) is nonoscillatory.14(λ − 1)14(1 − µ)Remark 1. It might seem that if assumption (5) is satisfied in the previous corollary thenassumption (2) is redundant. However, one can show that, under assumption (5), the functionc(· ; λ) possesses a limit for every λ > 1 and c(t; λ) > −∞. If this limit is equal to +∞,limt→+∞then system (1) is oscillatory according to Theorem 1. Therefore, assumption (2) in the previouscorollary is necessary in a certain sense also in the case where inequality (5) is supposed to besatisfied.48(5)

The next theorem deals with the upper limit of the sum on the left-hand side of inequality (4)and thus it complements Corollary 1 in a certain sense.Theorem 3. Let there exist λ > 1 and µ < 1 such that condition (2) holds andThen system (1) is oscillatory.or( ) λ 2lim sup Q(t; λ) + H(t; µ) >t→+∞4(λ − 1) + µ 24(1 − µ) .The following statement complements Corollary 2.Theorem 4. Let there exist λ > 1 and µ < 1 such that condition (2) holds and eitherλ(2 − λ)4(λ − 1) ≤ Q ∗(λ) ≤µ(2 − µ)4(1 − µ) ≤ H ∗(µ) ≤Then system (1) is oscillatory.14(λ − 1) , H∗ (µ) >14(1 − µ) , Q∗ (λ) >µ 24(1 − µ) + 1 + √ 1 − 4(λ − 1)Q ∗ (λ)2λ 24(λ − 1) − 1 − √ 1 − 4(1 − µ)H ∗ (µ).2At last, we present two results dealing with nonoscillation of system (1).Theorem 5. Let there exist λ > 1 such that condition (2) holds and eitheror(2λ − 3)(2λ − 1)− < Q ∗ (λ), Q ∗ (λ) 1. Therefore, assumption (2)in Theorems 5 and 6 is necessary also in the case where inequalities (6) and (7) are supposed tobe satisfied.49

An Infinite Dimensional Generalization of Weak Exponents forSolutions of Equations in Total DerivativesE. K. Makarov and E. A. ShakhInstitute of Mathematics of the National Academy of Sciences of Belarus, Minsk, BelarusE-mail: jcm@im.bas-net.byLet E and F be real Banach spaces, K ⊂ E be a closed convex cone with a bounded base.Consider a linear completely integrable equation in total derivativesy ′ h = A(x)hy, y ∈ F, h ∈ E, x ∈ E, (1)with a bounded continuous coefficient A : E → L(E, L(F, F )) (here and in the sequel, we usenotation and notions from [1]). Let E(y) be a set of all linear continuous functionals µ ∈ E ∗ suchthat the inequality lim sup ∥x∥ −1 (ln y(x) + µx) ≤ 0 holds.x→∞, x∈KIn [2], the interrelation is established between characteristic functionals and (weak) characteristicexponents of solutions of equation (1) for a finite-dimensional E in the form E(y) = E(exp ψ[y]),where ψ[y](x) := limt→+∞ t−1 ln y(tx) is a modified exponent of a solution y. This result is valid onlyfor a finite-dimensional E. Therefore it is necessary to generalize the above notions in order toobtain some analog of the statement in [2] for the settings of infinite-dimensional E.For this, we introduce new exponent ψ[y](φ) by means of the formulaψ[y](φ) :=limt→+∞ t−1 ln ∥ ∥y(φ(t)) ∥ ∥,as the functional on the space Φ of continuous functions φ : [0, +∞[ → K such that ∥φ(t)∥ → ∞,as t → +∞ and sup ∥φ(t)∥/t < +∞.Theorem. The inclusion λ ∈ E(y) holds if and only if for any φ ∈ Φ the inequality ψ[y](φ) +λ(φ) ≤ 0, where λ(φ) := lim t −1 λφ(t), holds.t→∞References[1] I. V. Gaǐshun, Linear total differential equations. (Russian) “Nauka i Tekhnika”, Minsk, 1989.[2] E. K. Makarov, On the interrelation between characteristic functionals and weak characteristicexponents. (Russian) Differentsial’nye Uravneniya 30 (1994), No. 3, 393–399; translation inDifferential Equations 30 (1994), No. 3, 362–367.50

A Certain Class of Discontinuous Dynamical Systems in the PlaneKateryna Mamsa and Yuriy PerestyukTaras Shevchenko National University of Kyiv, Kyiv, UkraineE-mail: Perestyuk@gmail.comIn the monographs [1]–[3], a theory of impulsive differential equations is built. Mainly, the mathematicalmodels of evolutionary processes that undergo impulsive perturbations at fixed momentsof time or at the moments, when the moving point meets the given hypersurfaces in the extendedphase space are considered. However, in the monographs [1]–[3], the importance of studying ofsystems with impulsive perturbations that occur at the moments, when the phase point meets thegiven sets in the phase space is emphasized. In this report, we investigate a linear differentialsystems in the plane that are subjected to impulsive perturbations on the given line, i.e. systemsof the formdxdt = Ax, ⟨a, x⟩ ̸= 0; ∆x∣ ∣⟨a,x⟩=0= Bx,where x ∈ R 2 , A and B are constant matricies, a is a constant vector.The motion of the phase point is defined by the differential system ẋ = Ax, when this point isoutside of the line ⟨a, x⟩ = 0 and immediately transfers to a point x + = (E + B)x(t ∗ ) at the timewhen phase point meets with the line ⟨a, x⟩ = 0.We have indicated the necessary and sufficient conditions for the existence of one-impulsive andtwo-impulsive discontinuous cycles of this system, as well as conditions for asymptotic stability ofthe zero equilibrium position.References[1] A. M. Samoǐlenko and N. A. Perestyuk, Impulsive differential equations. (Russian) VishchaShkola, Kiev, 1987.[2] A. M. Samoǐlenko and N. A. Perestyuk, Impulsive differential equations. World Scientific Serieson Nonlinear Science. Series A: Monographs and Treatises, 14. World Scientific PublishingCo., Inc., River Edge, NJ, 1995.[3] N. A. Perestyuk, V. A. Plotnikov, A. M. Samoǐlenko, and N. V. Skripnik, Impulsive differentialequations with a multivalued and discontinuous right-hand side. (Russian) Proceedings of Instituteof Mathematics of NAS of Ukraine. Mathematics and its Applications. 67. NatsionalnaAkademiya Nauk Ukraini, Institut Matematiki, Kiev, 2007.51

Generalized Linear Differential Equations in a Banach Space:Continuous Dependence on a ParameterG. A. MonteiroUniversidade de São Paulo, Instituto de Ciências Matemáticas e Computação, ICMC-USP, SãoCarlos, SP, BrazilE-mail: gam@icmc.usp.br.M. TvrdýMathematical Institute, Academy of Sciences of Czech Republic, Prague, Czech RepublicE-mail: tvrdy@math.cas.czIn what follows, X is a Banach space and L(X) is the Banach space of bounded linear operatorson X. By ∥ · ∥ X we denote the norm in a Banach space X. Further, BV ([a, b], X) is theset of X valued functions of bounded variation on [a, b] and G([a, b], X) is the set of X valuedfunctions having on [a, b] all one-sided limits (i.e. X valued functions regulated on [a, b]). A coupleP =(D, ξ), where D={α 0 , α 1 , . . . , α m } and ξ=(ξ 1 , . . . , ξ m )∈[a, b] m , is said to be a partition of [a, b]if a=α 0

∫ tx k (t) = ˜x k +ad[A k (s)]x k (s)+f k (t)−f k (a), t ∈ [a, b], k∈N. (1 k )The following assumptions are crucial for the existence of solutions to (1) and (1 k )[I − ∆ − A(t) ] −1 ∈L(X) for all t ∈ (a, b], (2)and[I − ∆ − A k (t) ] −1 ∈L(X) for all t ∈ (a, b], k∈N. (2k )For the basic properties of generalized linear differential equations in a Banach space, see [7].Our first result extends that by M. Ashordia [1] valid for the case X = R n .Theorem 1. Let A, A k satisfy (1) and (1 k ), and letA k ⇒ A on [a, b], (3)α ∗ ( ):= sup v arba A k

[4] Z. Halas, G. Monteiro, and M. Tvrdý, Emphatic convergence and sequential solutions ofgeneralized linear differential equations. (submitted).[5] Z. Opial, Continuous parameter dependence in linear systems of differential equations. J.Differential Equations 3 (1967), 571–579.[6]Š. Schwabik, Abstract Perron-Stieltjes integral. Math. Bohem. 121 (1996), No. 4, 425–447.[7]Š. Schwabik, Linear Stieltjes integral equations in Banach spaces. Math. Bohem. 124 (1999),No. 4, 433–457.54

Two-Point Boundary Value Problems for Strongly SingularHigher-Order Linear Differential Equations withDeviating ArgumentsSulkhan MukhigulashviliMathematical Institute, Academy of Sciences of the Czech Republic, Brno, Czech RepublicIlia State University, Tbilisi, GeorgiaE-mail: mukhig@ipm.czConsider the boundary value problemu (n) (t) =m∑p j (t)u (j−1) (τ j (t)) + q(t) for a < t < b (1)j=1with the two-point boundary conditionsu (i−1) (a) = 0 (j = 1, . . . , m), u (i−1) (b) = 0 (j = 1, . . . , n − m), (2)where n ≥ 2, m is the integer part of n/2, p j , q ∈ L lob (]a, b[) (j = 1, . . . , m), and τ j : ]a, b[ → ]a, b[are measurable functions. By u (j−1) (a) (by u (j−1) (b)) we denote the right (the left) limit of thefunction u (j−1) at the point a (at the point b).The Agarwal–Kiguradze type theorems [1] are obtained by us, which contains unimprovable ina certain sense conditions guaranteeing the unique solvability of problem (1), (2). The results below∑cover the strongly singular case, where m ∫ b(t − a) n−j (b − t) n−j |p j (t)| dt = +∞.We use the following notations.L α,β (]a, b[) (L 2 α,βj=1 a(]a, b[)) is the space of integrable (square integrable) with the weight(t − a) α (b − t) β functions y : ]a, b[ → R, with the norm ∥y∥ Lα,β =(∥y∥L 2 = ( ∫ b(s − a) α (b − s) β y 2 (s) ds ) 1/2 );α,β∫ tca∫ ba(s − a) α (b − s) β |y(s)| ds˜L 2 α,β (]a, b[) is the space of functions y ∈ L loc(]a, b[) such that ỹ ∈ L 2 α,β(]a, b[), where ỹ(t) =y(s) ds, c = (a + b)/2. The norm in ˜L 2 α,β(]a, b[) is defined by the equality{[ ∫t= max (s − a) α( ∫ t∥y∥˜L2 α,βa{[ ∫bs+ max (b − s) β( ∫ts] 2 1/2 }y(ξ) dξ)ds : a ≤ t ≤ c +t] 2 1/2 }y(ξ) dξ)ds : c ≤ t ≤ b . (3)˜C n−1,m (]a, b[) is the space of (n − 1)-times continuously differentiable functions y : ]a, b[ → Rsuch that∫ ba|u (m) (s)| 2 ds

By h j and f j (j = 1, . . . , m) we denote the functions and operator, respectively, defined by theequalitiesh 1 (t, s)=∣∫ ts(ξ−a) n−2m[ (−1) n−m p 1 (ξ) ] + dξ ∣ ∣∣∣,f j (c)(t, s) =∣∫ ts∫(ξ − a) n−2m |p j (ξ)|∣τ j (ξ)ξh j (t, s)=∣∫ ts(ξ−a) n−2m p j (ξ) dξ∣ (j =2, . . . , m),∣ ∣∣∣1/2 (ξ 1 − c) 2(m−j) dξ 1 dξ∣ (j = 1, . . . , m).Along with (1), we consider the corresponding homogeneous equationm∑u (n) (t) = p j (t)u (j−1) (τ j (t)). (1 0 )j=1Theorem 1. Let there exist a 0 ∈ ]a, b[ , b 0 ∈ ]a 0 , b[ and positive numbers l kj and γ kj (k = 0, 1,j = 1, . . . , m) such that(t − a) 2m−j h j (t, s) ≤ l 0j for a ≤ t ≤ s < a 0 , lim sup(t − a) m− 1 2 −γ 0jf j (a)(t, s) < +∞,t→a(b − t) 2m−j h j (t, s) ≤ l 1j for b 0 ≤ s ≤ t < b, lim sup(b − t) m− 1 2 −γ 1jf j (b)(t, s) < +∞,m∑j=1t→b(2m − j)2 2m−j+1(2m − 1)!!(2m − 2j + 1)!! l kj < 1 (k = 0, 1).If, moreover, problem (1 0 ), (2) has only the trivial solution in the space ˜C n−1,m (]a, b[), then problem(1), (2) is uniquely solvable in this space for every q ∈ ˜L 2n−2m−2,2m−2 (]a, b[).Theorem 2. Let there exist numbers t ∗ ∈ ]a, b[ , l kj > 0, l kj ≥ 0, and γ kj > 0 (k = 0, 1;j = 1, . . . , m) such that along with inequalitiesm∑ ((2m − j)2 2m−j+1 l 0j(2m − 1)!!(2m − 2j + 1)!! + 2 2m−j−1 (t ∗ − a) γ 0jl)0j(2m − 2j − 1)!!(2m − 3)!! √ < 1 2γ 0j 2 ,the conditionsj=1m∑ ((2m − j)2 2m−j+1 l 1j(2m − 1)!!(2m − 2j + 1)!! + 2 2m−j−1 (b − t ∗ ) γ 0jl)1j(2m − 2j − 1)!!(2m − 3)!! √ < 1 2γ 1j 2 ,j=1(t − a) 2m−j h j (t, s) ≤ l 0j , (t − a) m−γ 0j−1/2 f j (a)(t, s) ≤ l 0j for a < t ≤ s ≤ t ∗ ,(b − t) 2m−j h j (t, s) ≤ l 1j , (b − t) m−γ 1j−1/2 f j (b)(t, s) ≤ l 1j for t ∗ ≤ s ≤ t < bhold. Then for every q ∈ ˜L 2 2n−2m−2,2m−2 (]a, b[), (1), (2) is uniquely solvable in the space ˜C n−1,m (]a, b[).AcknowledgementsSupported by the Shota Rustaveli National Science Foundation (Project # GNSF/ST09 175 3-101)and the Academy of Sciences of the Czech Republic (Institutional Research Plan # AV0Z10190503).References[1] R. P. Agarwal and I. Kiguradze, Two-point boundary value problems for higher-order lineardifferential equations with strong singularities. Boundary Value Problems, 2006, 1–32; ArticleID 83910.56

Asymptotics of Solutions of a Linear Homogeneous System ofDifferential Equations in the Case of Asymptotically Equivalent,as t → +∞, Roots of a Characteristic EquationV. V. NikonenkoI. I. Mechnikov Odessa National University, Odessa, UkraineE-mail: a − kostin@ukr.netThe investigation of linear homogeneous systems (LHS) was carried out by the author by combiningthe method of generalized shearing transformations [1] and the method of L-diagonal systems[2]. The cases which are particular for the well-known methods are considered.We study the 2-dimensional linear homogeneous systemε(t)Y ′ = ( D −1 ΛD + α(t)B + Q(t) ) Y, (1)where t ∈ [t 0 , +∞[ = I, Y = (y 1 , y 2 ) T , the scalar functions ε(t) and α(t), the constant matrices D,Λ, B and the matrix Q(t) are, in a general case, complex, and the following conditions are fulfilled:1) 0 ≠ ε(t) ∈ C 1 (I),+∞ ∫t 0|ε −1 (t)| dt = +∞,2) 0 ≠ α(t) ∈ C 1 (I), α(+∞) = 0,( )d11 d3) D =12, detD ≠ 0,d 21 d 22( )λ eΛ = , λ ≠ 0, e ∈ {0, 1},0 λ( )b11 bB =12, B ≠ O (O is a zero matrix),b 21 b 22( )q11 (t) qQ(t) =12 (t), Q(t) ∈ C(I), Q(t) = o(|α(t)|),q 21 (t) q 22 (t)4) ∃ a finite or an infinite limit limt→+∞ ε(t) w′ (t)w(t) = a, where w = w(t) = α 1 2 (t) (for a specific choiceof root).Denote( )A = D −1 a11 aBD =12,a 21 a 22( )P (t) = D −1 p11 (t) pQ(t)D =12 (t), P (t) = o(|α(t)|).p 21 (t) p 22 (t)Theorem 1. Let e = 1, |a| < +∞, the conditions (1)–(4) be fulfilled, and4 + a 21 ≠ 0, |α ′ | + |w ′ | + ∣a 2) ′ ∣ ∣∣ + ∥P ′ ( P ) ′ ∥ ∥( ∥∥ ∥∥ P) ′ ∥ ∥∥w 2 ∥ + ∥ + ∈ wL1 (I),α(εw′57

Re ( wε (˜µ 1 − ˜µ 2 ) ) do not change its sign in I, where ˜µ i (t) (i = 1, 2) are the roots of the equation˜µ 2 − ˜µ(wa 12 − ε w′w 2 + p 22w + wa 11 + p ) (11− a 21 + p )21 (1 )+ αa12 + p 12 = 0.wwThen LHS (1) has two linearly independent solutionsY i (t) = e∫ t(λε(τ) + w′ (τ)ε(τ) ˜µ i(τ)t 0)dτ(d1i + o(1)·d 2i + o(1))(i = 1, 2). (2)If all d ik ≠ 0 (i, k = 1, 2), then asymptotics (2) are exact.The cases, where among d ik (i, k = 1, 2) there are those equal to zero, and also the case e = 1,a = ∞, are considered.Generalization of the obtained results to the case of LHS (1) of dimension n > 2 (Λ is Jordan’sblock) is also considered.References[1] V. V. Nikonenko, On an asymptotic representation of solutions of quasi-linear systems ofordinary differential equations. Dissertation for the degree of Candidate of the Phys.-Math.Sciences, Tbilisi, 1989, 169p.[2] M. V. Fedoryuk, Asymptotical methods in the theory of ordinary linear differential equations.Nauka, Moscow, 1980.58

On Oscillations of Solutions to Second-Order Linear DelayDifferential EquationsZdeněk OpluštilInstitute of Mathematics, Brno University of Technology, Brno, Czech RepublicJiří ŠremrInstitute of Mathematics, Academy of Sciences of the Czech Republic, branch in Brno, Brno,Czech RepublicE-mail: sremr@ipm.czOn the half-line R + = [0, +∞[ , we consider the second-order linear delay differential equationu ′′ (t) + p(t)u(τ(t)) = 0, (1)where p: R + → R + is a locally Lebesgue integrable function and τ : R + → R + is a measurablefunction such thatτ(t) ≤ t for a. e. t ≥ 0,lim ess inf{τ(s) : s ≥ t} = +∞.t→+∞Solutions to equation (1) can be defined in various ways. Since we are interested in propertiesof solutions in the neighbourhood of +∞, we introduce the following commonly used definition.Definition 1. Let t 0 ∈ R + and a 0 = ess inf{τ(t) : t ≥ t 0 }. A continuous function u: [a 0 , +∞[ →R is said to be a solution to equation (1) on the interval [t 0 , +∞[ if it is absolutely continuoustogether with its first derivative on every compact interval contained in [t 0 , +∞[ and satisfiesequality (1) almost everywhere in [t 0 , +∞[ . A solution u to equation (1) on the interval [t 0 , +∞[is called proper if the inequality sup{|u(s)| : s ≥ t} > 0 holds for t ≥ t 0 .Definition 2. A proper solution u to equation (1) is said to be oscillatory if it has a sequenceof zeros tending to infinity, and non-oscillatory otherwise.It is known that if the integral+∞ ∫0τ(s)p(s) ds is convergent, then equation (1) has propernon-oscillatory solutions. Therefore, we will assume throughout the paper thatLet us putG ∗ = lim inft→+∞∫+∞0∫1tsτ(s)p(s) ds,t0τ(s)p(s) ds = +∞. (2)G ∗ = lim supt→+∞∫1tsτ(s)p(s) ds.t0Proposition 1. Let condition (2) hold and G ∗ > 1. Then every proper solution to equation(1) is oscillatory.In view of Proposition 1, it is natural to suppose in the sequel thatG ∗ ≤ 1. (3)A Wintner type oscillation criterion is presented in the next theorem.59

Theorem 1. Let conditions (2) and (3) be fulfilled and let there exist λ < 1 and ε ∈ [0, 1[ suchthat∫+∞s λ( τ(s)) 1−εG∗p(s)ds= +∞. (4)sThen every proper solution to equation (1) is oscillatory.0Theorem 2. Let conditions (2) and (3) hold and let there exist ε ∈ [0, 1[ such thatlim supt→+∞∫1tsln t0( τ(s)Then every proper solution to equation (1) is oscillatory.In view of Theorem 1, we can assume in the sequel that∫+∞0s λ( τ(s)ss) 1−εG∗p(s)ds >14 . (5)) 1−εG∗p(s)ds < +∞ for all λ < 1, ε ∈ [0, 1[ .It allows one to define, for any λ < 1 and ε ∈ [0, 1[, the function∫+∞Q(t; λ, ε) := t 1−λ s λ( τ(s)) 1−εG∗p(s)ds for t > 0.sMoreover, for any µ > 1 and ε ∈ [0, 1[ , we putH(t; µ, ε) := 1t µ−1By using the lower and upper limitst∫tQ ∗ (λ, ε) = lim inf Q(t; λ, ε),t→+∞ Q∗H ∗ (µ, ε) = lim inf H(t; µ, ε),t→+∞ H∗0s µ( τ(s)) 1−εG∗p(s)ds for t > 0.s(λ, ε) = lim sup Q(t; λ, ε),t→+∞(µ, ε) = lim sup H(t; µ, ε),t→+∞we establish new Hille and Nehari type oscillation criteria, which coincide with the well-knownresults in the case of ordinary differential equations.Theorem 3. Let conditions (2) and (3) be fulfilled and let there exist λ < 1, µ > 1, andε ∈ [0, 1[ such that( ) λ 2lim sup Q(t; λ, ε) + H(t; µ, ε) >t→+∞4(1 − λ) + µ 24(µ − 1) . (6)Then every proper solution to equation (1) is oscillatory.As a corollary of Theorem 3 (with µ = 2 and λ = 0, respectively) we obtainCorollary 1. Let conditions (2) and (3) be fulfilled and let there exist ε ∈ [0, 1[ such thateither Q ∗ (λ, ε) >(2 − λ)24(1 − λ) for some λ < 1, or H∗ (µ, ε) >Then every proper solution to equation (1) is oscillatory.60µ 24(µ − 1)for some µ > 1.

The next theorem deals with the lower limit of the sum on the left-hand side of inequality (6)and thus it complements Theorem 3 in a certain sense.Theorem 4. Let conditions (2) and (3) be fulfilled and let there exist λ < 1, µ > 1, andε ∈ [0, 1[ such that( ) 1lim inf Q(t; λ, ε) + H(t; µ, ε) >t→+∞4(1 − λ) + 14(µ − 1) . (7)Then every proper solution to equation (1) is oscillatory.Theorem 4 yieldsCorollary 2. Let conditions (2) and (3) be fulfilled and let there exist ε ∈ [0, 1[ such thateither Q ∗ (λ, ε) >14(1 − λ) for some λ < 1, or H ∗(µ, ε) >14(µ − 1)for some µ > 1.Then every proper solution to equation (1) is oscillatory.Now we give a statement complementing Corollary 2.Theorem 5. Let conditions (2) and (3) be fulfilled and let there exist λ < 1, µ > 1, andε ∈ [0, 1[ such that eitherorH ∗ (µ, ε) >λ(2 − λ)4(1 − λ) ≤ Q ∗(λ, ε) ≤µ 24(µ − 1) − 1 214(1 − λ) , (8))(9)(1 − √ 1 − 4(1 − λ)Q ∗ (λ, ε)µ(2 − µ)4(µ − 1) ≤ H 1∗(µ, ε) ≤4(µ − 1) , (10)Q ∗ λ 2(λ, ε) >4(1 − λ) + 1 (1 + √ )1 − 4(µ − 1)H ∗ (µ, ε) .2(11)Then every proper solution to equation (1) is oscillatory.If both conditions (8) and (10) are satisfied then oscillation criteria (9) and (11) can be slightlyrefined as is presented in the last statement.Theorem 6. Let conditions (2) and (3) be fulfilled and let there exist λ < 1, µ > 1, andε ∈ [0, 1[ such that inequalities (8) and (10) are satisfied. If( )lim sup Q(t; λ, ε) + H(t; µ, ε) >t→+∞> Q ∗ (λ, ε) + H ∗ (µ, ε) + 1 (√1 − 4(1 − λ)Q∗ (λ, ε) + √ )1 − 4(µ − 1)H ∗ (µ, ε) ,2then every proper solution to equation (1) is oscillatory.τ(t)tRemark 1. If we assume, in addition, that there are numbers α > 0 and t 0 ≥ 0 such that≥ α for a. e. t ≥ t 0 , then we can put ε = 1 in all above presented statements.61

Solvability and Well-Posednessof Two-Point Weighted Singular Boundary Value ProblemsNino PartsvaniaA. Razmadze Mathematical Institute of I. Javakhishvili Tbilisi State University, Tbilisi, GeorgiaE-mail: ninopa@rmi.geIn an open interval ]a, b[, we consider the second order nonlinear differential equationu ′′ = f(t, u) (1)with two-point weighted boundary conditions of one of the following two types:andlim supt→a|u(t)|< +∞, lim sup(t − a) αt→blim supt→a|u(t)|< +∞ (2)(b − t) β|u(t)|< +∞, lim(t − a) α t→b u′ (t) = 0. (3)Here f :]a, b[×R → R is a continuous function, α ∈]0, 1[, and β ∈]0, 1[.Following R. P. Agarwal and I. Kiguradze [1,2] we say that Eq. (1) with respect to the timevariable has a strong singularity at the point a (at the point b) if for any t 0 ∈]a, b[ and x > 0∫t 0the condition (t − a) [ |f(t, x)|−f(t, x) sgn x ] dt = +∞ ( ∫ b(b − t) [ |f(t, x)|−f(t, x) sgn x ] = +∞ ) isat 0satisfied.Theorems obtained by us contain unimprovable in a certain sense sufficient conditions for thesolvability and well-posedness of the problem (1), (2) (of the problem (1), (3)), at that these theorems,unlike the previous well-known results, cover the case, where Eq. (1) with respect to thetime variable has strong singularities at the points a and b (has a strong singularity at the pointa).Before passing to the formulation of the main results, we introduce some notation.f ∗ (t, x) = max { |f(t, y)| : 0 ≤ y ≤ x } for a < t < b, x ≥ 0.By G 0 and G 1 we denote the Green functions of the problems u ′′ = 0; u(a) = u(b) = 0 andu ′′ = 0; u(a) = u ′ (b) = 0, respectively, i.e.,⎧(s − a)(t − b)⎪⎨G 0 (t, s) = b − a⎪⎩(t − a)(s − b)b − afor a ≤ s ≤ t ≤ b,for a ≤ t < s ≤ b,For any continuous function h :]a, b[→ R, we assume∫bν α,β (h) = sup{(t − a) −α (b − t) −β∫bν α (h) = sup{(t − a) −αaaand G 1 (t, s) ={a − s for a ≤ s ≤ t ≤ b,a − t for a ≤ t < s ≤ b.}|G 0 (t, s)h(s)| ds : a < t < b ,}|G 1 (t, s)h(s)| ds : a < t < b .62

Definition. The problem (1), (2) (the problem (1), (3)) is said to be well-posed if for anycontinuous function h :]a, b[→ R, satisfying the condition ν α,β (h) < +∞ ( ν α (h) < +∞ ) , theperturbed differential equationv ′′ = f(t, v) + h(t) (4)has a unique solution, satisfying the boundary conditions (2) (the boundary conditions (3)), andthere exists a positive constant r, independent of the function h, such that in the interval ]a, b[ theinequality |u(t) − v(t)| ≤ rν α,β (h)(t − a) α (b − t) β ( |u(t) − v(t)| ≤ rν α (h)(t − a) α) is satisfied, whereu and v are the solutions of (1),(2) and (4),(2) (of (1),(3) and (4),(3)), respectively.The following statements are valid.Theorem 1. Let there exist a constant l ∈ [0, 1[ and a continuous function q :]a, b[→ [0, +∞[such that( )α(1 − α)f(t, x) sgn x ≥ −l(t − a) 2 + 2αβ β(1 − β)+(t − a)(b − t) (b − t) 2 |x| − q(t) for a < t < b, x ∈ R,and ν α,β (q) < +∞. Then the problem (1), (2) has at least one solution.Theorem 2. Let there exist a constant l ∈ [0, 1[ such that( )α(1 − α)f(t, x) − f(t, y) ≥ −l(t − a) 2 + 2αβ β(1 − β)+(t − a)(b − t) (b − t) 2 (x − y) for a < t < b, x > y,and ν α,β (f(·, 0)) < +∞. Then the problem (1), (2) is well-posed.Theorem 3. Let there exist a constant l < α(1−α) and a continuous function q :]a, b[→ [0, +∞[such thatlf(t, x) sgn x ≥ − |x| − q(t) for a < t < b, x ∈ R(t − a) 2and ν α (q) < +∞. If, moreover, the condition∫ btholds, then the problem (1), (3) has at least one solution.f ∗ (s, x)ds < +∞ for a < t < b, x > 0 (5)Theorem 4. Let there exist a constant l < α(1 − α) such thatlf(t, x) − f(t, y) ≥ − (x − y) for a < t < b, x > y.(t − a) 2If, moreover, ν α (f(·, 0)) < +∞ and the condition (5) holds, then the problem (1), (3) is well-posed.AcknowledgementsSupported by the Shota Rustaveli National Science Foundation (Project # GNSF/ST09 175 3-101).References[1] R. P. Agarwal and I. Kiguradze, Two-point boundary value problems for higher-order lineardifferential equations with strong singularities. Boundary Value Problems, 2006, 1–32; ArticleID 83910.[2] I. Kiguradze, On two-point boundary value problems for higher order singular ordinary differentialequations. Mem. Differential Equations Math. Phys. 31 (2004), 101–107.63

On Invariant Sets of Impulsive Multi-Frequency SystemsMykola Perestyuk and Petro FeketaTaras Shevchenko National University of Kyiv, Kyiv, UkraineE-mail: pmo@univ.kiev.ua, petro.feketa@gmail.comWe consider a system of differential equations, defined in the direct product of an m-dimensionaltorus T m and an n-dimensional Euclidean space E n that undergo impulsive perturbations at themoments when the phase point φ meets a given set in the phase spacedφdt = a(φ),dx= A(φ)x + f(φ), φ ∉ Γ,dt∆x ∣ φ∈Γ= B(φ)x + g(φ),(1)where φ = (φ 1 , . . . , φ m ) T ∈ T m , x = (x 1 , . . . , x n ) T ∈ E n , a(φ) is a continuous 2π-periodic withrespect to each of the components φ v , v = 1, . . . , m vector function that satisfies a Lipschitzcondition with respect to φ. Functions A(φ), B(φ) are continuous 2π-periodic with respect toeach of the components φ v , v = 1, . . . , m square matrices; f(φ), g(φ) are continuous (piecewisecontinuous with first kind discontinuities in the set Γ) 2π-periodic with respect to each of thecomponents φ v , v = 1, . . . , m vector functions.We assume that the set Γ is a subset of the torus T m , which is a manifold of dimension m − 1defined by the equation Φ(φ) = 0 for some continuous scalar 2π-periodic with respect to each ofthe components φ v , v = 1, . . . , m function.Denote by t i (φ), i ∈ Z the solutions of the equation Φ(φ t (φ)) = 0 that are the moments ofimpulsive action in system (1). Let the function Φ(φ) be such that the solutions t = t i (φ) existsince otherwise system (1) would not be an impulsive system.We call a point φ ∗ an ω-limit point of the trajectory φ t (φ) if there exists a sequence {t n } n∈Nin R so thatlim t n = +∞,n→+∞lim φ t n(φ) = φ ∗ .n→+∞The set of all ω-limit points for a given trajectory φ t (φ) is called ω-limit set of the trajectory φ t (φ)and denoted by Ω φ . DenoteΩ =∪Ω φ ,φ∈T mand assume that the matrices A(φ) and B(φ) are constant in the domain Ω:A(φ) ∣ ∣φ∈Ω= Ã,B(φ)∣ ∣φ∈Ω= ˜B.We will obtain sufficient conditions for the existence and asymptotic stability of an invariant set ofthe system (1) in terms of the eigenvalues of the matrices à and ˜B. Denoteγ = max Reλj(Ã), j=1,...,n α2 = max λ (j (E + ˜B) T (E + ˜B) ) .j=1,...,nTheorem 1. Let the moments of impulsive perturbations {t i (φ)} be such that uniformly withrespect to t ∈ R there exists a finite limiti(t, t +lim˜T )= p. (2)˜T →∞ ˜T64

If the following inequality holdsthen system (1) has an asymptotically stable invariant set.γ + p ln α < 0, (3)Such approach may be extended to the nonlinear system of the formdφdt = a(φ),dxdt = A 0(φ)x + A 1 (φ, x)x + f(φ), φ ∉ Γ,∆x ∣ ∣φ∈Γ= B 0 (φ)x + B 1 (φ, x)x + g(φ).Theorem 2. Let the matrices A 0 (φ) and B 0 (φ) be constant in the domain Ω, uniformly withrespect to t ∈ R there exist a finite limit (2) and the inequality (3) hold. Then there exist sufficientlysmall constants a 1 and b 1 and sufficiently small Lipschitz constants L A and L B such that forany continuous 2π-periodic with respect to each of the components φ v , v = 1, . . . , m functions∥A 1 (φ, x) and B 1 (φ, x) such that maxA 1 (φ, x) ∥ ≤ a 1 ,φ∈T m,x∈ ¯J hx ′ , x ′′ ∈ ¯J h ,∥∥A 1 (φ, x ′ ) − A 1 (φ, x ′′ ) ∥ ≤ L A ∥x ′ − x ′′ ∥,system (4) has an asymptotically stable invariant set.(4)∥maxB 1 (φ, x) ∥ ≤ b 1 and for anyφ∈T m,x∈ ¯J h∥∥B 1 (φ, x ′ ) − B 1 (φ, x ′′ ) ∥ ∥ ≤ L B ∥x ′ − x ′′ ∥,In summary, we have obtained sufficient conditions for the existence and asymptotic stabilityof invariant sets of a linear impulsive system of differential equations defined in T m × E n that hasspecific properties in the ω-limit set Ω of the trajectories φ t (φ). We have proved that it is sufficientto impose some restrictions on system (1) only in the domain Ω to guarantee the existence andasymptotic stability of the invariant set.References[1] S. I. Doudzyany and M. O. Perestyuk, On stability of trivial invariant torus of a certain classof systems with impulsive perturbations. Ukrainian Math. J. 50 (1998), No. 3, 338–349.[2] M. O. Perestyuk and S. Ī. Baloha, Existence of an invariant torus for a class of systems ofdifferential equations. (Ukrainian) Nelīnīīnī Koliv. 11 (2008), No. 4, 520–529 (2009); Englishtransl.: Nonlinear Oscil. (N. Y.) 11 (2008), No. 4, 548–558.[3] M. O. Perestyuk and P. V. Feketa, Invariant manifolds of a class of systems of differentialequations with impulse perturbation. (Ukrainian) Nelīnīīnī Koliv. 13 (2010), No. 2, 240–252;English transl.: Nonlinear Oscil. (N. Y.) 13 (2010), No. 2, 260–273.[4] M. Perestyuk and P. Feketa, Invariant sets of impulsive differential equations with particularitiesin ω-limit set. Abstr. Appl. Anal. 2011, Art. ID 970469, 14 pp.[5] N. A. Perestyuk and V. A. Plotnikov, A. M. Samoilenko, and N. V. Skripnik, Differentialequations with impulse effects: multivalued right-hand sides with discontinuities. DeGruyterStudies in Mathematics 40, Walter de Gruyter Co., Berlin, 2011.[6] A. M. Samoǐlenko, Elements of the mathematical theory of multi-frequency oscillations. Translatedfrom the 1987 Russian original by Yuri Chapovsky. Mathematics and its Applications(Soviet Series), 71. Kluwer Academic Publishers Group, Dordrecht, 1991.[7] A. M. Samoǐlenko and N. A. Perestyuk, Impulsive differential equations. World Scientific Serieson Nonlinear Science. Series A: Monographs and Treatises, 14. World Scientific PublishingCo., Inc., River Edge, NJ, 1995.65

On the Well-Possedness of the Weighted Cauchy Problem forNonlinear Singular Differential Equations of Higher Orders withDeviating ArgumentsB. PůžaMasaryk University, Institute of Mathematics of the Academy of Sciences of the Czech Republic,Brno, Czech RepublicE-mail: puza@math.muni.czZ. SokhadzeAkaki Tsereteli State University, Kutaisi, GeorgiaE-mail: z.soxadze@gmail.comIn a finite interval ]a, b[ we consider the differential equation()u (n) (t) = f t, u(τ 1 (t)), . . . , u (n−1) (τ n (t))(1)with the weighted initial conditionslim supt→a( |u (i−1) (t)|ρ (i−1) (t))< +∞ (i = 1, . . . , n), (2)where f : ]a, b[ ×R n → R is a function, satisfying the local Carathéodory conditions, τ i : ]a, b[ → ]a, b](i = 1, . . . , n) are measurable functions, and ρ : [a, b] → [0, +∞[ is the (n − 1)-times continuouslydifferentiable function such thatρ (i−1) (a) = 0,ρ (i−1) (t) > 0 for a < t ≤ b (i = 1, . . . , n).Let q : ]a, b] → ]0, +∞[ be some nonincreasing function. The problems (1), (2) is said to bewell-posed with the weight q, if for any integrable with the weight q function h : ]a, b[ → R,satisfying the conditionthe differential equation{ ( ∫tν q,n (h; ρ) def= supa)}q(s)h(s) ds /q(t)ρ (n−1) (t) : a < t ≤ b < +∞,()u (n) (t) = f t, u(τ 1 (t)), . . . , u (n−1) (τ n (t)) + h(t)under the initial conditions (2) has a unique solution u h and there exists the positive not dependingon r function h, such that∣∣u (i−1)h(t) − u (i−1)0 (t) ∣ ≤ rν q,n (h; ρ)ρ (i−1) (t) for a ≤ t ≤ b (i = 1, . . . , n),where u 0 is a solution of the problem (1), (2).The problem (1), (2) is said to be well-posed if it is well-posed with the weight q(t) ≡ 1.Theorem 1. Let in the domain ]a, b[ ×R n the condition∣ f(t, x1 , . . . , x n ) − f(t, y 1 , . . . , y n ) ∣ ∣ ≤66n∑h i (t)|x i − y i |, (3)i=1

e fulfilled, where h i ∈ L loc (]a, b]) (i = 1, . . . , n). Let, moreover,{ ( ∫tsupand there exist a number γ ∈ ]0, 1[ such thatn∑∫ ti=1 aThen the problem (1), (2) is well-posed.a)}|f(s, 0, . . . , 0)| ds /ρ (n−1) (t) : a < t ≤ b < +∞ρ (i−1) (τ i (s))h i (s) ds ≤ γρ (n−1) (t) for a ≤ t ≤ b.Theorem 2. Let n ≥ 2, the function f 0 (t) def= f(t, 0, . . . , 0) be integrable with the weight q, andν n,q (f 0 ; ρ) < +∞, where q : ]a, b] → ]0, +∞[ is some nonincreasing function, satisfying the equality(lim q(t)ρ (n−1) (t) ) = 0.t→aLet, moreover, in the domain ]a, b[ ×R n the condition (3) be fulfilled, where h i ∈ L loc (]a, b]) (i =1, . . . , n), and there exist the numbers m ∈ {1, . . . , n − 1} and γ ∈ ]0, 1[ such that∫ ta(∑ mexp∫ ti=1 s(x − a) n−i(n − i)!)h i (x) dx ≤ q(s)q(t)[∑ m q(s) ∣ ρ (i−1) (τ i (s)) − ρ (i−1) (s) ∣ hi (s) +i=1n∑i=m+1≤ γq(t)ρ (n−1) (t) for a < t ≤ b.for a < s ≤ t ≤ b,]ρ (i−1) (τ i (s))h i (s) ds ≤Then the problem (1), (2) is well-posed with the weight q.The above-formulated theorems cover the case in which the equation (1) is strongly singular,i.e., the case, where{ ∣∣f(t,where f ∗ (t, x) = max x1 , . . . , x n ) ∣ n∑ :∫ ba(t − a) µ f ∗ (t, x) dt = +∞ for µ ≥ 0, x > 0,i=1}|x i | ≤ x . It should be also noted that the conditionγ ∈ ]0, 1[ in these theorems is unimprovable and it cannot be replaced by the condition γ = 1.AcknowledgementThis work was supported by the Education Ministry of Czech Republic (Project # MSM0021622409)and by the Shota Rustaveli National Science Foundation (Project # GNSF/ST09 175 3 101).67

Three Types of Solutions of Some Singular NonlinearDifferential EquationIrena RachůnkováPalacký University, Olomouc, Czech RepublicE-mail: irena.rachunkova@upol.czThe differential equation(p(t)u′ ) ′ = p(t)f(u) (1)is investigated on the positive half line under the assumptions (2)–(6):f ∈ Lip loc (R), ∃ L ∈ (0, ∞) : f(L) = 0, (2)∃ L 0 ∈ [−∞, 0) : xf(x) < 0, x ∈ (L 0 , 0) ∪ (0, L), (3)∃ ¯B ∈ (L 0 , 0) : F ( ¯B) = F (L),where F (x) = −∫ x0f(z) dz, x ∈ R, (4)p ∈ C[0, ∞) ∩ C 1 (0, ∞), p(0) = 0, (5)p ′ (t) > 0, t ∈ (0, ∞),Due to p(0) = 0, equation (1) has a singularity at t = 0.The following results are proved.p ′ (t)lim = 0. (6)t→∞ p(t)1. For each B < 0 equation (1) has a unique solution u B ∈ C 1 [0, ∞) ∩ C 2 (0, ∞) which satisfiesthe initial conditionsu(0) = B, u ′ (0) = 0. (7)2. A solution u B of problem (1), (7) satisfying sup{u(t) : t ∈ [0, ∞)} < L is called a dampedsolution. If M d is the set of all B < 0 such that u B is a damped solution, then M d is nonemptyand open in (−∞, 0).3. A solution u B of problem (1), (7) satisfying sup{u(t) : t ∈ [0, ∞)} > L is called an escapesolution. If M e is the set of all B < 0 such that u B is an escape solution, then M e is open in(−∞, 0). In addition, M e is nonempty provided one of the following additional assumptions (A1),(A2), (A3), or (A4) is valid:• (A1): L 0 ∈ (−∞, 0), f(L 0 ) = 0.• (A2):f(x) > 0 for x ∈ (−∞, 0) and0 ≤ lim supx→−∞f(x)|x|< ∞.• (A3):f(x) > 0 for x ∈ (−∞, 0) and there exists k ≥ 2 such thatlimt→0+p ′ (t)∈ (0, ∞).tk−2 Further, there exists r ∈ (1, k+2k−2) such that f fulfilsf(x)limx→−∞ |x| r ∈ (0, ∞).68

• (A4): ∫1dsp(s) < ∞.4. If M e is nonempty, then problem (1), (7) has a solution u such that0sup { u(t) : t ∈ [0, ∞) } = L.Such solution is called homoclinic. It is increasing and lim t→∞ u(t) = L.5. Some other additional conditions for p and f which give asymptotic formulas for dampedand homoclinic solutions are discussed.References[1] I. Rachůnková and J. Tomeček, Bubble-type solutions of nonlinear singular problems. Math.Comput. Modelling 51 (2010), No. 5-6, 658–669.[2] I. Rachůnková and J. Tomeček, Strictly increasing solutions of a nonlinear singular differentialequation arising in hydrodynamics. Nonlinear Anal. 72 (2010), No. 3-4, 2114–2118.[3] I. Rachůnková, L. Rachůnek, and J. Tomeček, Existence of oscillatory solutions of singularnonlinear differential equations. Abstr. Appl. Anal. 2011, Art. ID 408525, 20 pp.[4] I. Rachůnková and J. Tomeček, Superlinear singular problems on the half line. Bound. ValueProbl. 2010, Art. ID 429813, 18 pp.69

Slowly Growing Solutions of Singular Linear FunctionalDifferential SystemsAndras Rontó and Vita PylypenkoNational Technical University of Ukraine “KPI”, Kyiv, UkraineInstitute of Mathematics, Academy of Sciences of Czech Republic, Brno, Czech RepublicE-mail: ronto@ipm.czWe consider the system of linear functional differential equationsx ′ i(t) =n∑(l ik x k )(t) + q i (t), t ∈ [a, b), i = 1, 2, . . . , n, (1)k=1subjected to the initial conditionsx i (a) = λ i , i = 1, 2, . . . , n, (2)where −∞ < a < b < ∞, the functions q i , i = 1, 2, . . . , n, are locally integrable, l ik : C([a, b), R) →L 1; loc ([a, b), R), i, k = 1, 2, . . . , n, are linear mappings and L 1; loc ((a, b], R) is the set of functionsu : (a, b] → R such that u| [a+ε,b] ∈ L 1 ([a + ε, b], R) for any ε ∈ (0, b − a).Our aim is to find conditions sufficient for the existence and uniqueness of a slowly growingsolution of the initial value problem (1), (2). The “slow growth” of a solution x = (x i ) n i=1 : [a, b) →R n is understood in the sense that its components satisfy the conditionssup h i (t)|x i (t)| < +∞, i = 1, 2, . . . , n, (3)t∈[a,b)where h i : [a, b) → [0, +∞), i = 1, 2, . . . , n, are certain given continuous functions possessing thepropertieslim h i(t) = 0, i = 1, 2, . . . , n. (4)t→b−Solutions of system (1) are sought for in the class of functions that are only locally absolutelycontinuous and, in particular, may be unbounded in a neighbourhood of the point b.Definition. By a solution of the functional differential system (1), we mean a locally absolutelycontinuous vector function x = (x i ) n i=1 : [a, b) → Rn with components possessing the propertiesh i x ′ i ∈ L 1([a, b), R), i = 1, 2 . . . , n, and satisfying equalities (1) almost everywhere on the interval[a, b). We say that a solution x = (x i ) n i=1 : [a, b) → Rn of system (1) is slowly growing if it hasproperty (3).The theorem formulated below concerns the case where the right-hand side terms of equations(1) are determined by linear operators which are positive with respect to the pointwise partialorderings of the linear manifolds C([a, b), R) and L 1; loc ([a, b), R).Definition. An operator l : C([a, b), R) → L 1; loc ([a, b), R) is said to be positive if (lu)(t) ≥ 0for a. e. t ∈ [a, b) whenever u is non-negative on [a, b).Theorem ([1]). Let us assume that the linear mappings l ik : C([a, b), R) → L 1; loc ([a, b), R),i, k = 1, 2, . . . , n, are positive, the relationsn∑ ( 1)h k l ik ∈ L 1 ([a, b), R), i = 1, 2, . . . , n, (5)h kk=170

hold, and there exists a certain δ ∈ [0, 1) such that the inequalityn∑( n∑l ikj=1k=1∫·a( ) 1l kj)(s) ds (t) ≤ δh jn∑ ( 1l ik)(t) (6)h kis satisfied for a. e. t ∈ [a, b) and every i = 1, 2, . . . , n.Then the initial value problem (1), (2) has a unique slowly growing solution for arbitrary locallyintegrable functions q i : [a, b) → R, i = 1, 2, . . . , n, possessing the propertyk=1{h i q i | i = 1, 2, . . . , n} ⊂ L 1 ([a, b), R). (7)Furthermore, if q i and λ i , i = 1, 2, . . . , n, satisfy the condition−n∑λ k (l ik 1)(t) ≤ q i (t), t ∈ [a, b), i = 1, 2, . . . , n, (8)k=1then the unique solution of problem (1), (2), (3) has non-negative components.The symbol l ik 1 in (8) stands for the result of application of the operator l ik to the functionequal identically to 1.By virtue of the positivity of the mappings l ik , i, k = 1, 2, . . . , n, conditions (8) are satisfied, inparticular, if {λ i | i = 1, 2, . . . , n} ⊂ [0, +∞) and the functions q i , i = 1, 2, . . . , n, are non-negativealmost everywhere on [a, b).Remark. The condition δ < 1 on the constant δ appearing in assumption (6) of Theorem issharp and cannot be weakened.References[1] V. Pylypenko and A. Rontó, On slowly growing solutions of singular linear functional differentialsystems. Math. Nachr., 2011, Issue 17-18.71

On the Existence of Special Class Solutions of a Quasi-LinearDifferential System in One Particular CaseS. A. ShchogolevI. I. Mechnikov Odessa National University, Odessa, UkraineE-mail: shchogolevs@rambler.ruThe system of differential equationsdz jdt = (j − 1)b(t, ε)z 1 + (2 − j)εα(t, ε)z 2 + h j (t, ε, θ) + µZ j (t, ε, θ, z 1 , z 2 ), j = 1, 2 (1)is considered, where α ∈ S m−1 , b ∈ S m , h 1 , h 2 ∈ B m (the definitions of classes S m and B m aregiven in [1]), inf |b| > 0, z 1, z 2 ∈ D, the functions Z 1 and Z 2 belong to the class B m with respect toGt, ε, θ and have in D continuous partial derivatives with respect to x 1 , x 2 up to some order 2q + 3,inclusive, and if z 1 , z 2 ∈ B m , then these partial derivatives are also of the class B m . µ ∈ R + .Denote ∀ f ∈ B m :and introduce the functionsξ 10 (t, ε, θ) =ξ 20 (t, ε, θ) =∞∑n=−∞(n≠0)∞∑n=−∞(n≠0)Γ n (f) = 12π∫ 2π0Γ n (h 1 (t, ε, θ))inϕ(t, ε)( −b(t, ε)Γn (h 1 (t, ε, θ))n 2 ϕ 2 (t, ε)f(t, ε, θ)exp(−inθ)dθ, n ∈ Z,exp(inθ) − Γ 0(h 2 (t, ε, θ))b(t, ε)where the function N 0 (t, ε) is defined from the equationDenote, further, (Z) 0 = Z(t, ε, θ, ξ 10 , ξ 20 ),η 11 (t, ε, θ) =η 21 (t, ε, θ) =,+ Γ n(h 2 (t, ε, θ)))exp(inθ) + N 0 (t, ε),inϕ(t, ε)Q(t, ε, N 0 ) = Γ 0(Z1 (t, ε, θ, ξ 10 , ξ 20 ) ) = 0. (2)∞∑n=−∞(n≠0)∞∑n=−∞(n≠0)Q 1 (t, ε, N 0 ) = Γ 0( (∂Z1∂ξ 1)Γ n ((Z 1 ) 0 )inϕ(t, ε) exp(inθ) − Γ 0((Z 2 ) 0 ),b(t, ε)( −b(t, ε)Γn ((Z 1 ) 0 )n 2 ϕ 2 (t, ε)( η ∂Z111(t, ε, θ) +0+ Γ )n((Z 2 ) 0 )inϕ(t, ε)∂ξ 2)Lemma. Let the system (1) satisfy the following conditions:(1) Γ 0 (h 1 (t, ε, θ)) ≡ 0 ∀ t, ε ∈ G;(2) the equality (2) is fulfilled ∀ t, ε ∈ G and ∀ N 0 ;72)η 21(t, ε, θ) .0exp(inθ),

(3) the equation Q 1 (t, ε, N 0 ) = 0 has a root N 0 (t, ε) satisfying the condition∣ infdQ 1 (t, ε, N 0 ) ∣∣∣G∣> 0.dN 0Then for sufficiently small values of the parameter µ there exists transformation of the typez j = ˜ξ j (t, ε, θ, µ) +where ˜ξ j , ψ jk ∈ B m (j, k = 1, 2), reducing the system (2) to the formd˜z jdt = (j − 1)b(t, ε)˜z j ++ε2∑k=1( q∑l=12∑r jk (t, ε, θ, µ)˜z k + µ q+1k=12∑ψ jk (t, ε, µ)˜z k , j = 1, 2, (3)k=1a jkl (t, ε)µ l )˜z k + εc j (t, ε, θ, µ) + µ 2q (t, ε, θ, µ)+2∑w jk (t, ε, θ, µ)˜z k + µ ˜Z j (t, ε, θ, ˜z 1 , ˜z 2 , µ), j = 1, 2, (4)k=1where a jkl ∈ S m , c j , d j , r jk , w jk ∈ B m−1 , ˜Z j contain summands, not lower than of the 2nd orderwith respect to ˜z 1 , ˜z 2 .Introduce the matrices A l (t, ε) = (a jkl (t, ε)) j,k=1,2 , (l = 1, q) (a jkl are defined in Lemma 1),Theorem. Let:J(t, ε) =(0 0b(t, ε) 0), A ∗ (t, ε, µ) = J(t, ε) +q∑A l (t, ε)µ l .(1) eigenvalues λ ∗ j (t, ε, µ) (j = 1, 2) of the matrix A∗ (t, ε, µ) be such that infG |Reλ∗ j (t, ε, µ)| ≥ γ 0µ q 0(γ 0 > 0, 0 < q 0 ≤ q);(2) for the matrix A ∗ (t, ε, µ), there exist a matrix U(t, ε, µ) such that(a) inf |detU(t, ε, µ)| > 0;G(b) U −1 A ∗ U = Λ(t, ε, µ) is a diagonal matrix;(3) the conditions of the lemma be fulfilled.Then for sufficiently small values µ, ε/µ 2q 0−1 , the system (1) has a particular solution of the classB m−1 .l=1References[1] A. V. Kostin and S. A. Shchëgolev, On the stability of oscillations represented by Fourierseries with slowly varying parameters. (Russian) Differ. Uravn. 44 (2008), No. 1, 45–51, 142;translation in Differ. Equ. 44 (2008), No. 1, 47–53.73

Asymptotic Representation for Solutions of Systems ofDifferential Equations with Regularly andFast Varying NonlinearitiesO. R. ShlyepakovI. I. Mechnikov Odessa National University, Odessa, UkraineE-mail: oleg@gavrilovka.com.uaWe consider the system of differential equations{y ′ i = α ip i (t)φ i+1 (y i+1 ) (i = 1, n − 1),y ′ n = α n p n (t)φ 1 (y 1 ),(1)where α i ∈ {−1, 1} (i = 1, n), p i : [a, ω[ → ]0, +∞[ (i = 1, n) are continuous functions, −∞ < a

⎧1 − Λ i − γ i , if i ∈ I,⎪⎨ β lβ i =, if i ∈ {l + 1, . . . , n} \ I,Λ l · · · Λ i−1β l⎪⎩, if i ∈ {1, . . . , l − 1} \ I,Λ l · · · Λ n Λ 1 · · · Λ i−1where limits of integration A i ∈ {ω, a} are chosen in such a way that corresponding integral I i aimseither to zero, or to ∞ when t ↑ ω.{A ∗ 1, if A i = a,i =(i = 1, . . . , n).−1, if A i = ωTheorem. Let Λ i ∈ R\{0} (i = 1, n) and l = min I. Then for the existence of P ω (Λ 1 , . . . , Λ n )-solutions of (1) it is necessary and, if algebraic equationn∏ ( ∏i−1(1 − γ i )i=1j=1)Λ j + ν −n∏ ∏i−1Λ j = 0 (3)i=1 j=1does not have roots with zero real part, it is also sufficient that for each i ∈ {1, . . . , n}I i (t)I i+1 ′ lim(t)t↑ω Ii ′(t)I i+1(t) = Λ β i+1iβ iand following conditions are satisfied A ∗ i β i > 0 when Φ 0 i = +∞, A ∗ i β i < 0 when Φ 0 i = 0,sign [ α i A ∗ i β i]= sign φ′i (z). Moreover, components of each solution of that type admit followingasymptotic representation when t ↑ ωφ i (y i (t))φ ′ i (y i(t))φ i+1 (y i+1 (t)) = α iβ i I i (t)[1 + o(1)], if i ∈ I,φ i (y i (t))φ ′ i (y i(t))φ i+1 (y i+1 (t)) = α I i (t)iβ i [1 + o(1)], if i ∈ I,I l (t)and there exists the whole k-parametric family of these solutions if there are k positive roots(including multiple roots) among the solutions of (3), with signs of real parts different from thoseof the number A ∗ l β l.References[1] D. D. Mirzov, Asymptotic properties of solutions of a system of Emden-Fowler type. (Russian)Differentsial’nye Uravneniya 21 (1985), No. 9, 1498–1504, 1649.[2] V. M. Evtukhov, Asymptotic representations of regular solutions of a two-dimensional systemof differential equations. (Russian) Dopov. Nats. Akad. Nauk Ukr. Mat. Prirodozn. Tekh. Nauki2002, No. 4, 11–17.[3] E. Seneta, Regularly varying functions. (Russian) Nauka, Moscow, 1985.[4] V. Marić, Regular variation and differential equations. Lecture Notes in Mathematics, 1726.Springer-Verlag, Berlin, 2000.75

Limit Properties of Positive Solutions of Fractional BoundaryValue ProblemsSvatoslav StaněkPalacký University, Olomouc, Czech RepublicE-mail: svatoslav.stanek@upol.czWe discuss the set of the scalar fractional boundary value problemsc D α nu(t) + f(t, u(t), u ′ (t),c D µ nu(t)) = 0, n ∈ N, (1)u ′ (0) = 0, u(1) + au ′ (1) = 0, (2)where {α n } ⊂ (1, 2), {µ n } ⊂ (0, 1), limn→∞ α n = 2, limn→∞ µ n = 1, a ≥ 0, and f satisfies the conditions:(H 1 ) f ∈ C([0, 1] × D), D = R + × R 2 −, R + = [0, ∞), R − = (−∞, 0],(H 2 ) the estimate0 ≤ φ(t) ≤ f(t, x, y, z) ≤ ω(x, |y|, |z|),is fulfilled for (t, x, y, z) ∈ [0, 1] × D, where φ ∈ C[0, 1], φ(t 0 ) > 0 for some t 0 ∈ [0, 1],ω ∈ C(R 3 +), ω is nondecreasing in all its arguments, andω(x, x, x)lim= 0.x→∞ xHere c D is the Caputo fractional derivative. The Caputo fractional derivative c D γ x of order γ > 0of a function x : [0, 1] → R is defined as (see, e.g., [1], [2])⎧⎪⎨ 1 d n ∫tc D γ x(t) = Γ(n − γ) dt n (t − s) n−γ−1( n−1∑ x (k) (0)x(s) − s k) ds if γ ∉ N,k!0k=0⎪⎩x (γ) (t) if γ ∈ N,where n = [γ] + 1 and [γ] means the integral part of γ and where Γ is the Euler gamma function.We say that a function u ∈ C 1 [0, 1] is a positive solution of problem (1), (2) if u > 0 on [0, 1),c D αn u ∈ C[0, 1], u satisfies the boundary conditions (2) and equality (1) holds for t ∈ [0, 1].Together with equation (1) we investigate the differential equationu ′′ (t) + f(t, u(t), u ′ (t), u ′ (t)) = 0. (3)A function u ∈ C 2 [0, 1] is called a positive solution of problem (3), (2) if u > 0 on [0, 1), u satisfies(2) and equality (3) is fulfilled for t ∈ [0, 1].The following result is proved by the Leray–Schauder nonlinear alternative [3].Theorem 1. Let conditions (H 1 ) and (H 2 ) hold. Then for each n ∈ N problem (1), (2) has asolution u n and there exists a positive constant S independent of n such that the estimate(1 − t α∗−1 )Q ≤ u n (t) < (1 + a)S, 0 ≥ u ′ n(t) > −S, 0 ≥ c D µ u n (t) > − S ∆(4)is fulfilled for t ∈ [0, 1], whereQ = 1Γ(α)∫ 10(1 − s)φ(s) ds, ∆ = min{Γ(s) : 1 ≤ s ≤ 2}, α ∗ = inf{α n : n ∈ N}.76

Let ∥x∥ = max{|x(t)| : t ∈ [0, 1]} be the norm in C[0, 1] and let ∥x∥ C 1 = max{∥x∥, ∥x ′ ∥}. Thefollowing theorem states the relation between positive solutions of problem (1), (2) and positivesolutions of problem (3), (2).Theorem 2. Let (H 1 ) and (H 2 ) hold. Let u n be a positive solution of problem (1), (2) satisfyinginequality (4). Then there exist a subsequence {u kn } of {u n } and a positive solution u of problem(3), (2) such thatlim ∥u kn→∞ n− u∥ C 1 = 0,limn→∞ ∥c D µ kn ukn − u ′ ∥ = 0,limn→∞ ∥c D α kn un − u ′′ ∥ = 0.References[1] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and applications of fractional differentialequations. North-Holland Mathematics Studies, 204. Elsevier Science B.V., Amsterdam,2006.[2] K. Diethelm, The analysis of fractional differential equations. An application-oriented expositionusing differential operators of Caputo type. Lecture Notes in Mathematics, 2004. Springer-Verlag, Berlin, 2010.[3] K. Deimling, Nonlinear functional analysis. Springer-Verlag, Berlin, 1985.77

Optimization of Initial Data for Nonlinear DelayFunctional-Differential Equations with the MixedInitial ConditionTamaz TadumadzeI. Javakhishvili Tbilisi State University, Tbilisi, GeorgiaE-mail: tamaztad@yahoo.comAbdeljalil NachaouiJean Leray Laboratory of Mathematics, University of Nantes, Nantes, FranceE-mail: Abdeljalil.Nachaoui@univ-nantes.frNika GorgodzeA. Tsereteli State University, Kutaisi, GeorgiaE-mail: nika − gorgodze@yahoo.comIn the present paper, for initial data, the necessary conditions of optimality are obtained. Underinitial data we imply the collection of initial moment and constant delays, initial vector and function.Let Rx n be an n-dimensional vector space of points x = (x 1 , . . . , x n ) T , where T means transpose;P ⊂ Rp k and Z ⊂ Rzm be open sets and O = (P, Z) T = {x = (p, z) T ∈ Rx n : p ∈ P, z ∈ Z}, withk + m = n; next t 01 < t 02 < t 1 , 0 < τ 1 < τ 2 , 0 < σ 1 < σ 2 be given numbers, with t 1 − t 02 > τ 2 ; then-dimensional function f(t, x, p, z) be continuous on the set [t 01 , t 1 ] × O × P × Z and continuouslydifferentiable with respect to x, p, z; let P 0 ⊂ P be a compact convex set of initial vectors p 0 ; Φ andG be sets of continuous initial functions φ(t) ∈ P 1 , t ∈ [̂τ, t 02 ] and g(t) ∈ Z 1 , t ∈ [̂τ, t 02 ], respectively,where ̂τ = t 01 −max{τ 2 , σ 2 }, P 1 ⊂ P , Z 1 ⊂ Z are convex and compact sets. Suppose that functionsq i (t 0 , τ, σ, p, z, x), i = 0, l are continuously differentiable with respect to all arguments t 0 ∈ [t 01 , t 02 ],τ ∈ [τ 1 , τ 2 ], σ ∈ [σ 1 , σ 2 ], p ∈ P , z ∈ Z, x ∈ O.To each element w = (t 0 , τ, σ, p 0 , φ, g) ∈ W = (t 01 , t 02 ) × (τ 1 , τ 2 ) × (σ 1 , σ 2 ) × P 0 × Φ × G, weassign the delay functional-differential equationwith the mixed initial conditionẋ(t) = f ( t, x(t), p(t − τ), z(t − σ) ) (1)x(t) = (φ(t), g(t)) T , t ∈ [̂τ, t 0 ), x(t 0 ) = (p 0 , g(t 0 )) T . (2)Definition 1. Let w = (t 0 , τ, σ, p 0 , φ, g) ∈ W . A function x(t) = x(t; w) = (p(t; w), z(t; w)) T ∈O, t ∈ [̂τ, t 1 ], is called a solution of Eq. (1) with the initial condition (2) if it satisfies condition(2) and is absolutely continuous on the interval [t 0 , t 1 ] and satisfies Eq. (1) almost everywhere on[t 0 , t 1 ].Definition 2. An element w ∈ W is said to be admissible if the solution x(t) = x(t; w) isdefined on the interval [̂τ, t 1 ] and satisfies the conditionsWe denote the set of admissible elements by W 0 .q i( t 0 , τ, σ, p 0 , g(t 0 ), x(t 1 ) ) = 0, i = 1, l.Definition 3. An element w 0 = (t 00 , τ 0 , σ 0 , p 00 , φ 0 , g 0 ) ∈ W 0 is said to be optimal if for anyw = (t 0 , τ, σ, p 0 , φ, g) ∈ W 0 we haveq 0( t 00 , τ 0 , σ 0 , p 00 , g 0 (t 00 ), x 0 (t 1 ) ) ≤ q 0( t 0 , τ, σ, p 0 , g(t 0 ), x(t 1 ) ) , x 0 (t) = x(t; w 0 ).78

Theorem. Let w 0 = (t 00 , τ 0 , σ 0 , p 00 , φ 0 , g 0 ) be an optimal element and the following conditionhold: the functions φ 0 (t), g 0 (t) are continuously differentiable. Then there exist a non-zero vectorπ = (π 0 , . . . , π l ), where π 0 ≤ 0 and the solution Ψ(t) to the equation{ ()˙Ψ(t) = −Ψ(t)fx [t] − Ψ(t + τ 0 )f p [t + τ 0 ], Ψ(t + σ 0 )f z [t + σ 0 ] , t ∈ [t 00 , t 1 ],Ψ(t 1 ) = πQ 0x , Ψ(t) = 0, t > t 1such that the conditions listed below hold: the conditions for the function Ψ(t) = (ψ(t), χ(t)) =(ψ 1 (t), . . . , ψ k (t), χ 1 (t), . . . , χ m (t)) and vectors p 00 , g 0 (t 00 )(πQ 0p0 + ψ(t 00 ))p 00 = maxp 0 ∈P 1(πQ0p0 + ψ(t 00 ) ) p 0 ,(πQ 0z + χ(t 00 ))g 0 (t 00 ) = maxg∈Z 1(πQ0z + χ(t 00 ) ) g,where Q 0p0 = Q p0(t00 , τ 0 , σ 0 , p 00 , x 0 (t 1 ) ) , Q = (q 0 , . . . , q l ) T ;the integral maximum principles for the optimal initial functions φ 0 (t) and g 0 (t)∫t 00t 00 −τ 0∫t 00t 00 −σ 0Ψ(t + τ 0 )f p [t + τ 0 ]φ 0 (t) dt = maxΨ(t + σ 0 )f z [t + σ 0 ]g 0 (t) dt = max∫t 00φ(·)∈Φt 00 −τ 0∫t 00g(·)∈Gt 00 −σ 0Ψ(t + τ 0 )f p [t + τ 0 ]φ(t) dt,Ψ(t + σ 0 )f z [t + σ 0 ]g(t) dt;the condition for the optimal initial moment t 00πQ 0t0 + ( πQ 0z + χ(t 00 ) ) ġ(t 00 ) ={}= Ψ(t 00 )f[t 00 ] + Ψ(t 00 + τ 0 ) f[t 0 + τ 0 ; p 00 ] − f[t 0 + τ 0 ; φ 0 (t 00 )] ;the conditions for the optimal delays τ 0 , σ 0Here{}πQ 0τ = Ψ(t 00 + τ 0 ) f[t 0 + τ 0 ; p 00 ] − f[t 0 + τ 0 ; φ 0 (t 00 )] +πQ 0σ =∫ t 1t 00Ψ(t)f z [t]ż 0 (t − σ 0 ) dt.∫ t 1t 00Ψ(t)f 0p [t]ṗ 0 (t − τ 0 ) dt,f x [t] = f x (t, x 0 (t), p 0 (t − τ 0 ), z 0 (t − σ 0 )), f[t] = f(t, x 0 (t), p 0 (t − τ 0 ), z 0 (t − σ 0 )),f[t; p 0 ] = f(t, x 0 (t), p 0 , z 0 (t − σ 0 )).AcknowledgementsThe work was supported by the Shota Rustaveli National Science Foundation (Georgia), GrantNo. GNSF/STO8/3-399 and Centre National de la Recherche Scientifique CNRS (France), PICS-4962.79

On Asymptotics of Solutions of Cyclic Systems ofDifferential EquationsO. S. VladovaI. I. Mechnikov Odessa National University, Odessa, UkraineE-mail: lena@gavrilovka.com.uaWe consider the system of differential equations{y ′ i = α ip i (t)φ i+1 (y i+1 ) (i = 1, n − 1),y ′ n = α n p n (t)φ 1 (y 1 ),(1)where α i ∈ {−1, 1} (i = 1, n), p i : [a, ω[ → ]0, +∞[ (i = 1, n) are continuous functions, −∞ < a

where limits of integration A i ∈ {ω, a} are chosen in such a way that corresponding integral I i aimseither to zero, or to ∞ as t ↑ ω.{A ∗ 1, if A i = a,i =(i = 1, . . . , n).−1, if A i = ωTheorem. Let Λ i ∈ R\{0} (i = 1, n) and l = min I. Then for the existence of P ω (Λ 1 , . . . , Λ n )-solutions of (1) it is necessary and, if algebraic equationn∏i=1( i−1∏j=1)Λ j + ν −n∏i=1(σ ii−1∏ )Λ j = 0 (3)does not have roots with zero real part, it is also sufficient that for each i ∈ {1, . . . , n}j=1I i (t)I i+1 ′ lim(t)t↑ω Ii ′(t)I i+1(t) = Λ β i+1iβ iand following conditions are satisfied A ∗ i β i > 0 if Yi 0 = ±∞, A ∗ i β i < 0 if Yi 0 = 0, sign [ α i A ∗ i β i]=µ i . Moreover, components of each solution of that type admit following asymptotic representationwhen t ↑ ω:y i (t)φ i+1 (y i+1 (t)) = α iβ i I i (t)[1 + o(1)], when i ∈ I,y i (t)φ i+1 (y i+1 (t)) = α I i (t)iβ i [1 + o(1)], when i ∈ I,I l (t)and there exists the whole k- parametric family of these solutions if there are k positive roots(including multiple roots) among the solutions of (3) with signs of real parts different from thoseof the number A ∗ l β l.References[1] E. Seneta, Regularly varying functions. (Russian) Nauka, Moscow, 1985.[2] T. A. Chanturia, Oscillatory properties of systems of nonlinear ordinary differential equations.(Russian) Tbiliss. Gos. Univ. Inst. Prikl. Mat. Trudy 14 (1983), 163–206.[3] D. D. Mirzov, Asymptotic properties of solutions of a system of Emden-Fowler type. (Russian)Differentsial’nye Uravneniya 21 (1985), No. 9, 1498–1504, 1649.[4] D. D. Mirzov, Some asymptotic properties of the solutions of a system of Emden-Fowler type.(Russian) Differentsial’nye Uravneniya 23 (1987), No. 9, 1519–1532, 1651.[5] D. D. Mirzov, Asymptotic properties of solutions for systems of nonlinear non-autonomousordinary differential equations. Adygeen Book Publ. Maikop, 1993.[6] V. M. Evtukhov, Asymptotic representations of regular solutions of a two-dimensional systemof differential equations. (Russian) Dopov. Nats. Akad. Nauk Ukr. Mat. Prirodozn. Tekh. Nauki2002, No. 4, 11–17.[7] V. M. Evtukhov, Asymptotic representations of regular solutions of a semilinear twodimensionalsystem of differential equations. (Russian) Dopov. Nats. Akad. Nauk Ukr. Mat.Prirodozn. Tekh. Nauki 2002, no. 5, 11–17.[8] V. M. Evtukhov, On solutions vanishing at infinity of real nonautonomous systems of quasilineardifferential equations. (Russian) Differ. Uravn. 39 (2003), No. 4, 441–452, 573; translationin Differ. Equ. 39 (2003), No. 4, 473–484.81

C o n t e n t sM. AshordiaOn the Two Point Boundary Value Problem for Systems of Linear GeneralizedDifferential Equations with Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3I. AstashovaUniform Estimates and Existence of Solution with Prescribed Domain to aNonlinear Third-Order Differential Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5G. BerikelashviliOn the Convergence of Difference Schemes for Generalized BBM Equation . . . . . . . . . . . . . . 8M. O. BilozerovaAsymptotic Behavior of Solutions of Differential Equationsof the Type y (n) = α 0 p(t) n−1 ∏φ i (y (i) ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10A. Domoshnitskyi=0On Stability of Delay Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13Ph. DvalishviliOn Well-Posedness with Respect to Functional for an Optimal Control Problemwith Distributed Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15V. M. Evtukhov and A. M. KlopotAsymptotic Representations of Solutions of Ordinary Differential Equationsof N-th Order with Regularly Varying Nonlinears . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17J. GvazavaOn Mixed Type Quasi-Linear Equations with General Integrals,Represented by Superposition of Arbitrary Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19R. HaklFredholm Type Theorem for Systems of Functional-Differential Equationswith Positively Homogeneous Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21R. Hakl and M. ZamoraPeriodic Solution to Two-Dimensional Half-Linear System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2482

M. IordanishviliNecessary Conditions of Optimality for Optimal Problems with General VariableDelays and the Mixed Initial Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26N. A. Izobov and S. A. MazanikAsymptotic Equivalence of Linear Systems for Small Linear Perturbations . . . . . . . . . . . . . . 27O. JokhadzeThe Initial-Characteristic Problems for Wave Equationswith Nonlinear Damping Term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30S. KharibegashviliThe Multidimensional Darboux Problem for a Class of Nonlinear Wave Equations . . . . . . 32I. KiguradzeOn Two-Point Boundary Value Problems for Higher Order Quasi-HalflinearDifferential Equations with Strong Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34T. KiguradzeOn Conditional Well-Posedness of Nonlocal Problems for Fourth OrderSingular Linear Hyperbolic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36A. Yu. Kolesov and N. Kh. RozovRelaxational Oscillations and Diffusive Chaos in Belousov’s Reaction . . . . . . . . . . . . . . . . . . . 38A. V. Kostin, L. L. Kol’tsova, T. Yu. Kosetskaya, and T. V. KondratenkoOn Three Problems of the Qualitative Theory of Differential Equations . . . . . . . . . . . . . . . . 40A. A. LevakovInvestigation of Stability of Stochastic Differential Equationsby Using Lyapunov Functions of Constant Signs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42E. Litsyn, Yu. V. Nepomnyashchikh, and A. PonosovOn Hybrid Stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45A. Lomtatidze and J. ŠremrOn Oscillation and Nonoscillation of Two-Dimensional Linear Differential Systems . . . . . 4783

E. K. Makarov and E. A. ShakhAn Infinite Dimensional Generalization of Weak Exponents for Solutionsof Equations in Total Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50K. Mamsa and Yu. PerestyukA Certain Class of Discontinuous Dynamical Systems in the Plane . . . . . . . . . . . . . . . . . . . . . 51G. A. Monteiro and M. TvrdýGeneralized Linear Differential Equations in a Banach Space:Continuous Dependence on a Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52S. MukhigulashviliTwo-Point Boundary Value Problems for Strongly Singular Higher-OrderLinear Differential Equations with Deviating Arguments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55V. V. NikonenkoAsymptotics of Solutions of a Linear Homogeneous System of Differential Equationsin the Case of Asymptotically Equivalent, as t → +∞,Roots of a Characteristic Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57Z. Opluštil and J. ŠremrOn Oscillations of Solutions to Second-Order Linear Delay Differential Equations . . . . . . . 59N. PartsvaniaSolvability and Well-Posedness of Two-Point Weighted SingularBoundary Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62M. Perestyuk and P. FeketaOn Invariant Sets of Impulsive Multi-Frequency Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64B. Půža and Z. SokhadzeOn the Well-Possedness of the Weighted Cauchy Problem for Nonlinear SingularDifferential Equations of Higher Orders with Deviating Arguments . . . . . . . . . . . . . . . . . . . . . 66I. RachůnkováThree Types of Solutions of Some Singular Nonlinear Differential Equation . . . . . . . . . . . . . 68A. Rontó and V. PylypenkoSlowly Growing Solutions of Singular Linear Functional Differential Systems . . . . . . . . . . . 7084

S. A. ShchogolevOn the Existence of Special Class Solutions of a Quasi-Linear Differential Systemin One Particular Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72O. R. ShlyepakovAsymptotic Representation for Solutions of Systems of Differential Equationswith Regularly and Fast Varying Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74S. StaněkLimit Properties of Positive Solutions of Fractional Boundary Value Problems . . . . . . . . . . 76T. Tadumadze, A. Nachaoui, and N. GorgodzeOptimization of Initial Data for Nonlinear Delay Functional-Differential Equationswith the Mixed Initial Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78O. S. VladovaOn Asymptotis of Solutions of Cyclic Systems of Differential Equations . . . . . . . . . . . . . . . . 8085