FOR DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER

Boundary Value Problems for Differential Equations of Fractional Order 43Passing to investigation of structural and qualitative properties of thesolution of the Problem 2, we will notice that any solution of the equation(2.13) of the class C 2 ]0, r[ ∩C 1 [0, r[ ∩[0, r[ will be a solution of the equationτ(x) = τ ′ (0)x+τ(0)+µ β D −20x D1−2β 0tτ(ξ)+λD −20x τ(t)+D1−2β 0t ψ β (t). (2.17)The equation (2.16) is deduced from the equality (2.13) after applicationof the operator D 1−2β0t to both its parts.It is easy to see thatD0x −2 D1−2β 0tτ(ξ)=∫ x0(x−t) ∂ ∂t D−2β 0t τ(ξ) dt=D −10x D−2β 0tτ(ξ)=D −1In view of the latter from (2.16) by virtue of (2.12) we have0x D−2β 0tτ(t).ϕ r (0) − ϕ 0 (0) = rr ′ (0) + µ β D −1−2β0r τ(t) + D0r −2 ψ β(t). (2.18)Substituting the value of r ′ (0) (2.17) into the equation (2.16), we haveτ(x) = µ β D0x −α τ(t) + a 1xD0x −α τ(t) + λD−2 0x τ(t) + f 0(x), (2.19)where a 1 = −µ βr, α = 1 + 2β = 2(m + 1)/(m + 2),f β (x) = D −20x ψ β(t) + ϕ 0 (0) + x r[ϕr (0) − ϕ 0 (0) − D −20x ψ β(t) ] . (2.20)The equation (2.18) is an integral Fredholm’s equation of the secondkind and it is equivalent to Problem 1.2. Formulas for Calculation of Eigenvalues of a Boundary ValueProblemIntroduce some formulas from the theory of disturbance which we needin the further. Let T (χ, ε) be a linear operator bunch,T (χ, ε) = T + χT ′ + εT ′′ ,where T is a complete self-adjoined operator all eigenvalues of which areisolated and have multiplicity equal to 1; T ′ and T ′′ are defined in the sameHilbert space as T ; T is bounded.T (χ, ε) − ζ = T − ζ + χT ′ + εT ′′ = [ J + (χT ′ + εT ′′ )R(ζ, T ) ] (T − ζ)if ζ is not anigenvalue of T (χ, ε). Then, the resolventexists if the termR(ζ, x, ε) = R(ζ, T ) [ J − (χT ′ − εT ′′ )R(ζ, T ) ] −1[J − (χT ′ − εT ′′ )R(ζ, T ) ] −1may be determined as a Neumann’s series[J − (χT ′ − εT ′′ )R(ζ, T ) ] −1∑∞ [= − (χT ′ − εT ′′ )R(ζ, T ) ] n,n=0