FOR DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER

Boundary Value Problems for Differential Equations of Fractional Order 33∑a series f = ∞ ( ∞∑c k u k |c k | 2 < ∞ ) . It is known that if the spectrum ofn=1n=1a dissipative operator with completely continuous imaginary component “ispressed enough” to the real axis, then it is possible to form a basis of thelinear closed hull of eigenvectors of this operator.We need a theorem of Glazman.Theorem of Glazman. Let A be a linear bounded dissipative operatorwith completely continuous imaginary component, possessing an infinitesystem of eigenvectors {u k } ∣ ∞ k=1 , normalized by the condition (u k, u u ) = 1(k = 1, 2, 3, . . . ), and {λ k } ∣ ∞ be the corresponding sequence of differentk=1eigenvalues. If the condition∑ Im λj Im λ k< ∞|λ j − λ k |is satisfied, then the system {ϕ k } ∣ ∞ is a Bari–Riesz’s basis of the closedk=1linear hull. Since for 0 < ρ < 1/2 all eigenvalues of the operator A ρ are real,due to theorem of Glazman the proof of Theorem 1.6 follows. Analogously,it is possible to prove similar statements for more general problems. □5. Partial Problem of Eigenvalues for the Operator A ρ(0 < ρ < 1/2)In [1], [3], there have been determined areas in the complex plane wherethere are no eigenvalues of the operator A ρ for any ρ. Here, as above, wewill assume that 0 < ρ < 1/2. In applied problems the greatest interestrepresents usually determination of first eigenvalues, therefore we will solvethis problem for eigenvalues of the operator A ρ .Let’s consider the operatorA= 1 [ ∫xΓ(ρ −1 (x−t) 1/ρ u(t) dt−)0∫ 10]x 1/ρ−1 (1−t) 1/ρ−1 u(t) dt =A 0 u+A 1 u.As 0 < ρ < 1/ρ, the operator A 0 and A 1 are invertible. Therefore sp(A) =spA 0 + spA 1 .Clearly1spA 0 = 0, spA 1 =Γ(2 − 1/ρ) .Therefore the determinant∞∑D A (λ) = (1 − λµ j ), µ k = 1 ,λ kj=1is meaningful.Earlier it has been established that the number λ is an eigenvalue of theoperator A only if λ −1 is a zero of the function E ρ (λ; ρ −1 ). Since the entire