FOR DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER

Boundary Value Problems for Differential Equations of Fractional Order 31Thus we have found |ϕ ε − ϕ 0 | ≤ 1/2, and proved Theorem 1.4.□Theorem 1.5. Let u 0 (x), u 1 (x), . . . , u n (x), . . . be the eigenfunctionsof the operator A ρ , and λ 0 , λ 1 , . . . , λ n , . . . be the corresponding eigenvalues.Then the least frequency self-oscillation, i.e. ϕ 0 (x), has no units, i.e.u 0 (x) ≠ 0, 0 < x < 1.Proof. Let λ 0 be the least eigenvalue of the operator A ρ . Thenu 0 (x) = E ρ (−λ 0 x; ρ −1 ) =∞∑k=0(−λ 0 x) kΓ(ρ −1 + ρ −1k ) .Show that the function u 0 (x) has no zero in the interval (0, 1). Supposethat the function u 0 (x) in a point x 0 ∈ (0, 1) equals to zero, that is,u 0 (x 0 ) = E ρ (−λ 0 x 0 ; ρ −1 ) =∞∑k=0(−λ 0 x 0 ) kΓ(ρ −1 + ρ −1k ) = 0.That’s to say, the number −λ 0 x 0 is a zero point of E ρ (z, ρ −1 ). However,−λ 0 x 0 < λ 0 as x 0 ∈ (0, 1) while we have assumed that the least zero is λ 0 .The obtained contradiction proves Theorem 1.5.□Remark 1. It is also possible to show analogously that u 1 (x) in theinterval has exactly one zero, etc.Theorem of existence of the basis made of root spaces of the operatorA ρ (0 < ρ < 1/2) is connected with the problem of completeness of systemsof eigenfunctions of the operator induced by the differential expression⎧⎪⎨ 1 d n−1 ∫xu ′ (t)l(D) n = Γ(1 − γ 1 ) dx n−1dt − (λ + q(x))u,γ1(x − t)0⎪⎩u(0) = 0, D σ 1u ∣ x=0= 0, . . . , D σ n−2u ∣ x=0= 0, u(1) = 0which is studied in case where q(x) is a semi-function (see [3] and referencestherein). If completeness of system of eigenfunctions bounded is proved,then the question is: is it possible to make basis with the eigenfunctions ofthis operator. Let us give an answer to this question.Lemma 1.1. The operator A is dissipativeProof. We will consider the operatorAu =∫ x0(x − t) 1+ε u(t) dt −∫ 10x 1+ε (1 − t) 1+ε u(t) dt.