FOR DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER

Boundary Value Problems for Differential Equations of Fractional Order 29Similarly,N ε u =where∫ 10˜K(x, t)u(t) dt − εwhich proves Theorem 1.2.∫ 10∫1˜K 1 (x, t)u(t) dt − · · · − ε n⎧⎨x(1 − t) ln n (x − xt), t < 1,˜K n (x, t) =n!⎩0, t = 1,0˜K n (x, t)u(t) dt + · · · ,Theorem 1.3. All eigenvalues λ n (ε) of the operator A(ε) are real.Proof. We havewhereλ n (ε) = πn 2 + ελ 1 + ε 2 λ 2 + · · · , (1.9)ϕ n (ε) = sin nx + εϕ 1 + ε 2 ϕ 2 + · · · , (1.10)λ n =n∑(A k ϕ n−k , sin nx), (1.11)k=1ϕ n = Rn∑(λ k − A k )ϕ n−k . (1.12)k=1Here R is the resolvent of the operator A, corresponding to the eigenvalueπn 2 . This resolvent is an integral operator with the kernel[S(x, y)= − y 1]cos ny sin nx+1−x sin ny cos nx+n n 2n 2 sin ny sin nx , y ≤x(if y > x it is necessary to interchange y and x in the right part of thisformula).Clearly R transforms H 0 (H 0 is the orthogonal complement of the functionsin πnx) into itself and cancels sin πnx.From (1.11) it follows that λ 1 = (A 1 sin nx, sin x). As the kernel of theoperator A 1 assumes real values, we have sin λ 1 = 0. From (1.12) it followsthat ϕ 1 = R(nk 2 − A 1 ) sin nx and since the kernels of the operators R andA 1 assume real values, we have Im ϕ 1 = 0. So, successively it is possible toestablish that all λ i are real. Since ε is real, λu(ε) is real too. □Theorem 1.4. For the eigenvalues λ n (ε) and eigenfunctions ϕ n (ε) ofthe operator A(ε) there hold the estimates|λ n (ε) − πn 2 |