FOR DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER

24 T. S. Aleroev, H. T. Aleroeva, Ning-Ming Nie, and Yi-Fa TangThen denotingg(x) = dαdx α f(x),in the future we will mean byd αdx αthe fractional integral when α < 0 and the fractional derivative when α > 0.The fractional derivatived αd(1 − x) αof order α > 0 of a function f(x) ∈ L 1 (0, 1) , with ending point x = 1 isdefined in the similar way.Let {γ k } n 0 be any set of real numbers satisfying the condition 0 < γ j ≤ 1(0 ≤ j ≤ n). We denoteσ k =k∑γ j − 1; µ k = σ k + 1 =j=0and we assume that1ρ =n ∑j=0k∑j=0γ jγ j − 1 = σ n = µ n − 1 > 0.(0 ≤ k ≤ n),Following M. M. Dzhrbashyan (see [6]), we consider the integro-differentialoperatorsD (σ0) f(x) ≡ d−(1−γ0)dx f(x),−(1−γ 0)D (σ1) f(x) ≡ d−(1−γ 1)d γ 0dx −(1−γ 1) dx γ f(x),0D (σ2) f(x) ≡ d−(1−γ 2)d γ 1d γ 0f(x),dx −(1−γ2) dx γ1 γ0dx. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .D (σn) f(x) ≡ d−(1−γn)dx −(1−γ n)d γn−1dx γ · · · d γ0n−1 dx γ f(x).0Here we note that if γ 0 = γ 1 = · · · = γ n = 1, then obviouslyD (σ k) f(x) = f (k) (x) (k = 0, 1, 2, . . . , n).The objects of our investigation are boundary value problems for the followingequations:D (σ n) u − [ λ + q(x) ] u = 0, 0 < σ n < ∞, (1.1)u ′′ + D α 0xu + q(x)u = λu, 0 < α < 1. (1.2)