FOR DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER

Boundary Value Problems for Differential Equations of Fractional Order 637. Construction of a Biorthogonal SystemConsider the Sturm–Liouville problem for a fractional differential equation⎧⎪⎨ u ′′ + a m (x)u ′ + a i (x)D α i0x ω i(x)u + λu = 0,u(0) cos(α) + u⎪ ′ (0) sin(α) = 0,(2.47)⎩u(1) cos(β) + u ′ (1) sin(β) = 0and the problem⎧⎪⎨ z − [a m (x)z(x)] ′ − ω i (x)D αix1 a i(t)z(t) + λz = 0,z(0) cos(α) + z⎪⎩′ (0) sin(α) = 0,z(1) cos(β) + z ′ (1) sin(β) = 0.We will call the problems (2.47) and (2.48) mutually adjoint.Following M. M. Dzhrbashyan [10], we introduce the functionsω(λ) = u(1, λ) + u ′ (1, λ) sin(β),˜ω(λ ∗ ) = z(0, λ ∗ ) + z ′ (0, λ ∗ ) sin(α),where u(x, λ) is a solution of the following Cauchy problem(2.48)u ′′ + a m (x)u ′ + a i (x)D α i0x ω i(x)u + λu = 0, (2.49)u(0) = sin(α), u ′ (0) = − cos(α) (2.50)and z(x, λ ∗ ) is a solution of the problemz ′′ (x) + a m (x)z ′ (x) + ω i (x)D α ix1 a i(x)z + λ ∗ z = 0, (2.51)z(1) = sin(β), z ′ = − cos(β). (2.52)The existence and uniqueness of solutions of mutually adjoint problems(2.47) and (2.48) are already proved [3].Clearly if u(x, λ) is a solution of the problem (2.49)–(2.50), then anecessary and sufficient condition for it to be also a solution of the problem(2.47) isω(λ) = u(1, λ) cos(β) + u ′ (1, λ) sin(β).There is a similar statement for the solutions of the problems (2.51)–(2.52).For construction of an orthogonal system of eigenfunctions and associatedfunctions of mutually adjoint problems by method of M. M. Dzhrbashyanwe need the followingTheorem 2.11. Let a m (0) = a m (1) = 0. Then for any values of theparameters λ, λ ∗ the identityholds.(λ − λ ∗ )∫ 10u(x, λ)z(x, λ ∗ ) dx = ω(λ) − ˜ω(λ ∗ ) (2.53)