FOR DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER

64 T. S. Aleroev, H. T. Aleroeva, Ning-Ming Nie, and Yi-Fa TangProof. Due to the definition of the functions u(x, λ) and z(x, λ ∗ ) as solutionsof problems of Cauchy type, we have (2.49)-(2.50) andz ′′ (x, λ ∗ ) [ a m z(x, λ ′′ ) ] ′+ ωi (x)D αi0x a i(x) + λ ∗ z = 0,z(1, λ ∗ ) = sin β, z ′ (1, λ ∗ ) = − cos β.Multiplying both parts of the first equation of (2.47) by z(x, λ ∗ ) andintegrating it from 0 to 1, we have+∫ x0∫ x0= ω(λ) − ˜ω(λ ∗ )+∫ 10z(x, λ ∗ )u ′′ (x, λ) dx +∫ 10a m (x)z(x, λ ∗ )u ′ (x, λ) dx+∫xa i (x)z(x, λ ∗ )D αi0x u(x, λ) dx + λu(x, λ)z(x, λ ∗ ) dx =∫ x0u(x, λ)z ′′ (x, λ) dx −0∫ 1ω i (x)D α ix1 a i(t)z(x, λ ∗ )u(x, λ) dx + λ0∫ 10[am (x)z(x, λ ∗ ) ] ′u(x, λ)+u(x, λ)z(x, λ ∗ ) dx. (2.54)At the same time, we multiply both parts of the first equation of (2.48) byu(x, λ) and integrate it from 0 up to 1. We have+∫ 10∫ 10z(x, λ ∗ )u(x, λ) dx −∫ 1ω i (x)D αix1 a i(t)z(x, λ ∗ )u(x, λ) dx + λSubtracting (2.55) from (2.56), we obtain∫ 100[am (x)z(x, λ ∗ ) ] ′u(x, λ) dx+∫ 1z(x, λ ∗ ) dx = ˜ω(λ∗ ) − ω(λ)λ − λ ∗ .0u(x, λ)z(x, λ ∗ ) dx = 0. (2.55)So we obtain an analogue of identity of M. M. Dzhrbashyan.Theorem 2.12. The formulaholds if λ = λ ∗ .ω(λ) = ˜ω(λ) (2.56)The equality (2.56) follows from the identity (2.53).We needed to construct a biorthogonal system of eigenfunctions andassociated functions of mutually adjoint problems (2.47).□