Existence of Solutions for Nonlocal Boundary Value Problems of Higher-Order Nonlinear Fractional Differential Equations

and Applied Analysis 3 involving fractional differential equations the same applies to the boundary value problems of fractional differential equations . Moreover, the Caputo derivative for a constant is zero while the Riemann-Liouville fractional derivative of a constant is nonzero. For more details, see 17 . Lemma 2.4 see 28 . For q > 0, the general solution of the fractional differential equation Dx t 0 is given by x t c0 c1t c2t · · · cn−1tn−1, 2.4 where ci ∈ R, i 0, 1, 2, . . . , n − 1 (n q 1). In view of Lemma 2.4, it follows that IqDqx t x t c0 c1t c2t · · · cn−1tn−1, 2.5 for some ci ∈ R, i 0, 1, 2, . . . , n − 1 n q 1 . Now, we state a known result due to Krasnoselskii 29 which is needed to prove the existence of at least one solution of 1.1 . Theorem 2.5. Let M be a closed convex and nonempty subset of a Banach space X. Let A,B be the operators such that i Ax By ∈ M whenever x, y ∈ M, ii A is compact and continuous, iii B is a contraction mapping. Then there exists z ∈ M such that z Az Bz. To study the nonlinear problem 1.1 , we first consider the associated linear problem and obtain its solution. Lemma 2.6. For a given σ ∈ C 0, 1 , the unique solution of the boundary value problem, Dx t σ t , 0 < t < 1, q ∈ m − 1, m , m ∈ N, m ≥ 2, x 0 0, x′ 0 0, x′′ 0 0, . . . , x m−2 0 0, x 1 αx ( η ) , 0 < η < 1, αηm−1 / 1, α ∈ R, 2.6