In this paper, we study boundary-value problems for the following nonlinear fractional differential equations involving the Caputo fractional derivative:continuous function and m ∈ R, n -1 < α < n (n ≥ 2), 0 < β < 1 is a real number. By means of the Banach fixed-point theorem and the Schauder fixed-point theorem, some solutions are obtained, respectively. As applications, some examples are presented to illustrate our main results. MSC: 34A08; 34B10

In this paper, we establish some sufficient conditions for the existence of solutions to two classes of boundary value problems for fractional differential equations with nonlocal boundary conditions. Our goal is to establish some criteria of existence for the boundary problems with nonlocal boundary condition involving the Caputo fractional derivative, using Banach's fixed point theorem and Schaefer's fixed point theorem. Finally, we present four examples to show the importance of these results.

In this paper, we investigate the initial value problems for a class of nonlinear fractional differential equations involving the variable-order fractional derivative. Our goal is to construct the spectral collocation scheme for the problem and carry out a rigorous error analysis of the proposed method. To reach this target, we first show that the variable-order fractional calculus of non constant functions does not have the properties like the constant order calculus. Second, we study the existence and uniqueness of exact solution for the problem using Banach's fixed-point theorem and the Gronwall-Bellman lemma. Third, we employ the Legendre-Gauss and Jacobi-Gauss interpolations to conquer the influence of the nonlinear term and the variable-order fractional derivative. Accordingly, we construct the spectral collocation scheme and design the algorithm. We also establish priori error estimates for the proposed scheme in the function spaces L 2 [0, 1] and L ∞ [0, 1]. Finally, numerical results are given to support the theoretical conclusions. Keywords Spectral collocation method • Variable fractional order • Initial value problem • Convergence analysis Mathematics Subject Classification 65L60 • 41A05 • 41A10 • 41A25 Communicated by José Tenreiro Machado.

No abstract

No abstract