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.

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A general class of nonlinear fractional differential equations is considered. Some sufficient conditions for the existence and uniqueness of exact solution are established by using Weissinger's fixed point theorem and the Gronwall-Bellman lemma. A spectral collocation method based on the smoothing technique is presented to solve the problem numerically. Then the rigorous error estimates under the L 2 and L ∞ norms are derived. The most remarkable feature of the method is its capability to achieve spectral convergence for weakly singular solutions. Finally, numerical results are given to support the theoretical conclusions with smooth and weakly singular solutions.

In this paper, we study the existence of solutions for the boundary value problems of fractional perturbation differential equationssubject towhere 1 < α < 2, D α is the standard Caputo fractional derivatives. Using some fixed point theorems, we prove the existence of solutions to the two types. For each type we give an example to illustrate our results.

In this paper, we investigate a class of nonlinear fractional differential system supplemented with coupled strip and infinite point boundary conditions. Existence results for the given problem are obtained by using the Banach's fixed point theorem and the • e norm. The Lipschitz type conditions on nonlinearities are needed and it seems that the continuity assumptions used previously are not sufficient. The proposed problem is of quite a general nature as it covers several special cases. Finally, we present an example to illustrate our main results.