In this paper, we investigate the existence of an absolute continuous solution to a class of first-order nonlinear differential equation with integral boundary conditions (BCs) or with infinite-point BCs. The Liouville-Caputo fractional derivative is involved in the nonlinear function. We first consider the existence of a solution for the first-order nonlinear differential equation with m-point nonlocal BCs. The existence of solutions of our problems is investigated by applying the properties of the Riemann sum for continuous functions. Several examples are given in order to illustrate our results.

We investigate the existence and uniqueness of a solution for a Sturm-Liouville fractional differential equation with a multi-point boundary condition via the Caputo derivative; existence and uniqueness results for the given problem are obtained via the Banach fixed point theorem. Also we study its continuous dependence on coefficients of the nonlocal condition. We discuss our results for more general boundary conditions, we present the existence of solutions under nonlocal integral conditions and also extend our results to an ordinary Sturm-Liouville problem. Two examples illustrating the main results are also presented.

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In this paper, we establish sufficient conditions for the existence of a unique solution for a class of nonlinear non-autonomous system of Riemann-Liouville fractional differential systems with different constant delays and non-local condition is. The stability of the solution will be proved. As an application, we also give some examples to demonstrate our results.

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