New Existence Results for Nonlinear Fractional Integrodifferential Equations

Abstract

This paper discusses a boundary value problem of nonlinear fractional integrodifferential equations of order $1<\alpha \le 2$ and $1<\beta \le 2$ and boundary conditions of the form $x\left(0\right)=x\left(1\right)={{}^{c}D}^{\beta }x\left(1\right)={{}^{c}D}^{\beta }x\left(0\right)=0$ . Some new existence and uniqueness results are proposed by using the fixed point theory. In particular, we make use of the Banach contraction mapping principle and Krasnoselskii’s fixed point theorem under some weak conditions. Moreover, two illustrative examples are studied to support the results.