New Existence Results for Fractional Integrodifferential Equations with Nonlocal Integral Boundary Conditions

Abstract

We consider a boundary value problem of fractional integrodifferential equations with new nonlocal integral boundary conditions of the form: $x(0)=\beta x(\theta ),\hspace{0.17em}x(\xi )=\alpha {\int }_{\eta }^1\mathrm{‍}x(s)ds$, and . According to these conditions, the value of the unknown function at the left end point $t=0$ is proportional to its value at a nonlocal point $\theta$ while the value at an arbitrary (local) point $\xi$ is proportional to the contribution due to a substrip of arbitrary length $(1-\eta )$. These conditions appear in the mathematical modelling of physical problems when different parts (nonlocal points and substrips of arbitrary length) of the domain are involved in the input data for the process under consideration. We discuss the existence of solutions for the given problem by means of the Sadovski fixed point theorem for condensing maps and a fixed point theorem due to O’Regan. Some illustrative examples are also presented.