Existence of Solutions for Nonlocal Boundary Value Problems of Higher-Order Nonlinear Fractional Differential Equations

Abstract

We study some existence results in a Banach space for a nonlocal boundary value problem involving a nonlinear differential equation of fractional order $q$ given by ${}^c{D}^{q}x\left(t\right)=f\left(t,x\left(t\right)\right)$, $0<t<1$, $q\in \left(m-1,m\right]$, $m\in \mathbb{N}$, $m\ge 2$, $x\left(0\right)=0$, $x^{\prime }\left(0\right)=0,\hspace{0.17em}{x}^{\prime \prime }\left(0\right)=0,\dots ,x^{\left(m-2\right)}\left(0\right)=0$, $x\left(1\right)=\alpha x\left(\eta \right)$. Our results are based on the contraction mapping principle and Krasnoselskii's fixed point theorem.