Existence and Uniqueness of Solutions for Fractional Boundary Value Problems under Mild Lipschitz Condition

Abstract

This paper deals with the following boundary value problem $\left\{\begin{array}{c}{D}^{\alpha }u\left(t\right)=f\left(t,u\left(t\right)\right),t\in \left(0,1\right),\hfill \\ u\left(0\right)=u\left(1\right)=D^{\alpha -3}u\left(0\right)=u^{\prime }\left(1\right)=0,\hfill \end{array}\right.$ where $3<\alpha \le 4,$ $D^{\alpha }$ is the Riemann-Liouville fractional derivative, and the nonlinearity $f,$ which could be singular at both $t=0$ and $t=1,$ is required to be continuous on $\left(0,1\right)\times ℝ$ satisfying a mild Lipschitz assumption. Based on the Banach fixed point theorem on an appropriate space, we prove that this problem possesses a unique continuous solution $u$ satisfying $\left|u\left(t\right)\right|\le c\omega \left(t\right),\text{for}t\in \left[0,1\right]\text{and}c>0,$ where $\omega \left(t\right)≔t^{\alpha -2}{\left(1-t\right)}^2.$