Existence of Solutions for Riemann-Liouville Fractional Dirichlet Boundary Value Problem

In this paper, existence theorems of solutions for the Riemann-Liouville fractional Dirichlet

boundary value problem \(\begin{aligned} \left\{ \begin{aligned} {D_{0+}^{\alpha }}x(t)=f\left( t,x(t),{D_{0+}^{\alpha -1}}x(t)\right) , \ t\in (0,1),\\ x(0)=0, \ x(1)=B, \end{aligned}\right. \end{aligned}\)are obtained, where \(B\in {\mathbb {R}}\), \({D_{0+}^{\alpha }}x(t)\) is the Riemann-Liouville fractional derivative, \({\alpha }\in (1,2]\) is a real number, and \(f\in C\left( [0,1]\times {\mathbb {R}}^{2}, {\mathbb {R}}\right)\). We do not impose growth restrictions on nonlinear term f as many authors do but merely require that f satisfies sign conditions at the origin. Our analysis is based on the nonlinear alternative of Leray-Schauder. Finally, we provide an example to verify our main results.