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 \(\begin{aligned} \left\{ \begin{aligned} {D_{0+}^{\alpha }}x(t)=f\left( t,x(t),{D_{0+}^{\alpha -1}}x(t)\right) , \ t\in (0,1),\\ x(0)=0, \ x(1)=B, \end{aligned}\right. \end{aligned}\) $\begin{array}{c}\hfill \left\{\begin{array}{c}\hfill D_{0+}^{\alpha }x(t)=f\left(t,,,x,(t),,,D_{0+}^{\alpha -1},x,(t)\right),t\in (0,1),\\ \hfill x(0)=0,x(1)=B,\end{array}\right)\end{array}$ are obtained, where \(B\in {\mathbb {R}}\) $B\in \mathbb{R}$ , \({D_{0+}^{\alpha }}x(t)\) $D_{0+}^{\alpha }x(t)$ is the Riemann-Liouville fractional derivative, \({\alpha }\in (1,2]\) $\alpha \in (1,2]$ is a real number, and \(f\in C\left( [0,1]\times {\mathbb {R}}^{2}, {\mathbb {R}}\right)\) $f\in C\left([0,1],\times ,{\mathbb{R}}^2,,,\mathbb{R}\right)$ . 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.