Solvability of a Three-Point Fractional Nonlinear Boundary Value Problem

Abstract

In this paper we study the fractional boundary value problem $$\begin{array}{ll}& ^{c}D_{0^{+}}^{q}u \left( t \right) =f(t, u(t)),\quad 0 < t <1 \\ & u\left( 0 \right) = \alpha u^{\prime} \left( 0 \right) ,\quad u \left( 1\right) =\beta u^{\prime } \left( \eta \right) ,\end{array}$$ where \({1 < q < 2, \alpha , \beta \in IR}\) and \({^{c}D_{0^{+}}^{q}}\) denotes the Caputo’s fractional derivative. Using Banach contraction principle and Leray–Schauder nonlinear alternative we prove the existence and uniqueness of solutions. Some examples are given to illustrate our results.