Existence of positive solutions in the space of Lipschitz functions for a fractional boundary problem with nonlocal boundary condition

Abstract

In this paper, we study the existence of positive solutions for the following nonlinear fractional boundary value problem: D0+αu(t)+f(t,u(t),(Hu)(t))=0,0<t<1,u(0)=u′(0)=0,u′(1)=βu(ξ),}where 2 < α≤ 3 , 0 < ξ< 1 , 0 ≤ βξα-1< (α- 1) , H is an operator (not necessarily linear) applying C[0 , 1] into itself and D0+α denotes the standard Riemann–Liouville fractional derivative of order α. Our solutions are placed in the space of Lipschitz functions and the main tools used in the study are a sufficient condition for the relative compactness in Hölder spaces and the Schauder fixed point theorem. Moreover, we present one example illustrating our results.