Positive Solutions of Nonlinear Fractional Differential Equations with Integral Boundary Value Conditions

Abstract

We investigate the existence and uniqueness of positive solutions of the following nonlinear fractional differential equation with integral boundary value conditions${}^C{D}^{\alpha }u(t)+f(t,u(t))=0$, , $u(0)=u^{\mathrm{\prime \prime }}(0)=0,u(1)=\lambda {\int }_0^{1}u(s)ds$, where , and ${}^C{D}^{\alpha }$ is the Caputo fractional derivative and $f:[\mathrm{0,1}]\times [0,\infty )\to [0,\infty )$ is a continuous function. Our analysis relies on a fixed point theorem in partially ordered sets. Moreover, we compare our results with others that appear in the literature.