Lower and Upper Solutions Method for Positive Solutions of Fractional Boundary Value Problems

Abstract

We apply the lower and upper solutions method and fixed-point theorems to prove the existence of positive solution to fractional boundary value problem $D_{0^{+}}^{\alpha }u\left(t\right)+f\left(t,u\left(t\right)\right)=0$, , , $u\left(0\right)=u^{\prime }\left(0\right)=0$, $D_{0^{+}}^{\alpha -1}u\left(1\right)=\beta u\left(\xi \right)$, , where $D_{0^{+}}^{\alpha }$ denotes Riemann-Liouville fractional derivative, β is positive real number, $\beta {\xi }^{\alpha -1}\ge 2\mathrm{\Gamma }\left(\alpha \right)$, and $f$ is continuous on $\left[\mathrm{0,1}\right]\times \left[0,\left.\infty \right)\right.$. As an application, one example is given to illustrate the main result.