Existence of Positive Solutions for Fractional Differential Equation with Nonlocal Boundary Condition

Abstract

By using the fixed point theorem, existence of positive solutions for fractional differential equation with nonlocal boundary condition $D_{0+}^{\alpha }u\left(t\right)+a\left(t\right)f(t,u(t\left)\right)=0$, , $u\left(0\right)=0$, $u\left(1\right)={\sum }_{i=1}^{\infty }{\alpha }_{i}u\left({\xi }_i\right)$ is considered, where is a real number, $D_{0+}^{\alpha }$ is the standard Riemann-Liouville differentiation, and ${\xi }_i\in \left(\mathrm{0,1}\right),{\alpha }_i\in [0,\infty )$ with , $a\left(t\right)\in C\left(\right[\mathrm{0,1}],[0,\infty \left)\right),f(t,u)\in C\left(\right[\mathrm{0,1}]\times [0,\infty ),[0,\infty \left)\right)$.