Positive Solutions for Class of State Dependent Boundary Value Problems with Fractional Order Differential Operators

Abstract

We consider the following state dependent boundary-value problem $D_{0+}^{\alpha }y(t)-p{D}_{0+}^{\beta }g(t,y(\sigma (t)))+f(t,y(\tau (t)))=0$, ; $y(0)=0$, $\eta y(\sigma (1))=y(1),$ where $D^{\alpha }$ is the standard Riemann-Liouville fractional derivative of order , , $p\le 0$, , $\beta +1-\alpha \ge 0$ the function $g$ is defined as $g(t,u):[\mathrm{0,1}]\times [0,\mathrm{\infty })\to [0,\mathrm{\infty })$, and $g(\mathrm{0,0})=0$ the function $f$ is defined as $f(t,u):[\mathrm{0,1}]\times [0,\mathrm{\infty })\to [0,\mathrm{\infty })\sigma (t)$, $\tau (t)$ are continuous on $t$ and $0\le \sigma (t)$, $\tau (t)\le t$. Using Banach contraction mapping principle and Leray-Schauder continuation principle, we obtain some sufficient conditions for the existence and uniqueness of the positive solutions for the above fractional order differential equations, which extend some references.