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 D0+αy(t)-pD0+βg(t,y(σ(t)))+f(t,y(τ(t)))=0, 0<t<1; y(0)=0, ηy(σ(1))=y(1), where Dα is the standard Riemann-Liouville fractional derivative of order 1<α<2, 0<η<1, p≤0, 0<β<1, β+1-α≥0 the function g is defined as g(t,u):[0,1]×[0,∞)→[0,∞), and g(0,0)=0 the function f is defined as f(t,u):[0,1]×[0,∞)→[0,∞)σ(t), τ(t) are continuous on t and 0≤σ(t), τ(t)≤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.