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

We consider the following state dependent boundary-value problemD0+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),whereDαis the standard Riemann-Liouville fractional derivative of order1<α<2,0<η<1,p≤0,0<β<1,β+1-α≥0the functiongis defined asg(t,u):[0,1]×[0,∞)→[0,∞), andg(0,0)=0the functionfis defined asf(t,u):[0,1]×[0,∞)→[0,∞)σ(t),τ(t)are continuous ontand0≤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.