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

Dongyuan Liu, Zigen Ouyang, Huilan Wang
2015 Abstract and Applied Analysis  
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
more » ... obtain some sufficient conditions for the existence and uniqueness of the positive solutions for the above fractional order differential equations, which extend some references.
doi:10.1155/2015/263748 fatcat:4ex2j7xht5dq7lu3gm4ovqg3f4