Abstract
We consider the properties of Green’s function for the nonlinear fractional differential equation boundary value problem:D0+αu(t)+f(t,u(t))+e(t)=0,0<t<1,u(0)=u'(0)=⋯=u(n-2)(0)=0,u(1)=βu(η), wheren-1<α≤n,n≥3,0<β≤1,0≤η≤1,D0+αis the standard Riemann-Liouville derivative. Here our nonlinearityfmay be singular atu=0. As applications of Green’s function, we give some multiple positive solutions for singular boundary value problems by means of Schauder fixed-point theorem.
Highlights
Fractional differential equations have been of great interest recently
For the convenience of the reader, we present here the necessary definitions from fractional calculus theory
We establish the existence of positive solutions for equation
Summary
Fractional differential equations have been of great interest recently. This is due to the intensive development of the theory of fractional calculus itself as well as its applications. We give some existence of positive solutions for singular boundary value problems by means of Schauder fixed-point theorem γ∗ < 0 < γ∗. The Riemann-Liouville fractional derivative of order α > 0 of a continuous function y : (0, ∞) → R is given by Let us fix some notations to be used in the following.
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