The existence of a positive solution for nonlinear fractional differential equations with integral boundary value conditions

Abstract

In this paper, first, we consider the existence of a positive solution for the nonlinear fractional differential equation boundary value problem urn:x-wiley:mma:media:mma4105:mma4105-math-0002 where 0≤λ < 1,CDα is the Caputo's differential operator of order α, and f:[0,1] × [0,∞)→[0,∞) is a continuous function. Using some cone theoretic techniques, we deduce a general existence theorem for this problem. Then, we consider two following more general problems for arbitrary α, 1≤n < α≤n + 1:Problem 1: urn:x-wiley:mma:media:mma4105:mma4105-math-0003 where , 0≤λ < k + 1;Problem 2: urn:x-wiley:mma:media:mma4105:mma4105-math-0004 where 0≤λ≤α and Dα is the Riemann–Liouville fractional derivative of order α.For these problems, we give existence results, which improve recent results in the literature. Copyright © 2016 John Wiley & Sons, Ltd.