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

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, C D α 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 D α 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.