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

Abstract

© 2016 John Wiley & Sons, Ltd. In this paper, first, we consider the existence of a positive solution for the nonlinear fractional differential equation boundary value problemCDαu(t) + f(t,u(t)) = 0, 0 < t < 1, 2 < α ≤ 3, u′ (0) = u″ (0) = 0, u(1) = λ ∫01 u (s) ds, where 0 ≤ λ < 1,CDα˛ is the Caputo’s differential operator of order ˛, and f : [0,1] × [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: Problem1:CDαu(t) + f(t,u(t)) = 0, 0 < t < 1, u(f) (0) = 0, 0 ≤ i ≤ n, i ≠ k, u (1) = λ ∫01 u(s) ds, where k ∊ {0, 1,… n – 1}, 0 ≤ λ < k + 1; Problem2:CDαu(t) + f(t,u(t)) = 0, 0 < t < 1, u(i) (0) = 0, 0 ≤ i ≤ n – 1, u(1) = λ ∫01 u(s) ds, where 0 ≤ λ ≤ α and Dα and D˛ is the Riemann–Liouville fractional derivative of order ˛. For these problems, we give existence results, which improve recent results in the literature.