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 problemCD α u ( t ) + f ( t , u ( t ) ) = 0 , 0 < t < 1 , 2 < α ≤ 3 ,u ′ ( 0 ) = u ′′ ( 0 ) = 0 , u ( 1 ) = λ ∫ 0 1 u ( s ) d s ,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:CD α u ( t ) + f ( t , u ( t ) ) = 0 , 0 < t < 1 ,u ( i ) ( 0 ) = 0 , 0 ≤ i ≤ n , i ≠ k , u ( 1 ) = λ ∫ 0 1 u ( s ) d s ,where k ∈ { 0 , 1 , … , n − 1 } , 0≤ λ < k + 1; Problem 2:D α u ( t ) + f ( t , u ( t ) ) = 0 , 0 < t < 1 ,u ( i ) ( 0 ) = 0 , 0 ≤ i ≤ n − 1 ,u ( 1 ) = λ ∫ 0 1 u ( s ) d s ,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.