Existence and non-existence of positive solutions for a class of fractional boundary value problems

In this paper, we consider a boundary value problem consisting of the nonlinear fractional differential equation ?D?0+u + aD?0+u = f (t, u), 0 < t < 1, with nonlocal boundary conditions D? 0+u(0) = 0, D??? 0+ u(1) = au(1), u?(1) = 0, where, 2 < ? < ? ? 3, 0 ? ? < ? ? ?, 0 ? a < ?(? ? ? + 1) and f (t, u) ? C([0, 1] ? [0,?), [0,?)) and D?0 + is the standard Riemann-Liouville fractional derivative of order ?. The associated Green?s function is derived in terms of the generalized Mittag-Leffler functions and it is shown that it satisfies certain properties. An attempt has been made to establish the existence and non-existence of positive solutions by using Leggett- Williams fixed point theorems on a cone in a Banach space. The results obtained in this paper extended and generalizes the result of [R. Graef, L. Kong, Q. Kong and M.Wang, Positive solutions of nonlocal fractional boundary value problems, Discrete Contin. Dyn. Syst. 7 (4) (2013), 283?290]. Finally, we provide a couple of examples to illustrate the validation of established results.