Existence solutions for nonlocal fractional differential equation with nonlinear boundary conditions

In this paper, by employing the Guo-Krasnoselskii fixed point theorem in a cone, we study the existence of positive solutions to the following nonlocal fractional boundary value problems { c ??0+ ?? ??(??) = ??(??, ??(??)), ?? ? (0,1), ??(??) + ???(0) = 1 2 [??1(??(??)) + ? ??2 ?? (??, ??(??))????] , ??(1) + ???(1) = 0, where c??0+ ?? is the standard Caputo derivative of order ??, 1 < ?? < 2, ?? ? (0,1) is some measurable set. We provide conditions on ??, ??1, ??2 and ?? such that the problem exhibits at least one positive solution.