POSITIVE SOLUTIONS OF NONLINEAR FRACTIONAL THREE-POINT BOUNDARY-VALUE PROBLEM.

In this paper, we study the existence of positive solutions to the boundary-value problem with fractional order (^C_aD^αy)(t) +q(t)f(y) = 0, 0 ≤ a < t < b, 1 < α < 2, y(a) = 0, y(b) = βy(η), where a < η < b and β(η − a) − b + a ≠ 0. We prove the existence of at least one positive solution when f is either superlinear or sublinear using the well-known Guo-Lakshmikantham fixed point theorem in cones. Moreover, the convexity and concavity of the solutions are investigated with respect to the behavior of the function q. [ABSTRACT FROM AUTHOR]