Positive Solutions for a Higher-Order Semipositone Nonlocal Fractional Differential Equation with Singularities on Both Time and Space Variable. (3rd February 2019) - Zhang, Kemei
Abstract : In this paper, we consider the following higher-order semipositone nonlocal Riemann-Liouville fractional differential equationD 0 + α x ( t ) + f ( t, x ( t ), D 0 + β x ( t ) ) + e ( t ) = 0,     0 < t < 1, D 0 + β x ( 0 ) = D 0 + β + 1 x ( 0 ) = ⋯ = D 0 + n + β - 2 x ( 0 ) = 0, andD 0 + β x ( 1 ) = ∑ i = 1 m - 2 η i D 0 + β x ( ξ i ), whereD 0 + α andD 0 + β are the standard Riemann-Liouville fractional derivatives. The existence results of positive solution are given by Guo-krasnosel'skii fixed point theorem and Schauder's fixed point theorem.