Positive solutions for a class of fractional boundary value problems with fractional boundary conditions

Abstract

In this paper, we study the solvability of a nonlinear fractional differential equation under fractional integral boundary conditions. Via a mixed monotone operator method, some new results on the existence and uniqueness of a positive solution for the considered model are obtained. Moreover, we provide iterative sequences for approximating the solution. Some examples are also presented in order to illustrate the obtained result. Keywords:Fractional boundary value problem, fractional integral boundary condition, mixed monotone operator.

Fixed point theorems for a class of nonlinear sum-type operators and application in a fractional differential equation

“…By using the properties of cones and a fixed point theorem for mixed monotone operator, we obtain the existence and uniqueness of positive solutions. As applications, the above results on the sum-type nonlinear operators have been widely applied to study nonlinear differential and integral equations, see [11][12][13][14][15][16][17][18][19][20][21][22][23] and the references therein. In [12], the new fixed point theorems are used to prove positive solutions to a second order Neumann boundary value problem, a Sturm-Liouville boundary value problem, and a nonlinear elliptic boundary value problem for the Lane-Emden-Fowler equation.…”

“…In [12], the new fixed point theorems are used to prove positive solutions to a second order Neumann boundary value problem, a Sturm-Liouville boundary value problem, and a nonlinear elliptic boundary value problem for the Lane-Emden-Fowler equation. In [13,19,20], authors investigate the existence and uniqueness results for some kinds of fractional differential equations and nonlinear elastic beam equations via the fixed point theorems in [13]. Also, based on a method originally due to Zhai and Anderson [14], Feng et al present the existence and uniqueness of positive solutions for nonlinear elastic beam equations, Lane-Emden-Fowler equations, and a class of fractional differential equation with integral boundary conditions in [14,21].…”