The Fixed Point Theorems For Mixed Monotone Operators And Riemann-Liouville Fractional Differential Equation Boundary Value Problems

The fixed point theorems for mixed monotone operators and Riemann-Liouville frac-tional differential equation boundary value problems have been studied by many authors, and this paper will also consider these operators and problems. Firstly, by using the prop-erties of cones, a fixed point theorem for mixed monotone operators and the non-symmetric iteration method, this paper is to present some new fixed point theorems for mixed mono-tone operators. Secondly, by using new fixed point theorems for mixed monotone operators, Krasnoselskii’s fixed point theorem, the Banach contraction principle and Schauder fixed point theorem, this paper is to investigate existence or uniqueness of solutions or positive solutions of some classes of Riemann-Liouville fractional differential equation boundary value problems.This paper is mainly composed of three chapters:The first chapter is the introduction. We introduce the developments of theorems for nonlinear mixed monotone operators and Riemann-Liouville fractional differential equation boundary value problems and describe the basic concepts and notes.In the first section of the second chapter, by using the properties of cones and a fixed point theorem for mixed monotone operators, this section is to present some new fixed point theorems for mixed monotone operators with perturbation. In the second section, by using the non-symmetric iteration method, this paper is to present some new fixed point theorems for mixed monotone operators.In the first section of the third chapter, by using the first section of the second chapter of the fixed point theorems for mixed monotone operators with perturbation, this section is to concern with the existence and uniqueness of positive solutions of two classes for the following Riemann-Liouville fractional differential equation boundary value problems: andIn the second section of the third chapter, by using Krasnoselskii’s fixed point theorem, this section is to study the existence of at least one or two positive solutions to a coupled sys-tem of Riemann-Liouville fractional differential equation nonlinear boundary value problems given by:In the third section of the third chapter, by using the Banach contraction principle and Krasnoselskii’s fixed point theorem, this section is to investigate the second section of the third chapter of the existence and uniqueness of solutions to multi-point boundary value problems for a coupled system of Riemann-Liouville fractional differential equations.In the fourth section of the third chapter, by using Schauder fixed point theorem, this paper is to consider the existence of at least one positive solution to a coupled system of Riemann-Liouville fractional boundary value problems given by:...