Some Results On Mixed Monotone Operators And Fixed Points In Banach Spaces

Since mixed monotone operators were first proposed in1987by Guo Dajun, the existence and uniqueness of solutions have been investigated and many useful results are obtained.In this paper, we gain some new results by discussing the mixed monotone operators and fixed point theorems.To begin with, by using the cone theory and the monotonous iteration method, we obtain the existence and uniqueness of solutions of non-monotone binary system of equations without continuity and compactness conditions under more general conditions. The error estimates are also given. Our results here improve and generalize some known results.Secondly, by using a new method, we obtain some existence and uniqueness theorems of fixed points of mixed monotone operators with different types of con-vexity and concavity, which are more general than some known results. Moreover, some examples are given to demonstrate the applicability of our results.In the last section, a new fixed point theorem which is a generalization of Schauder’s fixed point theorem is established. By using it, a new fixed point theorem which generalizes the. well-known Darbo’s fixed point theorem is given. As applications, the existence of global solutions for first-order nonlinear integro-differential equations of mixed type in a real Banach space is investigated.