Discussion Of The Fixed-point Existence For The Decreasing Operators

Fixed point theorem is one of the important methods of the study for the existence and uniqueness of solutions of di?erential equations and integral equations. As well, the fixed point of multi-valued operators is of great value for nonlinear functional analysis. On the basis of previous research, we mainly study the existence of the fixed point of multivalued decreasing operators in this paper. Firstly, by using the Zorn lemma and the related maximal element existence theorems, we define the multi-valued decreasing operator and obtain its properties so that we receive new results together with inferences of the fixed point of multi-valued decreasing operators under tight conditions. Secondly, as the result of the weakness of the existence conditions of the fixed point by iterative techniques, we not only define the multi-valued quasi-decreasing operator and obtain its properties, but also receive new results of the fixed point of multi-valued decreasing operators under no tight conditions as well as verifying its correctness. In the end of this paper, for applying the thought of completion to the fixed point theory of decreasing operators, we continue to discuss the generalized fixed point of decreasing operators in addition to gain some relevant conclusions. During the study, we adopt several new methods to improve and extend previous results by using the theory of partially ordered methods, cone, successive iterative techniques and other related achievements. Principally, we obtain some new fixed point theorems of single-valued and multi-valued decreasing operators.