Abstract
In the paper, we introduce noncommutative Banach spaces which generalize the concept of Banach spaces, and the k-ordered contractive condition; we then discuss an ordered structure and several properties on noncommutative Banach spaces. Moreover, some fixed-point theorems for mappings with the k-ordered contractive condition on noncommutative Banach spaces are presented. In addition, we investigate the existence and uniqueness of fixed points for an integral equation of Fredholm type. MSC:47H10.
Highlights
The well-known fixed-point theorem of Banach [ ] is a very important tool for solving existence problems in many branches of mathematics and physics
Ran and Reurings [ ], O’Regan and Petruşel [ ] and others started the investigations concerning a fixed-point theory in ordered metric spaces. Many authors followed this concept by introducing and investigating the different types of contractive mappings, e.g., in [ ] Caballero et al considered contractive-like mappings in ordered metric spaces and applied their results in ordinary differential equations
The purpose of this paper is to present some fixed-point theorems for mappings satisfying the ordered contractive condition in the context of noncommutative metric spaces which are noncommutative sense of those in [ ]
Summary
The well-known fixed-point theorem of Banach [ ] is a very important tool for solving existence problems in many branches of mathematics and physics. Some interesting fixed-point theorems concerning partially ordered metric spaces can be found in [ , ]. The existence of fixed points for the given contractive type mappings in partially ordered cone metric spaces was investigated (see [ , ]).
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