Coupled Fixed-Point Theorems in Theta-Cone-Metric Spaces

Xiaolong Qin,Sahar Mohamed Ali Abou Bakr

doi:10.1155/2021/6617738

Abstract

This paper gives further generalizations of some well-known coupled fixed-point theorems. Specifically, Theorem 3 of the paper is the generalization of the Baskar–Lackshmikantham coupled fixed-point theorem, and Theorem 5 is the generalization of the Sahar Mohamed Ali Abou Bakr fixed-point theorem, where the underlying space is complete θ -cone-metric space.

Highlights

Introduction and PreliminariesSince 1922, the pioneering fixed-point principle of Banach [1] showed exclusive interest of researchers because it has many applications, including variational linear inequalities and optimization, and applications in differential equations, in the field of approximation theory, and in minimum norm problems.Since several types of contraction mappings have been introduced and many research papers have been written to generalize this Banach contraction principle.In 1987, Guo and Lakshmikantham [2] introduced one of the most interesting concepts of coupled fixed point.Definition 1
In 2006, Bhaskar and Lakshmikantham [3] introduced the concept of the mixed monotone property as follows
In 2006, Bhaskar and Lakshmikantham [3] proved the existence of coupled fixed points for mixed monotone mappings with weak contractivity assumption in a partialordered Banach space (E, ‖.‖, ≤ ) as follows

Summary

Introduction

Introduction and PreliminariesSince 1922, the pioneering fixed-point principle of Banach [1] showed exclusive interest of researchers because it has many applications, including variational linear inequalities and optimization, and applications in differential equations, in the field of approximation theory, and in minimum norm problems.Since several types of contraction mappings have been introduced and many research papers have been written to generalize this Banach contraction principle.In 1987, Guo and Lakshmikantham [2] introduced one of the most interesting concepts of coupled fixed point.Definition 1. An element (x, y) ∈ E × E is said to be a coupled fixed point of the mapping T: E × E ⟶ E if and only if T(x, y) x and T(y, x) y.

Results

Conclusion