Graph contractions in vector-valued metric spaces and applications

ABSTRACT

In this paper, we will present a study of the fixed point equation $x=f\left(x\right),x\in X$ (where $(X,d)$ is a generalized metric space in the sense that $d(x,y)\in {\mathbb{R}}_{+}^m$ and $f:X\to X$ is a graph contraction) by the following perspectives: existence, uniqueness, approximation, data dependence of the operator perturbation, well-posedness, and Ulam-Hyers stability. The non-self case is also discussed and some applications are given.

Keywords:

• Non-self graph (orbital) contraction
• vector-valued metric
• fixed point
• stability properties
• coincidence problem

2010 Mathematics Subject Classifications:

• 47H10
• 54H25

Acknowledgments

The authors are grateful to the anonymous referees for the useful remarks and suggestions that improved the quality of the paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

J.-C. Yao was partially supported by China Medical University Hospital [Grant MOST 108-2115-M-039-005-MY3].The first two authors extend their sincere appreciation to Professor J.-C. Yao for supporting, by the Grant MOST 108-2115-M-039-005-MY3, the scientific visit to NSYSU Kaohsiung, Taiwan.