Fixed point theorems without continuity in metric vector spaces

TL;DR

This paper establishes fixed point theorems in metric vector spaces without requiring continuity, broadening the scope of fixed point results and illustrating their necessity through examples and counterexamples.

Contribution

It introduces fixed point theorems in metric vector spaces that do not depend on continuity, expanding the theoretical framework.

Findings

Fixed point theorems without continuity assumptions

Concrete examples demonstrating the theorems

Counterexamples showing necessity of conditions

Abstract

In this oaper, we prove some fixed point theorems in metric vector spaces, in which the continuity is not required for the considered mappings to satisfy. We provide some concrete examples to demonstrate these theorems. We also give some counterexamples to show that every condition in the theorems is necessary for the considered mapping to have a fixed point.