Discontinuity at the fixed point in suprametric spaces

TL;DR

This paper extends fixed point theorems in complete suprametric spaces for convex contractions of order m, showing fixed points can exist under weaker conditions without requiring continuity.

Contribution

It introduces weaker conditions like $k$-continuity and $T$-orbitally lower semi-continuity to guarantee fixed points, addressing an open problem by Rhoades.

Findings

Fixed points exist under weaker conditions than continuity.

The class of convex contractions of order m is sufficient for fixed points.

Generalizations of Sehgal, Cirić, and Fisher's quasi-contraction results.

Abstract

The aim of this paper is to generalize some fixed point theorems in the class of convex contraction of order $m$ on a complete suprametric space. Then, we will prove that the class of convex contraction of order m is strong enough to generate a fixed point on a complete suprametric spaces but do not force the mapping to be continuous at the fixed point, and it can be replaced by relatively weaker conditions of $k$-continuity or $T$-orbitally lower semi-continuous. On this way a new and distinct solution to the open problem of Rhoades (Contemp Math 72:233-245,1988) is found. In sequel, we will prove some fixed point results in the setting suprametric spaces which are generalizations of the results regarding Sehgal, \'Ciri\'c and Fisher's quasi-contraction. Some examples and application will be approved our results.