Common Fixed Points of Cq-Commuting Maps via Generalized Gregus-Type Inequalities

TL;DR

This paper proves the existence of common fixed points for certain generalized mappings in normed spaces, extending classical fixed point theory with new set-distance and compatibility conditions, and applies these results to approximation problems.

Contribution

It introduces a generalized framework for fixed point existence involving $C_q$-commuting maps, set-distance constraints, and reciprocal continuity, broadening classical theorems.

Findings

Established existence of common fixed points under new conditions.

Extended fixed point theory to set-distance and reciprocal continuity.

Provided applications to invariant approximation theorems.

Abstract

We establish the existence of common fixed points for $C_q$-commuting self-mappings satisfying a generalized Gregus-type inequality with quadratic terms in $q$-starshaped subsets of normed linear spaces. Our framework extends classical fixed point theory through: (i) Set-distance constraints $\delta(\cdot, [q, \cdot])$ generalizing norm conditions (ii) Compatibility via $C_q$-commutativity without full affinity requirements (iii) Reciprocal continuity replacing full map continuity. Explicit examples (e.g., Example 2.6) demonstrate the non-triviality of these extensions. As applications, we derive invariant approximation theorems for best approximation sets. Our results generalize Nashine's work \cite{Nashine2007} and unify several known fixed point theorems.