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Common Fixed Point Theorems Of Weakly Compatible Maps Satisfying (f,g)-Weakly Contractive Condition And Invariant Approximation Results

Babu G. V. R., Ratna Babu D, Alemayehu Negash

TL;DR

The paper introduces $(f,g)$-weakly contractive mappings for three selfmaps on a metric space and proves existence and uniqueness of a common fixed point for $T,f,g$ under weak compatibility and invariance conditions, with $\\overline{T(X)}$ embedded in both $f(X)$ and $g(X)$ and forming a complete subspace. It further proves the convergence of the modified Mann and modified Ishikawa iterations to this unique fixed point in normed spaces under appropriate summability of parameters. Additionally, invariant-approximation results show that the common fixed point can be characterized as a best approximation from invariant compact subsets. Together, these results extend and unify prior fixed-point theorems by Beg–Abbas and Azam–Shakeel, and point to broader applicability in nonlinear analysis and equations.

Abstract

We prove the existence of common fixed points for three selfmaps $T,f$ and $g$ defined on a metric space $(X,d)$ satisfying, $T$ is $(f,g)$-weakly contractive; and the pairs $(T,f)$ and $(T,g)$ are weakly compatible. Also, for such $T,f$ and $g$, we prove the convergence of modified Mann iteration and modified Ishikawa iteration with respect to $T,f$ and $g$ to their common fixed point, in a normed space. Further, we obtain invariant approximation results from the set of best approximations to common fixed points of $T,f$ and $g$.

Common Fixed Point Theorems Of Weakly Compatible Maps Satisfying (f,g)-Weakly Contractive Condition And Invariant Approximation Results

TL;DR

The paper introduces -weakly contractive mappings for three selfmaps on a metric space and proves existence and uniqueness of a common fixed point for under weak compatibility and invariance conditions, with embedded in both and and forming a complete subspace. It further proves the convergence of the modified Mann and modified Ishikawa iterations to this unique fixed point in normed spaces under appropriate summability of parameters. Additionally, invariant-approximation results show that the common fixed point can be characterized as a best approximation from invariant compact subsets. Together, these results extend and unify prior fixed-point theorems by Beg–Abbas and Azam–Shakeel, and point to broader applicability in nonlinear analysis and equations.

Abstract

We prove the existence of common fixed points for three selfmaps and defined on a metric space satisfying, is -weakly contractive; and the pairs and are weakly compatible. Also, for such and , we prove the convergence of modified Mann iteration and modified Ishikawa iteration with respect to and to their common fixed point, in a normed space. Further, we obtain invariant approximation results from the set of best approximations to common fixed points of and .
Paper Structure (4 sections, 11 theorems, 72 equations)

This paper contains 4 sections, 11 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. Invariant Approximation Results
  4. Conclusion

Key Result

Theorem 1.4

Let $(X,d)$ be a complete metric space, $T$ a weakly contractive map. Then $T$ has a unique fixed point in $X$.

Theorems & Definitions (26)

  • Definition 1.1
  • Definition 1.2: Rhoades2001
  • Example 1.3
  • Theorem 1.4: Rhoades weakly contractive principle
  • Definition 1.5
  • Theorem 1.6: Beg and Abbas Beg2006, Theorem 2.5, page 4
  • Theorem 1.7: A. Azam and M. Shakeel Azam2008, Theorem 2.4, page 104
  • Definition 1.8
  • Example 1.9
  • Theorem 2.1
  • ...and 16 more