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On Necessary and Sufficient Conditions for Fixed Point Convergence: A Contractive Iteration Principle

Vasil Zhelinski

TL;DR

The paper extends Banach's fixed-point theory by introducing a universal iterative contraction framework that yields necessary and sufficient conditions for the existence of a unique fixed point $\alpha$ with $T^n x\to\alpha$ for all $x$, along with explicit error bounds $\rho(\alpha,T^{p_n}x)\le k^n M_x$ and $\sup_{i\ge p_n}\rho(\alpha,T^ix)\le 2k^n M_x$. It also provides a second theorem characterizing convergence of all iterates to fixed points that may not be unique, via an invariant closed cover where $T$ remains a universal iterative contraction. The work unifies and extends classical generalizations (Banach, Kannan, Chatterjea, Hardy–Rogers, Meir–Keeler, Guseman) by establishing necessary as well as sufficient conditions for convergence and by giving concrete error estimates. Four examples demonstrate the breadth of applicability, including cases where traditional results fail or are inconclusive, and show how the approach can both confirm convergence and, in some constructions, rule out convergence to any fixed point.

Abstract

While numerous extensions of Banach's fixed point theorem typically offer only sufficient conditions for the existence and uniqueness of a fixed point and the convergence of iterative sequences, this study introduces a generalization grounded in the iterative contraction principle in complete metric spaces. This generalization establishes both the necessary and sufficient conditions for the existence of a unique fixed point to which all iterative sequences converge, along with an accurate error estimate. Furthermore, we present and prove an additional theorem that characterizes the convergence of all iterative sequences to fixed points that may not be unique. Several examples are provided to illustrate the practical application of these results, including a case where the traditional and well-known generalizations of Banach's theorem, such as those by Banach, Kannan, Chatterjea, Hardy-Rogers, Meir-Keeler, and Guseman, are inapplicable.

On Necessary and Sufficient Conditions for Fixed Point Convergence: A Contractive Iteration Principle

TL;DR

The paper extends Banach's fixed-point theory by introducing a universal iterative contraction framework that yields necessary and sufficient conditions for the existence of a unique fixed point with for all , along with explicit error bounds and . It also provides a second theorem characterizing convergence of all iterates to fixed points that may not be unique, via an invariant closed cover where remains a universal iterative contraction. The work unifies and extends classical generalizations (Banach, Kannan, Chatterjea, Hardy–Rogers, Meir–Keeler, Guseman) by establishing necessary as well as sufficient conditions for convergence and by giving concrete error estimates. Four examples demonstrate the breadth of applicability, including cases where traditional results fail or are inconclusive, and show how the approach can both confirm convergence and, in some constructions, rule out convergence to any fixed point.

Abstract

While numerous extensions of Banach's fixed point theorem typically offer only sufficient conditions for the existence and uniqueness of a fixed point and the convergence of iterative sequences, this study introduces a generalization grounded in the iterative contraction principle in complete metric spaces. This generalization establishes both the necessary and sufficient conditions for the existence of a unique fixed point to which all iterative sequences converge, along with an accurate error estimate. Furthermore, we present and prove an additional theorem that characterizes the convergence of all iterative sequences to fixed points that may not be unique. Several examples are provided to illustrate the practical application of these results, including a case where the traditional and well-known generalizations of Banach's theorem, such as those by Banach, Kannan, Chatterjea, Hardy-Rogers, Meir-Keeler, and Guseman, are inapplicable.
Paper Structure (8 sections, 13 theorems, 31 equations, 1 figure)

This paper contains 8 sections, 13 theorems, 31 equations, 1 figure.

Table of Contents

  1. Introduction and Preliminaries
  2. Main Result
  3. Auxiliary Results
  4. proof of main result
  5. proof of Theorem \ref{['t1']}
  6. proof of Theorem \ref{['t3']}
  7. proof of Theorem \ref{['t2']}
  8. Examples

Key Result

Theorem 1

Consider a complete metric space $(X,\rho)$ and a mapping $T:X\to X$. Suppose there exist constants $0\leq k_i<1$ for $i\in \{1,2,3\}$ such that $k_1+k_2+k_3<1$, and the inequality holds for all $x,y\in X$. Then, $T$ possesses a unique fixed point within $X$, and for any $x\in X$, the sequence $\{T^nx\}_{n=1}^\infty$ converges to the fixed point of $T$.

Figures (1)

  • Figure 1: Above in left $y=1$. Above in right $y=0.5$. Below $y=0.1$.

Theorems & Definitions (28)

  • Theorem 1: HARDY-ROGERS
  • Theorem 2: MK
  • Theorem 3: CI-Introduction-1
  • Theorem 4: CI-Introduction-2
  • Definition 1
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Lemma 1
  • proof
  • ...and 18 more