Asia Mathematika, Vol 10, issue 1, pages: 35--48.

Revisiting Suzuki contractions under weak completeness structures

John Pamba
Department of Information and Communication Technology, Zambia University College of Technology, Ndola, Zambia.

Received: 24 Feb 2026 | Accepted: 04 May 2026 | Final Version: 30 May 2026

Abstract

This paper investigates Suzuki-type contractive mappings from the viewpoint of convergence structures and completeness requirements. While classical Suzuki fixed point theorems are formulated in complete metric spaces, we show that metric completeness is not intrinsic to the Suzuki contractive mechanism. We introduce a weak completeness framework that emphasizes operator-dependent convergence and prove that orbital completeness suffices for fixed point existence and uniqueness. The results establish fixed point principles that strictly refine the classical Suzuki theorem while preserving the Cauchy and convergence behavior of Picard iteration sequences. In addition, we analyze the structural dependence of Suzuki contractions on completeness properties and derive stability consequences, including well-posedness and data dependence estimates. These findings demonstrate that the essential convergence mechanisms are governed by iterative orbits rather than by global geometric completeness. Nontrivial examples illustrating spaces where metric completeness fails but orbital convergence remains valid are provided. The study yields a sharper interpretation of Suzuki-type contractions and broadens their analytical scope within modern fixed point theory and nonlinear analysis.

Keywords:
Suzuki contraction, weak completeness, orbital completeness, fixed point theorem, Picard iteration.

1. Introduction

Fixed point theory occupies a central position in modern nonlinear analysis due to its deep theoretical importance and broad applicability across mathematics and applied sciences. The subject is traditionally rooted in the Banach Contraction Principle [1], which established that a contraction mapping on a complete metric space admits a unique fixed point. This principle has inspired extensive research aimed at weakening contractive hypotheses and refining the structural requirements underlying convergence mechanisms.

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