Fixed Point Theorems for Paired Contractive Mappings in $b$-Metric Spaces

Abstract

In this paper, we develop a fixed point framework for paired contractive mappings in complete $b$-metric spaces. We introduce the class of paired-Chatterjea contractions and establish a corresponding fixed point theorem. We further show that every Chatterjea-type contraction is a paired-Chatterjea contraction, whereas the converse implication does not hold, demonstrating that the paired formulation properly extends the Chatterjea-type class. Additionally, we establish a fixed point theorem for Paired contractions under the condition $α\in[0,1)$, thereby extending the admissible range of the contraction constant from $[0,\frac1s)$ in Bouker et al.~(2026) to [0,1). Furthermore, we examine the Paired-Kannan fixed point result of Bouker et al.~(2026) and provide an example showing that the stated coefficient condition does not guarantee the claimed fixed point conclusion in $b$-metric spaces. Motivated by this observation, we establish a corrected Paired-Kannan fixed point theorem under a complete and appropriately formulated set of hypotheses. Under suitable contractive conditions, we prove that the contractions under consideration have fixed points if and only if they have no periodic point of prime period 2. Moreover, we show that each of these contractions has at most two fixed points. Several examples are presented to illustrate our results and distinguish the proposed paired contractive mappings from existing ones.