Coupled best proximity point theorems for $p$-cyclic $φ$-contraction and $p$-cyclic Kannan nonexpansive mappings

TL;DR

This work extends best proximity point theory by introducing two new cyclic mappings: $p$-cyclic $\phi$-contraction and $p$-cyclic Kannan nonexpansive mappings on $(A\times B)\cup (B\times A) \to A\cup B$, with $dist(A,B)$ denoting the distance between $A$ and $B$. By developing iterative estimates and leveraging uniform convexity, the authors prove the existence of coupled best proximity points for $p$-cyclic $\phi$-contractions and demonstrate existence (and sometimes uniqueness) results; the Kannan-type mappings are treated via a Zorn-type minimality argument, yielding a coupled best proximity point under suitable conditions. The paper also provides an example showing non-uniqueness in general for the coupled points and shows how these results extend previous theorems for $p$-cyclic contractions and cyclic Kannan nonexpansive mappings. Overall, the results offer a unified framework for optimal approximate solutions in split Banach-space settings and broaden the applicability of best proximity point theory.

Abstract

In this paper, the notions of $p$-cyclic $φ$-contraction and $p$-cyclic Kannan nonexpansive mappings are introduced, and the existence of coupled best proximity points for such mappings is established.

Theorems & Definitions (33)

• Lemma 1
• Definition 2
• theorem 3
• Definition 4
• theorem 5
• Definition 6
• theorem 7
• Definition 8
• theorem 9
• Definition 10