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Fixed point theorems for weak, partial, Bianchini and Chatterjea-Bianchini contractions in semimetric spaces with triangle functions

Ravindra K. Bisht, Evgen O. Petrov

TL;DR

The paper extends fixed point theory to semimetric spaces endowed with triangle functions $\Phi$, enabling generalized contractions beyond standard metric spaces. It develops a unified framework in complete semimetric spaces with a continuous triangle function $\Phi$ and proves fixed point results for partial, weak, Bianchini, and Chatterjea-Bianchini contractions under $\Phi$-based inequalities. The main contributions include existence (and where possible uniqueness) theorems for each contraction type under explicit $\Phi$-based conditions, applicable to metric, $b$-metric, ultrametric, and power-triangle distance spaces. These results broaden the applicability of fixed point theory to diverse spaces and offer potential impact on optimization and computation where non-metric distances arise, by providing stable solution guarantees under triangle-function frameworks.

Abstract

This paper advances a line of research in fixed point theory initiated by M. Bessenyei and Z. Páles, building on their introduction of the triangle function concept in [J. Nonlinear Convex Anal, Vol 18 (3), 515-524 (2017)]. By applying this concept, the study revises several well-known fixed point theorems in metric spaces, extending their applicability to semimetric spaces with triangle functions. The paper focuses on general theorems involving weak, partial, Bianchini and Chatterjea-Bianchini contractions, deriving corollaries relevant to metric spaces, $b$-metric spaces, ultrametric spaces, and distance spaces with power triangle functions. Notably, several new and interesting findings emerge in the context of weak and partial contractions.

Fixed point theorems for weak, partial, Bianchini and Chatterjea-Bianchini contractions in semimetric spaces with triangle functions

TL;DR

The paper extends fixed point theory to semimetric spaces endowed with triangle functions , enabling generalized contractions beyond standard metric spaces. It develops a unified framework in complete semimetric spaces with a continuous triangle function and proves fixed point results for partial, weak, Bianchini, and Chatterjea-Bianchini contractions under -based inequalities. The main contributions include existence (and where possible uniqueness) theorems for each contraction type under explicit -based conditions, applicable to metric, -metric, ultrametric, and power-triangle distance spaces. These results broaden the applicability of fixed point theory to diverse spaces and offer potential impact on optimization and computation where non-metric distances arise, by providing stable solution guarantees under triangle-function frameworks.

Abstract

This paper advances a line of research in fixed point theory initiated by M. Bessenyei and Z. Páles, building on their introduction of the triangle function concept in [J. Nonlinear Convex Anal, Vol 18 (3), 515-524 (2017)]. By applying this concept, the study revises several well-known fixed point theorems in metric spaces, extending their applicability to semimetric spaces with triangle functions. The paper focuses on general theorems involving weak, partial, Bianchini and Chatterjea-Bianchini contractions, deriving corollaries relevant to metric spaces, -metric spaces, ultrametric spaces, and distance spaces with power triangle functions. Notably, several new and interesting findings emerge in the context of weak and partial contractions.
Paper Structure (7 sections, 14 theorems, 71 equations)

This paper contains 7 sections, 14 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Partial contractions in semimetric spaces
  4. Weak contractions in semimetric spaces
  5. Bianchini's contractions in semimetric spaces
  6. Chatterjea-Bianchini contractions in semimetric spaces
  7. Conclusion

Key Result

Lemma 2.1

Let $(a_n)$ and $(b_n)$ be sequences such that $a_n, b_n \geqslant 0$, $a_n, b_n \to 0$ as $n \to \infty$, and let $(c_n)$ be any sequence with $c_n \geqslant 0$. Suppose the following conditions hold for the triangle function $\Phi$: where $l > 0$ is a positive real number, and for every sequence $(x_n)$ such that $x_n \to 0$ as $n \to \infty$ and every arbitrary sequence $(y_n)$ with $x_n, y_n

Theorems & Definitions (35)

  • Definition 1.1
  • Lemma 2.1
  • proof
  • Theorem 2.1
  • proof
  • Example 2.1
  • Remark 2.1
  • Lemma 2.2
  • proof
  • Remark 2.2: PSB24
  • ...and 25 more