On generalizations of some fixed point theorems in semimetric spaces with triangle functions
Evgeniy Petrov, Ruslan Salimov, Ravindra K. Bisht
TL;DR
The paper addresses extending fixed-point theory from metric spaces to semimetric spaces equipped with triangle functions $\Phi$, unifying Banach, Kannan, Chatterjea, and Ćirić-Reich-Rus contractions under a common semimetric framework. The authors introduce a key lemma: if a sequence satisfies $d(x_n,x_{n+1})\le \alpha d(x_{n-1},x_n)$ with $\alpha\in[0,1)$, then it is Cauchy, enabling convergence proofs for generalized contractions. They develop generalized theorems for Banach, Kannan, Chatterjea, and CRR mappings, plus mappings contracting perimeters of triangles, and show continuity of these mappings in semimetric spaces with triangle functions, with specializations to metric, ultrametric, and $b$-metric settings. The results yield existence and uniqueness of fixed points under broad contractive conditions and have potential applications in optimization and modeling, offering pathways for future work in applying the framework to other contractive mappings and real-world problems.
Abstract
In the present paper, we prove generalizations of Banach, Kannan, Chatterjea, Ćirić-Reich-Rus fixed point theorems, as well as of the fixed point theorem for mappings contracting perimeters of triangles. We consider corresponding mappings in semimetric spaces with triangle functions introduced by M. Bessenyei and Z. Páles. Such an approach allows us to derive corollaries for various types of semimetric spaces including metric spaces, ultrametric spaces, b-metric spaces etc. The significance of these generalized theorems extends across multiple disciplines, including optimization, mathematical modeling, and computer science. They may serve to establish stability conditions, demonstrate the existence of optimal solutions, and improve algorithm design.