Fixed Point Theory For Singh-Chatterjea Type Contractive Mappings
Zouaoui Bekri, Nicola Fabiano
TL;DR
This work addresses extending fixed point theory beyond classical contractions by introducing a Singh-Chatterjea contraction defined on the $p$-th iterate: $d(T^p x, T^p y) \leq \alpha(d(x, T^p y) + d(y, T^p x))$ with $p\in\mathbb{N}$ and $\alpha\in(0,1/2)$. The authors prove existence and uniqueness of fixed points for such mappings in complete metric spaces, using an iterative scheme on $S=T^p$ and showing geometric decay of successive differences, which yields a Cauchy sequence converging to a fixed point. They demonstrate that Banach contractions become Sing-Chatterjea contractions after a finite number of iterations, establish a non-Banach example that satisfies the Singh-Chatterjea condition, and discuss the universality and strictness of the framework. The results unify two classical contraction theories and extend applicability to a broader class of operators, with potential implications for nonlinear analysis and numerical methods.
Abstract
In this paper, we introduce a new contraction condition that combines the framework of Singh's extension with the classical Chatterjea contraction. This generalized form, called the Singh-Chatterjea contraction, is defined on the p-th iterate of a mapping. We establish fixed point theorems for such mappings in complete metric spaces and show that our results extend and unify both Singh's and Chatterjea's classical fixed point theorems. Illustrative examples and a simple numerical implementation are provided to demonstrate the applicability of the obtained results.