Non-stationary {\phi}-contractions and associated fractals

TL;DR

This paper generalizes fixed-point theory using non-stationary {}-contractions with variable Lipschitz constants, exploring their trajectories and attractors, which form complex fractals with multi-scale structures.

Contribution

It introduces non-stationary fixed-point results for {}-contractions using control functions, extending traditional fractal and iterated function system theories.

Findings

Established convergence of non-stationary trajectories.

Characterized attractors with multi-scale structures.

Extended fixed-point principles to generalized contractions.

Abstract

In this study we provide several significant generalisations of Banach contraction principle where the Lipschitz constant is substituted by real-valued control function that is a comparison function. We study non-stationary variants of fixed-point. In particular, this article looks into trajectories of maps defined by function systems which are regarded as generalizations of traditional iterated function system. The importance of forward and backward trajectories of general sequences of mappings is analyzed. The convergence characteristics of these trajectories determined a non-stationary variant of the traditional fixed point theory. Unlike the normal fractals which have self-similarity at various scales, the attractors of these trajectories of maps which defined by function systems that may have various structures at various scales. In this literature we also study the sequence of countable IFS having some generalized contractions on a complete metric space.