Weak θ-contractions and some fixed point results with applications to fractal theory

Abstract

In this paper, we define weak θ-contractions on a metric space into itself by extending θ-contractions introduced by Jleli and Samet (J. Inequal. Appl. 2014:38, 2014) and utilize the same to prove some fixed point results besides proving some relation-theoretic fixed point results in generalized metric spaces. Moreover, we give some applications to fractal theory improving the classical Hutchinson-Barnsley s theory of iterated function systems. We also give illustrative examples to exhibit the utility of our results.

Existence of Picard operator and iterated function system

“…Definition 3.1 (see [6,8,9]). Let θ : (0, ∞) → (1, ∞) be a function and consider the following conditions: Θ1 : θ is non-decreasing.…”

“…In this paper, we study the concept of weak θ -contraction used by Imdad and Alfaqih [8] which is an extension of θ -contraction (or JS contraction) introduced by Jleli and Samet [9]. We consider the family Θ 1,2,4 and introduce weak θ m -contraction and prove that every (continuous) weak θ m -contraction is a Picard operator in section 3.…”

Fixed Point Results for a New Class of Multi-Valued Mappings Under (θ; R)-Contractions With an Application

“…Imdad et al [12] noticed that Theorem 2.1 can be proved without the assumption θ 1 , from which they have introduced the notion of weak θ-contractions. Inspired by this, we deduce some relation-theoretic results (without assumption θ 1 ) for single-valued mappings.…”

Common Fixed Points of(α,η)−(θ,ϝ)Rational Contractions with Applications

“…For further results in this direction, we refer to [7][8][9][10]. Another variation of contraction mapping that can be found in literature is -contraction mapping.…”