Generalized IFSs on Noncompact Spaces

In this paper, we deal with the part of Fractal Theory related to finite families of (weak) contractions, called iterated function systems (IFS, herein). An attractor is a compact set which remains invariant for such a family. Thus, we consider spaces homeomorphic to at-tractors of either IFS or weak IFS, as well, which we will refer to as Banach and topological fractals, respectively. We present a collection of counterexamples in order to show that all the presented definitions are essential, though they are not equivalent in general.

Mağallaẗ baġdād li-l-ʿulūm, 2024

In certain mathematical, computing, economic, and modeling issues, the presence of a solution to a theoretical or real-world problem is synonymous with the presence of a fixed point (Fp) for an appropriate mapping. Consequently, Fp plays an essential role in a wide variety of mathematical and scientific contexts. In its own right, the theory is a stunning amalgamation of analysis (both pure and applied), geometry, and topology. Recent years have shown the theory of Fps is a highly strong and useful tool in the study of nonlinear events. Fp theorems are concerned with mappings f of a set 𝑋 into itself that, under particular conditions, permit a Fp, that is, a point 𝑥 ∈ 𝑋 such that𝑓(𝑥) = 𝑥. This work introduces and proves the Fp theorem for various kinds of contraction mappings in a fuzzy metric space (𝐹ℳ-space) namely almost 𝒁 ̂-contraction mapping and (Ψ ̃ , Φ ̃)almost weakly contraction mapping. At first, the concept of 𝐹ℳ-space and the terms used in the fuzzy setting are recalled. Then the concept of simulation function is given. The concept of simulation function is used to present the notion of almost 𝒁 ̂contraction mapping. In addition, this notion is used to prove the existence and uniqueness of the Fp for this kind of mapping. After that the notion of (Ψ ̃ , Φ ̃)-almost weakly contraction mapping is introduced in the framework of 𝐹ℳ-space, as well as the Fp theorem for this kind of mapping. At the end of the paper, some examples are given to support the results.

2017

In this paper, we establish the results on existence and uniqueness of fixed point for φ-contractive and generalized C-contractive mapping in the fuzzy metric space in the sense of George and Veeramani. We use the notion of altering distance for proving the results. Full text Acknowledgment: The work is partially supported by Council of Scientific and Industrial Research, India (Project No. 25(0168)/ 09/EMR-II). The first author gratefully acknowledges the support.

Applied Mathematics and Computation, 2012

Afterward Berinde [12, Theorem 3.4] generalized the above definition and proved the following fixed point result. Theorem 1.3 [12]. Let (X, d) be a complete metric space and T : X ? X a mapping for which there exist a 2 ]0, 1[ and some L P 0 such that for all x, y 2 X dðTx; TyÞ 6 aMðx; yÞ þ L minfdðx; TxÞ; dðy; TyÞ; dðx; TyÞ; dðy; TxÞg; ð1Þ where M(x, y) = max{d(x, y), d(x, Tx), d(y, Ty), d(x, Ty), d(y, Tx)}. Then (1) T has a unique fixed point, i.e., F(T) = {x ⁄ }; (2) for any x 0 2 X, the Picard iteration fx n g 1 n¼0 defined by (1.1) converges to some x ⁄ 2 F(T); (3) the prior estimate dðx n ; x Ã Þ 6 a n ð1ÀaÞ 2 dðx 0 ; x 1 Þ holds for n = 1,2,.. .; (4) the rate of convergence of Picard iteration is given by d(x n , x ⁄) 6 hd(x nÀ1 , x ⁄) for n = 1,2,.. .. The contractive condition (1) is termed as generalized condition (B). Recently, Abbas and Ilić in [1] introduced the following definition: Definition 1.4 [1]. Let T and f be two self maps of a metric space (X, d). A map T is called generalized almost f-contraction if there exists d 2 [0, 1[ and L P 0 such that for all x, y 2 X,

Mathematics, 2020

In this manuscript we discuss, consider, generalize, improve and unify several recent results for so-called F-contraction-type mappings in the framework of complete metric spaces. We also introduce ( φ , F ) -weak contraction and establish the corresponding fixed point result. Using our new approach for the proof that a Picard sequence is a Cauchy in metric space, our obtained results complement and enrich several methods in the existing literature. At the end we give one open question for F-contraction of Ćirić-type mapping.

Bulletin of The American Mathematical Society, 1968

Topological Methods in Nonlinear Analysis, 2015

In this paper we study the Hutchinson-Barnsley theory of fractals in the setting of multimetric spaces (which are sets endowed with point separating families of pseudometrics) and in the setting of topological spaces. We find natural connections between these two approaches.

2011

Afterward Berinde [12, Theorem 3.4] generalized the above definition and proved the following fixed point result. Theorem 1.3 [12]. Let (X, d) be a complete metric space and T : X ? X a mapping for which there exist a 2 ]0, 1[ and some L P 0 such that for all x, y 2 X dðTx; TyÞ 6 aMðx; yÞ þ L minfdðx; TxÞ; dðy; TyÞ; dðx; TyÞ; dðy; TxÞg; ð1Þ where M(x, y) = max{d(x, y), d(x, Tx), d(y, Ty), d(x, Ty), d(y, Tx)}. Then (1) T has a unique fixed point, i.e., F(T) = {x ⁄ }; (2) for any x 0 2 X, the Picard iteration fx n g 1 n¼0 defined by (1.1) converges to some x ⁄ 2 F(T); (3) the prior estimate dðx n ; x Ã Þ 6 a n ð1ÀaÞ 2 dðx 0 ; x 1 Þ holds for n = 1,2,.. .; (4) the rate of convergence of Picard iteration is given by d(x n , x ⁄) 6 hd(x nÀ1 , x ⁄) for n = 1,2,.. .. The contractive condition (1) is termed as generalized condition (B). Recently, Abbas and Ilić in [1] introduced the following definition: Definition 1.4 [1]. Let T and f be two self maps of a metric space (X, d). A map T is called generalized almost f-contraction if there exists d 2 [0, 1[ and L P 0 such that for all x, y 2 X,

arXiv (Cornell University), 2021

The underlying theme of this article is a class of sequences in metric structures satisfying a much weaker kind of Cauchy condition, namely quasi-Cauchy sequences (introduced in [7]) that has been used to define several new concepts in recent articles [9, 10]. We first consider a weaker notion of precompactness based on the idea of quasi-Cauchy sequences and establish several results including a new characterization of compactness in metric spaces. Next we consider associated idea of continuity, namely, ward continuous functions [8], as this class of functions strictly lies between the classes of continuous and uniformly continuous functions and mainly establish certain coincidence results. Finally a new class of Lipschitz functions called "quasi-Cauchy Lipschitz functions" is introduced following the line of investigations in [3, 4, 5, 12] and again several coincidence results are proved. The motivation behind such kind of Lipschitz functions is ascertained by the observation that every real valued ward continuous function defined on a metric space can be uniformly approximated by real valued quasi-Cauchy Lipschitz functions.

2018

In this paper, we present a common fixed point theorem for two pairs of self mappings satisfying a generalized almost contractive condition in metriclike spaces. We provide two examples to illustrate our obtained results, also an application to study the existence of solution for a system of integral equations is given.

Chaos, Solitons & Fractals, 2007

Some common fixed point theorems for multi-valued mappings under u-contraction condition have been studied by Rashwan [Rashwan RA, Ahmed MA. Fixed points for u-contraction type multivalued mappings. J Indian Acad Math 1995;17(2):194-204]. Butnariu [Butnariu D. Fixed point for fuzzy mapping. Fuzzy Sets Syst 1982;7:191-207] and Helipern [Hilpern S. Fuzzy mapping and fixed point theorem. J Math Anal Appl 1981;83:566-9] also, discussed some fixed point theorems for fuzzy mappings in the category of metric spaces. In this paper, we discussed some common fixed point theorems for fuzzy mappings in metric space under u-contraction condition. Our investigation are related to the fuzzy form of Hausdorff metric which is a basic tool for computing Hausdorff dimensions. These dimensions help in understanding e 1 -space [El-Naschie MS. On the unification of the fundamental forces and complex time in the e 1space. Chaos, Solitons & Fractals 2000;11:1149-62] and are used in high energy physics [El-Naschie MS. Wild topology hyperbolic geometry and fusion algebra of high energy particle physics. Chaos, Solitons & Fractals 2002;13:1935-45].

2015

The purpose of this paper is to obtain some results on existence of fixed points for contractive mappings in fuzzy metric spaces using control function. We prove our results on fuzzy metric spaces in the sense of George and Veeramani. Our results mainly generalize and extend the result of various authors, announced in the literature. As an application, a consequence theorem of integral type contraction is given in support of our result.

Advances in Difference Equations, 2019

In this paper, we prove some fixed point theorems for ψF-contractions in the framework of quasi-metric spaces generalizing and improving several similar results in metric spaces. At the same time, we consider iterated function systems consisting of ψF-contractions on quasi-metric spaces, and we give some sufficient conditions for the existence and uniqueness of their attractor which is, generally, a fractal. Some illustrative examples are provided.

Open Mathematics, 2020

In this work, we show that the existence of fixed points of F-contraction mappings in function weighted metric spaces can be ensured without third condition () F 3 imposed on Wardowski function (∞) → F: 0, R. The present article investigates (common) fixed points of rational type F-contractions for single-valued mappings. The article employs Jleli and Samet's perspective of a new generalization of a metric space, known as a function weighted metric space. The article imposes the contractive condition locally on the closed ball, as well as, globally on the whole space. The study provides two examples in support of the results. The presented theorems reveal some important corollaries. Moreover, the findings further show the usefulness of fixed point theorems in dynamic programming, which is widely used in optimization and computer programming. Thus, the present study extends and generalizes related previous results in the literature in an empirical perspective.

Open Mathematics

In this work, we show that the existence of fixed points of F-contraction mappings in function weighted metric spaces can be ensured without third condition ( F 3 ) (F3) imposed on Wardowski function F :(0, ∞ ) → ℜ F\mathrm{:(0,\hspace{0.33em}}\infty )\to \Re . The present article investigates (common) fixed points of rational type F-contractions for single-valued mappings. The article employs Jleli and Samet’s perspective of a new generalization of a metric space, known as a function weighted metric space. The article imposes the contractive condition locally on the closed ball, as well as, globally on the whole space. The study provides two examples in support of the results. The presented theorems reveal some important corollaries. Moreover, the findings further show the usefulness of fixed point theorems in dynamic programming, which is widely used in optimization and computer programming. Thus, the present study extends and generalizes related previous results in the literatur...