A generalization of Matkowski’s fixed point theorem and Istrăţescu’s fixed point theorem concerning convex contractions

Miculescu, Radu; Mihail, Alexandru

doi:10.1007/s11784-017-0411-7

Cited by 14 publications

(6 citation statements)

References 24 publications

(37 reference statements)

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“…for all x, y ∈ X. Then, based on (3.1), using Theorem 3.1 from [5], we infer that there exists a unique α ∈ X such that g(α) = α and lim n→∞ g [n] (x) = α for every x ∈ X.…”

Section: Resultsmentioning

confidence: 93%

Possibly infinite generalized iterated function systems comprising phi-max contractions

Urziceanu¹

2019

Stud. Univ. Babes-Bolyai Math.

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One way to generalize the concept of iterated function system was proposed by R. Miculescu and A. Mihail under the name of generalized iterated function system (for short GIFS). More precisely, given m ∈ N * and a metric space (X, d), a generalized iterated function system of order m is a finite family of functions f1,. .. , fn : X m → X satisfying certain contractive conditions. Another generalization of the notion of iterated function system, due to F. Georgescu, R. Miculescu and A. Mihail, is given by those systems consisting of ϕ-max contractions. Combining these two lines of research, we prove that the fractal operator associated to a possibly infinite generalized iterated function system comprising ϕ-max contractions is a Picard operator (whose fixed point is called the attractor of the system). We associate to each possibly infinite generalized iterated function system comprising ϕ-max contractions F (of order m) an operator HF : C m → C, where C stands for the space of continuous and bounded functions from the shift space on the metric space corresponding to the system. We prove that HF is a Picard operator whose fixed point is the canonical projection associated to F.

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“…for all x, y ∈ X. Then, based on (3.1), using Theorem 3.1 from [5], we infer that there exists a unique α ∈ X such that g(α) = α and lim n→∞ g [n] (x) = α for every x ∈ X.…”

Section: Resultsmentioning

confidence: 93%

Possibly infinite generalized iterated function systems comprising phi-max contractions

Urziceanu¹

2019

Stud. Univ. Babes-Bolyai Math.

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“…The next Lemma (see for example Miculescu and Mihail [103] and Suzuki [139]) opens the possibility of obtaining more general fixed point results than Theorem 4.1.…”

Section: Zbmathmentioning

confidence: 91%

"The early developments in fixed point theory on b-metric spaces: a brief survey and some important related aspects"

Berinde¹,

Pačurar²

2022

CJM

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"A very impressive research work has been devoted in the last two decades to obtaining fixed point theorems in quasimetric spaces (also called b-metric spaces). Some incorrect and incomplete references with respect to the early developments on fixed point theory in b-metric spaces are though perpetually taking over from the existing publications to the new ones. Starting from this fact, our main aim in this note is threefold: (1) to briefly survey the early developments in the fixed point theory on quasimetric spaces (b-metric spaces); (2) to collect some relevant bibliography related to this topic; (3) to discuss some other aspects of current interest in the fixed point theory on quasimetric spaces (b-metric spaces)."

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“…Theorem 2.6 (see Theorem 3.1 from [6]). Every continuous function f : X → X, where (X, d) is a complete metric space, is a Picard operator provided that there exist a comparison function ϕ…”

Section: Given a Functionmentioning

confidence: 99%

Iterated function systems consisting of phi-max-contractions have attractor

Georgescu¹,

Miculescu²,

Mihail³

2017

Preprint

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We associate to each iterated function system consisting of ϕmax-contractions an operator (on the space of continuous functions from the shift space on the metric space corresponding to the system) having a unique fixed point whose image turns out to be the attractor of the system. Moreover, we prove that the unique fixed point of the operator associated to an iterated function system consisting of convex contractions is the canonical projection from the shift space on the attractor of the system.

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A generalization of Matkowski’s fixed point theorem and Istrăţescu’s fixed point theorem concerning convex contractions

Cited by 14 publications

References 24 publications

Possibly infinite generalized iterated function systems comprising phi-max contractions

Possibly infinite generalized iterated function systems comprising phi-max contractions

"The early developments in fixed point theory on b-metric spaces: a brief survey and some important related aspects"

Iterated function systems consisting of phi-max-contractions have attractor

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