scholarly journals Generalizations of the Converse of the Contraction Mapping Principle

1966 ◽  
Vol 18 ◽  
pp. 1095-1104 ◽  
Author(s):  
James S. W. Wong
Keyword(s):  
Fixed Point ◽  
Metric Space ◽  
Contraction Mapping ◽  
Unique Fixed Point ◽  
Complete Metric Space ◽  
Contraction Mapping Principle ◽  
Converse Statement ◽  
Natural Formulation

This paper is an outgrowth of studies related to the converse of the contraction mapping principle. A natural formulation of the converse statement may be stated as follows: “Let X be a complete metric space, and T be a mapping of X into itself such that for each x ∈ X, the sequence of iterates ﹛Tnx﹜ converges to a unique fixed point ω ∈ X. Then there exists a complete metric in X in which T is a contraction.” This is in fact true, even in a stronger sense, as may be seen from the following result of Bessaga (1).

scholarly journals On iterative solution of nonlinear functional equations in a metric space

1983 ◽  
Vol 6 (1) ◽  
pp. 161-170
Author(s):  
Rabindranath Sen ◽  
Sulekha Mukherjee
Keyword(s):  
Metric Space ◽  
Contraction Mapping ◽  
Unique Fixed Point ◽  
Complete Metric Space ◽  
Iterative Solution ◽  
Contraction Mapping Principle ◽  
Iterative Sequence ◽  
Constructive Proof ◽  
Nonlinear Functional ◽  
Positive Integers

Given thatAandPas nonlinear onto and into self-mappings of a complete metric spaceR, we offer here a constructive proof of the existence of the unique solution of the operator equationAu=Pu, whereu∈R, by considering the iterative sequenceAun+1=Pun(u0prechosen,n=0,1,2,…). We use Kannan's criterion [1] for the existence of a unique fixed point of an operator instead of the contraction mapping principle as employed in [2]. Operator equations of the formAnu=Pmu, whereu∈R,nandmpositive integers, are also treated.

scholarly journals MULTIDIMENSIONAL FIXED POINT RESULTS FOR CONTRACTION MAPPING PRINCIPLE WITH APPLICATION

2021 ◽  
pp. 919
Author(s):  
Amrish Handa
Keyword(s):  
Integral Equation ◽  
Fixed Point ◽  
Metric Space ◽  
Existence And Uniqueness ◽  
Contraction Mapping ◽  
Metric Spaces ◽  
Complete Metric Space ◽  
Contraction Mapping Principle ◽  
Uniqueness Of The Solution ◽  
Isotone Mappings

The main aim of this article is to study the existence and uniqueness of fixed point for isotone mappings of any number of arguments under contraction mapping principle on a complete metric space endowed with a partial order. As an application of our result we study the existence and uniqueness of the solution to an integral equation. The results we obtain generalize, extend and unify several classical and very recent related results in the literature in metric spaces.

scholarly journals A discussion on a Pata type contraction via iterate at a point

Filomat ◽  
2020 ◽  
Vol 34 (4) ◽  
pp. 1061-1066
Author(s):  
Erdal Karapınar ◽  
Andreea Fulga ◽  
Vladimir Rakocevic
Keyword(s):  
Fixed Point ◽  
Metric Space ◽  
Unique Fixed Point ◽  
Complete Metric Space ◽  
Continuity Assumption ◽  
The Given ◽  
Type Contraction

In this paper, we introduce the notion of Pata type contraction at a point in the context of a complete metric space. We observe that such contractions possesses unique fixed point without continuity assumption on the given mapping. Thus, is extended the original results of Pata. We also provide an example to illustrate its validity.

scholarly journals Fixed Points Results in G-Metric Spaces

2019 ◽  
Vol 32 (1) ◽  
pp. 142
Author(s):  
Salwa Salman Abed ◽  
Anaam Neamah Faraj ◽  
Anaam Neamah Faraj
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Metric Space ◽  
Contraction Mapping ◽  
Fixed Point Theorems ◽  
Metric Spaces ◽  
Complete Metric Space ◽  
Closed Ball ◽  
Local Contraction

  In this paper, the concept of contraction mapping on a -metric space is extended with a consideration on local contraction.  As a result, two fixed point theorems were proved for contraction on a closed ball in a complete -metric space.

scholarly journals Fixed point theorem with contractive mapping on C*-Algebra valued G-Metric Space

2021 ◽  
Vol 2106 (1) ◽  
pp. 012015
Author(s):  
A Wijaya ◽  
N Hariadi
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Metric Space ◽  
Unique Fixed Point ◽  
Complete Metric Space ◽  
Contractive Mapping ◽  
C Algebra ◽  
Metric Function ◽  
Interesting Theorem

Abstract Banach-Caccioppoli Fixed Point Theorem is an interesting theorem in metric space theory. This theorem states that if T : X → X is a contractive mapping on complete metric space, then T has a unique fixed point. In 2018, the notion of C *-algebra valued G-metric space was introduced by Congcong Shen, Lining Jiang, and Zhenhua Ma. The C *-algebra valued G-metric space is a generalization of the G-metric space and the C*-algebra valued metric space, meanwhile the G-metric space and the C *-algebra valued metric space itself is a generalization of known metric space. The G-metric generalized the domain of metric from X × X into X × X × X, the C *-algebra valued metric generalized the codomain from real number into C *-algebra, and the C *-algebra valued G-metric space generalized both the domain and the codomain. In C *-algebra valued G-metric space, there is one theorem that is similar to the Banach-Caccioppoli Fixed Point Theorem, called by fixed point theorem with contractive mapping on C *-algebra valued G-metric space. This theorem is already proven by Congcong Shen, Lining Jiang, Zhenhua Ma (2018). In this paper, we discuss another new proof of this theorem by using the metric function d(x, y) = max{G(x, x, y),G(y, x, x)}.

scholarly journals A fixed point theorem with applications to convolution equations

1963 ◽  
Vol 3 (4) ◽  
pp. 385-395 ◽  
Author(s):  
R. E. Edwards
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Metric Space ◽  
Unique Fixed Point ◽  
Complete Metric Space ◽  
Banach Contraction Principle ◽  
Banach Contraction ◽  
Image Position ◽  
Convolution Equations

The well-known Banach Contraction Principle asserts that any self-map F of a complete metric space M with the property that, for some number k < 1, for all x, y,∈M, possesses a unique fixed point in M. some extensions and analogues have recently been given by Edelstein [1]. For the reader's convenlience we state here the result of Edelstein which we shall employ. It asserts that if F is a self-map of a metric space M having the property that for any two distinct points x and y of M, and if x0 is a point of M such that the sequence of iterates xn = Fn (x0) contains a subsequence which converges in M, then the limit of this subsequence is the unique fixed point of F.

scholarly journals Some fixed point theorems

1992 ◽  
Vol 53 (3) ◽  
pp. 304-312 ◽  
Author(s):  
P. V. Subrahmanyam ◽  
I. L. Reilly
Keyword(s):  
Fixed Point ◽  
Metric Space ◽  
Fixed Point Theorems ◽  
Unique Fixed Point ◽  
Complete Metric Space ◽  
Contraction Principle ◽  
Carry Over ◽  
Banach's Theorem ◽  
Banach’S Contraction Principle

AbstractBanach's contraction principle guarantees the existence of a unique fixed point for any contractive selfmapping of a complete metric space. This paper considers generalizations of the completeness of the space and of the contractiveness of the mapping and shows that some recent extensions of Banach's theorem carry over to spaces whose topologies are generated by families of quasi-pseudometrics.

scholarly journals Stability of Picard operators under operator perturbations

2018 ◽  
Vol 56 (2) ◽  
pp. 3-12
Author(s):  
Adrian Petruşel ◽  
Ioan A. Rus
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Metric Space ◽  
Unique Fixed Point ◽  
Complete Metric Space ◽  
Sufficient Conditions ◽  
Picard Operator

AbstractIn this paper we study the following problems: I. Let (M, d) be a complete metric space and f, g : M → M be two operators. We suppose that:(a) f is a Picard operator with its unique fixed point x *f;(b) there exists η > 0 such that d(f(x), g(x)) ≤ η, for every x ∈ M.The problem consists in estimating d(gn(x), x*f), for x ∈ M and n ∈ 𝕅*.II. Let B be a Banach space and f, g : B → B be two operators. We suppose that f is a Picard operator. The problem is to find sufficient conditions which guarantee that f + g is a Picard operator.

scholarly journals Fixed Point Theorems and Iterative Function System in G-Metric Spaces

2019 ◽  
Vol 27 (2) ◽  
pp. 329-340
Author(s):  
Salwa Salman Abed ◽  
Anaam Neamah Faraj
Keyword(s):  
Fixed Point ◽  
Metric Space ◽  
Fixed Point Theorems ◽  
Metric Spaces ◽  
Unique Fixed Point ◽  
Complete Metric Space ◽  
Generalized Contraction ◽  
Numerical Examples ◽  
Iterated Function ◽  
Self Similar

Iterated function space is a method to construct fractals and the results are self-similar. In this paper, we introduce the Hutchinson Barnsley operator (shortly, operator) on a  metric space and employ its theory to construct a fractal set as its unique fixed point by using Ciric type generalized -contraction in complete metric space. In addition, some concepts are illustrated by numerical examples.

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