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 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 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 Ulam-Hyers-Rassias stability for a class of nonlinear implicit Hadamard fractional integral boundary value problem with impulses

2021 ◽  
Vol 7 (2) ◽  
pp. 3169-3185
Author(s):  
Kaihong Zhao ◽  
◽  
Shuang Ma
Keyword(s):  
Contraction Mapping ◽  
Contraction Mapping Principle ◽  
Integral Boundary ◽  
Nonlinear Functional Analysis ◽  
Nonlinear Functional ◽  
Hadamard Fractional Differential Equations ◽  
Analysis Technique ◽  
Fractional Differential ◽  
Rassias Stability ◽  
Hadamard Fractional Integral

<abstract><p>This paper considers a class of nonlinear implicit Hadamard fractional differential equations with impulses. By using Banach's contraction mapping principle, we establish some sufficient criteria to ensure the existence and uniqueness of solution. Furthermore, the Ulam-Hyers stability and Ulam-Hyers-Rassias stability of this system are obtained by applying nonlinear functional analysis technique. As applications, an interesting example is provided to illustrate the effectiveness of main results.</p></abstract>

scholarly journals Banach contraction principle on cone rectangular metric spaces

2009 ◽  
Vol 3 (2) ◽  
pp. 236-241 ◽  
Author(s):  
Akbar Azam ◽  
Muhammad Arshad ◽  
Ismat Beg
Keyword(s):  
Metric Space ◽  
Contraction Mapping ◽  
Metric Spaces ◽  
Contraction Mapping Principle ◽  
Banach Contraction Principle ◽  
Space Setting ◽  
Banach Contraction ◽  
Rectangular Metric Space ◽  
Banach Contraction Mapping Principle ◽  
Banach Contraction Mapping

We introduce the notion of cone rectangular metric space and prove Banach contraction mapping principle in cone rectangular metric space setting. Our result extends recent known results.

Another Proof of the Contraction Mapping Principle

1968 ◽  
Vol 11 (4) ◽  
pp. 605-606
Author(s):  
D.W. Boyd ◽  
J. S. W. Wong
Keyword(s):  
Metric Space ◽  
Contraction Mapping ◽  
Contraction Mapping Principle ◽  
Alternative Proof ◽  
Intersection Theorem ◽  
Cantor’S Theorem ◽  
Special Case

In a recent note of Kolodner [2], the Cantor Intersection Theorem is used to give an alternative proof of the well known Contraction Mapping Principle. Kolodner applied Cantor's theorem first to a bounded metric space and then reduced the general case to this special case. Sometime ago, we found a somewhat different proof of the Contraction Mapping Principle using Cantor's theorem. Since our proof seems somewhat more direct we propose to present it here.

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)}.

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