Fixed points and continuity of contractive maps

Filomat ◽  
2017 ◽  
Vol 31 (11) ◽  
pp. 3501-3506 ◽  
Author(s):  
Abhijit Pant ◽  
R.P. Pant
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Open Problem ◽  
Contractive Mapping ◽  
Continuous Mappings ◽  
Discontinuous Mappings ◽  
Contractive Maps

The aim of this paper is to generalize celebrated results due to Boyd and Wong [2] and Matkowski [9] and also to provide yet new solutions to the once open problem on the existence of a contractive mapping which possesses a fixed point but is not continuous at the fixed point. Besides continuous mappings our results also apply to discontinuous mappings which include threshold operations that are integral part of many a phenomena.

scholarly journals On discontinuity at fixed point via power quasi contraction

2020 ◽  
Vol 108 (122) ◽  
pp. 5-11
Author(s):  
Ravindra Bisht ◽  
Narendra Singh ◽  
Vladimir Rakocevic ◽  
Brian Fisher
Keyword(s):  
Fixed Point ◽  
Open Problem ◽  
Fixed Point Theorems ◽  
Continuous Mappings ◽  
Discontinuous Mappings

We extend the scope of the study of fixed point theorems of power quasi contractions from the class of continuous mappings to a wider class of mappings which also include discontinuous mappings. As a by-product, we provide a new answer to an open problem posed by Rhoades.

Fixed Points for Dissipative-Repulsive Systems and Topological Dynamics of Mappings Defined on N-dimensional Cells

2005 ◽  
Vol 5 (3) ◽  
Author(s):  
Marina Pireddu ◽  
Fabio Zanolin
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Fixed Points ◽  
Point Theorem ◽  
Topological Dynamics ◽  
Continuous Mappings ◽  
Expansion Condition ◽  
Dimensional Cell

AbstractWe prove a fixed point theorem for continuous mappings which satisfy a compression-expansion condition on the boundary of a N-dimensional cell of ℝ

Perturbation of the fixed point problem for continuous mappings

2020 ◽  
pp. 241-249
Author(s):  
Zukhra T. Zhukovskaya ◽  
Sergey E. Zhukovskiy
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Fixed Points ◽  
Continuous Mapping ◽  
Fixed Point Problem ◽  
Point Problem ◽  
Bounded Subset ◽  
Completely Continuous ◽  
Continuous Map ◽  
Continuous Mappings

We consider the problem of a double fixed point of pairs of continuous mappings defined on a convex closed bounded subset of a Banach space. It is shown that if one of the mappings is completely continuous and the other is continuous, then the property of the existence of fixed points is stable under contracting perturbations of the mappings. We obtain estimates for the distance from a given pair of points to double fixed points of perturbed mappings. We consider the problem of a fixed point of a completely continuous mapping on a convex closed bounded subset of a Banach space. It is shown that the property of the existence of a fixed point of a completely continuous map is stable under contracting perturbations. Estimates of the distance from a given point to a fixed point are obtained. As an application of the obtained results, the solvability of a difference equation of a special type is proved.

scholarly journals On the weakly(α, ψ, ξ)-contractive condition for multi-valued operators in metric spaces and related fixed point results

2016 ◽  
Vol 14 (1) ◽  
pp. 167-180 ◽  
Author(s):  
Marwan Amin Kutbi ◽  
Wutiphol Sintunavarat
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Binary Relation ◽  
Metric Spaces ◽  
Contractive Mapping ◽  
Contractive Condition ◽  
Integral Type

AbstractThe aim of this paper is to introduce the concept of a new nonlinear multi-valued mapping so called weakly (α, ψ, ξ)-contractive mapping and prove fixed point results for such mappings in metric spaces. Our results unify, generalize and complement various results from the literature. We give some examples which support our main results while previous results in literature are not applicable. Also, we analyze the existence of fixed points for mappings satisfying a general contractive inequality of integral type. Many fixed point results for multi-valued mappings in metric spaces endowed with an arbitrary binary relation and metric spaces endowed with graph are given here to illustrate the results in this paper.

Fixed points of contractive maps on dcpo's

2013 ◽  
Vol 24 (1) ◽  
Author(s):  
E. COLEBUNDERS ◽  
S. DE WACHTER ◽  
R. LOWEN
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Fixed Point Theorems ◽  
Scott Topology ◽  
Study Approach ◽  
Special Cases ◽  
Monotone Maps ◽  
Contractive Maps ◽  
Approach Spaces

In this paper we study approach structures on dcpo's. A dcpo (X, ≤) will be endowed with several other structures: the Scott topology; an approach structure generated by a collection of weightable quasi metrics onX; and a collectionof weights corresponding to the quasi metrics. Understanding the interaction between these structures onXwill eventually lead to some fixed-point theorems for the morphisms in the category of approach spaces, which are called contractions. Existing fixed-point theorems on both monotone and non-monotone maps are obtained as special cases.

scholarly journals Existence of Fixed Point Results in -Metric Spaces

2009 ◽  
Vol 2009 ◽  
pp. 1-10 ◽  
Author(s):  
Zead Mustafa ◽  
Wasfi Shatanawi ◽  
Malik Bataineh
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Metric Space ◽  
Metric Spaces ◽  
Contractive Mapping ◽  
Weaker Conditions

The purpose of this paper is to prove the existence of fixed points of contractive mapping defined on -metric space where the completeness is replaced with weaker conditions. Moreover, we showed that these conditions do not guarantee the completeness of -metric spaces.

scholarly journals Fixed points and their approximation in Banach spaces for certain commuting mappings

1982 ◽  
Vol 23 (1) ◽  
pp. 1-6
Author(s):  
M. S. Khan
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Fixed Points ◽  
Banach Spaces ◽  
Fixed Point Theorems ◽  
Nonexpansive Mappings ◽  
Contraction Mappings ◽  
Continuous Mappings ◽  
Banach Contraction ◽  
The Fixed Point Theorem

1. Let X be a Banach space. Then a self-mapping A of X is said to be nonexpansive provided that ‖AX − Ay‖≤‖X − y‖ holds for all x, y ∈ X. The class of nonexpansive mappings includes contraction mappings and is properly contained in the class of all continuous mappings. Keeping in view the fixed point theorems known for contraction mappings (e.g. Banach Contraction Principle) and also for continuous mappings (e.g. those of Brouwer, Schauderand Tychonoff), it seems desirable to obtain fixed point theorems for nonexpansive mappings defined on subsets with conditions weaker than compactness and convexity. Hypotheses of compactness was relaxed byBrowder [2] and Kirk [9] whereas Dotson [3] was able to relax both convexity and compactness by using the notion of so-called star-shaped subsets of a Banach space. On the other hand, Goebel and Zlotkiewicz [5] observed that the same result of Browder [2] canbe extended to mappings with nonexpansive iterates. In [6], Goebel-Kirk-Shimi obtainedfixed point theorems for a new class of mappings which is much wider than those of nonexpansive mappings, and mappings studied by Kannan [8]. More recently, Shimi [12] used the fixed point theorem of Goebel-Kirk-Shimi [6] to discuss results for approximating fixed points in Banach spaces.

scholarly journals Fixed Points and Continuity for a Pair of Contractive Maps with Application to Nonlinear Volterra Integral Equations

2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
Santosh Kumar
Keyword(s):  
Fixed Point ◽  
Integral Equations ◽  
Existence And Uniqueness ◽  
Fixed Point Theorems ◽  
Metric Spaces ◽  
Volterra Integral Equations ◽  
Discontinuous Mappings ◽  
Contractive Maps ◽  
Uniqueness Of A Solution ◽  
Type Contraction

In this paper, we have established and proved fixed point theorems for the Boyd-Wong-type contraction in metric spaces. In particular, we have generalized the existing results for a pair of mappings that possess a fixed point but not continuous at the fixed point. We can apply this result for both continuous and discontinuous mappings. We have concluded our results by providing an illustrative example for each case and an application to the existence and uniqueness of a solution of nonlinear Volterra integral equations.

scholarly journals On contractive mappings and discontinuity at fixed points

2020 ◽  
Vol 14 (1) ◽  
pp. 33-54 ◽  
Author(s):  
Hiranmoy Garai ◽  
Lakshmi Dey ◽  
Yeol Cho
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Open Problem ◽  
Metric Spaces ◽  
Complete Metric Spaces ◽  
Compact Metric Spaces ◽  
Contractive Mappings ◽  
Interesting Open Problem

This paper deals with an interesting open problem of B.E. Rhoades (Contemporary Math. (Amer. Math. Soc.) 72(1988), 233-245) on the existence of general contractive conditions which have fixed points, but are not necessarily continuous at the fixed points. We propose some more solutions to this problem by introducing two new types of contractive mappings, that is, A-contractive and A`-contractive, which are, in some sense, more appropriate than those of the important previous attempts. We establish some new fixed point results involving these two contractive mappings in compact metric spaces and also in complete metric spaces and show that these contractive mappings are not necessarily continuous at their fixed points. Finally, we suggest an applicable area, where our main results may be employed.

scholarly journals Convergence theorems for generalized contractions and applications

Filomat ◽  
2020 ◽  
Vol 34 (3) ◽  
pp. 945-964
Author(s):  
Mudasir Younis ◽  
Deepak Singh ◽  
Stojan Radenovic ◽  
Mohammad Imdad
Keyword(s):  
Fixed Point ◽  
Graph Theory ◽  
Fixed Points ◽  
Open Problem ◽  
Fixed Point Theorems ◽  
Balance Model ◽  
Population Balance Model ◽  
New Techniques ◽  
New Class ◽  
Article Deal

The principal results in this article deal with the existence of fixed points of a new class of generalized F-contraction. In our approach, by visualizing some non-trivial examples we will obtain better geometrical interpretation. Our main results substantially improve the theory of F-contraction mappings and the related fixed point theorems. In section-4, application to graph theory is entrusted and proved results are endorsed by an example through graph. The presented new techniques give the possibility to justify the existence problems of the solutions for some class of integral equations. For the future aspects of our study, an open problem is suggested regarding discretized population balance model, whose solution may be derived from the established techniques.

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