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 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 A NOTE ON THE PAPER “ON COMPLETENESS IN METRIC SPACES AND FIXED POINT THEOREMS"

2020 ◽  
Vol 15 (1) ◽  
Author(s):  
Deepak Khantwal ◽  
Umesh Cuandra Gairola
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Fixed Point Theorems ◽  
Metric Spaces ◽  
Complete Metric Space ◽  
Continuous Mappings ◽  
Discontinuous Mappings ◽  
The Fixed Point Theorem ◽  
Common Fixed Point Theorem

In the present note, we show that the assumption of continuity used in the fixed point theorem of Gregori et al. (Results Math. 73 (2018), no. 4, Art. 142, 13) can be relaxed to some weaker version of continuity. More precisely, we prove a fixed point theorem for orbitally continuous and k-continuous mappings in weak G-complete metric space and provide an appropriate example to show that our result is not only valid for continuous mappings but also for some discontinuous mappings. Moreover, we apply our main result to establish a common fixed point theorem for two self-mappings

scholarly journals Common fixed point theorems in complete metric and probabilistic metric spaces

1987 ◽  
Vol 36 (1) ◽  
pp. 73-88 ◽  
Author(s):  
Mila Stojakovic
Keyword(s):  
Fixed Point ◽  
Common Fixed Point ◽  
Fixed Point Theorems ◽  
Metric Spaces ◽  
Probabilistic Metric Spaces ◽  
Continuous Mappings ◽  
Contraction Condition ◽  
Probabilistic Metric ◽  
Common Fixed Point Theorems

In this paper several common fixed point theorems for four continuous mappings in Menger and metric spaces are proved. These mappings are assumed to satisfy some generalizations of the contraction condition.

scholarly journals More general common fixed point theorems under a new concept

2016 ◽  
Vol 49 (1) ◽  
Author(s):  
H. Bouhadjera
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Common Fixed Point ◽  
Existence And Uniqueness ◽  
Fixed Point Theorems ◽  
Common Fixed Points ◽  
Implicit Function ◽  
Continuous Mappings ◽  
Common Fixed Point Theorems ◽  
Common Fixed Point Theorem

AbstractA general common fixed point theorem for two pairs of weakly subsequentially continuous mappings (recently introduced) satisfying a significant estimated implicit function is proved. An extension of this result is thereby obtained. Our results assert the existence and uniqueness of common fixed points in several cases.

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 Discontinuity at fixed point and metric completeness

2020 ◽  
Vol 21 (2) ◽  
pp. 349
Author(s):  
Ravindra K. Bisht ◽  
Vladimir Rakocevic
Keyword(s):  
Fixed Point ◽  
Metric Space ◽  
Open Problem ◽  
Fixed Point Theorems ◽  
Intersection Property ◽  
Contractive Mappings

<p>In this paper, we prove some new fixed point theorems for a generalized class of Meir-Keeler type mappings, which give some new solutions to the Rhoades open problem regarding the existence of contractive mappings that admit discontinuity at the fixed point. In addition to it, we prove that our theorems characterize completeness of the metric space as well as Cantor's intersection property.</p>

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 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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