Graphical structure of extended b-metric spaces: an application to the transverse oscillations of a homogeneous bar

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Mudasir Younis ◽  
Deepak Singh ◽  
Ishak Altun ◽  
Varsha Chauhan
Keyword(s):  
Metric Spaces ◽  
Fixed Point Theory ◽  
Directed Graphs ◽  
Point Theory ◽  
Graph Structure ◽  
Open Ball ◽  
Open Problems ◽  
Graphical Structure ◽  
State Of Art ◽  
Transverse Oscillations

Abstract The purpose of this article is to present the notion of graphical extended b-metric spaces, blending the concepts of graph theory and metric fixed point theory. We discuss the structure of an open ball of the new proposed space and elaborate on the newly introduced ideas in a novel way by portraying suitably directed graphs. We also provide some examples in graph structure to show that our results are sharp as compared to the results in the existing state-of-art. Furthermore, an application to the transverse oscillations of a homogeneous bar is entrusted to affirm the applicability of the established results. Additionally, we evoke some open problems for enthusiastic readers for the future aspects of the study.

Two open problems in the fixed point theory of contractive type mappings on quasimetric spaces

2017 ◽  
Vol 33 (2) ◽  
pp. 169-180
Author(s):  
MITROFAN M. CHOBAN ◽  
◽  
VASILE BERINDE ◽  
◽  
Keyword(s):  
Fixed Point ◽  
Metric Spaces ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Partial Answer ◽  
Open Problems ◽  
Complete Answer ◽  
Contractive Type ◽  
Generalized Distances

Two open problems in the fixed point theory of quasi metric spaces posed in [Berinde, V. and Choban, M. M., Generalized distances and their associate metrics. Impact on fixed point theory, Creat. Math. Inform., 22 (2013), No. 1, 23–32] are considered. We give a complete answer to the first problem, a partial answer to the second one, and also illustrate the complexity and relevance of these problems by means of four very interesting and comprehensive examples.

scholarly journals Applications of Graph Kannan Mappings to the Damped Spring-Mass System and Deformation of an Elastic Beam

2019 ◽  
Vol 2019 ◽  
pp. 1-9 ◽  
Author(s):  
Mudasir Younis ◽  
Deepak Singh ◽  
Adrian Petruşel
Keyword(s):  
Fixed Point ◽  
Graph Theory ◽  
Metric Spaces ◽  
Fixed Point Theory ◽  
Elastic Beam ◽  
Nonlinear Problems ◽  
Point Theory ◽  
Open Problems ◽  
Mass System ◽  
Fixed Point Result

The purpose of this article is twofold. Firstly, combining concepts of graph theory and of fixed point theory, we will present a fixed point result for Kannan type mappings, in the framework of recently introduced, graphical b-metric spaces. Appropriate examples of graphs validate the established theory. Secondly, our focus is to apply the proposed results to some nonlinear problems which are meaningful in engineering and science. Some open problems are proposed.

scholarly journals Some Fixed Point Results of Weak-Fuzzy Graphical Contraction Mappings with Application to Integral Equations

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 541
Author(s):  
Shamoona Jabeen ◽  
Zhiming Zheng ◽  
Mutti-Ur Rehman ◽  
Wei Wei ◽  
Jehad Alzabut
Keyword(s):  
Fixed Point ◽  
Integral Equations ◽  
Metric Spaces ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Continuity Condition ◽  
Graph Structure ◽  
Cone Metric Spaces ◽  
Contraction Mappings ◽  
Cone Metric

The present paper aims to introduce the concept of weak-fuzzy contraction mappings in the graph structure within the context of fuzzy cone metric spaces. We prove some fixed point results endowed with a graph using weak-fuzzy contractions. By relaxing the continuity condition of mappings involved, our results enrich and generalize some well-known results in fixed point theory. With the help of new lemmas, our proofs are straight forward. We furnish the validity of our findings with appropriate examples. This approach is completely new and will be beneficial for the future aspects of the related study. We provide an application of integral equations to illustrate the usability of our theory.

scholarly journals Nadler and Kannan Type Set Valued Mappings in M-Metric Spaces and an Application

Mathematics ◽  
2019 ◽  
Vol 7 (4) ◽  
pp. 373 ◽  
Author(s):  
Pradip R. Patle ◽  
Deepesh Kumar Patel ◽  
Hassen Aydi ◽  
Dhananjay Gopal ◽  
Nabil Mlaiki
Keyword(s):  
Fixed Point ◽  
Hausdorff Distance ◽  
Fixed Point Theorems ◽  
Metric Spaces ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Current State ◽  
Set Valued Mappings ◽  
State Of Art

This article intends to initiate the study of Pompeiu–Hausdorff distance induced by an M-metric. The Nadler and Kannan type fixed point theorems for set-valued mappings are also established in the said spaces. Moreover, the discussion is supported with the aid of competent examples and a result on homotopy. This approach improves the current state of art in fixed point theory.

Revisiting graphical rectangular b-metric spaces

2021 ◽  
pp. 2250072
Author(s):  
Mudasir Younis ◽  
Deepak Singh ◽  
Luoyi Shi
Keyword(s):  
Fixed Point ◽  
Directed Graph ◽  
Metric Spaces ◽  
Fixed Point Theory ◽  
Directed Graphs ◽  
Intimate Relationship ◽  
Point Theory ◽  
Science And Engineering ◽  
Novel Approach ◽  
Wide Range

Very recently, the notion of graphical rectangular [Formula: see text]-metric spaces has been introduced, where the weight of the symmetric edges endowing the directed graph is defined in terms of the underlying metric. Due to its intimate relationship with the directed graphs, such spaces have a wide range of applications in science and engineering. In this paper, we rectify some errors that arose in the paper [Younis et al., A novel approach of graphical rectangular [Formula: see text]-metric spaces with an application to the vibrations of a vertical heavy hanging cable, J. Fixed Point Theory Appl. 21(1) (2019) 33]. We show that the main result of the paper described above can be modified, and we present the same in this note.

scholarly journals Results on contractions of Reich type in graphical b-metric spaces with applications

Filomat ◽  
2019 ◽  
Vol 33 (17) ◽  
pp. 5723-5735 ◽  
Author(s):  
Mudasir Younis ◽  
Deepak Singh ◽  
Mehdi Asadi ◽  
Vishal Joshi
Keyword(s):  
Final Section ◽  
Metric Spaces ◽  
Existence Of Solutions ◽  
Directed Graphs ◽  
Nonlinear Problems ◽  
Graph Structure ◽  
Open Problems

The main purpose of this article is to present some results concerning Reich type contractions in the graph structure in the framework of recently introduced graphical b-metric spaces. Our results are significant extensions and generalizations of some pioneer results in the existing theory. Innovative examples along with directed graphs are propounded to support the newfangled results, making the established theory more comprehensible. Final section is devoted to apply our results to the existence of solutions of some nonlinear problems along with some open problems which may be fruitful for the further scope of the study.

scholarly journals Some Common Fixed-Point Theorems for Generalized-Contractive-Type Mappings on Complex-Valued Metric Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Chakkrid Klin-eam ◽  
Cholatis Suanoom
Keyword(s):  
Fixed Point ◽  
Common Fixed Point ◽  
Fixed Point Theorems ◽  
Metric Spaces ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Contractive Type ◽  
Complex Valued ◽  
Common Fixed Point Theorems

Fixed-point theory in complex valued metric spaces has greatly developed in recent times. In this paper, we prove certain common fixed-point theorems for two single-valued mappings in such spaces. The mappings we consider here are assumed to satisfy certain metric inequalities with generalized fixed-point theorems due to Rouzkard and Imdad (2012). This extends and subsumes many results of other authors which were obtained for mappings on complex-valued metric spaces.

scholarly journals Semicompatibility and fixed point theorems in an unboundedD-metric space

2005 ◽  
Vol 2005 (5) ◽  
pp. 789-801
Author(s):  
Bijendra Singh ◽  
Shishir Jain ◽  
Shobha Jain
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Metric Space ◽  
Fixed Point Theorems ◽  
Metric Spaces ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Contractive Condition ◽  
Restricted Domain

Rhoades (1996) proved a fixed point theorem in a boundedD-metric space for a contractive self-map with applications. Here we establish a more general fixed point theorem in an unboundedD-metric space, for two self-maps satisfying a general contractive condition with a restricted domain ofxandy. This has been done by using the notion of semicompatible maps inD-metric space. These results generalize and improve the results of Rhoades (1996), Dhage et al. (2000), and Veerapandi and Rao (1996). These results also underline the necessity and importance of semicompatibility in fixed point theory ofD-metric spaces. All the results of this paper are new.

EXISTENCE OF SOLUTION FOR A VOLTERRA TYPE INTEGRAL EQUATION USING DARBO-TYPE F-CONTRACTION

2021 ◽  
Vol 10 (6) ◽  
pp. 2687-2710
Author(s):  
F. Akutsah ◽  
A. A. Mebawondu ◽  
O. K. Narain
Keyword(s):  
Integral Equation ◽  
Fixed Point ◽  
Metric Spaces ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Existence Of Solution ◽  
Volterra Type ◽  
Complete Metric Spaces ◽  
Type Integral ◽  
New Type

In this paper, we provide some generalizations of the Darbo's fixed point theorem and further develop the notion of $F$-contraction introduced by Wardowski in (\cite{wad}, D. Wardowski, \emph{Fixed points of a new type of contractive mappings in complete metric spaces,} Fixed Point Theory and Appl., 94, (2012)). To achieve this, we introduce the notion of Darbo-type $F$-contraction, cyclic $(\alpha,\beta)$-admissible operator and we also establish some fixed point and common fixed point results for this class of mappings in the framework of Banach spaces. In addition, we apply our fixed point results to establish the existence of solution to a Volterra type integral equation.

scholarly journals A New Approach to the Solution of the Fredholm Integral Equation via a Fixed Point on Extended b-Metric Spaces

Symmetry ◽  
2018 ◽  
Vol 10 (10) ◽  
pp. 512 ◽  
Author(s):  
Erdal Karapınar ◽  
Panda Kumari ◽  
Durdana Lateef
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Nash Equilibria ◽  
Metric Spaces ◽  
Fixed Point Theory ◽  
Real Life ◽  
Point Theory ◽  
New Approach ◽  
Correctness Of Programs

It is very well known that real-life applications of fixed point theory are restricted with the transformation of the problem in the form of f ( x ) = x . (1) The Knaster–Tarski fixed point theorem underlies various approaches of checking the correctness of programs. (2) The Brouwer fixed point theorem is used to prove the existence of Nash equilibria in games. (3) Dlala et al. proposed a solution for magnetic field problems via the fixed point approach.

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