Discrete Dynamics in Nature and Society

Volume 2019, Article ID 1315387, 9 pages

https://doi.org/10.1155/2019/1315387

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

^{1}Department of Applied Mathematics, UIT-Rajiv Gandhi Technological University (RGPV), University of Technology of Madhya Pradesh, Bhopal 462033, India^{2}Department of Applied Sciences, National Institute of Technical Teachers’ Training and Research (Under Ministry of HRD, Govt. of India), Bhopal 462002, MP, India^{3}Faculty of Mathematics and Computer Science, Babeş-Bolyai University Cluj-Napoca, Romania^{4}Academy of Romanian Scientists, Bucharest, Romania

Correspondence should be addressed to Adrian Petruşel; or.julcbbu.htam@lesurtep

Received 5 March 2019; Accepted 6 May 2019; Published 9 June 2019

Academic Editor: Rodica Luca

Copyright © 2019 Mudasir Younis et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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 -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.

#### 1. Introduction

In several recent studies, graph theory engages an influential role especially for metric fixed point theory in numerous aspects. Jachymski [1] gave a unified version of Banach’s contraction principle, by considering the graph structure instead of an order structure on a metric space. This result has been extended and generalized by many researchers in different ways (see, e.g., [2–11]). Recently, Shukla et al. in [10] proposed the notion of graphical metric spaces as a graphical version of metric spaces. It is necessary to mention that the triangular inequality in [10] is required to hold only on those elements which are related to each other under the underlying graph structure rather than the whole space. Most recently, Singh et al. as a coauthor in [5, 12] extended the idea given in [10] and introduced graphical -metric spaces and graphical rectangular -metric spaces by showing suitable graphs. Kannan [13] was the first mathematician who derived, under some metrical conditions, a unique fixed point for a not necessary continuous mapping. Since then Kannan mappings have been the center of research for many authors from last few decades (see, e.g., [14–19] and the related references therein). Another charm of Kannan’s theorem is that one can characterize the metric completeness in terms of the fixed point of such mappings. For detailed discussion we refer the reader to [20]. Analytically, Kannan mappings play an important role in metric fixed point theory, but, interestingly, one can hardly find literature connecting Kannan type mappings to application part. In this paper efforts have been made to apply the abstract results invoking graph Kannan type mapping to some real world applications, such as damped spring-mass system of an automobile suspension system and nonlinear elastic beam equations, problems which play an indispensable role in mechanics and engineering. At last, the paper is concluded with some open problems which may lead to new results and applications. In the subsequent analysis the basic assumptions on the graphs is that they are directed graphs encompassing nonempty set of edges.

#### 2. Notations and Basic Facts

We will present first some notations and basic concepts which are important for the main part of the paper.

Following Jachymski [1], let us denote by the diagonal of , where is a nonempty set. Further, we suppose that is a directed graph possessing no parallel edges and let be the set of all vertices of , such that coincides with the set . Let be the set of all edges of such that it contains all loops (i.e., ). We will write that . If we reverse the direction of edges of , the obtained graph is denoted by . Furthermore, the letter denotes a directed graph with symmetric edges. More precisely, we define . Let . A path (or a directed path) of length between and in is defined to be a sequence of vertices with , , and for . If any two vertices of contain a path between them, then is called a connected graph. If there exists a path between every two vertices in a undirected graph , then is said to be weakly connected. We call a subgraph of if and . We will also use (see Shukla [10]) the following notation: Further, a relation on is defined as follows:and we will write if is contained in the path . For a sequence if for all , then we say that is a -termwise connected (in short -) sequence.

Singh et al. in [21, 22] proposed some notable real world problems from engineering science and invoked their results to find the solution of these problems. Recently in [5], the authors applied the connected graph theory in metric fixed point theory in a very interesting way and introduced graphical -metric space as a generalization of -metric space as follows.

*Definition 1 (see [5]). *A graphical -metric coefficient on a nonempty set is a mapping satisfying the following conditions:

if and only if ;

for all ;

.

The pair is called graphical -metric space (briefly ) space with coefficient .

*Example 2. *Let be endowed with graphical -metric defined bywhere is a constant. It is easy to show that is a with coefficient encompassing the graph equipped with and as displayed in Figure 1.