Research Article | Open Access
Applications of Graph Kannan Mappings to the Damped Spring-Mass System and Deformation of an Elastic Beam
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.
In several recent studies, graph theory engages an influential role especially for metric fixed point theory in numerous aspects. Jachymski  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  proposed the notion of graphical metric spaces as a graphical version of metric spaces. It is necessary to mention that the triangular inequality in  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  and introduced graphical -metric spaces and graphical rectangular -metric spaces by showing suitable graphs. Kannan  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 . 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 , 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 ) 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 , 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 ). 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.
Definition 3 (see ). Every open ball centered in and having radius in a is an open set. Moreover, the corresponding topological space is but it is not .
Definition 4 (see ). A sequence in a is said to be a convergent sequence if there exists such that as and it is a Cauchy sequence if as .
3. Main Results
Consider to be a subgraph of with and further assume to be a weighted graph. Let be the initial value of a sequence ; we say is a -Picard sequence - for a mapping if for all .
Furthermore, we say a graph satisfies the property , if a -- converging in ensures that there is a limit of and such that or .
We present now our main definition as follows.
Definition 5. Let be a . A self-mapping is said to be -graph Kannan mapping on if
for all if ;
there exists , with , we have
Example 6. Every Kannan mapping is an -graph Kannan mapping equipped with the graph defined by and
For instance, let via the graphical -metric defined by Obviously, is a with . The mapping defined by is a Kannan mapping for . Now examine the graph along with and . One can clearly see that is a -graph Kannan mapping; the same is validated in Figure 2.
Example 7. -graph Kannan mapping is not necessarily a Kannan mapping.
Let be endowed with the graphical -metric as defined in Example 6. Then is a with the coefficient . Define the mapping byNow consider the graph for which and . Then, is -graph Kannan mapping for . Figure 3 illustrates the directed graph associated with .
Now we present our main result concerning -graph Kannan mapping as follows.
Theorem 8. Let be -graph Kannan mapping on a -complete . The following conditions hold:
satisfies the property ;
there exists with for some .
Then, there exists such that the - with initial value is - and converges to .
Proof. Let be such that , for some . Since is a - starting from ; therefore, there exists a path such that , , and for . By hypothesis being -graph Kannan mapping, therefore from assertion we have for . This implies that is a path from to having length , and hence . Pursuing this process, we acquire that is a path from to of length ; i.e., is a path possessing length from to and hence , for all . Thus we attain that is a - sequence.
Now for and ; thanks to , we obtain This implies thatSet and notice that by putting different values of and . Thus, the above inequality can be written as Repeating this process, we getSince the sequence is - and is a subgraph of , using (11) and the triangle inequality, we obtainSet . Then, inequality (12) reduces to Again, is -, for ; we get Since , we infer that . Hence is a Cauchy sequence in . Also since is -complete, therefore converges in and, by hypothesis, there exists some such that or and which shows that converges to .
For the existence of a fixed point of the underlying mapping, Shukla [10, Theorem 3.10] and Chuensupantharat [5, Theorem 3.4] used the condition (S). However, we drop this condition and assume that the subgraph is weakly connected. This guarantees the fixed point of mapping along with its uniqueness.
Theorem 9. If the hypotheses embodied in Theorem 8 are satisfied and, additionally, we suppose that is weakly connected, then is the unique fixed point of .
Proof. Theorem 8 ensures that the -PS with initial value converges to . Since is weakly connected, therefore or , and hence we have Utilizing , we obtain It follows that as . Hence ; therefore, is the fixed point of .
Uniqueness: Let be another fixed point of . Assume ; then there exists a sequence such that , with . Since is -graph Kannan mapping, repeated use of gives . Making use of and proceeding along with the same lines as done in Theorem 8, we acquire Now utilizing , we haveSince and . Proceeding limit , we obtain . Hence admits exactly one fixed point.
To make our results explicit, we propose the following example.
Example 10. Let be endowed with a graph such that and .
Define the graphical -metric byIt is evident that is a with . Let the map be defined by , for all . One can easily find that there exists such that ; i.e., and mapping (4) is satisfied for ; thus is an -graph Kannan mapping on . Figure 4 authenticates the domination of RHS of graph mapping (4) over LHS for .
By routine calculations, one can see that all the conditions of Theorem 9 are contented and 0 is the desired fixed point of the mapping . Figure 5 exemplifies the weighted graph for , where the value of is equal to the weight of edge and .
In this section we show the importance and applicability of the obtained results.
Let = be the set of real continuous functions on . Consider ; and let the graph be defined by and
4.1. Application to Damped Spring-Mass System
In engineering problems, one of the realistic applications for the “spring-mass system" is an automobile suspension system. Consider the motion of a spring of a car when it moves along a rough and pitted road, where the forcing term is rough road and shock absorbers provide the damping. The external forces under which the system operates may be gravity, ground vibrations, earthquake, tension force, etc. Let be the mass of the spring and is the external force acting on it; then the critical damped motion of this system subjected to the external force is governed by the following initial value problem:where is the damping constant and is a continuous function.
The aforementioned problem is equivalent to the integral equationwhere the respective Green’s function with is given byDefine graphical metric byfor all . Then is a -complete .
Consider the operator given by Then is a solution of (24), iff is a fixed point of .
Now consider the following assertions:
is a lower solution of problem (24); i.e.,
is an increasing function on for every and choose suitably such that where
For each and , we have
Proof. Utilizing the hypothesis of the theorem under consideration, for with , we have It follows that In view of hypothesis, taking , we get Thus the contractive condition of Theorem 9 is established.
Next, for each such that , we obtain that and for all ; and by condition , we obtain , and This amounts to say that and . In accordance with condition , there exists a solution say such that , so that the condition of Theorem 9 is verified. Next, it is easy to see that other conditions of Theorem 9 are satisfied. Therefore, Theorem 9 guarantees that has a unique fixed point and hence problem (23) has a unique solution in .
4.2. Application to Deformation of Elastic Beam
A fourth-order two-point boundary value problem describes the deformations of elastic beam as per the controls at the ends of the beam. Due the nature and characterization of the equilibrium state, these higher order beam equations have significant applications pertaining to mechanics and engineering related to bridges, ships, automobiles, buildings, and aircraft. We refer readers interested in applications of elastic beams and related problems to [24, 25].
Consider the following functional equation describing the deformation of an elastic beam with one end free and other end cantilevered or fixed:where is a continuous function that appears in the study of elastic beam equations.
Solving problem (36) is equivalent to finding ; solution of the integral equation iswhere is the Green function given by Define graphical metric by for all . Clearly is a -complete .
Next theorem provides us a sufficient condition for the existence and uniqueness of solution of problem (36).
Theorem 13. Suppose the following conditions hold:
(1) is a lower solution of the problem (37); i.e.,(2) Choose suitably such that where
(3) For each and , we have where the operator is defined in (43).
Then the existence of solution for integral equation (37) provides a unique solution for the functional equation describing the deformation of an elastic beam (36).
Proof. Define the operator given byClearly operator is well defined. Utilizing condition hypothesis of the theorem under consideration, for with , we have It follows that In view of hypothesis, taking , we getThus the contractive condition of Theorem 9 is established.
Other conditions of Theorem 9 can be proved along similar lines as done in Theorem 12; for the sake of brevity, we omit the rest part of the proof.
Hence we conclude that the fourth-order boundary value problem (36) describing the deformation of an elastic beam has a unique solution in .
Open Problems(i) The governing second-order differential equation for an infinite rod with absorption describing the steady state concentration dispersing in an absorbing medium is given by where , is the density of the substance in the steady state, and the whole process is examined in an infinity long tube. Whether the existence of solution of the above second-order differential equation can be derived from results established in this note?(ii) Establish analogue results of Edelstein , Hardy-Rogers , Meir-Keelar , and Reich  type contractions in the underlying space.(iii) Apply Kannan type graph contractions or equivalent graph contractions to establish the existence of solution of the following beam equation where is a continuous function and .
This article implements the idea of graphical structure of Kannan mappings. Based on this structure, we showed that every Kannan mapping is graph Kannan but not conversely. We obtained the fixed point result by dropping the property as used in [5, 10]. Obtained results are validated by appropriate examples endowed with suitable graphs. Embellishing the applications of established results, two problems are considered: damped spring-mass system and deformation of an elastic beam. The results are important both theoretically and practically for fixed point theorists as well as engineers and physicists working in the related fields.
All the data related to this research work are included in the text.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
The authors are thankful to the learned referees for suggesting some improvements which improved the revised version of this article.
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