Research Article | Open Access

Binayak S. Choudhury, Nikhilesh Metiya, Pradip Debnath, "End Point Results in Metric Spaces Endowed with a Graph", *Journal of Mathematics*, vol. 2016, Article ID 9130107, 7 pages, 2016. https://doi.org/10.1155/2016/9130107

# End Point Results in Metric Spaces Endowed with a Graph

**Academic Editor:**Krassimir T. Atanassov

#### Abstract

We introduce the notion of end point of multivalued mappings in the setting of metric space endowed with a graph and prove some existence results in this context. The mappings are assumed to satisfy certain generalized multivalued almost -contractive type inequalities. Further, the consequences of the corresponding results in the cases of single-valued mappings are also discussed with examples.

#### 1. Introduction and Mathematical Preliminaries

One of the most famous fixed point theorems is the Banach contraction principle, which initiated a new era of research in fixed point theory due to its immense applicability in major areas of mathematics like numerical analysis and differential/integral equations. This important principle was used by Boyd and Wong [1] to investigate the fixed point results of nonlinear contraction maps.

Study of fixed point results in partially ordered sets has been a very well-motivated area of research because of its ease of compatibility in modelling various problems and in finding new convergence schemes. The first attempt in this direction was carried out by Ran and Reurings [2] where they combined the Banach Contraction Principle and the Knaster-Tarski fixed point theorem. Ran and Reurings considered a class of mappings , with as a complete metric space and a partial order . The mappings they considered were continuous, monotone with respect to the partial order . Those mappings also satisfy a Banach contraction inequality for every pair such that . When for some , the inequality is satisfied, they proved that the Picard sequence would converge to a fixed point of . Ran and Reurings also combined this interesting result with the Schauder fixed point theorem and applied it to obtain some existence and uniqueness results to nonlinear matrix equations.

Neito and Rodríguez-López ([3, 4]) extended the results of Ran and Reurings to the functions which were not necessarily continuous. They also applied their results to obtain a theorem on the existence of a unique solution for periodic boundary problems relative to ordinary differential equations.

The additional structure of ordering was introduced in metrizable uniform spaces by Turinici [5]. Later, fixed point studies were attempted in partially ordered metric spaces in a good number of works as, for instance, in [6–9]. Multivalued fixed point results were deduced in such contexts in works like [10–14].

The use of graph was introduced in metric fixed point theory by Jachymski [15]. It is in furtherance of metric fixed point theory in partially ordered spaces in that the partial orders themselves introduce a directed graph in metric spaces. A version of the Banach contraction mapping principle was established in these spaces where contraction inequality needs to be satisfied on the edges of the graph. In fact this is a general feature of the line of research which is that the contractions can be restricted to parts of spaces for the purpose of fixed points to exist. There are several results on fixed points of functions defined on these structures, some of the important references from these works being [16–20].

In the following we describe the mathematical background materials which are necessary for establishing the results in this papers.

Let be a metric space. We consider the following classes of subsets of the metric space : For , the functions and are defined as follows: If , then we write and . Also in addition, if , then and . Obviously, . For all , the definition of yields the following:(see [21]) -distance is not a metric like the Hausdorff distance [22] but shares all the properties of a metric except that it is possible to have .

*Definition 1. *Let be a nonempty set and a self-map on . By a fixed point of , we mean an element of such that . The set of all fixed points of is denoted as .

*Definition 2. *Let be a nonempty set and be multivalued mapping. Then is called a fixed point of if .

*Definition 3. *Let be a nonempty set and be multivalued mapping. Then is called an end point of if . The set of all end points of is denoted as .

*Remark 4. *Every end point of a mapping is a fixed point of the mapping but the converse is not true. So .

Fixed point theory for multivalued operators is an important topic of set-valued analysis. There are several works on fixed point theory of set-valued maps which have utilized -distance [10–13, 21, 23, 24].

Berinde [25] introduced a new class of self-mappings (usually almost contractions) that satisfy a simple but general contraction condition that includes any strict contraction, Kannan [26] and Zamfirescu [27] mappings, and a large class of quasi-contractions. Almost contractions and their generalizations were further considered in several works like [12, 13, 28–31]. Our aim in this paper is to deduce some fixed point theorems for certain multivalued almost contractions using -distance.

Let be a nonempty set and . Let be a directed graph such that its vertex set coincides with ; that is, and the edge set contains all loops; that is, . Assume that has no parallel edges. We can identify with the pair . By we denote the conversion of a graph , that is, the graph obtained from by reversing the directions of the edges. Thus we have Let denote the undirected graph obtained from by ignoring the direction of edges. Actually, it will be more convenient for us to treat as a directed graph for which the set of its edges is symmetric. Under this convention,A graph is called a subgraph of the graph if and .

*Definition 5. *If and are vertices in a graph , then a path in from to of length is a sequence of vertices such that , , and for .

A graph is connected if there is a path between any two vertices. is weakly connected if is connected. If is such that is symmetric and is a vertex in , then the subgraph consisting of all edges and vertices which are contained in some path beginning at is called the component of containing . In this case , where is the equivalence class of the following relation defined on by the rule:

*Definition 6. *Let be a mapping. Then , , , and are defined as follows:

*Definition 7. *Let be a multivalued mapping. Then , , , , and are defined as follows:

The essential feature of Jachymski’s work [15] is that the contraction inequality needs to be satisfied only on certain edges of the graph. This opened a new direction in fixed point theory in which a significant amount of works has appeared. Our work is in furtherance of this line of research.

#### 2. Main Results

We say a metric space is endowed with a directed graph , if is a directed graph such that and .

In the following we define generalized almost -contraction for single-valued mappings and generalized multivalued almost -contraction of types (A), (B), and (C). Assume that is a metric space, and is a directed graph such that and .

*Definition 8. *A mapping is a generalized almost -contraction if for all with (1),(2), , ,where and .

*Definition 9. *A mapping is a generalized multivalued almost -contraction of type (A) if for all with (i) implies that there exists such that ,(ii), , ,where and .

*Definition 10. *A mapping is a generalized multivalued almost -contraction of type (B) if for all with (i) implies that there exists such that ,(ii), + , ,where and .

*Definition 11. *A mapping is a generalized multivalued almost -contraction of type (C) if for all with (i) implies that there exists such that and implies that there exists such that ,(ii), + , ,where and .

Proposition 12. *If is generalized multivalued almost -contraction of type (B) then is generalized multivalued almost -contraction of type (A).*

*Proof. *Suppose that is generalized multivalued almost -contraction of type (B). Let with . Then .(1)By the condition (i) of the Definition 10, implies that there exists such that . Now implies that . Therefore, if with , then implies that there exists such that .(2)By the condition (ii) of the Definition 10, for all with where and .

Since implies that and and are symmetric, it follows that for all with where and .

Then it follows by Definition 9 that is generalized multivalued almost -contraction of type (A).

Proposition 13. *If is generalized multivalued almost -contraction of type (C) then is generalized multivalued almost -contraction of type (A).*

*Proof. *Suppose that is generalized multivalued almost -contraction of type (C). By Definitions 9 and 10 and Proposition 12 it follows that is both generalized multivalued almost -contraction of type (A) and generalized multivalued almost -contraction of type (A). Hence is generalized multivalued almost -contraction of type (A).

*Definition 14. *The triple is said to be regular if(1)for any sequence in with and for all , for all ,(2)for any sequence in with and for all , for all .

Theorem 15. *Let be a complete metric space endowed with a directed graph and be a generalized multivalued almost -contraction of type . Suppose that the triple is regular. Then the following statements hold:*(1)*For any , has an end point.*(2)*If and is weakly connected, then has an end point in .*(3)*If , then has an end point.*(4)* if and only if .*

*Proof. *(1) Let . By the definition of , there exists such that . Since is generalized multivalued almost -contraction of type (A), there exists such that . Again, by similar logic, there exists such that . Continuing this process we construct a sequence in such that Since and is generalized multivalued almost -contraction of type (A), we have, for all ,Since , it follows thatSuppose that , for some positive integer . So, .

Then it follows from (13) that which is a contradiction since . Therefore, , for all and hence we have from (13) thatBy repeated application of (15), we haveFor arbitrary with , which implies that is a Cauchy sequence. From the completeness of , there exists a such thatSince is a sequence in such that and for all , using the regular property of we have thatSince for all and is generalized multivalued almost -contraction of type (A), we haveTaking the limit as in the above inequality and using (18), we haveSince , the above inequality implies that or that . Moreover, is an end point of . By (19), it is obvious that and so . Hence has an end point.

(2) Let and is weakly connected. Since is weakly connected, for every . Since , there exists . Then . So by , has an end point in .

(3) Let . By and , has an end point.

(4) Let . Then there exists at least one element . Now means . So, . Now implies that . Therefore, we have and , which implies that . Hence . Conversely suppose that . Then by , .

Theorem 16. *Let be a complete metric space and be a generalized multivalued almost -contraction of type (B). Suppose that the triple is regular. Then the following statements hold:*(1)*For any , has an end point.*(2)*If and is weakly connected, then has an end point in .*(3)*If , then has an end point.*(4)* if and only if .*

*Proof. *Let . Then there exist such that . Now implies that . Since and for some , we have . Conversely, implies that . So . Therefore,(1) implies that ,(2) is weakly connected implies that is also weakly connected,(3),(4),(5)By Proposition 12, is a generalized multivalued almost -contraction of type (A).Let . Then by Theorem 15 we have the required proof.

Theorem 17. *Let be a complete metric space and be a generalized multivalued almost -contraction of type (C). Suppose that the triple is regular. Then the following statements hold:*(1)*For any , has an end point.*(2)*If and is weakly connected, then has an end point in .*(3)*If , then has an end point.*(4)* if and only if .*

*Proof. *Arguing similarly as in the proof of Theorem 16, we have that . Therefore,(1) implies that ,(2) is weakly connected implies that is connected,(3),(4)By Proposition 13, is a generalized multivalued almost -contraction of type (A).Let . Then by Theorem 15 we have the required proof.

*Example 18. *Let be a subset of ( is the set of all real numbers) and be a directed graph with and . Let be metric on defined as Let be defined as follows:Let and .

All the conditions of Theorem 15 are satisfied and it is seen that is an end point of .

#### 3. Consequences in Single-Valued Cases

In this section we obtain some consequences of the corresponding results of Section 2 in the cases of single-valued mappings.

Theorem 19. *Let be a complete metric space and be a generalized almost -contraction. Suppose that the triple is regular. Then following statements hold:*(1)*For any , has a fixed point.*(2)*If and is weakly connected, then has a fixed point in .*(3)*If , then has a fixed point.*(4)* if and only if .*

*Proof. *We know that, for every , . We define a multivalued mapping as , for .(1)Since is a generalized almost -contraction, it follows by Definitions 8 and 9 that is a generalized multivalued almost -contraction of type .(2). So, implies that .(3).(4)Every fixed point of is an end point of and every end point of is a fixed point of . Hence .So, all the conditions of Theorem 15 are satisfied and hence by application of the Theorem 15 we have required proof.

Theorem 20. *Let be a complete metric space and be a generalized almost -contraction. Suppose that the triple is regular. Then the following statements hold:*(1)*For any , has a fixed point.*(2)*If and is weakly connected, then has a fixed point in .*(3)*If , then has a fixed point.*(4)* if and only if .*

Theorem 21. *Let be a complete metric space and be a generalized almost -contraction. Suppose that the triple is regular. Then the following statements hold:*(1)*For any , has a fixed point.*(2)*If and is weakly connected, then has a fixed point in .*(3)*If , then has a fixed point.*(4)* if and only if .*

*Example 22. *Let , where with , and let be a directed graph with and . Let be metric the usual metric on . Let be defined as Let be such that and .Then all the conditions of Theorem 19 are satisfied and it is seen that the set of fixed points of is nonempty. Here .

*Remark 23. *The results we obtain here are for functions without any continuity requirement. We assume in our results that is regular. If we omit this condition in our theorem and, instead, assume the function to be continuous then also the conclusions of the theorem are valid.

#### Competing Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Authors’ Contributions

The authors contributed equally to this work.

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#### Copyright

Copyright © 2016 Binayak S. Choudhury 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.