Abstract

This paper proves a common fixed point theorem for single-valued and multivalued mappings by using ternary relation in -metric spaces. Henceforth, results obtained will be verified with the help of illustrative examples. Also, we demonstrate the results with an application.

1. Introduction

In 1969, Nadler [1] introduced multivalued contraction mappings using the Hausdorff metric and extended Banach’s contraction principle [2] from single-valued to multivalued mappings. Since then, several researchers have generalized these results for multivalued mappings in various spaces. As a result, the multivalued mapping theory has many applications in diverse areas, such as control theory, approximation theory, differential equations, and economics.

Definition 1 (see [1]). Let be a metric space. A map is said to be a multivalued contraction if there exists such that for all .

Kaneko and Sessa [3] extended the concept of compatible mappings due to Jungck [4] to include multivalued mappings as well as single-valued mappings. They followed the works of Kubiak [5] and Nadler [1] and proved coincidence and fixed point theorems for hybrid pair of compatible mappings. Pathak [6] extended the concept of compatible hybrid mappings to -weak compatible hybrid mappings and proved a coincidence theorem, an extension of the results of Kaneko and Sessa [3].

In 2006, Mustafa and Sims [7, 8] gave a generalization of -metric space to -metric space soon after identifying some shortcomings concerning the fundamental topological structure on -metric spaces. They defined several notions, such as continuity, completeness, compactness, convergence, and space product in the -metric space setting. Abbas and Rhoades [9] established the concept of a common fixed point in -metric spaces. In addition, Kaewcharoen and Kaewkhao [10] presented some common fixed points for single-valued and multivalued mappings in -metric spaces. At the same time, Tahat et al. [11] proved common fixed points for single-valued and multivalued maps satisfying a generalized contraction in -metric spaces. Recently, Mustafa et al. [12] gave the results in common fixed points for multivalued mappings in -metric spaces with applications using -contraction concept, and Shoaib and Shahzad [13] proved the common fixed point of multivalued mappings in ordered dislocated quasi -metric spaces.

On the other hand, Alam and Imdad [14] gave a generalization of the Banach contraction principle in a complete metric space equipped with binary relation. Their results show that the contraction condition holds only for those elements linked with the binary relation, not for every pair of elements. For more results on binary relation, one can see [1520] and the references therein. Perveen et al. [21] gave the prove in relation theoretic common fixed point results for generalized weak nonlinear contractions with an application. Hossain et al. [22] gave the study of relation-theoretic metrical fixed point theorem for rational type contraction mapping with an application. Further, Gaba et al. [23] extended the works of Alam and Imdad [14] by using the Banach contraction mapping principle in generalized metric spaces with a ternary relation. Furthermore, Radha and Singh [24] proved a novel approach to -metric spaces by using ternary relations. Badshah et al. [25] proved some common fixed point theorems for contractive mappings in cone 2-metric spaces equipped with a ternary relation.

This paper is aimed at proving a common fixed point theorem for single-valued and multivalued mappings by using the ternary relation concept in the -metric space setting. Specifically, we improve and extend the works due to Alam and Imdad [14], Ahmadullah et al. [15, 16], Eke et al. [18], and Radha and Singh [24]. In doing so, we will generalize several other works in the literature having the same directions.

2. Preliminaries

In this section, we introduce some definitions, theorems, and preliminary results, which will be helpful in developing the main result.

Motivated by Mustafa and Sims [8], we can formulate the axioms of -metric spaces as follows:

Definition 2 (see [8]). Let be a nonempty set and be a function satisfying the following conditions:
if
, for all with
for all with
(symmetry in all three variables)
, for all (rectangle inequality)

The function is called a generalized metric or -metric, and the pair is called a -metric space.

The following example satisfies the above axioms.

Example 1 (see [8]). Let be the set of all real number. Define by for all . Then, it is clear that is a -metric space with a -metric on .

Note that if then .

Mustafa and Sims [8] proved the following proposition satisfying a -metric properties.

Proposition 3 (see [8]). Let be a -metric space; then, the metric associated with satisfies the following: (i)(ii)(iii)for all .

Inspired by Mustafa and Sims [8], we established some topological properties such as convergence, completeness, and continuity in -metric spaces as follows:

Definition 4 (see [8]). Let be a -metric space. A sequence is said to be the following: (i)-convergent to if for any , there exists such that for all (ii)-Cauchy if for , there exists such that for all in is called Cauchy sequence if for each , there exists such that for all , i.e., as

Definition 5 (see [8]). A -metric space is said to be -complete if every -Cauchy sequence in is -convergent. Every -metric on defines a metric for all .

Proposition 6 (see [8]). Let be a -metric space. Then, the following properties are equivalent: (i) is -convergent to (ii) as (iii) as (iv) is a -Cauchy sequence(v)For every , there exists such that for

Definition 7 (see [8]). Let and be two -metric spaces, and let . Then, the map is said to be -continuous at if for , there exists such that for all and , we have . The function is -continuous if it is -continuous for each .

Proposition 8 (see [8]). Let and be two -metric spaces, and let . Then, the map is said to be -continuous at if and only if is sequentially continuous, i.e., whenever is -convergent to , the sequence is -convergent to .

Lemma 9 (see [8]). Let be a -metric space, then the function is jointly continuous in all three of its variables.

Later, Kaewcharoen and Kaewkhao [10] established the multivalued notion in -metric space. Let be a -metric space. Denote be the class of all nonempty, closed, and bounded subsets of . Let be the Haursdorff--distance on , that is, for , define where

Lemma 10 (see [10]). Let be a -metric space and . Then, for each , we have

Lemma 11 (see [11]). Let be a -metric space and ; then, for each , there exists such that

Lemma 12 (see [26]). Let be -sequence on a -metric space such that for all

Lemma 13 (see [26]). If is a -comparison function, for each

Definition 14 (see [10]). Let be a nonempty set. Assume and are two mappings. If for some , then is called a coincidence point of the pair and is a point of coincidence of and . The mapping and are called weakly compatible if for some implies .

Proposition 15 (see [10]). Let be a nonempty set. Assume and are weakly compatible mappings. If and have a unique point of coincidence , then is a unique common fixed point of and .

Definition 16 (see [27]). Let be a pair of self mappings on an ordered metric space with . For every , consider the sequence defined by , for all . Then, is called - sequence with initial point .

2.1. Ternary Relation-Theoretic in -Metric Spaces

In this part, we recall some definition, proposition, and preliminary results in -metric space related to ternary relation. We define , where is a set of natural numbers in .

Motivated by Nová and Novotný [28], NovotnỲ [29], Šlapal [30], and Alam and Imdad [14], we can introduce the property of ternary relation as follows:

Definition 17 (see [30]). Let be a set and . Then, is said to be ternary relation on . A ternary relation defined on a nonempty set is called (1)Reflexive if for all (2)Symmetric if and only if for any (3)Transitive if and only if , imply , for any in (4)Asymmetric if and only if (5)Irreflexive if (6)Irreversible if for any (7)Feebly regular if (8)Regular if , (9)Feebly translative if , (10)Translative if is feebly translative and or (11)Cyclic if for any (12)Complete if (13)Weakly complete if or ,

Definition 18 (see [14]). A relation among sets , , and is a subset of the Cartesian product , , and , .

Proposition 19 (see [24]). For any ternary relation defined on a nonempty set ,

Definition 20 (see [14]). Let be a nonempty set and a self- mapping on . A ternary relation defined on is called -closed if for ,

Motivated by Alam and Imdad [14] and Perveen et al. [21], we introduce the following definitions and proposition for multivalued mapping in the context of -metric space setting.

Definition 21. Let and be two multivalued mappings defined on a nonempty set . Then, a ternary relation defined on is called -closed if .

Proposition 22. Let be nonempty set, a ternary relation on , and two multivalued mappings on . If is -closed, then is -closed.

Definition 23. Let be a -metric space endowed with a ternary relation and . and are said to be -compatible if for any sequence such that and are -preserving and , we have

Inspired by Kolman et al. [19], we introduce the following definition in -metric space setting.

Definition 24. Let be a -metric space, a ternary relation defined on , and a pair of points in . Then, a finite sequence is said to be a path of length joining to in if and for each .

Next, we state some preliminary results which will be helpful to develop our main results.

Ahmadullah et al. [17] proved the results in metric-like space as well as partial metric spaces equipped with a binary relation as follows:

Theorem 25 (see [17]). Let be a metric-like spaces equipped with a binary relation defined on and a self-mapping on . Suppose that the following conditions are satisfied: (a)There exists a subset with such that is -complete(b)There exists such that (c) is -closed(d)Either is -continuous-like or is -self-closed(e)There exists a constant such that ( with )Then, has a fixed point. Moreover, if (f) is nonempty, for each . Then, has a unique fixed point

Radha and Singh [24] proved the following theorem in -metric spaces with ternary relation as follows:

Theorem 26 (see [24]). Let be a complete -metric space, a ternary relation on , and a self-mapping on . Suppose that the following conditions hold: (i) is nonempty(ii) is -closed(iii)Either is continuous or is -self-closed(iv)There exists a constant such that ( with )Then, has a fixed point. Moreover, if (v) is nonempty, for each . Then, has a unique fixed point

3. Main Results

In this section, we prove the following theorem which is a generalization and improvement of Theorem 26.

Theorem 27. Let be a complete -metric space equipped with a ternary relation defined on . Let and be a pair of hybrid multivalued mapping on . Suppose that the following conditions hold: (i)There exists such that is -complete(ii)There exists such that (iii) is nonempty(iv)For even and converge to a common fixed point(v)Either is -continuous- or is -self-closed(vi)There exists a constant such that ( with )where Then, and have a common fixed point. Moreover, if (vii) is nonempty, for each . Then, and have a unique common fixed point if and are coincidence points of and such that

Proof. Suppose that , is a -complete subspace of and is -complete, for with . We can construct a --sequence such that with initial point satisfying , such that , .
By (iii), let be an arbitrary element of , then . If , then is a common fixed point of and and proof is completed. On contrary, if , then . Now, we choose such that . Again, we can choose such that . Proceeding the same way and applying Definition 16, we can construct a sequence , such that which is equivalently to By substituting and in (18), we have where Applying in (24), we obtain By , we have Using (26) in (25), we get From (23), (27), and Lemma 11, we get Similarly, for , there exists such that Letting and (30), we have Continuing this process, we obtain by mathematical induction a sequence with such that It follows that By taking limits on both sides in (33), we obtain Using (32) and -symmetric properties, for all with , we obtain where and . This shows that the sequence is a -Cauchy sequence in the complete subspace .
The results obey a -Cauchy sequence properties of completeness. Hence, is -preserving Cauchy sequence. If the pair is closed and -compatible, using Definition 23, we have We can show that for , even and converge to a common fixed point of and . To see this, let , using (18) which gives where which implies that By using Lemma 10 and (39), we obtain Since is a complete -metric space and is closed and -compatible, from Definition 23, we get Assume that is a common fixed point of and such that . Thus, . So is a common fixed point of and .
Now we show that is the unique common fixed point of and . For the uniqueness, take as a common fixed point of and . Assume that and , and using in (3.1), we get where Consequently, Using Lemma 10 and (44), we get Hence, we get and we conclude that , which is a contradiction. Therefore, is a unique common fixed point of and .
Using the condition considered in Theorem 27, we prove assertion (vii) as follows: we observe that is nonempty, so let us take a pair of elements say in such that Next, we are required to show that . By observing the above assertion, there exists an -path (say, ) of length in from to , with such that for all .
And for every .
Define two constant sequences such that By using (49), for all , we have By usual substitution for for each , that is, Recall that and Lemma 12. Thus, we construct a sequence In general, We obtain for all . Corresponding to each , we have from (49), (50), and -compactness of , and we get for each .
Thus, is -closed, and we conclude that , for each and for all .
Otherwise, , for each and for all .
Define , for each and for all . We assert that . Assume that .
Since , either or .
If , for and , then applying the condition (vi), we have where By Lemma 10 and (60), we get Taking lim as and using , we get which is contradiction, and hence, The same for symmetric property , if , we have for .
Using (50), , and (), we have so that Therefore, Now, suppose that and , we claim that for the existence of unique fixed point . To see this, we have From we have which is a contradiction. Since , thus is a unique common fixed point of and , and we have . Thus, the proof is completed.

From Theorem 27, we can deduce the corollary as follows: (i) is closed in (ii)There exists a continuous comparison function such thatwhere

Corollary 28. Let be a -metric space and let and be pair of hybrid mapping such that the following conditions are hold:

Then, and have the unique common fixed point.

Proof. The proof of the above corollary follows similar steps of Theorem 27. Therefore, the proof is completed.

The following example is motivated from [31] for ternary relation.

Example 2. Consider , a ternary relation on defined by the equation endowed with complete -metric space, defined by -metric in with ternary relation. Since for , we need only to find solutions for and when . This yields the following:

We claim that is either -complete or -complete. To observe this, define by and by

Then, (i) is a -complete subspace in (ii)There exists a constant such that ( with )where

By using (i), we show that is a -complete subspace in . By Definition 5, we have

Using (79) and (79) in (3), we get

To prove (ii), let . If , then and . The proof is completed. On contrary, suppose that the values of are not all zero.

For , we have

By (4), we have

Since , then , using (5), and as a result,

Now for each and in (5) and (6), we have

Next, for each and in (5) and (6), we get

Likewise, for each and (5) and (6), it gives

Consequently,

On the other hand, we formulate the following -metrics. By (5) and (6), this yields to

In similar manner, we can calculate the following -metrics.

Applying the above equality, we obtain

From (88) and (91), it follows that

This shows that all and conditions imposed in Theorem 27 are satisfied. Hence, a pair of hybrid mapping and in -metric space has a unique common fixed point. Thus, is a unique common fixed point of and . Clearly, is -closed, and .

4. An Application to SODE in -Metric Space

Next, to validate the result of Theorem 27, we apply the second-order differential equation (SODE) by transforming it to the system of integral equations. The fixed point theory is involved in physics applications, specifically in the solution of electric equations, and this type of work is motivated from [22, 32, 33]. It is well known that an electric circuit can be represented by ternary relation, a resistor, an inductor, and a capacitor on an electromotive force in series. On the other hand, they can be presented as ternary relations with an inductive reactance, a capacitive reactance, and an impedance. If the rate of charge in condenser with respect to time is denoted by current , that is amount to say . We get the following ternary relations:

By Kirchhoff’s law, the sum of voltage drops across the circuit is equal to the supplied voltage. We consider the -circuit, and the differential equation for the charge on the condenser is where is the applied voltage at time . If the initial conditions are assumed to be

The differential equation (95) can be represented using the Green function given by

Consider the following systems of equations motivated by [21]. for all , where and . Let be a ternary relation on and on , set of all continuous mappings from . Consider the ternary relation on as for all . Now we define for all .

Let be the set of all continuous function defined on . Define a -metric on , by and

Then, is a -complete metric space.

Theorem 29. Suppose the following conditions holds: (i) and are continuous(ii)There exists some and such thatfor all . (iii)There exists some a function , for each or and with such thatwhere and (iv)There exists such thatwhere .

Then, the system of equations (95) has a common solution which is a solution of the integral equation (98).

Proof. For with , we claim that . Then, we have which is a contradiction. Hence, is a common fixed of and and also a solution to integral equation (98) and a second-order differential equation (95). Hence, we can conclude that all the conditions imposed in Theorems 27 and 29 are satisfied. Therefore, the proof is completed.

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Conflicts of Interest

The author declares no competing interests.

Authors’ Contributions

The author contributed fully and significantly in writing this article. The author read and approved the final manuscript.