Abstract

By combining the notions of -metric space and -metric space, in this paper, we present coincidence fixed-point theorems for -hybrid mappings in -metric spaces. An example is given to demonstrate the novelty of our main results. Henceforth, the illustrative applications are given by using nonlinear fractional differential equations.

1. Introduction

In 1922, Banach [1] initiated the study of constructive theory in metric space. The constructive theory is used for nonlinear functional analysis, approximation theory, optimization theory (saddle function), variation inequalities, game theory (Nash equilibrium), and economics (Black-Scholes theorem). In addition, it is used in many practical and research problems in various fields beyond mathematics. It can reduce to fixed-point problems, which include biology, chemistry, physics, computer science, economics, engineering, global analysis statistics, and operations research.

In 1969, Nadler [2] proved the multivalued version of Banach’s contraction principle [1]. Naimpally et al. [3] generalized Goebel’s [4] result in a hybrid contraction mapping. The method of hybrid fixed points can be used to derive another classical fixed-point theorem result. The concept of hybrid pair of mapping is very consequential for the theory of fixed point, and it has an important role in game theory, optimization theory, and differential equations.

Definition 1 (see [3]). Let and be such that is a complete subspace of and . Further, assume there exists , such that for every , Then, and have a coincidence; that is, there exists such that .

Later, Chauhan et al. [5] proved the results in unified common fixed-point theorems for a hybrid pair of mappings via an implicit relation involving altering distance function. Imdad et al. [6, 7] generalized the hybrid fixed-point theorems in symmetric spaces via common limit range property and joint common limit range property in metric spaces. Nashine et al. [8] gave the proof using (JCLR) property for hybrid fixed-point theorems via quasi -contractions. Wangwe and Kumar [9, 10] proved the common fixed-point theorem for a hybrid pair of mappings in weak partial -metric spaces and -metric space with some applications.

Bakhtin [11] and Czerwik [12] generalized the concept of metric space to -metric space due to some problems, especially the convergence issue of measurable functions to a measure that led to a generalization of a metric’s notion. Czerwik [12] established -metric spaces by introducing a parameter in the triangle inequality as a coefficient and generalized Banach’s contraction principle to these spaces. Later, Czerwik [13] proved the multivalued results in -metric spaces. These findings motivated several potential researchers to perform and analyze contraction condition variants using single- and multivalued maps in -metric space. One can refer to [1418] and the references therein.

The concepts of -metric space were initiated by Mustafa and Sims [19, 20] due to the shortcoming of the fundamental topological structure on -metric spaces. Also, they replaced the tetrahedral inequality with an inequality involving the repetition of indices. Further, several researchers generalized the results for single-valued mapping and multivalued mappings in -metric spaces. For more results, we refer the reader in [2125] and the references contained. Furthermore, Aghajani et al. [26] using both concepts of -metrics and -metrics initiated the results on -metric spaces. Since then, several results followed for single- and multivalued mappings for various abstract spaces. For more literature, we refer the reader to [2732] and the references contained.

This paper is aimed at proving a coincidence fixed-point theorem for -hybrid contraction mappings in -metric space with some application to the fractional differential equation. In particular, we modify and extend the works due to Karapinar et al. [33, 34], Wangwe and Kumar [10], and Aghajani et al. [26]. The results proved to have a novelty in the study of fixed-point theory.

2. Preliminaries

This part introduces some preliminary results of definitions and theorems, which will help develop the main result.

Bakhtin [11] and Czerwik [12] established a new metric on a nonempty set, known as a -metric.

Definition 2 (see [12]). Let be a nonempty set and be a given real number. Suppose that a function satisfies
(B1) iff
(B2)
(B3) , for all
Then, is said to be -metric and is a -metric space. is defined as a parameter of .

We give some examples which satisfy -metric space axioms.

Example 1 (see [11, 12]). Consider the set endowed with the function defined by for all . Thus, is a -metric space for .

Mustafa and Sims [20] gave the following axioms for -metric space.

Definition 3 (see [20]). Let represent a nonempty set with as a function which satisfies the following axioms:
(G1) for
(G2) if
(G3) if
(G4)
(G5) ,
Then, is called a metric and is a -metric space.

We give an example from [20].

Example 2 (see [20]). Let represent a set of real numbers. Define as follows: for all .
Recall that if , then .

Example 3 (see [20]). Let . Then, -metric is defined by for all .

On the other hand, Aghajani et al. combining the concepts from Bakhtin [11], Czerwik [12] and Mustafa and Sims [20], they established a new generalized space known as -metric space.

Definition 4 (see [26]). Let be a nonempty set and . Suppose that is a function satisfying the following conditions:
if
, for all with
for all with
(symmetry in all three variables)
, for all (rectangle inequality)

The distance metric is called a -metric, and is called a -metric space. The real number is called the coefficient of .

Let be the set of real numbers; then, the mapping is defined by for all , is a -metric.

Example 4 (see [26]). Let be a -metric space. Consider for all and . Therefore, is a -metric with .

Example 5 (see [26]). Let be a -metric space. Then, for , the -ball with center and radius is

For example, , and consider a -metric defined by for all . Then,

Aghajani et al. [26] gave the proposition below that satisfies -metric axioms.

Proposition 5 (see [26]). Let be a -metric space, ; we have the following: (i)If , then (ii)(iii)(iv)

Motivated by Aghajani et al. [26], we recall some properties in -metric spaces as follows.

Definition 6 (see [26]). Let be a -metric space. If then, we have the following: (i) is convergence to a point if, for each , there exists a positive integer such that, for all , (ii) is a Cauchy sequence if, for , a positive integer such that, for all ,

Proposition 7 (see [26, 27]). Let be a -metric space. Then, the function is given by for all . Define a -metric on . It is called a -metric induced by the -metric .

Proposition 8 (see [27]). Let be a -metric space. The properties below are similar: (i) is -convergence to (ii)(iii)(iv)

Proposition 9 (see [27]). Let be a -metric space. Therefore, the properties below hold: (i) is a -Cauchy sequence(ii) , there exists such that, ,

Definition 10 (see [26]). A -metric space is called -complete if every Cauchy sequence is a -convergent in .

Further, Makran et al. [29] extended the works due to Aghajani et al. [26] and Kaewcharoen and Kaewkhao [22] by introducing the multivalued versions in -metric spaces as follows:

Let be a -metric space. We shall denote as a nonempty, closed, and bounded subsets of . Let represent the Haursdorff--metric on , and define where

Lemma 11 (see [29]). Let be a -metric space with and . Then, for each , we have

Lemma 12 (see [29]). Let be a -metric space with . If and , then for each , there exists such that

Lemma 13 (see [35]). Let be a -metric space with , and suppose that and are -convergent to , , and , respectively. Then, we have

In particular, if , then we have

Definition 14 (see [22]). Let be a nonempty set. Assume and are two mappings. If for some , then is a coincidence point of two mapping . Then, the coincidence point of and is . On the other hand, the mappings and are said to be weakly compatible if for some consequently .

Proposition 15 (see [22]). Let be a nonempty set. Assume and are said to be weakly compatible mappings. If is a unique coincidence of and , is said to be a unique common fixed point of and .

Definition 16 (see [36]). Let be the two self-mappings on an ordered metric space with . For every , consider the sequence defined by , . A sequence is a - sequence starting at .

Wangwe and Kumar [10] gave the following definition and theorem.

Definition 17 (see [10]). Let be a -metric space and let be two hybrid mapping on this space for and , such that . We define the following expression:

Theorem 18 (see [10]). Let be a -metric space, and suppose is a -hybrid mapping with almost altering distance satisfying the following conditions: (a) and are weakly compatible(b) and satisfy property(c) (d) is a -complete subspace of (e), for all and Therefore, and admit a unique common fixed point in .

3. Main Results

We commence this section by extending Definition 17 to -metric space setting.

Definition 19. Let be a -metric space, be a pair of hybrid mapping, and with and , such that . Then, we define the following expression: for

Now, we are equipped to prove the following theorem.

Theorem 20. Let be a complete -metric space, and suppose and are -hybrid contraction mapping on and satisfying the following conditions: (i) , since is complete(ii) and are weakly compatible(iii) is a -complete subspace of (iv)For , and converge to a common fixed point(v) a constant , , and such that ; we have Then, and pose a unique coincidence fixed point.

Proof. Assume that and is a -complete subspace of . We can construct a --sequence such that with initial point satisfying , such that , .
Let be an arbitrary element in . If , then is a coincidence point of and . Therefore, our proof is completed. Otherwise, for , it implies that . Now, we choose such that . Again, we can choose such that . By repeating the above procedure and applying Definition 16, we formulate a sequence , such that Equivalently, Using Lemmas 11 and 12, we obtain that there exists Consequently, we have Apply , , and in (20); we get where By (), we have Applying (29) in (28), we obtain Using (9) and (31) in (26), we get Let ; we have By repeating the above procedure, we construct a sequence such that . In order to simplify the above equation, let . Thus, by (34), we have for all .
By taking limits as in (35), we obtain Therefore, converges.
Using (34) and -symmetric properties, for all with , we obtain On the other hand, using (35), we obtain where . This proves that the sequence satisfies the -Cauchy sequence conditions on complete subspace . Henceforth, is a Cauchy sequence.
Let be closed and weakly compatible mappings. From Definition 14, we have Now, we find that such that . We will show that . For each , using (20), it follows that where Taking the limit as in (40) and (41) with , we obtain This shows that . That is, and have a point of coincidence.
Next, we prove the uniqueness of the point of coincidence of and . Let and . Assume that . Using (20), we get where From the symmetric properties, using (43) and (44) with , we obtain We can conclude that . Suppose that and are weakly compatible. By applying Proposition 15 and Definition 14, we obtain that and have a unique coincidence fixed point, which is a contradiction. Thus, is a coincidence point of and , for . The proof is completed.

Inspired by the idea of Theorem 20, we can deduce the corollary as follows:

Corollary 21. Let be a -metric space, and let and be a -hybrid mapping which satisfies the following hypotheses: (i) (ii) a continuous function and such that where Then, and have the unique coincidence fixed point.

Proof. We prove the above corollary by following similar steps of Theorem 20. Therefore, the proof is completed.

Next, we demonstrate with an example for Theorem 20.

Example 6. Let be endowed with the usual ordering on and -metric on be given by , where and with , , and .

Consider to be -complete. Define a self-map as by and by

Therefore, (i) is a -complete subspace in (ii)for in (20), we have where for all

Applying (i), we prove that is a -complete subspace in . By Proposition 7, we have

By (52) and ((53)) in (9), we obtain

From (ii), assume that . If , then and . The proof is completed. Otherwise, we suppose that the value of are not all zero.

For , we get

By (10), (55) is equivalent to

Since , then , using (11) yields

Now, for each and in (11) and (12), we have

Next, for each and in (11) and (12), we get