#### Abstract

In this paper, we present a common fixed point result for a pair of mappings defined on a b-metric space, which satisfies quasi-contractive inequality with nonlinear comparison functions. An application in solving a class of integral equations will support our results.

#### 1. Introduction

In 1974, Ćirić presented the first fixed point result for quasi-contractive mappings. This Ćirić’s theorem is one of the most general results with linear comparison function in classical metrical fixed point theory (see [1, 2]). The existence and uniqueness of fixed point for mappings defined on metric spaces, which satisfies a quasi-contractive inequality with a nonlinear comparison function, were considered by Danes [3], Ivanov [4], Aranđelović et al. [5], and Bessenyei [6]. Alshehri et al. [7] proved a fixed point theorem for quasi-contractive mappings, defined by linear quasi-contractive conditions on b-metric spaces. Common fixed point generalizations of Ćirić result was obtained by Das and Naik [8], with linear comparison functions and by Di Bari and Vetro [9], with a nonlinear comparison function.

The notion of symmetric spaces, which is the oldest and one of the most important generalizations of metric spaces, was introduced by Fréchet [10]. He used the name E-space for a symmetric space. In the last 50 years, many authors (see [11–16]) called them semimetric (in German *halb-metrisher*) spaces. Now, the term symmetric space is usual. After 1955, the term semimetric space is widely used to denote a symmetric space in which the closure operator is idempotent, which started the papers of Heath, Brown, Mc Auley, Jones, and Burke (see [17, 18]). Fixed point investigation was started by Cicchese [19] and Jachymski et al. [18] on semimetric spaces and by Hicks and Rhoades [20] on symmetric spaces.

In [10], Fréchet also considered the class of E-spaces with regular écart which include the class of -metric spaces. Important examples of -metric spaces are quasi-normed spaces introduced by Bourgin [21] and Hyers [22] and spaces of homogeneous type which have many applications in the theory of analytic functions (see Coifman and Weiss [23]). First, fixed point results on -metric spaces were presented by Bakhtin [24] and Czerwik [25].

In this paper, we present a common fixed point result for a pair of mappings defined on a -metric space, which satisfies a quasi-contractive inequality with a nonlinear comparison function.

#### 2. Symmetric Spaces and -Metric Spaces

The ordered pair , where is a nonempty set and , is a symmetric space, if and only if it satisfies: (W1) if and only if (W2) for any

The difference between symmetric spaces and more convenient metric spaces is in the absence of triangle inequality, but many notions in symmetric spaces are defined similar to those in metric spaces. For instance, in symmetric space , the limit point of a sequence is defined by

Also, we say that a sequence is a Cauchy sequence, if for any given , there exists a positive integer such that for every . If each Cauchy sequence in symmetric space is convergent, then we say that is a complete symmetric space.

By we indicate the diameter of the set .

Let be a symmetric space. We can introduce the topology by defining the family of all closed sets as follows: a set is closed if and only if for each , implies , where

The convergence of a sequence in the topology need not imply , but the converse is true.

Let be a symmetric space, and . We considered the following seven properties as partial replacements for the triangle inequality:

(W3)

(W4)

(HE)

(CC)

(W)

(JMS) ;

(SC) implies

The property (W3) has been introduced by Fréchet [10]; (W4), (HE), and (W) by Pitcher and Chittenden [14]; (CC) by Sims [15]; (JMS) by Jachymski et al. [18]; and (SC) by Aranđelović and Kečkić [17]. Note that , , , , and (see [17, 26]).

In [26], the authors give examples for the following relationships: , , , , , , , , , and . The fact that has been proved in [27].

*Definition 1. *Let be a nonempty set, . is said to be a *-*metric space if there exists such that
(1) if and only if (2) for any (3)for all .Any which satisfies inequality (3) of Definition 1 for all , where is a -metric space, is said to be the constant of space . It is clear that if , then is a metric space.

Lemma 2. *Let be a -metric space with constant . Then, .*

*Proof. *Let . Then, , which implies that .

In [17], the following result was proved.

Lemma 3 (see [17]). *Let be a -metric space. Then, is a symmetric space which satisfies the properties (W3), (W4), (HE), (W), and (JMS).*

#### 3. Comparison Functions

Let be a function such that if and only if . Define: (1) if and only if for each (2) if and only if for each (3) if and only if for any (4) if and only if for any (5) if and only if for all (6) if and only if (7) if and only if is a strictly increasing surjection(8) if and only if is bounded for every (9) if and only if is monotone nondecreasing

If , then we say that is a comparison function.

If , then is continuous from the right on .

If , then is upper semicontinuous on .

The class of has been applied in the theory of nonlinear quasi-contractions by Danes [18], by Ivanov [4], by Aranđelović et al. [5] and Di Bari and Vetro [9], by Aranđelović et al. [5], and the class of by Bessenyei [6].

Note that , , and . Some further inclusion between different classes of comparison functions will be presented in the next statements.

Proposition 4. *If , then .*

*Proof. *For any , from , we get that , because is monotone nondecreasing. So, we obtain that , for every .

Proposition 5. *.*

*Proof. *Let . If there exists such that is unbounded; then, for every , there exists such that . So,
which is a contradiction with .

Let . Suppose that there exists an increasing sequence such that and such that , for each . Hence, . So, is unbounded which implies that .

Proposition 6. *If , then .*

*Proof. *Let . Suppose that there exists a strictly increasing sequence such that and such that , for each positive integer . So, for any , we have , because there exists such that , which implies that is not a surjection. Hence, .

Two following two lemmas have been proved in [5].

Lemma 7. *Let . Then, there exists such that
for each .*

Lemma 8. *Let . Then, there exists such that
for each and .*

#### 4. Main Results

First, recall some standard terminology and notations from the fixed point theory.

Let be a nonempty set, and let be an arbitrary mapping.

Let and be nonempty sets, , and . Choose a point such that . Continuing this process, having , we obtain such that . is called a Jungck sequence with an initial point . Note that a Jungck sequence might not be determined by its initial point .

Let be a nonempty set and . and are called weakly compatible if they commute at their coincidence points.

Lemma 9 (see [28]). *Let be a nonempty set and let be weakly compatible self mappings. If and have a unique point of coincidence , then is the unique common fixed point of and .**Now, we present our main result. Before stating the result, we make a convention to abbreviate and in order to avoid too much parenthesis.*

Theorem 10. *Let be a -metric space with constant and let be two mappings. Suppose that the range of contains the range of and that is a complete subspace of . If there exist such that
for any , then there exists which is the limit of every Jungck sequence defined by and . Further, is the unique point of coincidence of and . Moreover, if and , are weakly compatible, then is the unique common fixed point for and .*

*Proof. *We shall, first, reduce the statement to the case and . Indeed, from Lemma 7, it follows that there exist functions such that and
for each and for all , whereas from Lemma 8, it follows that there exists a real function such that and
which implies
Thus, we can assume that for all and .

Let be arbitrary and let be an arbitrary sequence such that is a Jungck sequence with an initial point .

Let . We will prove that there exists a real number such that:
Consider the set which is nonempty, since as . Also, if and imply , and hence, is an unbounded interval. Set . For each positive integer , there is such that , and therefore, there is such that . Since is nondecreasing, we have which implies that . Taking the limit as , we get .

For any , define and . Also, let denote the diameter of .

Next, we prove that
for all positive integer .

Since is nondecreasing, it commutes with , and for all , we have
By induction, from (13), we obtain that
For , we have , and hence, by (13)
Therefore, there is such that
Hence, we get
which implies that , and hence
Hence, all Jungck sequences defined by and are bounded.

Now, we shall prove that our Jungck sequence is a Cauchy sequence. Let be positive integers. Then, . Using (15) (with ) and (19), we get
as . Since , and is complete, it follows that is convergent. Let be its limit.

Clearly, . So, there is such that . Let us prove that is also equal to . By (8), we have
If , then the left-hand side in the previous inequality tends to , and the first, the second, and the fifth argument of tend to , whereas the third and the fourth tend to . Thus, we have
which is impossible, unless .

Finally, we prove that the point of coincidence is unique. Suppose that there is two points of coincidence and obtained by and , i.e., and . Then, by (8) we have
unless . Since every Jungck sequence converges to some point of coincidence, and the point of coincidence is unique, it follows that all Jungck sequences converge to the same limit.

Let and let , be weakly compatible. By Lemma 3, we get that which is the unique common fixed point of and .

The previous theorem extended earlier results for nonlinear contractions on metric spaces obtained by Danes [3], Ivanov [4], Aranđelović et al. [5], and Bessenyei [6] and common fixed point results of Das and Naik [8] and Di Bari and Vetro [9]. It also generalizes the fixed point theorem of Aleksić et al. [7] which proved the fixed point theorems for quasi-contractive mappings on -metric spaces, defined by linear quasi-contractive conditions.

*Example 1. *Let be equipped with the following -metric by

It is easy to see that is a complete -metric space with .

Define the self-maps and by
We see that .

Define by . One can easily check that satisfies condition (8). Indeed, we have some cases as follows:
(1) Then,
(2) Then,
(3) Then,
Thus, all the conditions of Theorem 10 are satisfied, and hence, and have a common fixed point. Indeed, is the unique common fixed point of and .

#### 5. Application

The existence of the solution for the following integral equation is the main purpose in this section.

where

We will ensure such an existence by applying Theorem 10.

Let be the space of all real, bounded and continuous functions on the interval . We endow it with the -metric where

Theorem 11. *Suppose that the following assumptions are satisfied:
*(i)* are continuous functions so that
*(ii)*The function is continuous so that
*(iii)*For all and where is continuous.*(iv)* and **Then, the integral equation (28) admits at least one solution in the space .*

*Proof. *Let us consider the operator defined by
In view of the given assumptions, we infer that the function is continuous for arbitrarily . Now, we show that is bounded in . As
we have
Thus,
From the above calculations, we have
Due to the above inequality, the function is bounded.

Now, we show that satisfies all the conditions of Theorem 10. Let be some elements of . Then, we have
where is defined by
Thus, we obtain that
Using Theorem 10, we obtain that the operator admits a fixed point. Thus, the functional integral equation (28) admits at least one solution in .

#### Data Availability

No data were used to support this study.

#### Conflicts of Interest

The author declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

#### Authors’ Contributions

All authors read and approved the final manuscript.

#### Acknowledgments

The second author was supported by the Ministry of Education, Science and Technological Development of Serbia, Grant no. 451-03-68 / 2020-14 / 200105.