#### Abstract

Motivated by the ideas of -weak contractions and -contractions, the notion of -contractions is introduced and studied in the present paper. The idea is to establish some interesting results for the existence and uniqueness of a coincidence point for these contractions. Further, using an additional condition of weakly compatible mappings, a common fixed-point theorem and a fixed-point result are proved for -contractions in metric spaces equipped with a transitive binary relation. The results are elaborated by illustrative examples. Some consequences of these results are also deduced in ordered metric spaces and metric spaces endowed with graph. Finally, as an application, the existence of the solution of certain Voltera type integral equations is investigated.

#### 1. Introduction and Preliminaries

In the development of the metric fixed-point theory, one of the main pillar is the Banach contraction principle [1], which states that every contraction on a complete metric space has a unique fixed point. Due to its extensive application potential, this concept has been observed in various forms over the years (see [2–9]).

The concept of -contractions was introduced by Wardowski [10]. He proved some new fixed-point results for such kind of contractions. He built these results in a different way rather than traditional ways as done by many authors. Later on, fixed points for -contractions were proved by Secelean [11] using an iterated function. Abbas et al. [12] extended the work of Wardowski and established various results of fixed points using -contraction mappings. For further related works on -contractions, see [13–16].

The idea of -contractions was established by Sawangsup et al. [17]. They used this idea to demonstrate some fixed-point consequences using a binary relation. It is further investigated by Imdad et al. [18]. In present paper, we study the results presented by Alfaqih et al. [19] and we define -contractions. We also prove similar results for -contractions.

Recall that a binary relation on nonempty set is said to be a partial order if it is reflexive, antisymmetric, and transitive. Moreover, the inverse or transpose or dual relation of , denoted by , is defined by

The symmetric closure of , denoted by , is defined as the set , that is, In fact, is the smallest symmetric relation on containing .

Notice that there is another binary relation on , which is defined by , whenever and .

*Definition 1 [10]. *Let be the set of functions such that

() is strictly increasing;

() For every sequence , ;

() There is so that

The following functions are in :

Many papers in literature deal with the concept of -contractions (see [20–22]). Throughout this work, the set of all continuous functions verifying is denoted by .

*Definition 2. *Let and be a binary relation on . A sequence is such that , then it is called an preserving sequence.

*Definition 3. *Consider a metric space with a binary relation . Then, is called complete if each preserving Cauchy sequence is convergent in .

*Definition 4 [23]. *Let be a metric space and be a binary relation on , and . We say that is -continuous at if for each -preserving sequence so that , we have . Also, is named to be -continuous if it is -continuous at any element of .

*Definition 5 [23]. *Let be a metric space and be a binary relation on and and . We say that is -continuous at if for each sequence so that is -preserving and , we have . Also, is named to be -continuous if it is -continuous at any element of .

*Definition 6 [24]. *For , a path of length in from to is a finite sequence such that , and for every . Also, a subset is called connected if for any two elements , there is a path from to in .

*Definition 7 [23]. *Let be a metric space and be a binary relation on and . The pair is -compatible if for each sequence so that and are -preserving and ,

*Definition 8. *Let and be self-maps of a set . If for some , then is said to a common fixed point of and .

*Definition 9 [25]. *Let . If for some , then is said to be a coincidence point of and , and is said to be a point of coincidence of and .

and are said to be weakly compatible if they commute at their coincidence point, i.e., if for some , then .

*Definition 10 [26]. *Let be a metric space endowed with a binary relation . Such a is named to be -self closed if for each -preserving sequence so that , there is of so that .

*Definition 11 [23]. *Let be a nonempty set and . A binary relation on is called closed if for any , yields that .

Lemma 12 [27, 28]. *Consider a metric space and a sequence in X. If is not Cauchy in , then are and and of so that
*

*Moreover, if is so that , then*

Lemma 13 [29]. *Let be a nonempty set and . Then, there is a subset so that and is one to one.*

#### 2. Main Results

We begin this section by introducing the idea of -contractions as follows.

*Definition 14. *Consider a metric space endowed with a transitive binary relation on and . Then, is called an -contractions if there exist and such that
for all with and .

*Remark 15. *Every contraction is an contraction, but the converse of statement is not true.

The following result is easy to prove. We omit it.

Proposition 16. *Let be a metric space endowed with a transitive binary relation . Given . Then, for each , we have equivalence of the two following statements:
*(a)* so that and *(b)* such that either , or , *

Theorem 17. *Consider a metric space equipped with (where is a transitive binary relation) and . Assume that:
*(1)*there exists such that *(2)* is -closed*(3)* is an -contraction*(4)*(a) A subset of exists such that and is -complete(b)One of the subsequent conditions is fulfilled:
(i) is -continuous, or(ii) and are continuous, or(iii) is -self closed in condition that (6) holds for all with and *

*or on the other hand:*

*() (*

_{1}) a subset of such that and is - complete,*(*

_{2}) is an -compatible pair,*(*

_{3}) and are -continuous.*Then, the pair admits a coincidence point.*

*Proof. *In the above two cases (11) and , note that . Using assumption (6), we get . If , then a coincidence point of is . This completes the proof. Suppose that . Since , there must exist such that . Similarly, there is such that . Proceeding in this way, we can construct a sequence such that
Now, we will prove is an -preserving sequence, that is,
By using induction, we will prove this claim. If we put in (9) and use condition (6), we get . This implies that the above statement holds for . Suppose that (10) is accurate for , that is, . Since is -closed, we get , and so .

Hence, our claim is true for all . By using (9) and (10), we can conclude that is also an -preserving sequence, that is,
If for some , then is a coincidence point of .

Suppose on the contrary that for al . With the help of (9), (10), (11), and condition (10), we can see that
Now, cannot be . Otherwise,
which is a contradiction. Hence, Therefore,
Take . With the help of above condition, we obtain
By using and taking in above inequality, we obtain
This together with imply that
Now, we will show that is a Cauchy sequence. We argue by contradiction. In this case, Lemma 12 guarantees the existence of and two subsequences and of such that
with
This implies that there is so that .

Since is transitive, one writes
Using condition (10), we have for all ,
Denote
If or it is equal to then taking and using (20), we get
Since is continuous, letting in (22) and using (20) and (24), we get
which is a contradiction. On the other hand, if or it is equal to then letting in (22), using continuity of and (20) together with condition , we get which is again a contradiction. Thus, is a Cauchy sequence.

Let the condition (11) hold. With the help of (9), we obtain . Therefore, is -preserving Cauchy in . By utilizing -completeness of , there is so that . As , there is so that . Hence, by using (2),
In order to prove that is coincidence point of , we will use three different cases of condition . First of all, suppose that is -continuous. By utilizing (10) and (26), we get
By utilizing (26) and (27), we get This shows that is a coincidence point of .

Now, suppose the second case of , that is, and are continuous. Since and , by using Lemma 13, there is so that and is one-one. Define a mapping by
Recall that is one-one and , so is well-defined mapping. As and are continuous, is also continuous. Now, utilizing the fact that , we can rewrite condition as , so that, without loss of generality, we can select a sequence in and . By using (26), (28), and continuity of , we have
Finally, assume that condition (iii) of (b) holds, which implies that is -self closed and (2.1) detain , with and . As , is preserving due to (10) and with the help of (26) . So, there is a subsequence such that
Utilizing condition (b) and (30), one writes
Now, let . If the set is infinite, then has a subsequence , such that . This implies that . By using (26), we have . So we obtain .

If the set is finite, then has a subsequence such that . Next, we will show that . With the help of (30), (31) and , we have
Now, with the help of (32), (33), Proposition 16 and the fact that (2.1) is satisfied, we get
Denote
If then, we have
By using (26), (*F*_{2}) and taking , we get If then, we have
By using (26), (*F*_{2}) and taking , we get Now, if then
By using (26), () and taking , we get If then, we have
By using (26), () and taking , we get
From (26) and (40), we obtain Hence, when the set is finite or infinite, is a coincidence point of and . Now, if holds, then , and hence is an -preserving Cauchy sequence in . Since is -complete, there is so that
Using Equations (9) and (41), one gets
Now, with the help of (10), (41), and continuity of , we have
Utilizing (11), (42) and continuity of to find
As and are -preserving due to (10), (11) and
Now, using (41), (42), and condition (),
Next, we will demonstrate that is a coincidence point of . Making use of (10), (41) and the -continuity of , we get
With the use of (44), (46), and (47), we get
This implies that is a coincidence point of .☐

Theorem 17 does not guarantee the uniqueness of a coincidence point. The following theorem guarantees that coincidence point is unique.

Theorem 18. *Suppose all hypothesis of Theorem 17 are true except and assume that and are -comparable for all coin , and one of or is one-one, then there is a unique coincidence point of .*

*Proof. *The set is nonempty, because of Theorem 17. Consider two elements , then by definition of , we have and , . This implies .

Now, if , we obtain , and hence, , because one of and is one-one.

If , then by utilizing condition (10) and Proposition 16, we get
Since , our assumption is false. Therefore, a unique coincidence point of exists.☐

Theorem 19. *Consider above theorem and add a condition that is a weakly compatible pair, then a unique common fixed point of exists.*

*Proof. *Above theorem assures that the pair has a unique coincidence point. Let be the common coincidence point and suppose be such that
The weak compatibility of and leads to . That is, is a coincidence point of and . Since is unique, one writes . That is, the uniqueness of a common fixed point. Since all the assumptions of Theorem 18 are true, the set is nonempty.☐

*Example 1. *Let and define *by*. Then, is a complete metric space.

Consider the sequence which is defined by

Define the binary relation on by

Define by and

Observe that if and , then and for Further, by choosing and , we have

Now, for and for , we have

Therefore,

Moreover, all the assumptions of Theorem 19 are true, and is the unique common fixed point of

On setting in Theorem 19, we obtain the following result.

Theorem 20. *Consider a self-mapping and let be a metric space with a transitive binary relation . Assume that:
*(1)* such that *(2)* is Q-closed*(3)* is an -contraction*(4)*() a subset of such that and is -complete,**() one of these conditions hold:
*(i)* is -continuous, or*(ii)* is -self closed on condition that (1.1) with binary relation holds with and **Then, a fixed point of exists. Furthermore, if**() **Then, such fixed point of is unique.*

Theorem 21. *Replace condition () of above theorem by:**() is -connected,**then has a unique fixed point.*

*Proof. *Assume on contrary that has more than one fixed point, say and with . Then, there exists a path . As it is from to of length , let us denote the path by such that for each where . If , it is a contradiction. Hence,
As , so for each . With the help of condition , we obtain
That is,
Since , our supposition is not true. Hence, has a unique fixed point.☐

In the next section, we are presenting a significance of our results in ordered metric spaces.

#### 3. Some Consequences in Ordered Metric Spaces

*Definition 22. *Let be a metric space and be an ordered set, then the triplet is known as an ordered metric space.

*Definition 23. *Consider self-mappings and an ordered set . If, for any , implies that . Then, is -increasing.

*Remark 24. *Notice that the notion of is -increasing is equal to say that is -closed.

Taking in Theorem 17 to 19 and with the help of Remark 24, we state the following result.

Corollary 25. *Consider self-mappings and an ordered metric space . Assume that:
*(a)* such that *(b)*-increasing*(c)*There are and so that
*(d)* a subset of such that and is -complete*(e)*Either and are continuous, or is -continuous. Then, a coincidence point of exists. Additionally, we suppose *(f)* and are -comparable for all distinct coincidence points , then pair has a unique coincidence point**Furthermore, if and are weakly compatible, then has a unique common fixed point.**Taking in Theorem 20 and with the help of Remark 24, we conclude the result given below.*

Corollary 26. *Consider an ordered metric space and mapping . Suppose the that conditions given below are fulfilled:
*(a)* such that *(b)* is -increasing*(c)* and such that
*(d)*A subset of exists such that and is -complete*(e)* is -continuous. Then a fixed point of exists. Furthermore,*(f)*if for any two fixed points we have , then has a unique fixed point*

#### 4. Applications to Metric Spaces Endowed with a Graph

Jachymski [30] in 2008 has instituted the idea of metric spaces endowed with a graph in order to generalize the idea of a partial ordering and specified the Banach contraction principle in metric spaces and partially ordered metric spaces. In this section, we are going to present an application of our results in the situating of complete metric spaces endowed with a graph.

Corollary 27. *Consider self-mappings on a metric space endowed with a graph . Define on as if and only if there is an edge between and . Assume that all the conditions given in Corollary 25 are satisfied. Then a coincidence point of exists. Further, if we suppose that and are comparable on edges for all distinct coincidence points , then the pair has a unique coincidence point.*

Furthermore, a unique common fixed point of exists if and are weakly compatible.

Corollary 28. *Consider a metric space endowed with a graph and a mapping . Define on as if and only if there is an edge between and . Suppose that conditions given in Corollary 27 are fulfilled. Then, a fixed point of exists. Furthermore, if are such that there is an edge between and , then a unique fixed point of exists.*

#### 5. Applications to Integral Equations

In this section, we present an application of Theorem 21 by finding a solution of the integral equation of Volterra type given below:

Here, and

Let be the Banach space of all continuous functions Define a norm on as follows.

. Then, the metric on is defined as