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1 Introduction

In 2015, Khojasteh et al. [16] introduced the notion of simulation functions and employ it to unify several fixed point results in the existence literature including Banach contraction principle. Thereafter, several authors studied and extended this notion enlarging such class of auxiliary functions. In this regard, in 2017, Roldán and Samet [10] bring in the concept of an extended simulation function and proved some fixed point results utilizing there extended notion. One year later, Liu et al. [19] obtained a new generalization of simulation functions using the class of C-function (the class of C-functions initiated by Ansari [2] in 2014) called -simulation functions. In [31], the author successively extended the fixed point results from the metric setting to the partial metric setting. In [29], the ordered approach was involved to fixed point results. In [30], the author used the fixed point result to solve a first-order periodic differential problem.

Very recently, Chanda et al. [6] bring in the notion of extended -simulation functions, which generalized several notions such as simulation functions, extended simula-tion functions and -simulation functions.

On the other hand, the notion of fixed point (a fixed point that belongs to the zero set of a given function was introduced by Jleli and Samet [12] to establish some φ-fixed point theorems on a metric space , which has been used to deduce some fixed point results on partial metric space.

For more details, we refer the reader to [3, 7, 8, 11, 13–15, 17, 18, 20, 25–28] and references cited therein.

Motivated by the above research work, in this paper, we use the idea of extended -simulation functions to study the existence and uniqueness of point of coincidence and common fixed point for two self-mappings defined on complete metric and partial metric spaces. The obtained results extend and generalize several results as shown in the following diagram:

2 Preliminaries

With a view to have a self-contained presentation, we collect the relevant background material (basic notions, definitions, and fundamental results) starting with the definition of simulation functions, which runs as follows.

Definition 1. (See [16].) A simulation function is a mapping satisfying the following conditions:

Roldan et al. [9] modified Definition 1 in order to enlarge the class of simulation functions by sharping the condition as follows:

are sequences in such that and , then .

Several examples of simulation functions can be found in [16]. Let us denote by the class of all simulation functions.

Roldán and Samet [10] extended the notion of simulation functions as under.

Definition 2. (See [10].) Afunction is said to be an extended simulation function if the following conditions hold:

Proposition 1. (See [10, Ex. 2.6].) Every simulation function is an extended simulation function, but the converse is not true in general.

For basic examples and more details about extended simulation functions, we refer the reader to [10]. The family of all extended simulation functions will be denoted by . Ansari [2] introduced the family of C-class functions as below.

Definition 3. (See [2].) A continuous function is said to be a C-class function if it satisfies the following conditions (for all ):

1. ,

2. implies that either or .

The family of all C-class functions will be denoted by C.

Definition 4.(See [19].) A function has a property if there exists a constant such that

(G1) implies ,

(G2) for all .

Liu et al. [19] defined -simulation functions as follows.

Definition 5. (See [19].) A functionis said to be a -simulation function if the following conditions are satisfied:

For basic examples of -simulation functions, we refer the reader to [19]. Let us denote by the family of all -simulation functions.

Chanda et al. [6] extended the notion of -simulation functions as under.

Definition 6. (See [6].) A function is said to be an extended -simulation function if the following conditions are hold:

Let us denote by the family of all extended -simulation functions.

Remark 1. Every simulation function, -simulation function, an extended simulation function is an extended -simulation function (see [6, Props. 3.3, 3.4 and 3.5]). The converse is not true in general (see Example 1).

In support of Remark 1, the following example is given in [6].

Example 1. Let be a function defined by

For all , , and let with . Then , but θ does not belong to , and .

In the present paper, is a nonempty set, and the following notions are used:

• Fix ,

• Pcoin ,

• Com ,

• , where is a given function}.

Now, we present the notion of fixed point, which runs as follows.

Definition 7. (See [12].) Let T be a self-mapping on and a given function. An element is said to be fixed point of T if and only if it is a fixed point of T and , that is, .

Let T and S be two self-mappings defined on .

• A sequence is called a Picard–Jungck sequence of T and S based on if for all .

• T and S are said to be weakly compatible if they are commute at their coincidence points, that is, for all such that .

Proposition 2. (See [1].) Let T and S be two weakly compatible self-mappings defined on If T and S have a unique point of coincidence u, then u is a unique common fixed point of T and S.

Let be the set of all functions satisfying the following conditions for all :

1. (F1),

2. (F2),

3. (F3) is continuous.

The following functions belong to:

1. 1. ,
2. 2. ,
3. 3.

3 Main results

At the beginning of this section, we define the notions of point of coincidence and common fixed point of the self-mappings T and S defined on a nonempty set .

Definition 8. Let S and T be two self-mapping on , and let be a given function. An element z in is said to be

• point of coincidence of T and S if and only if it is a point of coincidence of T and S and , that is, ;

• common fixed point of T and S if and only if it is a common fixed point of T and S such that that is, .

Now, we prove the following proposition.

Proposition 3. Let T and S be two weakly compatible self-mappings defined on . Suppose that T and S have unique point of coincidence u, then u is a unique common fixed point of T and S.

Proof. Suppose that u is a unique point of coincidence of the mappings T and S, that is, u is a unique point of coincidence of T and S with . Then it follows from Proposition 2 that u is a unique common fixed point of the mappings T and S and hence a unique common fixed point (as ).

Let be a metric space. For a given three functions , and , we consider the self-mappings that satisfy the following contractive condition:

for all such that , where

Before formulating our main results, we prove some auxiliary results as under.

Lemma 1. Let T and S be two self-mappings defined on a metric space . Assume that there exist three functions , and such that (1) holds. If is aPicard–Jungck sequence of the pair based on such that for all , then

1. ,

2. is a Cauchy sequence.

Proof. Let be an arbitrary point and be the Picard–Jungck sequence of the pair based on , that is, for all . Assume that for all .

(i) In view of (F1), we have

Now, we show that. For simplicity, let , for all . Then

Setting and for all in (1) and utilizing (θ1), we get

which follows from (G1) that , that is, . for all . This implies that the sequence of real numbers is decreasing and bounded below by zero. Therefore, there exists such that

Our claim is . On the contrary, suppose that and consider two sequences

and

for all . Then . As is strictly decreasing, then for all , and hence, condition (θ2) implies that

which is a contradiction. So, we conclude that

Using condition (F1), we have

and

Lettingin the above two inequalities and using (2), we deduce that

(ii) Let us assume that the sequence is not Cauchy. Then (due to Lemma 13 of [5]) there exist and two subsequences and of with for all such that

with

Let for all . Using (3),(4),(F2),part (i) of Lemma 1 and the continuity of , one easily can show that

Making use of (F1), we have

for all . Applying (θ2), we obtain

which is a contradiction. Hence, we must have that is a Cauchy sequence.

Lemma 2. Let T and S be two self-mappings defined on a metric space . Assume that there exist three functions , and such that (1) holds. Then the point of coincidence of T and S is unique, provided it exists.

Proof. For the seek of contradiction, we suppose that T and S have two distinct points of coincidence u and v, that is, for some and . In view of (F2), we have

Again, in view of (F2), we also have in, fact.

Setting and in (1) and utilizing (θ1) and (G2), we get

which is a contradiction. Hence, the point ofcoincidence of T and S is unique.

Now, we are equipped to state and prove our main results starting with the following one.

Theorem 1. Let T and S bet wo self-mappings defined on a metric space . Suppose that there exists a Picard–Jungck sequence of T and S, and the following conditions are satisfied:

1. there exist and a lower semicontinuous function such that (1) holds,

2. (or ) is complete.

Then

• Pcoin , and the pair has a unique point of coincidence.

• Com . Moreover, if is weakly compatible pair, then it has a unique common fixed point.

Proof. (a) Firstly, we show that Pcoin . To do so, let Pcoin , that is, for some . Since

therefore, on using (1) with , we get

Now, we claim that . On contrary, let . In view of (5), (θ1), and (G2), we have

a contradiction. Therefore, we must have . Now, employing (F1), we obtain

which implies that , and hence, Pcoin .

Secondly, we show that T and S have a point of coincidence. Let be an arbitrary point, and let be the Picard–Jungck sequence of T and S based at , that is, for all . If for some , then is a coincidence point of T and S. Therefore, T and S have a point of coincidence and hence a point of coincidence (as Pcoin ), which is unique (due to Lemma 2). Now, suppose that for all . Then by Lemma 1, the sequence is Cauchy. Assume that is complete, then there exists (for some ) such that

Since is lower semicontinuous, therefore, in view of (6) and part (i) of Lemma 1, we have

which implies that

Now we prove that u is appoint of coincidence. On contrary, assume that u is not a point of coincidence for . We distinguish the following two cases:

Case 1. Assume that for all . Let and for all . Then, in view of (F1), we have

Using the continuity of , (6), and part (i) of Lemma 1, we have

Observe that

Owing to the continuity of, we get

As a consequence, we can find such that

Therefore, using (1), (9), and ( θ3), we obtain (for all with

which contradicts (8). Therefore, u must be a point of coincidence of the pair .

Case 2. Assume that . This assumption contradicts Eq. (7). Therefore,again u must be a point of coincidence of the pair .

Similarly, if we assume that is complete, then we again reach to a contradiction. Therefore, these contradictions in all cases show that u is a point of coincidence of T and S, which is unique (due to Lemma 2).

(b) Following a similar argument used in part (a), one can easily prove that Com . Now, as T and S are weakly compatible mappings, in view of Lemma 2 and Proposition 3, the mappings T and S have a unique common fixed point. This completes the proof.

For a given three functions , and , let the contractive condition (1) in Theorem 1 be replaced by the following one:

for all such that _. Then the proof of the following theorem is similar and much easier than that in the proof of Theorem 1, so the proof is omitted. Notice that there is no direct relation between these theorems as the extended -simulation function need not be monotone in its second argument.

Theorem 2. Let T and S be two self-mappings defined on a metric space . Assume that there exists a Picard–Jungck sequence of T and S, and the following conditions are satisfied:

1. there exist and a lower semicontinuous function such that (10) holds,

2. (or ) is complete.

Then

• Pcoin , and the pair has a unique point of coincidence.

• Com . Moreover, if is weakly compatible pair, then it has a unique common fixed point.

The following example shows that Theorem 1 is a genuine extension of [24,Thm.2.2] and [10, Thm. 3.1].

Example 2. Consider the metric space , where is the space of all bounded sequences of complex numbers, and d is defined by

Let ,where is the zero sequence, and is the sequence whose ith term equals to 4 and all other terms are zeros. It is clear that the pair is a complete metric space. Define two mappings by

First, we show that [24, Thm. 2.3] is not applicable in this example. In fact, on contrary, assume that there exists . such that for all such that with . Then, taking and using and (G2), we obtain

which is a contradiction. This contradiction ensures that there is no such that . Therefore, [24, Thm. 2.3] is not applicable. Now, to show the applicability of Theorem 1, we define two essential functions: and by

It is easy to see that , and φ is a lower semicontinuous function. Now, consider the extended -simulation function given by

We have to prove that the contractive condition (1) holds for all such that . For this purpose, we consider three cases:

Case 1. If and , then and , and hence, we have

Case 2. If and , then and , and hence, we have

Case 3. If , then and , and hence, we have

Therefore, in all cases, the contractive condition (1) is satisfied. Also, observe that T and S are weakly compatible and TX is complete subspace of . Hence, all the hypotheses of Theorem 1 are satisfied, and consequently, the mappings . and . have a unique common fixed point (namely, ).

As consequences of our newly proved results, we deduce several corollaries, which can be viewed as generalizations of various results in the existing literature. Putting , the identity mapping on, in Theorems 1 and 2 and taking to the account that every -simulation function is an extended -simulation function, we deduce the following two corollaries, which seem to be new to the existing literature.

Corollary 1.Let T be a self-mapping defined on a metric space . Suppose that there exist , and a lower semicontinuous function such that

where

Then and T has a unique fixed point.

Corollary 2.Let T be a self-mapping on a metric space . Suppose that there exist , and a lower semicontinuous function such that

Then , and T has a unique fixed point.

Since every simulation function (also, extended simulation function) is an extended -simulation function, then from Theorems 1 and 2 we deduce the following two corollaries, which also seem to be new to the existing literature.

Corollary 3.Let T and S be two self-mappings defined on a metric space . Assume that there exists a Picard–Jungck sequence of T and S, and the following conditions are satisfied:

(i) there exist and a lower semicontinuous function such that

where

(ii) (or is complete.

Then

• Pcoin , and the pair has a unique point of coincidence.

• Com . Moreover, if is weakly compatible pair, then it has a unique common fixed point.

Corollary 4. Let T and S be two self-mappings defined on a metrics pace . Assume that there exists a Picard–Jungck sequence of T and S, and the following conditions are satisfied:

(i)there exist , and a lower semicontinuous function such that

(ii) (or ) is complete.

Then

• Pcoin , and the pair has a unique point of coincidence.

• Com . Moreover, if is weakly compatible pair, then it has a unique common fixed point.

4 Applications

In this section, we employ our main results obtained in metric spaces (Theorems1 and 2) to deduce some related results in partial metric spaces besides proving an existence and uniqueness result on the solution of system of functional equations.

4.1 Application to partial metric spaces

In 1994, Matthews [21] introduced the notion of partial metric spaces as below.

Definition 9. (See [21].) Let be a nonempty set. A partial metric is a mapping satisfying the following conditions:

(P1)

(P2)

(P3)

(P4)

for all . The pair is called a partial metric space.

Observe that, in the setting of partial metric spaces, the distance from a point to itself need not to be zero.

In the following definition, we present some well-known basic notions related to partial metric spaces.

Definition10. (See [21].) Let be a partial metric space.

1. A sequence in X is called convergent and converges to x in if .

2. A sequence { is said to be a Cauchy sequence if exists and is finite.

3. A partial metric space is called a complete partial metric space if every Cauchy sequence in converges to appoint x in such that .

For a partial metric p on a nonempty set , the function given by

remains a standard metric on .

Lemma 3. (see [21,23].) Let be a partial metric space. Then

1. is a Cauchy sequence in if and only if is a Cauchy sequence in the metric space .

2. If the metric space is complete, then the partial metric space is also complete and vice versa. Furthermore, if and only if .

Lemma 4. (see [22].) Let be a partial metric space, and let a function defined by for all . Then is lower semicontinuous in .

From Theorem 1 we deduce the following fixed point result in the setting of partial metric spaces.

Theorem 3. Let T and S be two self-mappings defined on a partial metric space . Assume that there exists a Picard–Jungck sequence of T and S, and the following conditions are satisfied:

1. there exists a function such that

2. (or ) is complete.

Then T and S have a unique point of coincidence u. Moreover, if T and S are weakly compatible, then u is a unique common fixed point with .

Proof. Consider the metric on defined as

where is given in (11). Due to Lemma 3, forms a complete metric space. Define two functions and by

Observe that is lower semicontinuous (due to Lemma 4) and . Now, using (13) and (14) in (12), we get

where

Therefore, all the hypotheses of Theorem 1 are satisfied, and hence, the result follows, which completes the proof.

Similarly, from Theorem 2 we deduce the following related result in partial metric spaces.

Theorem 4.Let T and S be two self-mappings defined on partial metric space . Assume that there exists a Picard–Jungck sequence of T and S, and the following conditions are satisfied:

1. there exists a function such that

2. (or ) is complete.

Then T and S have a unique point of coincidence u. Moreover, if T and S are weakly compatible, then u is a unique common fixed point with .

Proof. The proof follows on the similar lines of proof of Theorem 3.

Taking , the identity mapping on , in Theorems 3 and 4 and taking to the account that every -simulation function is an extended -simulation function, we deduce the following two corollaries, which seem to be new to the existing literature.

Corollary 5. Let T be a self-mapping defined on a partial metric space . Suppose that there exists such that

Then T has a unique fixed point u with .

Corollary 6.Let T be a self-mapping defined on a partial metric space . Suppose that there exists such that

Then T has a unique fixed point _u with .

4.2 Application to system of integral equations

In this section, to highlight the applicability of Theorem 2, we investigated the existence and uniqueness of a common solution of the following system of integral equations:

where are given functions. Let denotes the set of all real valued continuous functions defined on [0,1].

For any arbitrary , define a norm . Let be endowed with the metric

Then is a Banach space.

Now, we are equipped to state and prove our result in this section as under.

Theorem 5. Consider the system of Eqs.(15) and (16). Assume that the following conditions are satisfied:

1. and g are continuous functions,

2. are two mappings defined by

with the property that for all such that ,

1. for all and , we have

Then the system of the integral equations (15) and (16) have a unique common solution.

Proof. For all , we have

which on taking supremum leads to

or

Now, we define two essential functions and as

and

Hence, the above inequality can be written as

Thus, the contractive condition (10) is satisfied with and . Therefore, all the hypotheses of Theorem 2 are satisfied. Hence, the result is established.

Acknowledgments

The authors thanks anonymous referees for their remarkable comments, suggestion, and ideas that help to improve this paper.

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