Research Article | Open Access

Abdolsattar Gholidahneh, Shaban Sedghi, Vahid Parvaneh, "Some Fixed Point Results for Perov-Ćirić-Prešić Type F-Contractions with Application", *Journal of Function Spaces*, vol. 2020, Article ID 1464125, 9 pages, 2020. https://doi.org/10.1155/2020/1464125

# Some Fixed Point Results for Perov-Ćirić-Prešić Type F-Contractions with Application

**Academic Editor:**Syed Abdul Mohiuddine

#### Abstract

Ćirić and Prešić developed the concept of Prešić contraction to Ćirić-Prešić type contractive mappings in the background of a metric space. On the other hand, Altun and Olgun introduced Perov type F-contractions. In this paper, we extend the concept of Ćirić-Prešić contractions to Perov-Ćirić-Prešić type F-contractions. Our results modify some known ones in the literature. To support our main result, an example and an application to nonlinear operator systems are presented.

#### 1. Introduction

The Banach contraction principle (BCP) [1] is one of the powerful results in nonlinear analysis. It has many applications in the background of ODE and PDE.

Theorem 1 [1]. *Let be a complete metric space and let so that
where . Then, there is a unique in such that . Also, for each , the sequence converges to .*

The BCP has been extended and generalized in many directions (see [2–4]).

Prešić [5] gave the following result.

Theorem 2 [5]. *Let be a complete metric space and let ( is a positive integer). Suppose that
for all in , where and . Then has a unique fixed point (that is ). Moreover, for all arbitrary points in the sequence defined by , converges to .*

It is obvious that for , Theorem 2 coincides with the BCP.

Theorem 2 was generalized by Ćirić and Prešić [6] as follows.

Theorem 3 [6]. *Let be a complete metric space and ( is a positive integer). Suppose that
for all in , where . Then has a fixed point . Also, for all points the sequence defined by , converges to . The fixed point of is unique if
for all with .*

For more details on Prešić type contractions, we refer the reader to [2, 5, 7–11].

In this paper, , denotes the set of real matrices, will be the set of real matrices with elements in , denotes the zero matrix, denotes the set of all matrices with elements in , and will be the zero matrix, by the identity matrix. If , then states the transpose matrix of . Let , then by (resp. ), we suppose (resp. ) for each . Also, and will mean the same.

Let be a nonempty set and let be a function. is called a vector-valued metric, and is called a vector-valued metric space, if (1) if and only if (2)(3)for all

*Example 1 (Example 1.3. of [12]). *Let be usual metrics on .

Then, the mapping defined by is a VVM on .

From now on, we apply VVMS instead of a vector-valued metric space.

The concepts of convergence, Cauchyness, and completeness in a VVMS will be similar as in a usual metric case. Perov [13] stated the contraction mapping principle in the setting of VVMSs. Before stating this theorem, we must remember the following facts:

Let Then is said to converge to zero if and only if as (see [14]).

Perov [13] proved the following interesting extension of BCP (see more results in [15–20]).

Theorem 4 [13]. *Let be a VVMS and be a mapping such that there exists a matrix such that
for all . If is convergent to zero, then
*(1)* has a unique fixed point in *(2)*for all , the sequence defined by is convergent to *(3)

In this paper, considering the recent approach of Wardowski [21], we present a generalization of Perov fixed point theorem and Ćirić-Prešić fixed point theorem. Some generalization of Wardowski results can be found in [22, 23].

As in [24], let be a function. Let (F1) be strictly increasing in each variable, i.e., ; then, for all and ,(F2)For each sequence of for each , where (F3)There exists such that for each where

We denote by the set of all functions satisfying (*F*1)–(*F*3)

*Example 2 [24]. *Define by
then .

Note that we can define by which we can treat it as the inverse of multivariable function .

Note that from now on, is a continuously differentiable function from all open sets of , and the Jacobian determinant of at every is nonzero; then, according to inverse function theorem, is invertible near .

*Example 3 [24]. *Define by
then .

*Example 4 [24]. *Define by
then .

Considering the class , Altun and Olgun [24] introduced the concept of Perov type -contraction as follows:

*Definition 5 [24]. *Let be a VVMS and be a map. If there exist and such that
for all with then is called a Perov type -contraction.

If we consider by then (13) turns to Perov contraction [24].

We can present new type contractions in a VVMS, via considering some function in (13).

Theorem 6 [24]. *Let be a complete VVMS and let be a Perov type -contraction. Then admits a unique fixed point.*

In this paper, we introduce the concept of Perov-Ćirić-Prešić type F-contractions. An illustrative example and an application are given to support our main result.

#### 2. Main Results

In this section, combining the ideas of Perov, Wardowski, and Ćirić-Prešić, we obtain a new extension of BCP.

Our main result is as follows:

Theorem 7. *Let be a complete VVMS and let ( is a positive integer). Assume that there exist and satisfying
for all with . Moreover, let there exists a sequence in such that and , for all . Also, if , then , for all . Then, the sequence converges to a fixed point of . Moreover, if for all with ,
then the fixed point of is unique.*

*Proof. *For any , we have

Therefore, where . Now,

Continuing this approach, we have

Continuing this process, we get

Now, taking and , we obtain that where and . Therefore,

Passing to the limit, we get . Therefore, for all . Thus, . From (*F*3), there exists such that

From (23),

Therefore,

Thus, for all . So, for any , there exists such that , for all . Thus, , for all . Putting , we have for all and all . We claim that is a Cauchy sequence. Consider two elements so that . Then, there are and such that , , and . Now, we have

As , we have . Thus, the last term in (27) converges to , and so is a Cauchy sequence in . Since is a complete VVMS, there is so that . Now, we shall prove that is a fixed point of . To see this, we have as . Thus,

Therefore, . Suppose that are two distinct fixed points of . From our hypothesis, which is a contradiction. Thus, the fixed point of is unique.

Note that by taking the above theorem reduces to the following theorem.

Theorem 8. *Let be a complete VVMS and ( is a positive integer). Suppose that there exist and satisfying
where
*

Let the sequence in be such that and , for all . Also, if , then , for all . Then, the sequence converges to a fixed point of . Also, if for all with , then the fixed point of is unique.

We present an example to support our main result.

*Example 5. *Let , , and define by

Firstly, note that for all with , from Example 2.3 of [25], we have

As we know, , for all and

Also, and

Define by

Obviously, . Also, take . We have for any . Now, Let , , and . If , then

So, we may assume that either or . We consider the following cases:

*Case 1. *. Let . If , then
If , then
and if , then

*Case 2. *. Here, if , then
and
and if , then
and

*Case 3. *. In this case, if , then
and
and if , then.
and