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 [24]).

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, 711].

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 [1520]).

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 (F1)–(F3)

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 (F3), 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

Case 4. . Here, if , then and and if , then and

Also, let with . Without loss of any generality, let with . If , then and if , then