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 (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
We see that all of the conditions of Theorem 2 are satisfied. Thus, has a unique fixed point. Here, , and is the unique fixed point.
We present an example in an infinite dimensional sequence space which is adapted from the above example, and so, we leave the details for the reader.
Let be the space of all convergent sequences for which ( is an arbitrary natural number) for exactly one and for other indices.
Let , , and define by
Define by
Obviously, . Also, take .
Reviewing the above example, we can show that all of the conditions of Theorem 2 are satisfied. Thus, has a unique fixed point. Here, and is the unique fixed point.
3. Application
Let be a Banach space and be nonlinear operators. In this section, motivated by the work in [26], we will present a result on existence of a solution for the following semilinear operator system:
Similar systems which appear in various branches of mathematics could be seen in [27].
Let and define , for , by . Evidently, is a complete VVMS.
If we define a mapping by then the system (61) can be written as a fixed point problem such as in the space . Therefore, applying Theorem 2, we investigate the sufficient hypothesis which leads to the existence of a solution of problem (63).
Theorem 9. Assume that there exist positive real numbers () such that for all , with . Then, the system (61) has a unique solution in .
Proof. By the inequality (64), we have for all . Hence, we get
Taking the function as , the above inequality can be written as or, equivalently, where . Thus, applying Theorem 2, possesses a unique fixed point in , or, equivalently, the semilinear operator system (61) has a unique solution in .
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no competing interests.
Authors’ Contributions
All authors read and approved the manuscript.