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## Recent Advances in Fixed Point Theory in Abstract Spaces

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Research Article | Open Access

Volume 2021 |Article ID 5526413 | https://doi.org/10.1155/2021/5526413

Marwan Amin Kutbi, Jamshaid Ahmad, Muhammad Imran Shahzad, "On Some New Fixed Point Results with Applications to Matrix Difference Equations", Journal of Mathematics, vol. 2021, Article ID 5526413, 9 pages, 2021. https://doi.org/10.1155/2021/5526413

# On Some New Fixed Point Results with Applications to Matrix Difference Equations

Academic Editor: Huseyin Isik
Accepted07 May 2021
Published01 Jun 2021

#### Abstract

The aim of this article is to discuss the convergence of iterative sequences of the Prešić type involving new classes of operators satisfying Prešić type -contractive condition in the context of metric spaces. Some examples are also provided to show the significance of the investigation of finding fixed points. Some convergence results for a class of matrix difference equations will be derived as application.

#### 1. Introduction and Preliminaries

Banach’s contraction principle [1] is one of the decisive results of fixed point theory. It states that if we have a self mapping on a complete metric space and a constant such thatholds, , then there exists a unique such that .

Because of its substance and accessibility, many authors have established various fascinating supplements and extensions of this principle (see [121] and references therein). From now to onward, we will consider as positive integer and as complete metric space.

In 1965, Prešić [2] generalized the famous Banach contraction principle and applied the obtained results to secure the convergence of a specific type of sequences. Prešić established the following theorem.

Theorem 1. (see [2]). Let be a mapping satisfying the following contractive condition:for every , where are non-negative constants such that . Then, such that and it is unique. Moreover if are arbitrary points in and for ,then the sequence is convergent and .

A mapping satisfying inequality (2) is said to be a Prešić operator. A point is called a fixed point of if . If in Theorem 1, then we get the Banach contraction principle as a specific case.

Ćirić et al. [3] established the following theorem.

Theorem 2. (see [3]). Assume that satisfiesfor any , where . Then, such that . Moreover, for any arbitrary points , the sequence given in (3) is convergent andIf in addition,holds for all , with , then the fixed point in is unique.

Pwfâcurar [4] obtained a convergence theorem for Prešić–Kannan contraction in this way.

Definition 1. (see [4]). Assume that . If with such thatholds for all . Then,(i) has a unique fixed point .(ii)For any arbitrary points , the sequence given by (3) converges to . Supplementary in this direction, we refer the readers to [58, 10].Very recently, Jleli et al. [11] gave a new variety of contraction called -contraction and established generalized results for these contractions.

Definition 2. Let be a mapping satisfying the following:(i) is nondecreasing.(ii) , .(iii) and such that .A mapping is called a -contraction if there exist some and a function satisfying such that.

Theorem 3. (see [11]). Let be a -contraction; then, has a unique fixed point.

We represent by the set of all mappings satisfying consistent with Samet et al. [11] . For more details, we refer the following [12, 1416, 1820] to the readers.

In this article, we discuss the convergence of given by (3), where is a Prešić type -contraction. The given results unify and generalize various existing results of the literature.

#### 2. Main Results

Motivated by the work of Samet et al. [11], we give the following definition.

Definition 3. A mapping is called a Prešić type -contraction if there exists some such that with .
Note that for , Prešić type -contractive condition is reduced to, .
Furthermore, for such that , inequality (4) is also satisfied, i.e., is a Ćirić–Prešić contraction.

Remark 1. Every Prešić type -contraction is a Prešić operator by and (7), that is,, . Thus, each Prešić type contraction is a continuous function.
Now we present our main result in this way.

Theorem 4. Assume that a Prešić type contraction. Then, for any arbitrary points , the sequence given by (3) converges to in and is a fixed point of . Additionally, if implies that with , then there exists unique such that .

Proof. Let be arbitrary elements in . Define in byfor . If for some , we have , thenthat is, is a fixed point of and we have nothing to prove. So, we assume that for all . Represent for and ; then, we obtain , , and . Thus, for , we getand so on. Hence,for . Letting , we get by thatBy , and such thatLet and . By definition of the limit, such that. It shows that. Then,, where . Now we assume that . Let . By definition of the limit, such that. This implies that, where . Thus, in all cases, there exist and such thatfor all . Thus, by (10), we getLetting , we getThus, such that. Therefore, we getfor all . Now we show that is a Cauchy sequence. For , we haveThis proves that is Cauchy in . As is complete, in such thatNow as is continuous, we getNow we show the uniqueness of fixed point of mapping . We suppose on the contrary that so that and with . Thus, . Hence, by given assumption, we havea contradiction as . Therefore, .

Example 1. Consider the sequence as follows:Let and . Then, becomes a complete metric space. Consider byFor , we getNowThus,does not hold for . Thus, the main hypothesis of Theorem 2 in [3] is not satisfied. Now, by considering the mapping defined bywe can easily show that and is Prešić type contraction. Indeed, the following holds:for , . and for some . The above condition is equivalent toSo, we have to check thatfor some . We discuss these two cases.

Case 1. For , we have

Case 2. For , we havewith .
Thus, all the conditions of Theorem 4 are satisfied and is the unique fixed point of .

Example 2. Let and be such thatand be defined byDefine the mapping given byIt is given in [11] that . Now for , we haveWe havewith . Moreover, for all with , we havewith . Hence, all the hypotheses of Theorem 4 are satisfied. Furthermore, for some arbitrary , the sequence defined by (3) converges to , which is the unique fixed point of mapping .
The upcoming result is an instant consequence of Theorem 4 by taking .

Corollary 1. Let be a given mapping. Assume that there exists some such thatfor all with . Then, for any arbitrary points , the sequence given by (3) converges to , and is a fixed point of , that is, . Moreover, ifholds for all with , then is the unique fixed point of .

Corollary 2. Let be a given mapping. Suppose that there exist non-negative constants with such thatfor all with . Then, for any arbitrary points , the sequence given by (3) converges to , where is the unique fixed point of .

Proof. Evidently, (16) (15) with . Now, let with . From (16), we havewhere . Hence, all the hypotheses of Corollary 1 are satisfied.
Now consider the family which contains large class of functions. For example, ifwhere and , we can obtain the following theorem from our main theorem.

Theorem 5. Let be a given mapping. If there exist a mapping and a constant such thatfor all with , then for any arbitrary points , the sequence given by (3) converges to , and is a fixed point of , that is, . Moreover, ifholds with , then is the unique fixed point of .(1)Theorem 1.3 in [3] and Theorem 1.2 in [2] are direct generalizations of Theorem 4.(2)Corollary 1 of Samet et al. in [11] can be deduced by putting in Theorem 4.(3)Banach contraction principle [1] can be deduced from Corollaries 1 and 2 by taking .

#### 3. Applications

We start this section with the definition of equilibrium point as follows.

Definition 6. Let . For given , consider the recursive sequence defined byfor all . A point is said to be an equilibrium point of equation (17) if the following condition is satisfied:

Definition 7. An equilibrium point is said to be global attractor if for all , we have as .
If we have the recursive sequence defined byfor all , then we explore the global attractivity of  (59) in this application. In this application, we explore the global attractivity of (18). Here (for ) denotes family of Hermitian positive definite matrices, is an Hermitian positive semidefinite matrix, is an nonsingular matrix, is the conjugate transpose of , and is a function from to .
We first analyze the Thompson metric on , which is defined byfor , where , the maximal eigenvalue of . Here, means that is positive semidefinite and means that is positive definite. Nussbaum [21] proved that is a complete metric space regarding andwhere is a spectral norm [21, 22]. Now we directly initiate the graceful properties of , i.e.,for any nonsingular matrix . The second valuable result is the nonpositive curvature property of in this way.In accordance with (62) and (63), we getfor all .

Lemma 1. (see [23]). For all , we have