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

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 [1–21] 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 [5–8, 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, 14–16, 18–20] 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

Moreover, for all positive semidefinite and , we have

Let be a -contraction regarding . For , consider defined by (59).

Theorem 8. Equation (59) has a unique equilibrium point . Furthermore, is global attractor.

Proof. Define byfor all .
Let . By Lemma 1, we getDenote . Then, using again Lemma 1, we havefor . As is nonsingular, is also nonsingular. By (19), , we get