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

Authors: Marwan Amin Kutbi; Jamshaid Ahmad; Muhammad Imran Shahzad
Journal: Journal of Mathematics
 (2021)

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## 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 typeΘ-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.


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## Body

## 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 ℒ on a complete metric space S,σ and a constant λ∈0,1 such that(1)σℒa,ℒb≤λσa,b,holds, ∀a,b∈S, then there exists a unique a∗∈S such that a∗=ℒa∗.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 k as positive integer and S,σ 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 ℒ:Sk⟶S be a mapping satisfying the following contractive condition:(2)σℒa1,a2,…,ak,ℒa2,…,ak,ak+1≤λ1σa1,a2+λ2σa2,a3+⋯+λkσak,ak+1,for every a1,…,ak+1∈S, where λ1,λ2,…,λk are non-negative constants such that λ1+λ2+⋯+λk<1. Then, ∀a∗∈S such that ℒa∗,…,a∗=a∗ and it is unique. Moreover if a1,…,ak are arbitrary points in S and for n∈ℕ,(3)an+k=ℒan,an+1,…,an+k−1,then the sequence an is convergent and a∗=liman=ℒliman,liman,…,liman.A mappingℒ:Sk⟶S satisfying inequality (2) is said to be a Prešić operator. A point a∗∈S is called a fixed point of ℒ if a∗=ℒa∗,…,a∗. If k=1 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 ℒ:Sk⟶S satisfies(4)σℒa1,a2,…,ak,ℒa2,…,ak,ak+1≤λmaxσa1,a2,σa2,a3,…,σak,ak+1,for any a1,…,ak+1∈S, where 0<λ<1. Then, ∃a∗∈S such that ℒa∗,…,a∗=a∗. Moreover, for any arbitrary points a1,…,ak∈S, the sequence given in (3) is convergent and(5)limn⟶∞an=ℒlimn⟶∞an,…,limn⟶∞an.
If in addition,(6)σℒa∗,…,a∗,ℒa/,…,a/<σa∗,a/,holds for all a∗,a/∈S, with a∗≠a/, then the fixed point in S is unique.Pwfâcurar [4] obtained a convergence theorem for Prešić–Kannan contraction in this way.Definition 1.
(see [4]). Assume that ℒ:Sk⟶S. If ∃λ∈ℝ with 0<λkk+1<1 such that(7)σℒa1,…,ak,ℒa2,…,ak+1≤λ∑i=1k+1σai,ℒai,…,ai,holds for all a1,…,ak+1∈Sk+1. Then,(i)
ℒ has a unique fixed point a∗∈S.(ii)
For any arbitrary pointsa1,…,ak∈S, the sequence an given by (3) converges to a∗. 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 Θ-contraction and established generalized results for these contractions.Definition 2.
LetΘ:0,∞⟶1,∞ be a mapping satisfying the following:(i)
Θ1Θ is nondecreasing.(ii)
Θ2∀tn⊆R+, limn⟶∞Θtn=1⟺limn⟶∞tn=0.(iii)
Θ3∃0<h<1 and l∈0,∞ such that limt⟶0+Θt−1/th=l.
A mappingℒ:S⟶S is called a Θ-contraction if there exist some λ∈0,1 and a function Θ satisfying Θ1–Θ3 such that(8)ℒa≠ℒb⟹Θσℒa,ℒb≤Θσa,bλ,∀a,b∈S.Theorem 3.
(see [11]). Let ℒ:S⟶S be a Θ-contraction; then, ℒ has a unique fixed point.We represent by theΩ set of all mappings Θ:0,∞⟶1,∞ satisfying Θ1–Θ3 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 ofan given by (3), where ℒ:Sk⟶S is a Prešić type Θ-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 mappingℒ:Sk⟶S is called a Prešić type Θ-contraction if there exists some λ∈0,1 such that(9)Θσℒa1,…,ak,ℒa2,…,ak+1≤Θmaxσai,ai+1:1≤i≤kλ,∀a1,…,ak+1∈Sk+1 with ℒa1,…,ak≠ℒa2,…,ak+1.
Note that forΘt=et, Prešić type Θ-contractive condition is reduced to(10)σℒa1,…,ak,ℒa2,…,ak+1≤λ2maxσai,ai+1:1≤i≤k,∀a1,…,ak+1∈Sk+1, ℒa1,…,ak≠ℒa2,…,ak+1.
Furthermore, fora1,…,ak+1∈Sk+1 such that ℒa1,…,ak=ℒa2,…,ak+1, inequality (4) is also satisfied, i.e., ℒ is a Ćirić–Prešić contraction.Remark 1.
Every Prešić typeΘ-contraction ℒ is a Prešić operator by Θ1 and (7), that is,(11)σℒa1,…,ak,ℒa2,…,ak+1<maxσai,ai+1:1≤i≤k,∀a1,…,ak+1∈Sk+1, ℒa1,…,ak≠ℒa2,…,ak+1. Thus, each Prešić type Θ−contraction ℒ is a continuous function.
Now we present our main result in this way.Theorem 4.
Assume thatℒ:Sk⟶S a Prešić type Θ−contraction. Then, for any arbitrary points a1,…,ak∈S, the sequence an given by (3) converges to a∗ in S and a∗ is a fixed point of ℒ. Additionally, if σℒa,…,a≠ℒb,…,b implies that(12)Θσℒa,…,a,ℒb,…,b≤σa,bλ,∀a,b∈S with a≠b, then there exists unique a∗∈S such that ℒa∗=a∗.Proof.
Leta1,…,ak be arbitrary k elements in S. Define an in S by(13)an+k=ℒan,an+1,…,an+k−1,for n∈ℕ. If for some n0∈1,2,3,…,k, we have an0=an0+1, then(14)an0+k=ℒan0,an0+1,…,an0+k−1=ℒan0+k,an0+k,…,an0+k,that is, an0+k is a fixed point of ℒ and we have nothing to prove. So, we assume that an+k≠an+k+1 for all n∈ℕ. Represent βn+k=σan+k,an+k+1 for n∈ℕ and μ=maxσa1,a2,σa2,a3,…,σak,ak+1; then, we obtain βn+k>0, ∀n∈ℕ, and μ>0. Thus, for n≤k, we get(15)1<Θβk+1=Θσak+1,ak+2=Θσℒa1,a2,…,ak,ℒa2,a3,…,ak+1≤Θmaxσai,ai+1:1≤i≤kλ=Θμλ,1<Θβk+2=Θσak+2,ak+3=Θσℒa2,a3,…,ak+1,ℒa3,a4,…,ak+2≤Θmaxσai,ai+1:2≤i≤k+1λ≤Θμλ2⋮,and so on. Hence,(16)1<Θβk+n=Θσan+k,an+k+1=Θσℒan,an+1,…,an+k−1,ℒan+1,an+2,…,an+k≤Θμλn,for n≥1. Letting n⟶∞, we get by Θ2 that(17)limn⟶∞Θβk+n=1⟺limn⟶∞βk+n=0.
ByΘ3, ∃0<h<1 and l∈0,∞ such that(18)limn⟶∞Θβk+n−1βk+nh=l.
Letl<∞ and ω=l/2>0. By definition of the limit, ∃n1∈ℕ such that(19)Θβk+n−1βk+nh−l≤ω,∀n>n1. It shows that(20)Θβk+n−1βk+nh≥l−ω=l2=ω,∀n>n1. Then,(21)nβk+nh≤ϖnΘβk+n−1,∀n>n1, where ϖ=1/ω. Now we assume that l=∞. Let ω>0. By definition of the limit, ∃n1∈ℕ such that(22)ω≤Θβk+n−1βk+nh,∀n>n1. This implies that(23)nβk+nh≤ϖnΘβk+n−1,∀n>n1, where ϖ=1/ω. Thus, in all cases, there exist ϖ>0 and n1∈ℕ such that(24)nβk+nh≤ϖnΘβk+n−1,for all n>n1. Thus, by (10), we get(25)nβk+nh≤ϖnΘμλn−1.
Lettingn⟶∞, we get(26)limn⟶∞nβk+nh=0.
Thus,∃n2∈ℕ such that(27)nβk+nh≤1,∀n>n2. Therefore, we get(28)βk+n≤1n1/h,for all n>n2. Now we show that an is a Cauchy sequence. For m>n>n2, we have(29)σak+n,ak+m=σℒan,…,ak+n−1,ℒam,…,ak+m−1≤σℒan,…,ak+n−1,ℒan+1,…,ak+n+σℒan+1,…,ak+n,ℒan+2,…,ak+n+1+⋯+σℒam−1,…,ak+m−2,ℒam,…,ak+m−1=σan+k,an+k+1+σan+k+1,an+k+2+⋯+σam+k−1,am+k=βn+k+βn+k+1+⋯+βm+k−1=∑i=nm−1βi+k<∑i=n∞βi+k≤∑i=n∞1i1/h<∞.
This proves thatan is Cauchy in S,σ. As S,σ is complete, ∃a∗ in S such that(30)limn,m⟶∞σan,am=limn⟶∞σan,a∗=0.
Now asℒ is continuous, we get(31)a∗=limn⟶∞an+k=limn⟶∞ℒan,an+1,…,an+k−1=ℒlimn⟶∞an,limn⟶∞an+1,…,limn⟶∞an+k−1=ℒa∗,a∗,…,a∗.
Now we show the uniqueness of fixed point of mappingℒ. We suppose on the contrary that ∃a∗,a′∈S so that a∗=ℒa∗,a∗,…,a∗ and a′=ℒa′,a′,…,a′ with a∗≠a′. Thus, ℒa∗,a∗,…,a∗≠ℒa′,a′,…,a′. Hence, by given assumption, we have(32)Θσa∗,a′=Θσℒa∗,a∗,…,a∗,ℒa′,a′,…,a′≤Θσa∗,a′λ,a contradiction as λ∈0,1. Therefore, a∗=a′.Example 1.
Consider the sequencean as follows:(33)a1=1a2=1+5⋮an=1+5+9+⋯+4n−3=n2n−1.
LetS=an:n∈ℕ and σa∗,a′=a∗−a′. Then, S,σ becomes a complete metric space. Consider ℒ:S2⟶S by(34)ℒan∗,an′=an−1∗+an−1′2,forn>1,a1∗+a1′2,otherwise.
Forn>3, we get(35)σℒan−2,an−1,ℒan−1,an=σan−3+an−22,an−2+an−12=12n−32n−7+n−22n−5−n−22n−5+n−12n−3=122n2−13n+21+2n2−9n+10−2n2−9n+10+2n2−5n+3=128n−18=4n−9,maxσan−2,an−1,σan−1,an=maxn−22n−5−n−12n−3,n−12n−3−n2n−1max4n−7,4n−3=4n−3.
Now(36)limn⟶∞σℒan−2,an−1,ℒan−1,anmaxσan−2,an−1,σan−1,an=limn⟶∞4n−94n−3=1.
Thus,(37)σℒan−2,an−1,ℒan−1,an≤λmaxσan−2,an−1,σan−1,andoes not hold for λ∈0,1. Thus, the main hypothesis of Theorem 2 in [3] is not satisfied. Now, by considering the mapping Θ:0,∞⟶1,∞ defined by(38)Θt=etet,we can easily show that Θ∈Ω and ℒ is Prešić type Θ−contraction. Indeed, the following holds:(39)eσℒai,ai+1,ℒai+1,ai+2eσℒai,ai+1,ℒai+1,ai+2≤eλσai,ai+1,ai+1,ai+2eσai,ai+1,ai+1,ai+2,for ℒai,ai+1≠ℒai+1,ai+2, i=1,2,…,. and for some λ∈0,1. The above condition is equivalent to(40)σℒai,ai+1,ℒai+1,ai+2⋅eσℒai,ai+1,ℒai+1,ai+2≤λ2maxσai,ai+1,σai+1,ai+2⋅emaxσai,ai+1,σai+1,ai+2.
So, we have to check that(41)σℒai,ai+1,ℒai+1,ai+2⋅eσℒai,ai+1,ℒai+1,ai+2maxσai,ai+1,σai+1,ai+2⋅emaxσai,ai+1,σai+1,ai+2≤λ2,for some λ∈0,1. We discuss these two cases.Case 1.
Fori=n=1, we have(42)σℒa1,a2,ℒa2,a3eσℒa1,a2,ℒa2,a3maxσa1,a2,σa2,a3emaxσa1,a1,σa2,a3=σa1+a1/2,a1+a2/2eσa1+a1/2,a1+a2/2maxσa1,a2,σa2,a3emaxσa1,a2,σa2,a3=σ1,7/2eσ1,7/2maxσ1,6,σ6,15emaxσ1,6,σ6,15=518e−13/2<e−2.Case 2.
Fori=n>1, we have(43)σℒan,an+1,ℒan+1,an+2⋅eσℒan,an+1,ℒan+1,an+2maxσan,an+1,σan+1,an+2⋅emaxσan,an+1,σan+1,an+2=σan−1+an/2,an+an+1/2⋅eσan−1+an/2,an+an+1/2maxσan,an+1,σan+1,an+2⋅emaxσan,an+1,σan+1,an+2=1/24n2−6n+3−4n2+2n+1⋅e1/24n2−6n+3−4n2+2n+1max4n+1,2n+2⋅emax4n+1,2n+2=−4n+14n+1e−8n<e−2,with λ=e−1.
Thus, all the conditions of Theorem4 are satisfied and 1,1 is the unique fixed point of ℒ.Example 2.
LetS=0,1 and σ be such that(44)σa∗,a′=a∗−a′,and ℒ:Sk⟶S be defined by(45)ℒa1,…,ak=a1+ak4kfor alla1,…,ak∈S.
Define the mappingΘ:0,∞⟶1,∞ given by(46)Θt=et.
It is given in [11] that Θ∈Ω. Now for a1,a2,…,ak+1∈S, we have(47)σℒa1,…,ak,ℒa2,…,ak+1>0.
We have(48)Θσℒa1,…,ak,ℒa2,…,ak+1=eσℒa1,…,ak,ℒa2,…,ak+1=e1/4ka1−a2+ak−ak+1=e1/2ka1−a2+ak−ak+1≤e1/2maxσa1,a2,σak,ak+1≤e1/2maxσaj,aj+1:1≤j≤k=Θmaxσaj,aj+1:1≤j≤kλ.with λ=1/2. Moreover, for all a∗,a′∈S with a∗≠a′, we have(49)σℒa∗,a∗,…,a∗,ℒa′,a′…,a′=a∗−a′2k>0,Θσℒa∗,a∗,…,a∗,ℒa′,a′…,a′=Θa∗−a′2k=ea∗−a′/2k=e1/2ka∗−a′≤e1/2a∗−a′=Θσa∗,a′λ,with λ=1/2. Hence, all the hypotheses of Theorem 4 are satisfied. Furthermore, for some arbitrary a1,…,ak∈S, the sequence an defined by (3) converges to a∗=0, which is the unique fixed point of mapping ℒ.
The upcoming result is an instant consequence of Theorem4 by taking Θt=et.Corollary 1.
Letℒ:Sk⟶S be a given mapping. Assume that there exists some λ∈0,1 such that(50)σℒa1,…,ak,ℒa2,…,ak+1≤λ2maxσai,ai+1:1≤i≤k,for all a1,…,ak+1∈Sk+1 with ℒa1,…,ak≠ℒa2,…,ak+1. Then, for any arbitrary points a1,…,ak∈S, the sequence an given by (3) converges to a∗, and a∗ is a fixed point of ℒ, that is, a∗=ℒa∗,…,a∗. Moreover, if(51)σℒa,…,a,ℒb,…,b≤λ2σa,b,holds for all a,b∈S with a≠b, then a∗ is the unique fixed point of ℒ.Corollary 2.
Letℒ:Sk⟶S be a given mapping. Suppose that there exist λ1,λ2…,λk non-negative constants with λ1+λ2+⋯+λk<1 such that(52)σℒa1,…,ak,ℒa2,…,ak+1≤λ1σa1,a2+λ2σa2,a3+⋯+λkσak,ak+1,for all a1,…,ak+1∈Sk+1 with ℒa1,…,ak≠ℒa2,…,ak+1. Then, for any arbitrary points a1,…,ak∈S, the sequence an given by (3) converges to a∗, where a∗ is the unique fixed point of ℒ.Proof.
Evidently, (16) ⟹ (15) with λ2=λ1+λ2+⋯+λk. Now, let a,b∈S with a≠b. From (16), we have(53)σℒa,a,…,a,ℒb,b,…,b≤σℒa,…,a,ℒa,…,a,b+σℒa,…,a,b,ℒa,…,a,b,b+⋯+σℒa,b,…,b,ℒb,b,…,b≤λk+λk−1+⋯+λ1σa,b=λ2σa,b,where λ2=λk+λk−1+⋯+λ1∈0,1. Hence, all the hypotheses of Corollary 1 are satisfied.
Now consider the familyΩ which contains large class of functions. For example, if(54)Θt=2−2πarctan1tβ,where 0<β<1 and t>0, we can obtain the following theorem from our main theorem.Theorem 5.
Letℒ:Sk⟶S be a given mapping. If there exist a mapping Θ∈Ω and a constant β,λ∈0,1 such that(55)2−2πarctanfrac1σℒa1,…,ak,ℒa2,…,ak+1β≤2−2πarctan1maxσai,ai+1:1≤i≤kβλ,for all a1,…,ak+1∈Sk+1 with ℒa1,…,ak≠ℒa2,…,ak+1, then for any arbitrary points a1,…,ak∈S, the sequence an given by (3) converges to a∗, and a∗ is a fixed point of ℒ, that is, a∗=ℒa∗,…,a∗. Moreover, if(56)2−2πarctan1σℒa,…,a,ℒb,…,bβ≤2−2πarctan1σa,bβλ,holds ∀a,b∈S with a≠b, then a∗ is the unique fixed point of ℒ.(1)
Theorem 1.3 in [3] and Theorem 1.2 in [2] are direct generalizations of Theorem 4.(2)
Corollary1 of Samet et al. in [11] can be deduced by putting k=1 in Theorem 4.(3)
Banach contraction principle [1] can be deduced from Corollaries 1 and 2 by taking k=1.


## 3. Applications

We start this section with the definition of equilibrium point as follows.Definition 6.
Letℒ:Sk⟶S. For given a1,a2,…,ak∈S, consider the recursive sequence an⊂S defined by(57)an+k=ℒan,an+1,…,an+k−1,for all n∈ℕ. A point a¯ is said to be an equilibrium point of equation (17) if the following condition is satisfied:(58)a¯=ℒa¯,…,a¯.Definition 7.
An equilibrium pointa¯ is said to be global attractor if for all a1,a2,…,ak∈S, we have σan,a¯⟶0 as n⟶∞.
If we have the recursive sequenceSn⊂PN defined by(59)Sn+k=Q+1k∑i=0k−1E∗ψSn+iE,for all n∈ℕ, then we explore the global attractivity of  (59) in this application. In this application, we explore the global attractivity of (18). Here PN (for N≥2) denotes family of N×N Hermitian positive definite matrices, Q is an N×N Hermitian positive semidefinite matrix, E is an N×N nonsingular matrix, E∗ is the conjugate transpose of E, and ψ is a function from PN to PN.
We first analyze the Thompson metricσ on PN, which is defined by(60)σE1,E2=maxlogME1E2,logME2E1,for E1,E2∈PN, where ME1/E2=infδ>0:E1≤δE2=δ+E2−1/2E1−1/2E2, the maximal eigenvalue of E2−1/2E1−1/2E2. Here, E1≤E2 means that E2−E1 is positive semidefinite and E1<E2 means that E2−E1 is positive definite. Nussbaum [21] proved that PN is a complete metric space regarding σ and(61)σE1,E2=lnE1−1/2E2E1−1/2,where ⋅ is a spectral norm [21, 22]. Now we directly initiate the graceful properties of σ, i.e.,(62)σE1,E2=σE1−1,E2−1=σM∗E1M,M∗E2M,for any nonsingular matrix M. The second valuable result is the nonpositive curvature property of σ in this way.(63)σE1r,E2r≤cσE1,E2,c∈0,1.
In accordance with (62) and (63), we get(64)σM∗XrM,M∗YrM≤cσX,Y,c∈−1,1,for all E1,E2∈PN.Lemma 1.
(see [23]). For all E1,E2,E3,E4∈PN, we have(65)σE1+E2,E3+E4≤maxσE1,E3,σE2,E4.Moreover, for all positive semidefiniteE1 and E2,E3∈PN, we have(66)σE1+E2,E1+E3≤σE2,E3.Letψ:PN⟶PN be a Θ-contraction regarding σ. For X1,X2,…,Xk∈PN, consider Xn⊂PN defined by (59).Theorem 8.
Equation (59) has a unique equilibrium point X¯∈PN. Furthermore, X¯ is global attractor.Proof.
Defineℒ:PNk⟶PN by(67)ℒM1,M2,…,Mk=Q+1kE∗ψM1E+E∗ψM2E+⋯+E∗ψMkE,for all M1,M2,…,Mk∈PN.
LetM1,M2,…,Mk+1∈PN. By Lemma 1, we get(68)σℒM1,M2,…,Mk,ℒM2,M3,…,Mk+1=σQ+1k∑i=1kE∗ψMiE,Q+1k∑j=2k+1E∗ψMjE≤σ1k∑i=1kE∗ψMiE,1k∑j=2k+1E∗ψMjE=σ∑i=1k1/kE∗ψMi1/kE,∑j=2k+11/kE∗ψMj1/kE.
DenoteO=1/kE. Then, using again Lemma 1, we have(69)σℒM1,M2,…,Mk,ℒM2,M3,…,Mk+1≤σ∑i=1kO∗ψMiO,∑j=2k+1O∗ψMjO=σO∗ψM1O+O∗ψM2O+⋯+O∗ψMkO,O∗ψM2O+O∗ψM3O+⋯+O∗ψMk+1O≤maxσO∗ψM1O,O∗ψM2O,σO∗ψM2O,O∗ψM3O,…,σO∗ψMkO,O∗ψMk+1O=maxσO∗ψMjO,O∗ψMj+1O,for j=1,2,…,k. As A is nonsingular, O is also nonsingular. By (19), ∀j=1,2,…,k, we get(70)σO∗ψMiO,O∗ψMi+1O=σψMi,ψMi+1.
Now asψ is a Θ-contraction, for all j=1,2,…,k, we have(71)ΘσO∗ψMiO,O∗ψMi+1O≤ΘσMi,Mi+1λ,for some λ∈0,1. Thus, we have(72)ΘσℒM1,M2,…,Mk,ℒM2,M3,…,Mk+1≤ΘmaxσMi,Mi+1:i=1,2,…,kλ,for all M1,M2,…,Mk+1∈PN. Thus, by Theorem 4, there exists a global attractor equilibrium point X¯∈PN.
Now forM1,M2∈PN such that ℒM1,M1,…,M1≠ℒM2,M2,…,M2, we have(73)ΘσℒM1,M1,…,M1,ℒM2,M2,…,M2=ΘσQ+E∗ψM1E,Q+E∗φM2E≤ΘσE∗ψM1E,E∗ψM2E=ΘσψM1,φM2≤ΘσM1,M2λ.
Again, by Theorem4, we get the unique equilibrium point.


## 4. Conclusion

Jleli and Samet [11] very recently exploited the idea of Θ-contraction and established some generalized theorems for these contractions in generalized metric spaces. We continued their investigations and defined Prešić type Θ-contraction. In this paper, we studied the convergence of iterative sequences of the Prešić type involving new classes of operators satisfying Prešić type Θ-contractive condition in the context of metric spaces. Some examples are also provided to show the significance of the investigation of finding fixed points. As application, we derived some convergence results for a class of matrix difference equations.



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