# Hybrid Coupled Fixed Point Theorems in Metric Spaces with Applications

Authors: Muhammad Shoaib; Muhammad Sarwar; Thabet Abdeljawad
Journal: Journal of Function Spaces
 (2019)

---

## Abstract

In this manuscript, using CLR property, coupled coincidence and common coupled fixed point results for two-hybrid pairs satisfying(F,φ)- contraction are demonstrated. Using the established results existence of solution to the coupled system of functional and nonlinear matrix equations is also discussed. We provide examples where the main theorem is applicable but most current relevant results in literature fail to have a common coupled fixed point.


---

## Body

## 1. Introduction and Preliminaries

The existence and uniqueness of solution of a nonlinear matrix equations and functional equations are very interesting research area. Metric fixed point theory provides beneficial and best techniques for the existence of the above-mention equations. In [1–6] the authors worked on matrix equations and demonstrated the existence and uniqueness in the form of their positive definite solutions. Matrix equations and functional equations often arise from various areas, such as ladder networks [7, 8], control theory [9, 10], and dynamic programming [11–14].Banach [15] has sorted out successful and well-known result; such consequence was later on named as Banach contraction principle (BCP). The Banach principle has been generalized in various spaces. Nadler [16] in 1969 further modified and elaborated the Banach contraction principle (BCP) to set-valued mapping using the Hausdorff metric known as Nadler contraction principle (NCP). Later, thousands of articles appeared in literature to generalize the BCP. Very recently, some authors proved the contraction principle in metric in controlled metric type spaces where the triangle inequality possess control functions (see [17–19] and the references therein).Aamri and Moutawakil [20] defined (E.A) property for self-mappings which contained the class of compatible and noncompatible mappings and proved common fixed point results under strict contractive conditions. Kamran [21] demonstrated the (E.A) property for hybrid pair and established fixed point and coincidence points results with hybrid strict contractions. Liu et al. [22] introduced common (E.A) property for hybrid pairs of single and multivalued mappings and presented new common fixed point theorems using hybrid contractive conditions. Sintunavarat and Kumam [23] brought together the idea of common limit range (CLR) property for single-valued mappings and displayed its superiority over the property (E.A). Imdad et al. [24] defined common limit range property for a hybrid pair of mappings and demonstrated fixed point results in the symmetric (semimetric) spaces. These concepts were converted by Abbas et al. [25] to multivalued mappings and formulated coupled coincidence point and common coupled fixed point theorems linking hybrid pair of mappings satisfying generalized contractive conditions. Deshpande and Handa [26, 27] defined (E.A) property and occasional w-compatibility for hybrid (pair) coupled maps and also presented common (E.A) property for two hybrid coupled mappings.In 2012, Wardowski [28] introduced a new type of contraction called F-contraction. In this way Wardowski generalized the Banach contraction principle (BCP) in different manner from the known results of literature. Following this direction Sgroi and Vetro [14] studied multivalued F-contractions and discussed their application on certain functional and integral equations. Recently, Nashine et al.[29] introduced generalized (F,φ)-contractions and studied common fixed point results for a hybrid pair under common limit range property with applications to certain system of functional equations and Volterra integral inclusion.Coupled fixed points for several type contraction mappings were studied by many authors in different type metric spaces [30–32]. For more details see [25, 33–37]. For other types of common fixed point results we refer to [38–40] and the references therein.Motivated by the above results, we studied common coupled fixed point results by defining the concept of CLR property for two hybrid pairs of mapping via generalized(F,φ) type contraction. Using the established results we also studied the existence of solution for the coupled system of functional and coupled system of nonlinear matrix equations. All over the paper R+, N, and N0 represent the set of all positive real numbers, the set of positive integers, and the set of nonnegative integers, respectively.Definition 1.
SupposeΘ is nonempty set and let d:Θ×Θ→R+ be a function satisfying the conditions (1)
d(κ,ξ)=0 if and only if κ=ξ for all κ,ξ∈Θ;(2)
d(κ,ξ)=d(ξ,κ), where κ,ξ∈Θ;(3)
d(κ,ξ)≤d(κ,z~)+d(ξ,z~) for all κ,ξ,z~∈Θ.Thend is a metric on Θ and the pair (Θ,d) is called metric space.Definition 2 (see [23]).
Functionsf,g:Θ→Θ are said to satisfy the common limit range property of f w.r.t g (shortly, the (CLRf)-property w.r.t g) if there exists a sequence {ξn} in Θ such that, for some u∈Θ, limn→∞fξn=limn→∞gξn=fu.Definition 3 (see [41]).
Supposef:Θ→Θ, S:Θ→CB(Θ) are defined on a metric space (Θ,d). Then f and S are said to satisfy the common limit range property of f w.r.t S (shortly, (CLRf)-property w.r.t S) if there exists a sequence {ξn} in Θ and Ω1∈CB(Θ) such that, for some u∈Θ, limn→∞fξn=fu∈Ω1=limn→∞Sξn.Definition 4 (see [41]).
Functionsf,g:Θ→Θ and S,T:Θ→CB(Θ) defined on a metric space (Θ,d), are to satisfy the common limit in the range of f w.r.t S (shortly, (CLRf)-property w.r.t to S) if there exist sequences {ξn} and {ζn} in Θ and Ω1,Ω2∈CB(Θ) such that, for some u∈Θ, we have limn→∞Sξn=Ω1, limn→∞Tζn=Ω2, and limn→∞fξn=limn→∞gζn=fu∈Ω1∩Ω2.Remark 5.
Clearly, iff=g and S=T in Definition 4 then we reobtain Definition 3.Definition 6 (see [25]).
Letf:Θ→Θ and F:Θ×Θ→CB(Θ) be mappings. (1)
A point(x,y)∈Θ×Θ is called a coupled coincidence point of f and F if f(x)∈F(x,y) and f(y)∈F(y,x).(2)
A point(x,y)∈Θ×Θ is called a coupled common point of f and F if x=f(x)∈F(x,y) and y=f(y)∈F(y,x).Definition 7 (see [25]).
Letf:Θ→Θ and F:Θ×Θ→CB(Θ) be mappings. The mapping f is called F- weakly commuting at some point, point (x,y)∈Θ×Θ if f2(x)∈F(fx,fy) and f2(y)∈F(fy,fx).Definition 8 (see [27]).
Mappingsf:Θ→Θ and S:Θ×Θ→CB(Θ) on metric space (Θ,d) are said to have the E.A property if there exist sequences {ξn} and {ζn} in Θ and Ω1,Ω2∈CB(Θ) such that for some u∈Θlimn→∞fξn=u∈Ω1=limn→∞S(ξn,ζn), limn→∞fζn=v∈Ω2=limn→∞S(ζn,ξn).Now, we recall some definitions for multivalued mappings defined in a metric space(Θ,d). Recall the Hausdorff metric H:CB(Θ)×CB(Θ)→R+ for Ω1,Ω2∈CB(Θ) by (1)HΩ1,Ω2=maxsupς∈Ω1dς,Ω2,supζ∈Ω2dζ,Ω1,where (2)dξ,Ω1=inf⁡dξ,ζ:ζ∈Ω1,δΩ1,Ω2=sup⁡dς,ζ:ς∈Ω1,ζ∈Ω2and (3)DΩ1,Ω2=inf⁡dς,ζ:ς∈Ω1,ζ∈Ω2.Lemma 9 (see [42]).
Let(Θ,d) be a metric space. For any Ω1,Ω2∈CB(Θ). We have d(ξ,Ω2)≤H(Ω1,Ω2), for all ξ∈Ω1.Lemma 10 (see [16]).
Assume(Θ,d) is a metric space and Ω1,Ω2∈CB(Θ). Then for every λ>1 and for each ς∈Ω1 there exists ζ(ς)∈Ω2 such that d(ς,ζ)≤λH(Ω1,Ω2).In [16] it was shown that the above lemma is also true for λ≥1. In fact we have the following.Lemma 11.
Assume(Θ,d) is a metric space and Ω1,Ω2∈CB(Θ). Then for every λ≥1 and for each ς∈Ω1 there exists ζ(ς)∈Ω2 such that d(ς,ζ)≤λH(Ω1,Ω2).Definition 12 (see [28]).
LetFs represent the family of all functions F:R+→R, with the following conditions (1)
F is continuous and strictly increasing;(2)
limn→∞αn=0⇔limn→∞F(αn)=-∞;(3)
for{αn}⊂R+, limn→∞αn=0, there exists k∈(0,1) such that limα→0+(αn)kF(αn)=0.Theorem 13 (see [27]).
Let(Θ,d) be a metric space. Assume f,g:Θ→Θ and F,G:Θ×Θ→CB(Θ) to be a mapping satisfying the following.(a)
(F,f) and (G,g) satisfy the common (EA) property.(b)
For allx,y,u,v∈Θ, there exist some φ∈Φ and some ψ∈Ψ such that (4)φHFx,y,Gu,v+HFy,x,Gv,u2≤φdfx,gu+dfy,gv2-ψdfx,gu+dfy,gv2.(c)
f(Θ) and g(Θ) are closed subsets of Θ. Then
(A1)
(g,G) have coupled coincidence point.(A2)
(f,F) have coupled coincidence point.(A3)
Ifg is G- weakly commuting at (w1,w2) and g2w1=gw1, g2w2=gw2 for (w1,w2)∈C(G,g), then G and g have a common coupled fixed point.(A4)
Iff is F- weakly commuting at (z1,z2) and f2z1=fz1, f2z2=fz2 for (z1,z2)∈C(F,f), then F and f have a common coupled fixed point.(A5)
F,G,f, and g have common coupled fixed point if (A3) and (A4) are true.Theorem 14 (see [27]).
Let(Θ,d) be a metric space. Assume f,g:Θ→Θ and F,G:Θ×Θ→CB(Θ) to be mappings satisfying (a) and (b) of Theorem 13 and(1)
(F,f) and (G,g) are w-compatible.(2)
Suppose that eitherg(Θ) is closed subset of Θ or G(Θ×Θ)⊆f(Θ) or f(Θ) is closed subset of Θ and F(Θ×Θ)⊆g(Θ). Then F,G,f,g have a common coupled fixed point.


## 2. Main Results

We define the CLR property for the study of common coupled fixed point in the following way in metric space.Definition 15.
Mappingsf:Θ→Θ and S:Θ×Θ→CB(Θ) on metric space (Θ,d) are to satisfy the common limit in the range of f with respect to S (shortly, the (CLRf)-property with respect to S) if there exist sequences {ξn} and {ζn} in Θ and Ω1,Ω2∈CB(Θ) such that, for some u∈Θ, we have limn→∞fξn=u=fξ∈Ω1=limn→∞S(ξn,ζn), limn→∞fζn=v=fζ∈Ω2=limn→∞S(ζn,ξn).Definition 16.
Letf,g:Θ→Θ and T,G:Θ×Θ→CB(Θ) be mappings on metric space (Θ,d). Then (T,f) and (G,g) have (CLR)-property, if there exist sequences {xn}, {yn}{un} and {vn} and C1,C2,D1,D2∈CB(Θ) such that (5)limn→∞Txn,yn=C1,limn→∞Gun,vn=C2,limn→∞fxn=fz1∈C1,limn→∞gun=gz1∈C2,limn→∞Tyn,xn=D1,limn→∞Gvn,un=D2,limn→∞fyn=fz2∈D1,limn→∞gvn=gz2∈D2,for some z1,z2∈Θ.Example 17.
LetΘ=[0,∞) with the usual metric. Define f,g:Θ→Θ and F,G:Θ×Θ→CB(Θ) by f(x)=1+x, g(x)=x/2, F(x,y)=[1,2+2x+y], G(x,y)=[1,2+3x+y/4], ∀x,y∈X.
Consider the sequences{xn}={1+1/n}, {yn}={4+1/n}, {un}={1-1/n}, {vn}={4-1/n}.
Clearlylimn→∞F(xn,yn)=[1,8], limn→∞f(xn)=2=f(1)∈[1,8], limn→∞G(un,vn)=[1,15/4], limn→∞g(un)=1/2=g(1)∈[1,15/4]. Further, limn→∞F(yn,xn)=[1,11], limn→∞f(yn)=5=f(4)∈[1,11], limn→∞G(vn,un)=[1,21/4], limn→∞g(vn)=2=g(4)∈[1,21/4]. Therefore (f,F) and (g,G) satisfy CLR property.Example 18.
AssumeΘ=[0,1) to be endowed through usual metric and f,g:Θ→Θ, T,S:Θ×Θ→CB(Θ), define by (6)fx=34if0≤x<1279100if12≤x<1,gx=34if0≤x<121120if12≤x<1,Tx,y=12,34if0≤x,y<1234,45Otherwise,Sx,y=35,45if0≤x,y<1212,35Otherwise.Let {xn}=1-1/n, where n=3,4,5,6..., {yn}=1-1/n3, where n=4,5,6...{un}={1/9-1/2n}, where n=3,4,5,6..., and {vn}={1/4-1/n}, where n=4,5,6....
Clearlylimn→∞T(xn,yn)=[3/4,4/5], limn→∞f(xn)=79/100=f(1/2)∈[3/4,4/5], limn→∞S(un,vn)=[3/5,4/5], limn→∞g(un)=11/20=g(1/2)∈[3/5,4/5]. Further, limn→∞T(yn,xn)=[3/4,4/5], limn→∞f(yn)=79/100=f(4/5)∈[3/4,4/5], limn→∞S(vn,un)=[3/5,4/5], limn→∞g(vn)=11/20=g(4/5)∈[3/5,4/5]. Therefore (f,F) and (g,G) satisfy CLR property.
Throughout the paperCB(Θ) denote the set of all closed and bounded subsets of Θ and(7)Φ=φ:R+→R+,increasing,uppersemicontineoussuchthatlims→t+φs<φt,φt<t,forallt>0.Theorem 19.
Letf,g:Θ→Θ and T,G:Θ×Θ→CB(Θ) be maps on metric space (Θ,d). Suppose that (T,f) and (G,g) have (CLR)-property and furthermore assume that (8)τ+FHpTx,y,Gu,v≤FφΘx,y,u,v,where H(T(x,y),G(u,v))>0 and (9)Θx,y,u,v=αdpfx,gu+βdpfy,Ty,xdpgv,Gv,u1+dpfx,gu+γdpfx,Tx,y+dpgu,Gu,v+σdpfy,Ty,x+ηdpgu,Gu,v.Here, τ∈R+, α+β+γ+σ+η≤1, p≥1, F∈Fs and φ∈Φ. Then the following holds. (A1)
(g,G) have coupled coincidence point.(A2)
(f,T) have coupled coincidence point.(A3)
Ifg is G- weakly commuting at (w1,w2) and g2w1=gw1, g2w2=gw2 for (w1,w2)∈C(G,g), then G and g have a common coupled fixed point.(A4)
Iff is T- weakly commuting at (z1,z2) and f2z1=fz1, f2z2=fz2 for (z1,z2)∈C(T,f). Then T and f have a common coupled fixed point.(A5)
T,G,f, and g have common coupled fixed point if (A3) and (A4) are true.Proof.
Since(T,f) and (G,g) have (CLR)-property, therefore there exist sequences {xn}, {yn}{un} and {vn} and C1,C2,D1,D2∈CB(Θ) such that (10)limn→∞Txn,yn=C1,limn→∞Gun,vn=C2,limn→∞fxn=fz1∈C1,limn→∞gun=gz1∈C2,limn→∞Tyn,xn=D1,limn→∞Gvn,un=D2,limn→∞fyn=fz2∈D1,limn→∞gvn=gz2∈D2.Putting x=xn,y=yn, u=un, v=vn in inequality (8), we have (11)τ+FHpTxn,yn,Gun,vn≤FφΘxn,yn,un,vn,where (12)Θxn,yn,un,vn=αdpfxn,gun+βdpfyn,Tyn,xndpgvn,Gvn,un1+dpfxn,gun+γdpfxn,Txn,yn+dpgun,Gun,vn+σdpfyn,Tyn,xn+ηdpgun,Gun,vn.Applying limit to Θ, we have (13)limn→∞Θxn,yn,un,vn=αdpfz1,gz1+βdpfz2,D1dpgz2,D21+dpfz1,gz1+γdpfz1,C1+dpgz1,C2+σdpfz2,D1+ηdpgz1,C2,which implies that (14)limn→∞Θxn,yn,un,vn=αdpfz1,gz1.Applying limit to (11) and using (14), we have (15)τ+FHpC1,C2≤Fφαdpfz1,gz1,which implies that (16)FHpC1,C2≤Fφαdpfz1,gz1.Using definitions of F and φ, we have (17)HpC1,C2≤αdpfz1,gz1.But α≤1 and using Lemma 11(18)dpfz1,gz1≤HpC1,C2<dpfz1,gz1,which is contradiction. Hence, dp(fz1,gz1)=0. Therefore (19)fz1=gz1.Putting x=yn,y=xn, u=vn, v=un in inequality (8), we have (20)τ+FHpTyn,xn,Gvn,un≤FφΘyn,xn,vn,un,where (21)Θyn,xn,vn,un=αdpfyn,gvn+βdpfxn,Txn,yndpgun,Gun,vn1+dpfyn,gvn+γdpfyn,Tyn,xn+dpgvn,Gvn,un+σdpfxn,Txn,yn+ηdpgvn,Gvn,un.Applying limit to Θ, we have (22)limn→∞Θyn,xn,vn,un=αdpfz2,gz2+βdpfz1,C1dpgz1,C21+dpfz2,gz2+γdpfz2,D1+dpgz2,D2+σdpfz1,C1+ηdpgz2,D2.(23)limn→∞Θyn,xn,vn,un=αdpfz2,gz2.Applying limit to (20) and using (23), we have(24)fz2=gz2.Putting x=yn,y=xn, u=z1, v=z2 in inequality (8), we have (25)τ+FHpTxn,yn,Gz1,z2≤FφΘxn,yn,z1,z2,where (26)Θxn,yn,z1,z2=αdpfxn,gz1+βdpfyn,Tyn,xndpgz2,Gz2,z11+dpfxn,gz1+γdpfxn,Txn,yn+dpgz1,Gz1,z2+σdpfyn,Tyn,xn+ηdpgz1,Gz1,z2.Applying limit to Θ, we have (27)limn→∞Θxn,yn,z1,z2=αdpfz1,gz1+βdpfz2,D1dpgz2,Gz2,z11+dpfz1,gz1+γdpfz1,C1+dpgz1,Gz1,z2+σdpfz2,D1+ηdpgz1,Gz1,z2,which implies that (28)limn→∞Θxn,yn,z1,z2=γdpgz1,Gz1,z2+ηdpgz1,Gz1,z2,and we get (29)limn→∞Θxn,yn,z1,z2=γ+ηdpgz1,Gz1,z2.Applying limit to (25) and using (29), we get (30)τ+FHpC1,Gz1,z2≤Fφγ+ηdpgz1,Gz1,z2,which implies that(31)FHpC1,Gz1,z2≤Fφγ+ηdpgz1,Gz1,z2.Using definitions of F and φ and using Lemma 9, we have (32)HpC1,Gz1,z2≤γ+ηdpgz1,Gz1,z2,which implies that (33)gz1∈Gz1,z2.
Puttingx=yn,y=xn, u=z2, v=z1 in inequality (8), we have(34)τ+FHpTyn,xn,Gz2,z1≤FφΘyn,xn,z2,z1,where (35)Θyn,xn,z2,z1=αdpfyn,gz2+βdpfxn,Txn,yndpgz1,Gz1,z21+dpfyn,gz2+γdpfyn,Tyn,xn+dpgz2,Gz2,z1+σdpfxn,Txn,yn+ηdpgz2,Gz2,z1.Applying limit to Θ, we have (36)limn→∞Θyn,xn,z2,z1=αdpfz2,gz2+βdpfz1,C1dpgz1,Gz1,z21+dpfz1,gz1+γdpfz1,C1+dpgz2,Gz2,z1+σdpfz1,C1+ηdpgz2,Gz2,z1,which implies that (37)limn→∞Θyn,xn,z2,z1=γdpgz2,Gz2,z1+ηdpgz2,Gz2,z1,(38)limn→∞Θyn,xn,z2,z1=γ+ηdpgz2,Gz2,z1.Applying limit to (25) and using (38), we get (39)τ+FHpC1,Gz2,z1≤Fφγ+ηdpgz2,Gz2,z1,which implies that(40)FHpC1,Gz2,z1≤Fφγ+ηdpgz2,Gz2,z1.Using definitions of F and φ and using Lemma 9, we have (41)HpC1,Gz2,z1≤γ+ηdpgz2,Gz2,z1,and we obtained (42)gz2∈Gz2,z1.
Similarly by puttingx=z1,y=z2 and u=un,v=vn and x=z2,y=z1 and u=vn,v=un we can obtained (43)fz1∈Tz1,z2and (44)fz2∈Tz2,z1.
Sinceg and G-weakly are commuting then g2z1∈G(fz1,gz2), g2z2∈G(fz2,fz1). Since g2z1=gz1, g2z2=gz2. Thus (gz1,gz2) is a common fixed point. A similar argument proves (A4). Then using (A3) and (A4), (A5) hold immediately.Theorem 20.
Letf,g:Θ→Θ and T,G:Θ×Θ→CB(Θ) are mapping on metric space (Θ,d). Furthermore assume that (T,f) and (G,g) have (CLR)-property and (45)τ+FHpTx,y,Gu,v≤FφΘx,y,u,v,where H(T(x,y),G(u,v))>0 and (46)Θx,y,u,v=max⁡dpfx,Tx,y,dpgu,Gu,v,dpfx,gu,dpfx,Tx,y+dpgu,Gu,v2,dpfy,Ty,xdpgv,Gv,u1+dpfx,gu,dpgv,Gv,udpfy,Ty,x1+dpfx,fu,dpfx,Tx,ydpgu,Gu,v1+DpTx,y,Gu,v.Here, τ∈R+, p≥1, F∈Fs, and φ∈Φ. Then the following holds. (A1)
(g,G) have coupled coincidence point.(A2)
(f,T) have coupled coincidence point.(A3)
Ifg is G weakly commuting at (w1,w2) and g2w1=gw1, g2w2=gw2 for w1,w2∈CG,g, then G and g have a common coupled fixed point.(A4)
Iff is T weakly commuting at (z1,z2) and f2z1=fz1, f2z2=fz2 for (z1,z2)∈C(T,f), then T and f have a common coupled fixed point.(A5)
T,G,f, and g have common coupled fixed point if (A3) and (A4) are true.Proof.
Since(T,f) and (G,g) have (CLR)-property, therefore there exist sequences {xn}, {yn}{un} and {vn} and C1,C2,D1,D2∈CB(Θ) such that (47)limn→∞Txn,yn=C1,limn→∞Gun,vn=C2,limn→∞fxn=fz1∈C1,limn→∞gun=gz1∈C2,limn→∞Tyn,xn=D1,limn→∞Gvn,un=D2,limn→∞fyn=fz2∈D1,limn→∞gvn=gz2∈D2.Putting x=xn,y=yn, u=un, v=vn in inequality (45), we get (48)τ+FHpTxn,yn,Gun,vn≤FφΘxn,yn,un,vn,where (49)Θxn,yn,un,vn=max⁡dpfxn,Txn,yn,dpgun,Gun,vn,dpfxn,gun,dpfxn,Txn,yn+dpgun,Gun,vn2,dpfyn,Tyn,xndpgvn,Gvn,un1+dpfxn,gun,dpgvn,Gvn,undpfyn,Tyn,xn1+dpfxn,fun,dpfxn,Txn,yndpgun,Gun,vn1+DpTxn,yn,Gun,vn.Applying limit to Θ, we have (50)limn→∞Θxn,yn,un,vn=max⁡dpfz1,C1,dpgz2,C2,dpfz1,gz1,dpfz1,C1+dpgz1,C22,dpfz2,D1dpgz2,D21+dpfz1,gz2,dpgz2,D2dpfz2,D11+dpfz1,gz2,dpfz1,C1dpgz2,C21+DpC1,C2.(51)limn→∞Θxn,yn,un,vn=dpfz1,gz1.
Applying limit to (48) and using (51) we can deduce that (52)τ+FHpC1,C2≤Flimn→∞φΘxn,yn,un,vn,
which implies that(53)FHpC1,C2≤Fdpfz1,gz1.Using definitions of F and φ, we have (54)HpC1,C2≤φdpfz1,gz1.Using Lemma 11, we have (55)dpfz1,gz1≤HpC1,C2≤φdpfz1,gz1.Thus, we have (56)fz1=gz1.
Similarly by takingx=yn,y=xn, u=vn, v=un in inequality (45) we get (57)τ+FHpTyn,xn,Gvn,un≤FφΘyn,xn,vn,un,where (58)Θyn,xn,vn,un=max⁡dpfyn,Tyn,xn,dpgvn,Gvn,un,dpfyn,gvn,dpfyn,Tyn,xn+dpgvn,Gvn,un2,dpfxn,Txn,yndpgun,Gun,vn1+dpfyn,gvn,dpgun,Gun,vndpfxn,Txn,yn1+dpfyn,fvn,dpfyn,Tyn,xndpgvn,Gvn,un1+DpTyn,xn,Gvn,un.Applying limit to Θ we have (59)limn→∞Θyn,xn,vn,un=max⁡dpfz2,D1,dpgz2,D2,dpfz2,gz2,dpfz2,D1+dpgz2,D22,dpfz1,C1dpgz2,C21+dpfz2,gz2,dpgz2,C2dpfz1,C11+dpfz2,gz2,dpfz2,D1dpgz2,D21+DpD1,D2,(60)limn→∞Θyn,xn,vn,un=dpfz2,gz2.
Applying limit to (57) and using (60) we can deduce that (61)fz2=gz2.
By takingx=xn,y=yn, u=z1, v=z2 in inequality (45), we get (62)τ+FHpTxn,yn,Gz1,z2≤FφΘxn,yn,z1,z2,where (63)Θxn,yn,z1,z2=max⁡dpfxn,Txn,yn,dpgz1,Gz1,z2,dpfxn,gz1,dpfxn,Txn,yn+dpgz1,Gz1,z22,dpfyn,Tyn,xndpgz2,Gz2,z11+dpfxn,gz1,dpgz2,Gz2,z1dpfyn,Tyn,xn1+dpfxn,fz1,dpfxn,Txn,yndpgz1,Gz1,z21+DpTxn,yn,Gz1,z2.Applying limit to Θ, we have (64)limn→∞Θxn,yn,z1,z2=max⁡dpfz1,C1,dpgz1,Gz1,z2,dpfz1,gz1,dpfxn,Txn,yn+dpgz1,Gz1,z22,dpfyn,Tyn,xndpgz2,Gz2,z11+dpfxn,gz1,dpgz2,Gz2,z1dpfyn,Tyn,xn1+dpfxn,fz1,dpfxn,Txn,yndpgz1,Gz1,z21+DpTxn,yn,Gz1,z2,(65)limn→∞Θxn,yn,z1,z2=dpgz1,Gz1,z2.
Applying limit to (62) using (65), we have (66)gz1∈Gz1,z2.By taking x=yn,y=xn, u=z2, v=z1 in inequality (45) we get (67)τ+FHpTyn,xn,Gz2,z1≤FφΘyn,xn,z2,z1where(68)Θyn,xn,z2,z1=max⁡dpfyn,Tyn,xn,dpgz2,Gz2,z1,dpfyn,gz2,dpfyn,Tyn,xn+dpgz2,Gz2,z12,dpfxn,Txn,yndpgz1,Gz1,z21+dpfyn,gz2,dpgz1,Gz1,z2dpfxn,Txn,yn1+dpfyn,fz2,dpfyn,Tyn,xndpgz2,Gz2,z11+DpTyn,xn,Gz2,z1.Applying limit to Θ, we have (69)limn→∞Θyn,xn,z2,z2=max⁡dpfz2,C2,dpgz2,Gz2,z1,dpfz2,gz2,dpfz2,C2+dpgz2,Gz2,z12,dpfz1,C1dpgz1,Gz1,z21+dpfz2,gz2,dpgz1,Gz1,z2dpfz1,C11+dpfz2,fz2,dpfz2,C2dpgz2,Gz2,z11+DpC2,Gz1,z2,(70)limn→∞Θxn,yn,z1,z2=dpgz2,Gz2,z1.
Applying limit to (67) using (70), we have (71)gz2∈Gz2,z1.Following the similar line of Theorem 19 we can obtain that T,G,f, and g have common coupled fixed point.Example 21.
LetΘ=(-10,10) with the usual metric. Define T,G:Θ×Θ→CB(Θ), f,g:Θ→Θ, φ:R+→R+ and F:R+→R by T(x,y)=[-5,1+αx/4], G(x,y)=[-5,1+α/6x], f(x)=x/2, g(x)=x/3∀x,y∈Θ, φ(t)=αt, 0<α<1 and F=ln⁡(x).
Consider the sequences{xn}={1+1/n}, {yn}={2-3/n2}, {un}={1-1/n}, {vn}={2-2/n}.
Now,(72)limn→∞Txn,yn=-5,1+α4,limn→∞fxn=12=f1∈-5,1+α4.limn→∞Gun,vn=-5,1+α6,limn→∞gun=13=g1∈-5,1+α6.limn→∞Tyn,xn=-5,1+α2,limn→∞fyn=1=f2∈-5,1+α2,limn→∞Gvn,un=-5,1+α3,limn→∞gvn=23=g2∈-5,1+α3.Therefore (f,T) and (g,G) satisfy CLR property. Now, (73)HTx,y,Gu,v=H-5,1+αx4,-5,1+αu6=max⁡d-5,1+αx4,-5,1+αu6,d-5,1+αx6,-5,1+αu4,=max⁡αx6-αu4,0,=α2dgx,fu=12φdgx,fu≤12φΘx,y,u,v≤e-1/6φΘx,y,u,v.Taking logarithm on both sides and p=1, we conclude that all the other conditions of Theorem 19 are satisfied. Therefore T,f and G,g have common coupled fixed point.Example 22.
LetΘ=(-1,∞) with the usual metric. Define T,G:Θ×Θ→CB(Θ), f,g:Θ→Θ, φ:R+→R+, F:R+→R by T(x,y)=[-0.5,12+αx2], G(x,y)=[-0.5,12+3α/2x2], f(x)=2x2, g(x)=3x2∀x,y∈Θ, φ(t)=αt, 0<α<1 and F=ln⁡(x).
Consider the sequences{xn}={1+1/n}, {yn}={2-3/n2}, {un}={1-1/n}, {vn}={2-2/n}.
Now,(74)limn→∞Txn,yn=-0.5,12+α,limn→∞fxn=2=f1∈-0.5,12+α.limn→∞Gun,vn=-0.5,12+32α,limn→∞gun=3=g1∈-0.5,12+32α.limn→∞Tyn,xn=-0.5,12+4α,limn→∞fyn=8=f2∈-0.5,12+4α,limn→∞Gvn,un=-0.5,12+6α,limn→∞gvn=12=g2∈-0.5,12+6α.Therefore (f,T) and (g,G) satisfy CLR property. Now, (75)HTx,y,Gu,v=H-0.5,12+αx2,-0.5,12+3α2u2=max⁡d-0.5,12+αx2,-0.5,12+3α2u2,d-0.5,12+3α2x2,-0.5,12+αu2,=max⁡3α2u2-αx2,0,=α2dgu,fx=12φdgu,fx≤12φΘx,y,u,v≤e-1/6φΘx,y,u,v.Taking logarithm on both sides and p=1, we conclude that all the other conditions of Theorem 19 are satisfied. Therefore, T,f and G,g have common coupled fixed point.Remark 23.
From the above examples the following is clear.(i)
Theorem13 is not applicable to Example 21 because f(Θ)=(-5,5) nor g(Θ)=-10/3,10/3 are closed.(ii)
Theorem14 is not applicable to Example 22, because neither T(Θ×Θ)⊆f(Θ) nor G:(Θ×Θ)⊆g(Θ).(iii)
Similarly the main results of [43] Theorem 2.1 and Theorem 2.6 are not applicable to the above examplesNext, we explain Example 27 of [27] to which our Theorem 19 is also applicable.Example 24.
LetΘ=[0,1], equipped with the metric d:Θ×Θ→[0,∞) define by (76)dx,y=max⁡x,y,forallx≠y∈Θdx,y=0,forallx=y∈Θ.Define T,G:Θ×Θ→CB(Θ), f,g:Θ→Θ, φ:R+→R+ and F:R+→R by (77)Tx,y=0,ifx,y=10,x2+y24ifx,y∈0,1,Gx,y=0,ifx,y=10,x+y8ifx,y∈0,1,fx=x2,ifx≠132,ifx=1,gx=x2,ifx≠11,ifx=1,φt=t2.And (78)F=ln⁡x3Consider the sequences {xn}=1/n, yn=3/n2, {un}=1/n2, {vn}={2/n}.
Now,(79)limn→∞Txn,yn=0,limn→∞fxn=0=f0∈0.limn→∞Gun,vn=0,limn→∞gun=0=g0∈0.limn→∞Tyn,xn=0,limn→∞fyn=0=f0∈0,limn→∞Gvn,un=0,limn→∞gvn=0=g0∈0.Therefore (f,T) and (g,G) satisfy CLR property.
Now forx,y,u,v∈[0,1), we discuss the following cases.Case 1.
Ifx2+y2/4=u+v/8, then (80)13HTx,y,Gu,v=13u+v8≤1314max⁡x2,u2+max⁡v2,u+v8=131/2dfx,gu+1/2dgv,Gu,v2=13φ12dfx,gu+12dgv,Gu,v=13φΘx,y,u,v≈e-0.00000000113φΘx,y,u,v.Case 2.
Ifx2+y2/4≠u+v/8 and x2+y2/4<u+v/8, then (81)13HTx,y,Gu,v=13u+v8≤1314max⁡x2,u2+max⁡v2,u+v8=131/2dfx,gu+1/2dgv,Gu,v2=13φ12dfx,gu+12dgv,Gu,v=13φΘx,y,u,v≈e-0.00000000113φΘx,y,u,v.Case 3.
Ifx2+y2/4>u+v/8, then (82)13HTx,y,Gu,v=13x2+y24≤1314max⁡x2,u2+max⁡y2,x2+y24=131/2dfx,gu+1/2dfy,Ty,x2=13φ12dfx,gu+12dgv,Gu,v=13φΘx,y,u,v≈e-0.00000000113φΘx,y,u,v,Similarly it is easy to show the same result for x,y∈[0,1) and u,v=1 and for x,y,u,v=1. Taking logarithm on both sides and p=1. we conclude that all conditions of our Theorem 19 are satisfied. Therefore T,f and G,g have common coupled fixed point.Remark 25.
(i) From Example24 it is clear that all conditions of our Theorem 19 are satisfied for Example 27 of [27] and hence the corresponding conclusions hold.


## 3. Applications to System of Functional Equations

In this section, we discuss common solution for two coupled functional equations with the help of Theorem19. Throughout this unit Z^ and Y^ stand for Banach spaces, the state space is E~⊂Z^, the decision space is F~⊂Y^, and the space of all bounded real-valued functions on E~ is Θ=B(E~) which is Banach space.Defined:Θ×Θ→R+, by (83)du´1,u´2=supx∈E~u´1x-u´2x=u´1-u´2.Here, (84)u´=supu´x:κ11∈E~∀u´∈BE~.Consider the following system(85)g1κ11=supκ22∈F~μκ11,κ22+Ψ1κ11,κ22,g1b1κ11,κ22,g2b2κ11,κ22∀κ11∈E~,g2κ11=supκ22∈F~μκ11,κ22+Ψ2κ11,κ22,g2b2κ11,κ22,g1b1κ11,κ22∀κ11∈E~,where μ:E~×F~→R, bi:E~×F~→E~, Ψi:E~×F~×R→R for i=1,2 and κ11, κ22 denote the state vectors and decision vectors, respectively, b1,b2 signify the transformations of the process, and g1(κ11),g2(κ11) symbolized the sup return functions under the initial state κ11.LetT,G:B(E~)×B(E~)→B(E~), defined by (86)Tg1,g2κ11=supκ22∈F~μκ11,κ22+Ψ1κ11,κ22,g1b1κ11,κ22,g2b2κ11,κ22,Gg1,g2κ11=supκ22∈F~μκ11,κ22+Ψ2κ11,κ22,g2b2κ11,κ22,g1b1κ11,κ22.Theorem 26.
AssumeT,G:B(E~)×B(E~)→B(E~) to be maps given by (86) which holds the following conditions. (1)
μ and Ψi, for i=1,2, are bounded.(2)
For(κ11, κ22)∈E~×F~ and g1,g1′,g2,g2′∈B(E~)(87)Ψ1κ11,κ22,g1b1κ11,κ22,g2b2κ11,κ22-Ψ2κ11,κ22,g2′b2κ11,κ22,g1′b1κ11,κ22≤e-τφΘx,y,u,v.
Here,(88)Θx,y,u,v=αdpx,u+βdpy,Ty,xdpv,Gv,u1+dpx,u+γdpx,Tx,y+dpu,Gu,v+σdpx,Tx,y+ηdpu,Gu,v.Then, system (85) has a common solution in B(E~).Proof.
Letλ be an arbitrary positive real number and there exist g1,g1′,g2,g2′∈B(E~), for arbitrary κ11∈E~, κ22∈F~ such that (89)Tg1,g2κ11≤μκ11,κ22+Ψ1κ11,κ22,g1b1κ11,κ22,g2b2κ11,κ22+λ,(90)Gg1′,g2′κ11≤μκ11,κ22+Ψ2κ11,κ22,g2′b2κ11,κ22,g1′b1κ11,κ22+λ.
From definition ofT and G we have (91)Tg1,g2κ11>μκ11,κ22+Ψ1κ11,κ22,g1b1κ11,κ22,g2b2κ11,κ22,(92)Gg1′,g2′κ11>μκ11,κ22+Ψ2κ11,κ22,g2′b2κ11,κ22,g1′b1κ11,κ22.Next, from (89) and (92) we have (93)Tg1,g2κ11-Gg1′,g2′κ11≤μκ11,κ22+Ψ1κ11,κ22,g1b1κ11,κ22,g2b2κ11,κ22-μκ11,κ22-Ψ2κ11,κ22,g2′b2κ11,κ22,g1′b1κ11,κ22+λ≤Ψ1κ11,κ22,g1b1κ11,κ22,g2b2κ11,κ22-Ψ2κ11,κ22,g2′b2κ11,κ22,g1′b1κ11,κ22+λ≤e-τφΘx,y,u,v+λ.Similarly from (90) and (91) we get (94)Gg1′,g2′κ11-Tg1,g2κ11≤e-τφΘx,y,u,v+λ.Combining (93) and (94) we conclude that (95)Gg1′,g2′κ11-Tg1,g2κ11≤e-τφΘx,y,u,v+λ.By taking f(x)=g(x)=x, F=ln⁡(x) and p=1 in Theorem 19. Then we deduce that the mappings T,G have a common coupled fixed point in B(E~); that is the system (85) has a solution.


## 4. Applications to Matrix Equations

In this section, we study the nonlinear matrix equations with the help of Theorem20. (96)Δ1=Q+∑i=1mAi∗G1Δ1Ai-∑j=1kBj∗K1Δ1Bj,(97)Δ2=Q+∑i=1mAi∗G2Δ1Ai-∑j=1kBj∗K2Δ1Bj.Here Q is a positive definite matrix, Ai,Bj are arbitrary n×n matrices, and continuous order preserving maps are G1,G2,K1,K2 defined from H(n) into P(n) such that G1(0)=G2(0)=K1(0)=K2(0)=0.In this unit we will use the following notations:M ( n ) symbolizes the set of all n×n complex matrices, H(n)⊂M(n) the set of all n×n Hermitian matrices, and P(n)⊂H(n) is the set of all n×n positive definite matrices. As a replacement for of Δ1∈P(n) we will also write Δ1>0. Similarly, positive semidefinite matrix Δ1 is denoted by Δ1≥0. We also signify by . the spectral norm, i.e., (98)B=λ+B∗B,where the biggest eigenvalue of B∗B is λ+(B∗B). We will use the metric induced by the trace norm .1 defined by B1=∑q=1n‍Sq(B), where Sq(B),q=1,…,n are the singular values of B. The set H(n) is a complete metric space endowed with this norm.The following lemma which is taken from [6] will be useful in the study of the matrix equations.Lemma 27.
LetA≥0 and B≥0 be n×n matrices; then 0≤tr(AB)≤Atr(B).In this section, we define the mappingT,G:H(n)×H(n)→H(n) by(99)TΔ1,Δ2=Q+∑i=1mAi∗G1Δ1Ai-∑j=1kBj∗K1Δ2Bj,(100)GΔ1,Δ2=Q+∑i=1mAi∗G2Δ1Ai-∑j=1kBj∗K2Δ2Bj.Here Q∈P(n), Ai,Bj∈M(n), and G1,G2,K1,K2 are continuous order-preserving maps. In the following theorem we first discuss the existence of common coupled fixed point of T and G in H(n)×H(n).Theorem 28.
LetQ∈P(n) such that (1)
for every(Δ1,Δ2),(Δ3,Δ4)∈H(n)×H(n), and (101)trG1Δ3-G2Δ1≤e-τtrΔ3-Δ1,and(102)trK1Δ2-K2Δ4≤e-τtrΔ2-Δ4,(2)
Δ2-Δ4≤Δ3-Δ1,(3)
∑i=1m‍AiAi∗<1/4 and ∑j=1k‍BjBj∗<1/4.
Then, there existΔ1∗,Δ2∗∈H(n) such that T(Δ1∗,Δ2∗)=Δ1∗ and T(Δ2∗,Δ1∗)=Δ2∗. G(Δ1∗,Δ2∗)=Δ1∗ and G(Δ2∗,Δ1∗)=Δ2∗.Proof.
Let(Δ1,Δ2),(Δ3,Δ4)∈H(n)×H(n); then (103)TΔ3,Δ4-GΔ1,Δ21=trTΔ3,Δ4-GΔ1,Δ2=∑i=1mtrAi∗G1Δ3-G2Δ1Ai+∑j=1ktrBj∗K1Δ2-K2Δ4Bj=∑i=1mtrAiAi∗G1Δ3-G2Δ1+∑j=1ktrBjBj∗K1Δ2-K2Δ4=tr∑i=1mAiAi∗G1Δ3-G2Δ1+tr∑j=1kBjBj∗K1Δ2-K2Δ4≤∑i=1mAiAi∗trG1Δ3-G2Δ1+∑j=1kBjBj∗trK1Δ2-K2Δ4≤∑i=1mAiAi∗e-τΔ3-Δ11+∑j=1kBjBj∗e-τΔ2-Δ41≤∑i=1mAiAi∗e-τΔ3-Δ11+∑j=1kBjBj∗e-τΔ3-Δ11≤12e-τΔ3-Δ11,≤e-τφΘΔ1,Δ2,Δ3,Δ4.Thus, the contractive condition of Theorem 20 is satisfied for all (Δ1,Δ2),(Δ3,Δ4)∈H(n)×H(n). By taking f(x)=g(x)=x, φ(t)=1/2t, F=ln⁡(x) and p=1 in Theorem 20. From Theorem 20, T and G have a common coupled fixed point.



---

*Source: 6870308-2019-04-01.xml*