# Some Fixed Point Results for Perov-Ćirić-Prešić Type F-Contractions with Application

Authors: Abdolsattar Gholidahneh; Shaban Sedghi; Vahid Parvaneh
Journal: Journal of Function Spaces
 (2020)

---

## 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.


---

## Body

## 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].
LetΔ,d be a complete metric space and let ϒ:Δ⟶Δ so that
(1)dϒι,ϒκ≤γdι,κforallι,κ∈Δ,where γ∈0,1. Then, there is a unique σ in Δ such that σ=ϒσ. Also, for each ζ0∈Δ, the sequence ζn+1=ϒζn converges to σ.The BCP has been extended and generalized in many directions (see [2–4]).Prešić [5] gave the following result.Theorem 2 [5].
LetΔ,d be a complete metric space and let ϒ:Δk⟶Δ (k is a positive integer). Suppose that
(2)dϒζ1,⋯,ζk,ϒζ2,⋯,ζk+1≤∑i=1kλidζi,ζi+1,for all ζ1,⋯,ζk+1 in Δ, where λi≥0 and ∑i=1kλi∈0,1. Then ϒ has a unique fixed point ζ∗ (that is ϒζ∗,⋯,ζ∗=ζ∗). Moreover, for all arbitrary points ζ1,⋯,ζk+1 in Δ, the sequence ζn defined by ζn+k=ϒζn,ζn+1,⋯,ζn+k−1, converges to ζ∗.It is obvious that fork=1, Theorem 2 coincides with the BCP.Theorem2 was generalized by Ćirić and Prešić [6] as follows.Theorem 3 [6].
LetΔ,d be a complete metric space and ϒ:Δk⟶Δ (k is a positive integer). Suppose that
(3)dϒζ1,⋯,ζk,ϒζ2,⋯,ζk+1≤λmaxdζi,ζi+1:1≤i≤k,for all ζ1,⋯,ζk+1 in Δ, where λ∈0,1. Then ϒ has a fixed point ζ∗∈Δ. Also, for all points ζ1,⋯,ζk+1∈Δ, the sequence ζn defined by ζn+k=ϒζn,ζn+1,⋯,ζn+k−1, converges to ζ∗. The fixed point of ϒ is unique if
(4)dϒρ,⋯,ρ,ϒρ,⋯,ρ<dρ,ρ,for all ρ,ϱ∈Δ with ρ≠ϱ.For more details on Prešić type contractions, we refer the reader to [2, 5, 7–11].In this paper,ℝ+=0,∞, ℝm denotes the set of m×1 real matrices, ℝ+m will be the set of m×1 real matrices with elements in 0,∞, θ denotes the zero m×1 matrix, Mm,mℝ+ denotes the set of all m×m matrices with elements in 0,∞, and Θ will be the zero m×m matrix, by I the identity m×m matrix. If A∈Mm,mℝ+, then AT states the transpose matrix of A. Let π=πii=1m,ϖ=ϖii=1m∈ℝm, then by π≺¯ϖ (resp. π≺ϖ), we suppose πi≤ϖi (resp. πi<ϖi) for each i∈1,2,⋯,m. Also, π≺¯ϖ and ϖ≻¯π will mean the same.LetΔ be a nonempty set and let V:Δ×Δ⟶ℝm be a function. V is called a vector-valued metric, and Δ,V is called a vector-valued metric space, if
(1)
Vι,κ=θ if and only if ι=κ,(2)
Vι,κ=Vκ,ι,(3)
Vι,κ≺¯ι,σ+Vσ,κ,for all ι,κ,σ∈Δ.Example 1 (Example 1.3. of [12]).
LetD1,D2,⋯,Dn be usual metrics on X.Then, the mappingd:X×X⟶Rn defined by dx,y=D1x,y,D2x,y,⋯,Dnx,y is a VVM on X.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:LetA∈Mm,mℝ+. Then A is said to converge to zero if and only if An⟶Θ as n⟶∞ (see [14]).Perov [13] proved the following interesting extension of BCP (see more results in [15–20]).Theorem 4 [13].
LetΔ,V be a VVMS and ϒ:Δ⟶Δ be a mapping such that there exists a matrix A∈Mm,mℝ+ such that
(5)Vϒι,ϒκVAVι,κ,for all ι,κ∈Δ. If A is convergent to zero, then
(1)
ϒ has a unique fixed point σ in Δ(2)
for allx0∈Δ, the sequence xn defined by xn=ϒnx0 is convergent to σ(3)
Vxn,σVAnI−A−1Vx0,ϒx0.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 F:ℝ+m⟶ℝm be a function. Let
(F1)
F be strictly increasing in each variable, i.e., π≺ϖ; then, Fπ≺Fϖ, for all π=πii=1m and ϖ=ϖii=1m∈ℝ+m,(F2)
For each sequenceπn=πn1,πn2,⋯,πnm of ℝ+m(6)limn→∞πni=0ifandonlyiflimn→∞ϖni=−∞,for each i∈1,2,⋯m, where
(7)Fπn1,πn2,⋯,πnm=ϖn1,ϖn2,⋯,ϖnm.(F3)
There existsk∈0,1 such that limπi→0+πikϖi=0 for each i∈1,2,⋯m, where(8)Fπ1,π2,⋯,πm=ϖ1,ϖ2,⋯,ϖm.We denote byFm the set of all functions F satisfying (F1)–(F3)Example 2 [24].
DefineF:ℝ+m⟶ℝm by
(9)Fπ1,π2,⋯,πm=lnπ1,lnπ2,⋯,lnπm,then F∈Fm.Note that we can defineG:ℝm⟶ℝ+m by
(10)Gϖ1,ϖ2,⋯,ϖm=eϖ1,eϖ2,⋯,eϖm,which we can treat it as the inverse of multivariable function F.Note that from now on,F is a continuously differentiable function from all open sets of A⊆ℝn⟶ℝn, and the Jacobian determinant of F at every p∈A⊆ℝn is nonzero; then, according to inverse function theorem, F is invertible near p.Example 3 [24].
DefineF:ℝ+2⟶ℝ2 by
(11)Fπ1,π2=lnπ1,π2+lnπ2,then F∈F2.Example 4 [24].
DefineF:ℝ+3⟶ℝ3 by
(12)Fπ1,π2,π3=lnπ1,π2+lnπ2,−1+π1π3,then F∈F3.Considering the classFm, Altun and Olgun [24] introduced the concept of Perov type F-contraction as follows:Definition 5 [24].
LetΔ,V be a VVMS and ϒ:Δ⟶Δ be a map. If there exist F∈Fm and ς=ςii=1m∈ℝ+m such that
(13)ς+FVϒι,ϒyVFVι,κ,for all ι,κ∈Δ with Vϒι,ϒκ≻θ, then ϒ is called a Perov type F-contraction.If we considerF:ℝ+m⟶ℝm by
(14)Fπ1,π2,⋯,πm=lnπ1,lnπ2,⋯,lnπm,then (13) turns to Perov contraction [24].We can present new type contractions in a VVMS, via considering some functionF∈Fm in (13).Theorem 6 [24].
LetΔ,V be a complete VVMS and let ϒ:Δ⟶Δ be a Perov type F-contraction. Then ϒ 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.
LetΔ,V be a complete VVMS and let ϒ:Δk⟶Δ (k is a positive integer). Assume that there exist F∈Fm and ς=ςii=1m∈ℝ+m satisfying
(15)ς+FVϒζ1,⋯,ζk,ϒζ2,⋯,ζk+1≤FsupVζ1,ζ2,Vζ2,ζ3,⋯,Vζk,ζk+1,for all ζ1,⋯,ζk+1∈Δ with Vϒζ1,⋯,ζk,ϒζ2,⋯,ζk+1≻θ. Moreover, let there exists a sequence ζn in Δ such that ζn+k=ϒζn,ζn+1,⋯,ζn+k−1 and Vζn+k,ζn+k+1≻θ, for all n∈ℕ. Also, if ζn⟶v, then Vζn,v≻θ, for all n∈ℕ. Then, the sequence ζn converges to a fixed point of ϒ. Moreover, if for all ρ,ϱ∈Δ with ρ≠ϱ,
(16)Vϒρ,⋯,ρ,ϒρ,⋯,ρ<Vρ,ρ,then the fixed point of ϒ is unique.Proof.
For anyn∈ℕ, we have
(17)FVζn+k,ζn+k+1=FVϒζn,⋯,ζn+k−1,ϒζn+1,⋯,ζn+k≤FsupVζn,ζn+1,Vζn+1,ζn+2,⋯,Vζn+k−1,ζn+k−ς.Therefore,(18)FVζk+1,ζk+2=FVϒζ1,⋯,ζk,ϒζ2,⋯,ζk+1≤FsupVζ1,ζ2,Vζ2,ζ3,⋯,Vζk,ζk+1−ς=FΛ−ς,where Λ=supVζ1,ζ2,Vζ2,ζ3,⋯,Vζk,ζk+1. Now,
(19)FVζk+2,ζk+3=FVϒζ2,⋯,ζk+1,ϒζ3,⋯,ζk+2≤FsupVζ2,ζ3,Vζ3,ζ4,⋯,Vζk+1,ζk+2−ς≤FmaxΛ,F−1FΛ−ς−ς=FΛ−ς.Continuing this approach, we have(20)FVζ2k,ζ2k+1=FVϒζk,⋯,ζ2k−1,ϒζk+1,⋯,ζ2k≤FsupVζk,ζk+1,Vζk+1,ζk+2,⋯,Vζ2k−1,ζ2k−ς≤FmaxΛ,F−1FΛ−ς−ς=FΛ−ς,FVζ2k+1,ζ2k+2=FVϒζk+1,⋯,ζ2k,ϒζk+2,⋯,ζ2k+1≤FsupVζk+1,ζk+2,Vζk+2,ζk+3,⋯,Vζ2k,ζ2k+1−ς≤FF−1FΛ−ς−ς=FΛ−2ς,FVζ3k,ζ3k+1=FVϒζ2k,⋯,ζ3k−1,ϒζ2k+1,⋯,ζ3k≤FsupVζ2k,ζ2k+1,Vζ2k+1,ζ2k+2,⋯,Vζ3k−1,ζ3k−ς≤FmaxF−1FΛ−ς,F−1FΛ−2ς−ς=FΛ−2ς,FVζ3k+1,ζ3k+2=FVϒζ2k+1,⋯,ζ3k,ϒζ2k+2,⋯,ζ3k+1≤FsupVζ2k+1,ζ2k+2,Vζ2k+2,ζ2k+3,⋯,Vζ3k,ζ3k+1−ς≤FF−1FΛ−2ς−ς=FΛ−3ς.Continuing this process, we get(21)FVζpk+i,ζpk+i+1≤FΛ−pς,forallp∈Vandi∈1,2,⋯,k.Now, takingVζn,ζn+1=πn1,πn2,⋯,πnm and Fπn1,πn2,⋯,πnm=ϖn1,ϖn2,⋯,ϖnm, we obtain that
(22)ϖpk+i1,ϖpk+i2,⋯,ϖpk+im=Fπpk+i1,πpk+i2,⋯,πpk+im≤FΛ−pς=r1−pς1,r2−pς2,⋯,rm−pςm,where Λ=Λii=1m and FΛii=1m=rii=1m. Therefore,
(23)ϖpk+ij≤rj−pςjforallj∈1,2,⋯,m.Passing to the limit, we getlimp→∞ϖpk+ij=−∞. Therefore, limp→∞πpk+ij=0 for all j∈1,2,⋯,m. Thus, limp→∞Vζpk+i,ζpk+i+1=θ. From (F3), there exists λ∈0,1 such that
(24)limp→∞πpk+ijλϖpk+ij=0,forallj∈1,2,⋯,m.From (23),
(25)πpk+ijλϖpk+ij≤πpk+ijλrj−pπpk+ijλςjforallj∈1,2,⋯,m.Therefore,(26)πpk+ijλϖpk+ij−πpk+ijλrj≤−pπpk+ijλςj≤0forallj∈1,2,⋯,m.Thus,limp→∞pπpk+ijλ=0 for all j∈1,2,⋯,m. So, for any j∈1,2,⋯,m, there exists pj∈ℕ such that pπpk+ijλ≤1, for all p≥pj. Thus, πpk+ij≤1/p1/λ, for all p≥pj. Putting p0=maxpj:1≤j≤m, we have πpk+ij≤1/p1/λ for all p≥p0 and all i∈1,2,⋯,k. We claim that ζn is a Cauchy sequence. Consider two elements m,n∈ℕ so that p0k≤n<m. Then, there are p,q∈ℕ and i,j∈1,2,⋯,k such that p0≤p≤q, n=pk+i, and m=qk+j. Now, we have
(27)Vζn,ζm=Vζpk+i,ζqk+j≤∑r=pq∑l=1kVζrk+l,ζrk+l+1≤∑r=pqk1r1/λ,⋯,1r1/λ≤k∑r=pq1r1/λ,⋯,∑r=pq1r1/λ.Asn,m⟶∞, we have p,q⟶∞. Thus, the last term in (27) converges to θ, and so ζn is a Cauchy sequence in Δ,V. Since Δ,V is a complete VVMS, there is v∈Δ so that limn→∞ζn=v. Now, we shall prove that v is a fixed point of ϒ. To see this, we have
(28)Vζn+k,ϒv,⋯,v=Vϒζn,ζn+1⋯,ζn+k−1,ϒv,⋯,v≤Vϒζn,ζn+1,⋯,ζn+k−1,ϒζn+1,ζn+2,⋯,ζn+k−1,v+Vϒζn+1,ζn+2,⋯,ζn+k−1,v,ϒζn+2,ζn+3,⋯,ζn+k−1,v,v+⋯+Vϒζn+k−1,v,⋯,v,ϒv,v,⋯,v≤F−1FmaxVζn,ζn+1,⋯,Vζn+k−2,ζn+k−1,Vζn+k−1,v−ς+F−1FmaxVζn+1,ζn+2,⋯,Vζn+k−2,ζn+k−1,Vζn+k−1,v−ς⋯+F−1FVζn+k−1,v−ς⟶θ,as n⟶∞. Thus,
(29)Vv,ϒv,⋯,v=limn→∞Vζn+k,ϒv,⋯,v=θ.Therefore,v=ϒv,⋯,v. Suppose that u,v are two distinct fixed points of ϒ. From our hypothesis,
(30)Vu,v=Vϒu,⋯,u,ϒv,⋯,v<Vu,v,which is a contradiction. Thus, the fixed point of ϒ is unique.Note that by taking(31)Fπ1,π2,⋯,πm=lnπ1,lnπ2,⋯,lnπm,the above theorem reduces to the following theorem.Theorem 8.
LetΔ,V be a complete VVMS and ϒ:Δk⟶Δ (k is a positive integer). Suppose that there exist F∈Fm and ς=ςii=1m∈ℝ+m satisfying
(32)Vϒζ1,⋯,ζk,ϒζ2,⋯,ζk+1≤AsupVζi,ζi+1:i=1,⋯,k,where
(33)A=e−ς10⋯00e−ς2⋯0⋮⋮⋱⋮00⋯e−ςmm×m.Let the sequenceζn in Δ be such that ζn+k=ϒζn,ζn+1,⋯,ζn+k−1 and Vζn+k,ζn+k+1≻θ, for all n∈ℕ. Also, if ζn⟶v, then Vζn,v≻θ, for all n∈ℕ. Then, the sequence ζn converges to a fixed point of ϒ. Also, if
(34)Vϒρ,⋯,ρ,ϒρ,⋯,ρ<Vρ,ρ,for all ρ,ϱ∈Δ with ρ≠ϱ, then the fixed point of ϒ is unique.We present an example to support our main result.Example 5.
LetΔ=ςn=1/n2:n=1,2,⋯∪ς0=0, Vρ,ϱ=∣ρ−ϱ∣,∣ρ−ϱ∣, and define ϒ:Δ2⟶Δ by
(35)ϒςn,ςm=ςmaxm,n+1,n,m≥1,0,n=0orm=0.Firstly, note that for allm,n∈ℕ∪0 with m≠n, from Example 2.3 of [25], we have
(36)ςn+1−ςm+11/∣ςn+1−ςm+1∣ςn−ςm−1/∣ςn−ςm∣≤12,1∣ςn+1−ςm+1∣−1∣ςn−ςm∣≥1.As we know,ςn+2=minςn+1,ςn+2=ϒςn,ςn+1, for all ℕ and
(37)Vςn+2,ςn+3=∣ςn+2−ςn+3∣,∣ςn+2−ςn+3∣=∣1n+22−1n+32∣,∣1n+22−1n+32∣≻0,0=θ.Also,ςn=1/n2⟶0 and
(38)Vςn,0=∣ςn−0∣,∣ςn−0∣=1n2,1n2≻0,0=θ.DefineF:ℝ+2⟶ℝ2 by
(39)Fπ1,π2=lnπ1π1,−1π2,π1≤eπ1ee,−1π2,π1>e.Obviously,F∈F2. Also, take ς=ς1,ς2=ln2,1. We have
(40)ς+FVϒε,ε,ϒε,σ≤FmaxVε,ε,Vε,σ⇔ln2,1+F∣ϒε,ε−ϒε,σ∣,∣ϒε,ε−ϒε,σ∣≤Fmax∣ε−ε∣,∣ε−σ∣⇔ln2,1+ln∣ϒε,ε−ϒε,σ∣∣ϒε,ε−ϒε,σ∣,−1∣ϒε,ε−ϒε,σ∣≤lnmax∣ε−ε∣,∣ε−σ∣max∣ε−ε∣,∣ε−σ∣,−1max∣ε−ε∣,∣ε−σ∣⇔ln2+ln∣ϒε,ε−ϒε,σ∣∣ϒε,ε−ϒε,σ∣≤lnmax∣ε−ε∣,∣ε−σ∣max∣ε−ε∣,∣ε−σ∣,1+−1∣ϒε,ε−ϒε,σ∣≤−1max∣ε−ε∣,∣ε−σ∣⇔∣ϒε,ε−ϒε,σ∣1/∣ϒε,ε−ϒε,σ∣max∣ε−ε∣,∣ε−σ∣−1/max∣ε−ε∣,∣ε−σ∣≤12,1∣ϒε,ε−ϒε,σ∣−1max∣ε−ε∣,∣ε−σ∣≥1,for any ε,ε,σ∈Δ. Now, Let ε=ςn, ε=ςm, and σ=ςp. If m≥maxn,p, then
(41)Vϒε,ε,ϒε,σ=Vϒςn,ςm,ϒςm,ςp=Vςm+1,ςm+1=0,0=θ.So, we may assume that eitherm<n or m<p. We consider the following cases:Case 1.
n≤m<p. Let n=0. If n=m=0, then
(42)Vϒε,ε,ϒε,σ=0,0=θ.
If0=n<m, then
(43)∣ϒε,ε−ϒε,σ∣1/ϒε,ε−ϒε,σmax∣ε−ε∣,∣ε−σ∣1/max∣ε−ε∣,∣ε−σ∣=ςp+11/ςp+1ςm1/ςm=1p+12p+11m2−m≤1p+12p+11p2−p≤1p+12≤121∣ϒε,ε−ϒε,σ∣−1max∣ε−ε∣,∣ε−σ∣=1ςp+1−1ςm=p+1−m≥p+1−p=1,and if n>0, then
(44)∣ϒε,ε−ϒε,σ∣1/∣ϒε,ε−ϒε,σ∣max∣ε−ε∣,∣ε−σ∣−1/max∣ε−ε∣,∣ε−σ∣≤ςm+1−ςp+11/ςm+1−ςp+1ςm−ςp−1/ςm−ςp≤121∣ϒε,ε−ϒε,σ∣−1max∣ε−ε∣,∣ε−σ∣=1ςp+1−1ςm=p+1−m≥p+1−p=1.Case 2.
m<p≤n. Here, if m=0, then
(45)∣ϒε,ε−ϒε,σ∣1/∣ϒε,ε−ϒε,σ∣max∣ε−ε∣,∣ε−σ∣−1/max∣ε−ε∣,∣ε−σ∣=ςn+11/ςn+1ςn−1/ςn=1n+12n+11n2−n≤1n+12≤12,and
(46)1∣ϒε,ε−ϒε,σ∣−1max∣ε−ε∣,∣ε−σ∣=1ςn+1−1ςn=n+1−n≥n+1−n=1,and if m>0, then
(47)∣ϒε,ε−ϒε,σ∣1/∣ϒε,ε−ϒε,σ∣max∣ε−ε∣,∣ε−σ∣−1/max∣ε−ε∣,∣ε−σ∣≤ςn+1−ςp+11/ςn−ςpςn−ςp−1/ςn−ςp≤12,and
(48)1∣ϒε,ε−ϒε,σ∣−1max∣ε−ε∣,∣ε−σ∣=1ςn+1−ςp+1−1ςn−ςp≥1.Case 3.
m≤n<p. In this case, if m=0, then
(49)∣ϒε,ε−ϒε,σ∣1/∣ϒε,ε−ϒε,σ∣max∣ε−ε∣,∣ε−σ∣−1/max∣ε−ε∣,∣ε−σ∣≤ςp+11/ςp+1ςp−1/ςp=1p+12p+11p2−p≤1p+12≤12,and
(50)1∣ϒε,ε−ϒε,σ∣−1max∣ε−ε∣,∣ε−σ∣=1ςp+1−1ςp=p+1−p=1,and if m>0, then.
(51)∣ϒε,ε−ϒε,σ∣1/∣ϒε,ε−ϒε,σ∣max∣ε−ε∣,∣ε−σ∣−1/max∣ε−ε∣,∣ε−σ∣≤ςn+1−ςp+11/ςn+1−ςp+1ςn−ςp−1/ςn−ςp≤12,and
(52)1∣ϒε,ε−ϒε,σ∣−1max∣ε−ε∣,∣ε−σ∣=1ςn+1−ςp+1−1ςn−ςp≥1.Case 4.
p≤m<n. Here, if p=0, then
(53)∣ϒε,ε−ϒε,σ∣1/∣ϒε,ε−ϒε,σ∣max∣ε−ε∣,∣ε−σ∣−1/max∣ε−ε∣,∣ε−σ∣≤ςn+11/ςn+1ςn−1/ςn=1n+12n+11n2−n≤1n+12≤12,and
(54)1∣ϒε,ε−ϒε,σ∣−1max∣ε−ε∣,∣ε−σ∣=1ςn+1−1ςn=n+1−n=1,and if p>0, then
(55)∣ϒε,ε−ϒε,σ∣1/∣ϒε,ε−ϒε,σ∣max∣ε−ε∣,∣ε−σ∣−1/max∣ε−ε∣,∣ε−σ∣≤ςn+1−ςm+11/ςn+1−ςm+1ςn−ςm−1/ςn−ςm≤12,and
(56)1∣ϒε,ε−ϒε,σ∣−1max∣ε−ε∣,∣ε−σ∣=1ςn+1−ςm+1−1ςn−ςm≥1.Also, letρ,ρ∈Δ with ρ≠ρ. Without loss of any generality, let ρ=ςn,ρ=ςm with n<m. If n=0, then
(57)Vϒρ,ρ,ϒρ,ρ=Vϒς0,ς0,ϒςm,ςm=V0,ςm+1=ςm+1,ςm+1≺ςm,ςm=V0,ςm=Vρ,ρ,Vς1,ςm−1=mm−12−1<mm+12−1=Vς1,ςm=Vρ,ρ,and if n>0, then
(58)Vϒρ,ρ,ϒρ,ρ=Vϒςn,ςn,ϒςm,ςm=Vςn+1,ςm+1=∣ςn+1−ςm+1∣,∣ςn+1−ςm+1∣=1n+12−1m+12,1n+12−1m+12≺1n2−1m2,1n2−1m2=Vςn,ςm=Vρ,ρ.We see that all of the conditions of Theorem2 are satisfied. Thus, ϒ has a unique fixed point. Here, ϒς0,ς0=ς0, and ς0 is the unique fixed point.We present an example in an infinite dimensional sequence spaceℓ1 which is adapted from the above example, and so, we leave the details for the reader.LetA be the space of all convergent sequences an for which ai=1/n2 (n is an arbitrary natural number) for exactly one i and aj=0 for other indices.LetΔ=A∪ς0=0,0,0,⋯, Vanm=1∞,bnn=1∞=Σm,n=1∞∣an−bn∣,Σm,n=1∞∣an−bn∣, and define ϒ:Δ2⟶Δ by
(59)ϒann=1∞,bnn=1∞=0,⋯,0,amaxi,j+1=1n2,0,⋯,an≠0,0,0,⋯andbn≠0,0,0,⋯,0,0,0,⋯,an=0,0,0,⋯orbn=0,0,0,⋯.DefineF:ℝ+2⟶ℝ2 by
(60)Fπ1,π2=lnπ1π1,−1π2,π1≤eπ1ee,−1π2,π1>e.Obviously,F∈F2. Also, take ς=ς1,ς2=ln2,1.Reviewing the above example, we can show that all of the conditions of Theorem2 are satisfied. Thus, ϒ has a unique fixed point. Here, ϒ0,0,0,⋯,0,0,0,⋯=0,0,0,⋯ and 0,0,0,⋯ is the unique fixed point.


## 3. Application

LetE,⋅E be a Banach space and A1,⋯,Ak:Ek⟶E be k 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:
(61)A1ι1,ι2,⋯,ιk=ι1,⋮Akι1,ι2,⋯,ιk=ιk.Similar systems which appear in various branches of mathematics could be seen in [27].LetΔ=Ek and define V:Δ×Δ⟶ℝk, for u=ι1,⋯,ιk, v=ε1,⋯,εk∈Δ by Vu,v=ι1−ε1E,⋯,ιk−εkE. Evidently, Δ,V is a complete VVMS.If we define a mappingϒ:Δk⟶Δ by
(62)ϒu,u,⋯,u=A1ι1,ι2,⋯,ιk,⋯,Akι1,ι2,⋯,ιk,then the system (61) can be written as a fixed point problem such as
(63)ϒu,u,⋯,u=ι1,ι2,⋯,ιk=u,in the space Δ. 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 numbersςi (i=1,⋯,k) such that
(64)Aiι1,ι2,⋯,ιk−Aiε1,ε2,⋯,εkE≤e−ςiιi−εiE,for all u=ι1,⋯,ιk, v=ε1,⋯,εk∈Ek with ιi≠εi. Then, the system (61) has a unique solution in Ek.Proof.
By the inequality (64), we have
(65)ςi+lnAiι1,ι2,⋯,ιk−Aiκ1,κ2,⋯,κkE≤lnιi−κiE,for all i=1,⋯,k. Hence, we get
(66)ς1+lnA1u1−A1u2E,⋯,ςk+lnAku1−Aku2E≺¯lnι1−κ1E,⋯,lnιk−κkE.Taking the functionF∈Fk as Fπ1,⋯,πk=lnπ1,⋯,lnπk, the above inequality can be written as
(67)ς1,⋯,ςk+FA1u1−A1u2E,⋯,Aku1−Aku2E≤Fι1−κ1E,⋯,ιk−κkE=FVu1,u2,VFsupVu1,u2,Vu1,u2,⋯,Vu1,u2,or, equivalently,
(68)ς1,⋯,ςk+FVϒu1,⋯,uk,ϒu2,⋯,uk+1≺¯FsupVu1,u2,Vu2,u3,⋯,Vuk,uk+1,where ς=ς1,⋯,ςk. Thus, applying Theorem 2, ϒ possesses a unique fixed point in Δ=Ek, or, equivalently, the semilinear operator system (61) has a unique solution in Ek.



---

*Source: 1464125-2020-08-28.xml*