# Some Fixed-Point Theorems over a GeneralizedF-Metric Space

Authors: Kushal Roy; Hossein Alaeidizaji; Mantu Saha; Babak Mohammadi; Vahid Parvaneh
Journal: Advances in Mathematical Physics
 (2021)

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## Abstract

In this article, the concept of sequentialF-metric spaces has been introduced as a generalization of usual metric spaces, b-metric spaces, JS-metric spaces, and mainly F-metric spaces. Some topological properties of such spaces have been discussed here. By considering this notion, we prove fixed-point theorems for some classes of contractive mappings over such spaces. Examples have been given in order to examine the validity of the underlying space and in support of our fixed-point theorems. Moreover, our fixed-point theorem is applied to obtain solution of a system of linear algebraic equations.


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## Body

## 1. Introduction and Preliminaries

Two distance-controlled functions have been used extensively by the researchers working on fixed-point theory for obtaining fixed points of mappings such as contractive or expansive mappings in nature. Also, the polygonal inequality involved in a metric-like structure plays vital role for defining the topology on such space. But nowadays, after the introduction ofJS-metric space, the latest fashion is to define a metric-type space which does not involve any type of polygonal inequality (see [1–3]). There is an immense literature in fixed-point theory and applications. For instance, in [4], a class of generalized ψ,α,β-weak contraction has been introduced, and some fixed-point theorems in the framework of partially ordered metric spaces have been proved. The authors also applied their results to a first-order ordinary differential equation.Now, we remember some efforts onF-metric spaces.In [5], Asif and Nazam noticed that the existence of fixed points of F-contractions, in an F-metric space, can be ensured with restricted conditions on the Wardowski function F:0,∞⟶R. They obtained some fixed-point results for both single and set-valued Reich-type F-contractions in F-metric spaces. To show the usability of our results, we present two examples. Also, an application to functional equations is presented.In [6], Jahangir et al. investigated some properties of F-metric spaces. They presented a simple proof to show that the natural topology induced by an F-metric is metrizable. They also presented a method to construct F-metric spaces from bounded metric spaces. They also showed that F-metrics are not necessarily jointly continuous functions. They showed that the Nadler fixed-point theorem and, therefore, the Banach contraction principle in the framework of F-metric spaces, the Schauder fixed-point theorem in F-normed spaces, and also some related F-metric fixed-point results can be reduced to their original metric versions.We now give some definitions of generalized metric-type spaces which are relevant to our research work.Definition 1 (b-metric space) [7, 8].
LetΔ be a nonempty set and h be a real number satisfying h≥1. A function σb:Δ×Δ⟶ℝ+ is a b−metric on Δ provided that
(1)
σba,c=0 if and only if a=c(2)
σba,c=σbc,a for all a,c∈Δ(3)
σba,c≤hσba,e+σbe,c for all a,c,e∈Δ.
The spaceΔ,σb is called a b-metric space.
LetΔ be a nonempty set and σg:Δ×Δ⟶0,∞ be a mapping. For any a∈Δ, let us define the set
(1)Cσg,Δ,a=an⊂Δ:limn⟶∞σgan,a=0.Definition 2 (JS-metric space) [9].
Letσg:Δ×Δ⟶0,∞ be a mapping such that
(1)
σga,c=0 implies a=c(2)
for everya,c∈Δ, we have σga,c=σgc,a(3)
ifa,c∈Δ×Δ and an∈Cσg,Δ,a, then σga,c≤llimsupn⟶∞σgan,c, for some l>0.
The pairΔ,σg is called a generalized metric space, usually known as JS-metric space (JSMS).Definition 3 (F-metric space) [10].
LetΔ be a nonempty set. A mapping σf:Δ×Δ⟶0,∞ is said to be an F-metric on Δ, if for all a,c∈Δ, σf satisfies
(1)
σfa,c=0 if and only if a=c(2)σfa,c=σfc,a(2)
For everym∈ℕ with m≥2, and for every ui1m with u1,um=a,c, we have
(3)Ωσfa,c≤Ω∑i=1m−1σfui,ui+1+k,a≠c,where Ω:0,∞⟶−∞,∞ is an increasing function such that Ωtn⟶−∞ for all 0-convergent sequence tn and k∈0,∞.
The pairΔ,σf is called an F-metric space.Motivated from the previous definitions and based on these ideas, we now define a new generalized metric-type space in our next section.


## 2. SequentialF-Metric Spaces

In this section, we introduce the concept of sequentialF-metric space. To develop such a notion, first we define Sσ,Δ,a≔an⊂Δ:limn⟶∞σan,a=0, where σ:Δ×Δ⟶0,∞ is a given mapping.Definition 4.
LetΔ be a nonempty set. A mapping σ:Δ×Δ⟶0,∞ is said to be a sequential F-metric if for all a,b∈Δ
(F1) σa,b=0 implies a=b
(F2) σa,b=σb,a
(F3) Ωσa,b≤Ωlimsupn⟶∞σan,b+p, for all an∈Sσ,Δ,a, where Ω:0,∞⟶−∞,∞ is an increasing function with Ωt=∞ iff t=∞, Ωtn⟶−∞ for all 0-convergent sequence tn and p≥0.The tripletΔ,σ,Ω is called a sequential F-metric space (SFMS). A SFMS indicated simply as Δ,σ.Example 5.
LetΛ=N and the metric σ:Λ2⟶0,∞ be defined by
(4)σ1,1=0;σn,n=e−1,forn≥2;σ1,n=σn,1=cosh1n+1−1,forn≥2;σn,m=σm,n=coshmn−1,foralln,m≥2withn≠m.
Also, letΩx=1−1/x.
Forn≥2, Sσ,Λ,n=∅. Let nk∈Sσ,Λ,1. If all but finitely many terms of nk are 1, then we have nothing to prove. So, suppose that nk only have finitely many 1’s. Without loss of generality, we can exclude such 1’s, and then, we get limsupk⟶∞σm,nk=limk⟶∞coshmnk−1=∞. Therefore, 1−1/σ1,m≤1−1/limsupk⟶∞σnk,m for all m≥2.
Hence,σ is a sequential p-metric on Λ for Ωx=1−1/x for all x>0 and p=1.
Note that takinga=3, b=2, and c=1, we see that σa,b−σa,c−σc,b=cosh6−1−cosh1/3−1+cosh1/4−1=198.190377258. So, σ is not a usual metric. Now, if a,b are sufficiently large and c=1, again, the left-hand side in triangular inequality in a b-metric space is greater than the right-hand side. So, σ is not also a b-metric.
To show thatσ is not an F-metric space, it is sufficient to take m=3, a,b are sufficiently large and c=1. So
(5)Ωσfa,c=Ωcoshmn−1⟶∞,Ω∑i=12σfui,ui+1+k=Ωcosh1n+1−1+cosh1n+1−1+k⟶k.Proposition 6.
AnyF-metric space Δ,D is a SFMS.Proof.
SinceΔ,D is an F-metric space, then for every a,b∈Δ2, for every m∈ℕ with m≥2 and for every ui1m with u1,um=a,b, we have
(6)fDa,b≤f∑i=1m−1Dui,ui+1+k,a≠b;wheref,k∈F×0,∞.
Therefore, it follows that(7)fDa,b≤fDa,c+Dc,b+k,fora≠bandforanyc∈Δ.
Thus, for anya,b∈Δ, if we take an∈Sσ,Δ,a, then we see that
(8)f∗Da,b≤f∗Da,an+Dan,b+kforalln≥1,where f∗:0,∞⟶−∞,∞ is defined by f∗0=−∞, f∗∞=∞, and f∗t=ft for all 0<t<∞. So
(9)f∗Da,b≤f∗limsupn⟶∞Dan,b+k,and therefore, D also satisfies condition (F3). Hence, D is a sequential F-metric on Δ for the mapping f∗ and k∈0,∞.Proposition 7.
AnyJS-metric space Δ,d¯ is a SFMS.Proof.
Sinced¯ is a JS-metric, then there exists l>0 such that for all a,b∈Δ(10)d¯a,b≤llimsupn⟶∞d¯an,bforanyan∈Sd¯,Δ,a.
Thus(11)Ωd¯a,b≤Ωlimsupn⟶∞d¯an,b+loglforanyan∈Sd¯,Δ,a,for all a,b∈Δ. So, d¯ also satisfies the third condition of Definition 4 for the function Ωt=logt for all 0<t<∞ and Ω0=−∞, Ω∞=∞, and p=logl. Hence, d¯ is a sequential F-metric on Δ.Remark 8.
(i)
A metric space,b-metric space [7, 8], metric-like space [11], and modular metric space with the Fatou property [12] are JS-metric spaces. Therefore, these spaces are also SFMS(ii)
Anys−relaxedp metric space [13] is an F-metric space and therefore is also a SFMS. There exist b-metric spaces which are not F-metric spaces (see [10]); therefore, our SFMS is a stronger concept than the concept of F-metric space.
The following is an example of a SFMS which is not anF-metric space as well as not a b-metric space.Example 9.
LetΔ=ℕ and σ:Δ2⟶0,∞ be defined by σ1,1=0, σn,n=2, σ1,n=σn,1=1/n2 for all n≥2, and σn,m=σm,n=1+1/m+n for all m,n∈ℕ\1 with m≠n. Then for all n≥2, Sσ,Δ,n=∅. Let nk∈Sσ,Δ,1. If all but finitely many terms of nk are 1, then we are done. So, let nk only have finitely many 1’s. Without loss of generality, we can ignore such 1’s, and then, we get limsupk⟶∞σm,nk=limk⟶∞1+1/m+nk=1. Therefore, σ1,m≤limsupk⟶∞σm,nk for all m≥2. So, Δ,σ is a SFMS for Ωt=logt for all 0<t<∞, Ω0=−∞, Ω∞=∞ and p=0. If it is an F-metric space, then there exists f,k∈F×0,∞ such that
(12)fσn,m≤fσn,1+σ1,m+kforalln,mn≠m≥2.
Takingn,m⟶∞, we see that f1+1/n+m⟶−∞, a contradiction. Hence, σ is not an F-metric. In a similar way, we can show that σ is not a b-metric on Δ.Definition 10.
LetΔ,σ be a SFMS. Also let an be a sequence in Δ and a∈Δ.(i)
ifan∈Sσ,Δ,a, an is called convergent and converges to a(ii)
iflimn,m⟶∞σan,am=0,an is called Cauchy(iii)
if any Cauchy sequence inΔ is convergent, Δ is called complete.Definition 11.
LetΔ,σ and Y,σ∗ be two SFMS. If for any ε>0 there exists δε>0 such that for any u∈Δ, σ∗Su,Sa<ε whenever σu,a<δε, then S:Δ⟶Y is called continuous at a point a∈Δ. S is said to be continuous on Δ if S is continuous at each point of Δ.Proposition 12.
LetΔ,σ be a SFMS and an be a convergent sequence converging to some a,b∈Δ; then, a=b.Proof.
If possible, leta≠b. Then
(13)Ωσa,b≤Ωlimsupn⟶∞σan,b+p⟶−∞,since limn⟶∞σan,b=0, a contradiction. Hence, the result.Proposition 13.
LetΔ,σ be a SFMS. If an converges to some a∈Δ, then σa,a=0.Proof.
From the condition (F3) of Definition 4, we have
(14)Ωσa,a≤Ωlimsupn⟶∞σan,a+p⟶−∞,implying that σa,a=0.Proposition 14.
Letan be a Cauchy sequence in a SFMS Δ,σ. If an has a convergent subsequence ank which converges to a∈Δ, then an also converges to a∈Δ.Proof.
From condition (F3) of Definition 4, we have
(15)Ωσam,a≤Ωlimsupk⟶∞σam,ank+p,foranym≥1.
Takingm⟶∞, we get Ωσam,a⟶−∞ as m⟶∞. Therefore, σam,a⟶0 as m⟶∞, that is, am⟶a as m⟶∞.Remark 15.
Here is an example ofJS-metric space which is given by Senapati et al. [14]. Let Δ=ℝ+∪0,∞ and d¯:Δ2⟶0,∞ be defined by
(16)d¯a,b=a+b,eithera=0orb=0;1+a+b,otherwise.(i)
In this space, we see that the sequence1/n converges to 0, but it is not a Cauchy sequence. Since any JS-metric space is a SFMS also, therefore, in a SFMS, a convergent sequence is not necessarily Cauchy(ii)
Also, in a SFMS, ifan and bn are two sequences convergent to a and b, respectively, then σan,bn may not be convergent to σa,b. For this, let us consider two sequences 1/2nn≥1 and 1/2n+1n≥1, and then, both the sequences converge to 0 but d¯1/2n,1/2n+1=1+1/2n+1/2n+1⟶1≠0=d¯0,0.Proposition 16.
In a SFMSΔ,σ, if a self mapping T is continuous at a∈Δ, then an∈Sσ,Δ,a implies that Tan∈Sσ,Δ,Ta.Proof.
Letε>0 be given. Since T is continuous at a, then for any ε>0, there exists δε>0 such that σu,a<δε, u∈Δ, implies σTu,Ta<ε.
Letan∈Sσ,Δ,a. Since an converges to a; for δε>0, there exists N∈ℕ such that σan,a<δε for all n≥N. Therefore, for any n≥N, σTan,Ta<ε, and thus, Tan⟶Ta as n⟶∞, that is, Tan∈Sσ,Δ,Ta.
LetΔ,σ be a SFMS with supporting function Ω and p≥0. Define
(17)Ba,ζ≔b∈Δ:σa,b<σa,a+ζ,Ba,ζ≔b∈Δ:σa,b≤σa,a+ζ,for all a∈Δ and ζ>0.Remark 17.
The family(18)τσ≔∅∪U≠∅⊂Δ:foranya∈U,thereexistsζ>0suchthatBa,ζ⊂Uforms a topology on Δ.Definition 18.
If there exists an open setU⊂Δ such that F=Δ\U in a SFMS Δ,σ, then F is said to be closed.Proposition 19.
LetΔ,σ be a SFMS and F⊂Δ be closed. Let an⊂F such that an⟶a as n⟶∞. Then, a∈F.Proposition 20.
LetΔ,σ be a complete SFMS and F⊂Δ be closed. Then, the subspace F,σ is also complete.Definition 21.
In a SFMSΔ,σ, for A⊂Δ, we define
(19)diamA≔supσa,b:a,b∈A.Theorem 22.
LetΔ,σ be a complete SFMS and Fn be a decreasing sequence of nonempty closed subsets of Δ such that diamFn⟶0 as n⟶∞. Then, ∩n=1∞Fn=a.


## 3. Some Fixed-Point Theorems

Theorem 23 (Banach-type fixed-point theorem).
LetΔ,σ be a complete SFMS and ϖ:Δ⟶Δ be a mapping which satisfies the following conditions:
(i)
σϖa,ϖb≤μσa,b for all a,b∈Δ and for some μ∈0,1(ii)
there existsa0∈Δ such that δσ,ϖ,a0≔supσϖia0,ϖja0:i,j=1,2,⋯<∞.
Then,ϖ has at least one fixed-point u in Δ. Moreover, if v and w are two fixed points of ϖ in Δ with σv,w<∞, then v=w.Proof.
Defineδσ,ϖp+1,a0≔supσϖp+ia0,ϖp+ja0:i,j≥1 for every p≥0. Since δσ,ϖ,a0<∞, then δσ,ϖp+1,a0<∞ for all p∈ℕ∪0. Now
(20)σϖp+ia0,ϖp+ja0≤μσϖp−1+ia0,ϖp−1+ja0≤μδσ,ϖp,a0,for all i,j≥1 and p≥1.
Therefore,δσ,ϖp+1,a0≤μδσ,ϖp,a0 for all p=1,2,3,⋯, from which it follows that limp⟶∞δσ,ϖp,a0=0. Now, for any 1≤n<m, we have
(21)σϖna0,ϖma0=σϖn−1+1a0,ϖn−1+m−n+1a0≤δσ,ϖn,a0⟶0asn⟶∞.
So,ϖna0 is Cauchy in Δ, and so, there exists some u∈Δ such that ϖna0⟶u as n⟶∞. Thus, σϖn+1a0,ϖu≤μσϖna0,u⟶0 as n⟶∞. From Proposition 12, it follows that ϖu=u and u is a fixed point of ϖ.
Now, ifv and w are two fixed points of ϖ in Δ with σv,w<∞, then we have σv,w=σϖv,ϖw≤μσv,w which gives σv,w=0 implying that v=w.Example 24.
LetΔ=ℝ+∪0 endowed with the distance function:
(22)σa,b=ea+b,ifa≠b;0,ifa=b.
Then,Δ,σ is a SFMS for Ωt=−1/t and p=1. Now, let us define ϖ:Δ⟶Δ as follows:
(23)ϖa=0,if0≤a≤1;a−1,ifa>1.
Then,ϖ has all contractive conditions of Theorem 23 for μ=exp−1 and also satisfies all other additional conditions. Here, ϖ has a unique fixed-point 0 in Δ.Theorem 25 (Reich-type fixed-point theorem).
LetΔ,σ be a complete SFMS and ϖ:Δ⟶Δ satisfy
(i)
σϖa,ϖb≤λσa,b+μσa,ϖa+νσb,ϖb for all a,b∈Δ and for λ,μ,ν∈0,1 with λ+μ+ν<1(ii)
there existsa0∈Δ such that δσ,ϖ,a0≔supσϖia0,ϖja0:i,j=1,2,⋯<∞.
Then, the Picard iterating sequencean, an=ϖna0 for all n∈ℕ converges to some u∈Δ. If σu,ϖu<∞ and Ωνt+p<Ωt for all t>0 then u∈Δ is a fixed point of ϖ. Moreover, if v is a fixed point of ϖ in Δ such that σu,v<∞ and σv,v<∞, then u=v.Proof.
Let us defineδσ,ϖp+1,a0≔supσϖp+ia0,ϖp+ja0:i,j≥1 for every p≥0. Since δσ,ϖ,a0<∞, then δσ,ϖp+1,a0<∞ for all p∈ℕ∪0. Now
(24)σϖp+ia0,ϖp+ja0≤λσϖp−1+ia0,ϖp−1+ja0+μσϖp−1+ia0,ϖp+ia0+νσϖp−1+ja0,ϖp+ja0foralli,j≥1,≤rδσ,ϖp,a0forallp≥1,wherer=λ+μ+ν<1.
Proceeding in a similar way as in Theorem23, it follows that ϖna0, that is, an is Cauchy in Δ, and so, there exists some u∈Δ such that an⟶u as n⟶∞. Therefore, we get
(25)σan+1,ϖu≤λσan,u+μσan,ϖan+νσu,ϖuforanyn≥1,which implies that limsupn⟶∞σan+1,ϖu≤νσu,ϖu.
Thus, using conditionF3 of Definition 4, we have
(26)Ωσu,ϖu≤Ωlimsupn⟶∞σan+1,ϖu+p≤Ωνσu,ϖu+p.
Now, ifσu,ϖu>0, then by the assumed condition of the theorem, we see that Ωσu,ϖu≤Ωνσu,ϖu+p<Ωσu,ϖu, a contradiction. Thus, σu,ϖu=0 gives ϖu=u, and u is a fixed point of ϖ.
Now, ifv is a fixed point of ϖ in Δ with σu,v<∞ and σv,v<∞, then we have σu,v=σϖu,ϖv≤λσu,v+μσu,ϖu+νσv,ϖv=λσu,v, as σv,v=0, implying that σu,v=0, that is, u=v.Corollary 26.
LetΔ,σ be a complete SFMS and ϖ:Δ⟶Δ satisfies
(i)
σϖa,ϖb≤νσa,ϖa+σb,ϖb for all a,b∈Δ and for ν∈0,1/2(ii)
there existsa0∈Δ such that δσ,ϖ,a0≔supσϖia0,ϖja0:i,j=1,2,⋯<∞.
Then the Picard iterating sequencean, an=ϖna0 for all n∈ℕ converges to some u∈Δ. If σu,ϖu<∞ and Ωνt+p<Ωt for all t>0, then u∈Δ is a fixed point of ϖ. Moreover, if v is a fixed point of ϖ in Δ such that σu,v<∞ and σv,v<∞, then u=v.Proof.
If we takeλ=0 and μ=ν, then this corollary follows from our Theorem 25.Example 27.
LetΔ=0,1 endowed with the distance function:
(27)σa,b=a−b+a−b2,ifa≠b;0,ifa=b.
ThenΔ,σ is a SFMS for Ωt=logt and p=log2. Now, let ϖ:Δ⟶Δ be defined as
(28)ϖa=a4,if0≤a<12;a5,if12≤a≤1.
Then,ϖ satisfies the contractive condition of Corollary 26 for ν=1/3. Here, all other additional conditions are also satisfied. We see that ϖ has a unique fixed-point 0 in Δ.Theorem 28 (Chatterjea-type fixed-point theorem).
LetΔ,σ be a complete SFMS and ϖ:Δ⟶Δ be a mapping satisfying
(i)
σϖa,ϖb≤χσa,ϖb+σb,ϖa for all a,b∈Δ and for some χ∈0,1/2(ii)
there existsa0∈Δ such that δσ,ϖ,a0≔supσϖia0,ϖja0:i,j=1,2,⋯<∞.
Then the Picard iterating sequencean, an=ϖna0 for all n≥1 converges to some u∈Δ. If limsupn⟶∞σan,ϖu<∞, then u∈Δ is a fixed point of ϖ. Also, if v is a fixed point of ϖ in Δ such that σu,v<∞, then u=v.Proof.
By similar argument as in Theorem23, an is a Cauchy sequence in Δ, and by the completeness of Δ, it converges to an element say u∈Δ.
Now, for alln∈ℕ∪0, σan+1,ϖu=σϖan,ϖu≤χσan,ϖu+σan+1,u, which implies that limsupn⟶∞σan+1,ϖu≤χlimsupn⟶∞σan,ϖa, and therefore, limsupn⟶∞σan,ϖu=0. Thus, we have
(29)Ωσu,ϖu≤Ωlimsupn⟶∞σan,ϖu+p=−∞,which gives σu,ϖu=0, that is, ϖu=u, and u is a fixed point of ϖ.
Ifv is a fixed point of ϖ in Δ with σu,v<∞, then we have σu,v=σϖu,ϖv≤χσu,ϖv+σv,ϖu=2χσu,v. Consequently, σu,v=0, that is, u=v.


## 4. An Application to the System of Linear Algebraic Equations

An application of Theorem23 for solving a system of linear algebraic equations has been presented in this section.Consider the following system ofn linear algebraic equations with n unknowns:
(30)p11λ1+p12λ2+⋯+p1nλn+c1=0,p21λ1+p22λ2+⋯+p2nλn+c2=0,⋮pn1λ1+pn2λ2+⋯+pnnλn+cn=0,where pij,ci∈ℝ for all 1≤i,j≤n. We can write the system of linear equations in matrix notation as PΔ+C=O, where P=pijn×n, Λ=λ1,λ2,⋯,λn, C=c1,c2,⋯,cn, and O=0,0,⋯,0. To find a solution of the system of linear Equations (30), we have to find a fixed point of the mapping g:ℝn⟶ℝn defined by gΛ=QΛ+C, where Q=P+In, that is, Q=qijn×n with qij=pij if i≠j and qii=pii+1 for all i=1,⋯,n.Now, we defineσ:ℝn2⟶0,∞ by
(31)σx,y=max1≤i≤nλi−λi′+λi−λi′2,wherex=λiandy=λi′.Then,σ is a sequential F-metric for Ωt=logt and p=log2.Theorem 29.
If(32)∑j=1n∣qij∣+∑j=1n∣qij∣2≤μ<1forall1≤i≤n,then the system of linear Equations (30) has a unique solution in ℝn,σ.Proof.
To find a unique solution of (30), we show that the mapping g:ℝn⟶ℝn defined by gx=Qx+C for all x∈ℝn, where Q=P+In, that is, Q=qijn×n with qij=pij if i≠j and qii=pii+1 for all i=1,⋯,n, satisfies the contractive condition of Theorem 23. Now, for any x=λi and y=λi′ in ℝn, we have
(33)σgx,gy=max1≤i≤n∑j=1nqijλj−λj′+∑j=1nqijλj−λj′2≤max1≤i≤n∑j=1nqijλj−λj′+∑j=1nqijλj−λj′2≤max1≤i≤n∑j=1nqijσx,y+∑j=1nqijσx,y2≤max1≤i≤n∑j=1nqijσx,y+max1≤i≤n∑j=1nqij2σx,y=max1≤i≤n∑j=1nqij+max1≤i≤n∑j=1nqij2σx,y≤μσx,y.
Sinceℝn,σ is complete, therefore, due to Theorem 23, g has a unique fixed point, that is, the system of linear Equations (30) has a unique solution in ℝn.We now give a numerical example in respect of Theorem23.Example 30.
Let us consider the following system of linear algebraic equations in three variables:(34)0.9λ1+0.1λ2+0.15λ3+1=0,0.1λ1+0.85λ2+0.1λ3+2=0,0.1λ1+0.05λ2+0.8λ3+3=0.
Then, the system of linear algebraic Equations (34) has a unique solution.Solution.
LetΛ=ℝ3 be the SFMS endowed with the metric σ:Λ2⟶0,∞ defined by
(35)σx,y=max1≤i≤3λi−λi′+λi−λi′2,forallx=λiandy=λi′inΛ.
We can write the above system of linear algebraic Equations (34) as
(36)−0.9λ1−0.1λ2−0.15λ3−1=0,−0.1λ1−0.85λ2−0.1λ3−2=0,−0.1λ1−0.05λ2−0.8λ3−3=0.
Here,p11=−0.9, p12=−0.1, p13=−0.15, p21=−0.1, p22=−0.85, p23=−0.1, p31=−0.1, p32=−0.05, p33=−0.8, c1=−1, c2=−2, and c3=−3.
Thus,q11=0.1, q12=−0.1, q13=−0.15, q21=−0.1, q22=0.15, q23=−0.1, q31=−0.1, q32=−0.05, and q33=0.2. Also, we see that
(37)∑j=13qij=0.35,i.e.,∑j=13qij+∑j=13qij2=0.4725<1forall1≤i≤3.
Hence, from the Theorem23, it follows that the system of linear algebraic Equations (34) has a unique solution in ℝ3, which is given by λ1≃−0.3018, λ2≃−1.894, and λ3≃−3.593.



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*Source: 5570653-2021-05-06.xml*