# A Tripled Fixed Point Theorem inC∗-Algebra-Valued Metric Spaces and Application in Integral Equations

Authors: Vahid Parvaneh; Samira Hadi Bonab; Hasan Hosseinzadeh; Hassen Aydi
Journal: Advances in Mathematical Physics
 (2021)

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## Abstract

Our aim is to establish a tripled fixed and coincidence point result on generalizedC∗-algebra-valued metric spaces. We present an example on matrices. At the end, we give an application on integral equations.


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## Body

## 1. Introduction

The Banach contraction principle (BCP) was considered by Perov [1] on spaces equipped with vector-valued metrics. The result of Perov has been generalized in [2], and its related fixed point property on generalized metric spaces was investigated.LetA be a unital algebra with the unit I and θ be its zero element. An involution on A is a conjugate linear map ι↦ι∗ on A so that for all ι,κ∈A,ι∗∗=ιandικ∗=κ∗ι∗. The pair A,∗ is named as an ∗-algebra. A Banach ∗-algebra is an ∗-algebra A with the complete submultiplicative norm so that ι∗=ι for all ι∈A. A C∗-algebra is a Banach ∗-algebra such that ι∗ι=ι2 for all ι∈A. Let H be a Hilbert space and BH be the family of all bounded linear operators on H; then, BH is a C∗-algebra with the operator norm. Let Asa be the family of all self-adjoint elements in A, and define the spectrum of ι∈A as σι=λ∈C:λI−ιisnotinvertible. An element ι∈A is positive (denoted by ι≥θ) if ι∈Asa and σι⊆ℝ+. Take A+=ι∈A:ι≥θ, then A+=ι∗ι:ι∈A (see [3]). One can define a partial ordering ≺¯ on Asa as ι≺¯κiffκ−ι≻¯θ. If ι,κ∈Asa and c∈A, then ι≤κ⇒c∗ιc≤c∗κc, and if ι,κ∈A+ are invertible, then ι≤κ⟹θ≤κ−1≤ι−1.Definition 1 (see [4]).
LetX be a nonempty set. If the function ℧:X×X→A is so that for all υ,τ,η∈X:
(i)
θ≤℧υ,τ and ℧υ,τ=θ iff υ=τ(ii)
℧υ,τ=fτ,υ(iii)
℧υ,τ≤℧υ,η+℧η,τthen X,A,℧ is named as a C∗-algebra-valued metric space.In this article, denote byMq,qA the set of all q×q matrices with coefficients in A. Note that Θ=thezeromatrix and I=theidentitymatrix.LetA∈Mq,qA, then A is said to be convergent to zero, iff An goes to θ as n⟶∞. See [5–8] for more details.Denote byZM the family of all matrices A∈Mq,qA so that An⟶θ. We provide the following examples.Example 1.
(1)A=141111,B=161111
are inZM. We have A+B2I−A2−1∈ZM.Example 2.
(2)A=130013,B=013130
are inZM. Clearly, A+B2I−A2−1∈ZM.Example 3.
A=αI and B=I−α3−αI are in ZM. Then, for α∈1/4,1/5,1/7,1/8, one gets A+B2I−A2−1∈ZM.Definition 2 (see [9]).
An elementℓ1,ℓ2∈X2 is named to be a coupled fixed point of F:X2→X if Fℓ1,ℓ2=ℓ1 and Fℓ2,ℓ1=ℓ2.Definition 3 (see [10]; see also [11]).
GivenF:X2→X and g:X→X. An element ℓ1,ℓ2∈X2 so that Fℓ1,ℓ2=gℓ1 and Fℓ2,ℓ1=gℓ2 is named as a coupled coincidence point of F and g. gℓ1,gℓ2 is called a coupled point of coincidence.Definition 4 (see [12]).
An elementℓ1,ℓ2,ℓ3∈X3 is named to be a tripled fixed point of F:X3→X if Fℓ1,ℓ2,ℓ3=ℓ1,Fℓ2,ℓ1,ℓ2=ℓ2 and Fℓ3,ℓ2,ℓ1=ℓ3.In this manuscript, we investigate a tripled common fixed point result for a sequence of mappingsTn:X3→X and g:X→X in the class of complete C∗-algebra-valued metric spaces. An example and an application are presented.


## 2. Main Results

Our main result is as follows.Theorem 5.
LetX,A,℧ be a complete C∗-algebra-valued metric space. Let g:X→X and Tii≥0 be a sequence of mappings from X3 into X so that
(3)℧Tiℓ1,ℓ2,ℓ3,Tju1,u2,u3≤A℧gℓ1,Tiℓ1,ℓ2,ℓ3+℧gu1,Tju1,u2,u3A∗+B℧gu1,gℓ1B∗,where I≠A=aij∈Mq,qA, I≠B=bij∈Mq,qA+ with A+B2I−A2−1∈ZM. If TiX3⊆gX and gX is complete in X, then Tii≥0 and g have a tripled coincidence point. Further, if Tii≥0 and g are w-compatible, then they have a unique tripled common fixed point in X.Proof.
Takeℓ01,ℓ02,ℓ03∈X, and let
(4)gℓ11=T0ℓ01,ℓ02,ℓ03,gℓ12=T0ℓ02,ℓ01,ℓ02,gℓ13=T0ℓ03,ℓ02,ℓ01.
Continuing this technique, we get(5)gℓn+11=Tnℓn1,ℓn2,ℓn3,gℓn+12=Tnℓn2,ℓn1,ℓn2,gℓn+13=Tnℓn3,ℓn2,ℓn1,foralln≥0.
By (3), we get
(6)℧gℓn1,ℓn+11=℧Tn−1ℓn−11,ℓn−12,ℓn−13,Tn−1ℓn1,ℓn2,ℓn3≤A℧gℓn−11,Tn−1ℓn−11,ℓn−12,ℓn−13+℧gℓn−11,Tnℓn1,ℓn2,ℓn3A∗+B℧gℓn1,gℓn−11B∗=A℧gℓn−11,gℓn1+℧gℓn1,gℓn+11A∗+B℧gℓn1,gℓn−11B∗=A℧gℓn−11,gℓn1A∗+A℧gℓn1,gℓn+11A∗+B℧gℓn1,gℓn−11B∗=A℧gℓn−11,gℓn1A∗+A℧gℓn1,gℓn+111/2℧gℓn1,gℓn+111/2A∗+B℧gℓn1,gℓn−11B∗=A℧gℓn−11,gℓn1A∗+A℧gℓn1,gℓn+111/22A∗+B℧gℓn1,gℓn−11B∗=A+B℧gℓn−11,gℓn11/2℧gℓn−11,gℓn11/2A+B∗+A℧gℓn1,gℓn+111/22A∗=A+B℧gℓn−11,gℓn11/22+A℧gℓn1,gℓn+111/22A∗.
It follows that(7)℧gℓn1,gℓn+11≤A+B2I−A2−1℧gℓn−11,gℓn1.
Similarly,(8)℧gℓn1,gℓn+11≤A+B2I−A2−1℧gℓn−11,gℓn1,(9)℧gℓn3,gℓn+13≤A+B2I−A2−1℧gℓn−13,gℓn3.
Adding (7), (8), and (9), we have
(10)δn≔℧gℓn1,gℓn+11+℧gℓn2,gℓn+12+℧gℓn3,gℓn+13≤A+B2I−A21℧gℓn−11,gℓn1+℧gℓn−12,gℓn2+℧gℓn−13,gℓn3=A+B2I−A2−1δn−1.
PutC=A+B2I−A2−1. Then, for all n≥2,
(11)Θ≤δn≤Cδn−1≤C2δn−2≤⋯≤Cnδ0.
Using the triangle inequality, for allp≥1,
(12)℧gℓn1,gℓn+p1+℧gℓn2,gℓn+p2+℧gℓn3,gℓn+p3≤℧gℓn1,gℓn+11+℧gℓn2,gℓn+12+℧gℓn3,gℓn+13+℧gℓn+11,gℓn+21+℧gℓn+12,gℓn+22+℧gℓn+13,gℓn+23+⋯+℧gℓn+p−11,gℓn+p1+℧gℓn+p−12,gℓn+p2+℧gℓn+p−13,gℓn+p3=δn+δn+1+⋯δn+p−1≤Cn+Cn+1+⋯+Cn+p−1δ0≤CnI+C+⋯+Cp−1+⋯δ0=CnI−C−1δ0.
We have(13)℧gℓn1,gℓn+p1+℧gℓn2,gℓn+p2+℧gℓn3,gℓn+p3≤CnI−C−1δ0=A+B2I−A2−1nI−A+B2I−A2−1−1δ0.
Now, taking the limit asn→+∞, we conclude
(14)℧gℓn1,gℓn+p1+℧gℓn2,gℓn+p2+℧gℓn3,gℓn+p3≤A+B2I−A2−1nI−A+B2I−A2−1−1δ0→0.
This implies that(15)℧gℓn1,gℓn+p1=℧gℓn2,gℓn+p2=℧gℓn3,gℓn+p3=0,and gℓn1,gℓn2, and gℓn3 are Cauchy sequences in gX, which is complete; there are ℘1,℘2,℘3∈X3 so that
(16)limn→+∞gℓn1=g℘1≔ℓ1,limn→+∞gℓn2=g℘2≔ℓ2,limn→+∞gℓn3=g℘3≔ℓ3.
We have(17)℧Tiℓ1,ℓ2,ℓ3,gℓ1≤℧Tiℓ1,ℓ2,ℓ3,gℓn+11+℧gℓn+11,gℓ1=℧Tiℓ1,ℓ2,ℓ3,Tngℓn1,gℓn2,gℓn3+℧gℓn+11,gℓ1≤A℧gℓ1,Tiℓ1,ℓ2,ℓ3+℧gℓn1,Tngℓn1,gℓn2,gℓn3A∗+B℧gℓn1,gℓ1B∗+℧gℓn+11,gℓ1.
Taking the limit asn→+∞ in the above relation, we obtain gℓ1=Tiℓ1,ℓ2,ℓ3. Similarly, gℓ2=Tiℓ2,ℓ1,ℓ2 and gℓ3=Tiℓ3,ℓ2,ℓ1. Therefore, ℓ1,ℓ2,ℓ3 is a tripled coincidence point of Tii∈ℕ and g.
Letℓ1,ℓ2,ℓ3 and ℘1,℘2,℘3 be tripled coincidence points, then
(18)℧gℓ1,g℘1=℧Tiℓ1,ℓ2,ℓ3,Tj℘1,℘2,℘3≤A℧gℓ1,Tiℓ1,ℓ2,ℓ3+℧g℘1,Tj℘1,℘2,℘3A∗+B℧g℘1,gℓ1B∗.
That is,(19)I−B2℧gℓ1,g℘1≤A2℧gℓ1,Tiℓ1,ℓ2,ℓ3+℧g℘1,Tj℘1,℘2,℘3,so
(20)℧gℓ1,g℘1≤A∗I−B2−1℧gℓ1,Tiℓ1,ℓ2,ℓ3+℧g℘1,Tj℘1,℘2,℘3,which further induces that
(21)℧gℓ1,g℘1≤A2I−B2−1℧gℓ1,Tiℓ1,ℓ2,ℓ3+℧g℘1,Tj℘1,℘2,℘3.
Therefore,℧gℓ1,g℘1=Θ, that is, gℓ1=g℘1. Similarly, we can prove that gℓ2=g℘2andgℓ3=g℘3. So, gℓ1=gℓ2=gℓ3=g℘1=g℘2=g℘3. Therefore, Tii∈ℕ and g have a unique tripled coincidence point. gℓ1,gℓ1,gℓ1. Now, set gℓ1=u, then u=gℓ1=Tiℓ1,ℓ1,ℓ1. By w-compatibility of Tii∈ℕ and g,
(22)gu=ggℓ1=gTiℓ1,ℓ1,ℓ1=Tigℓ1,gℓ1,gℓ1=Tiu,u,u=gℓ1.
Then,gu,gu,gu is a tripled coincidence point of Tii∈ℕ and g. By the uniqueness, we know gu=gℓ1, which yields that u=gu=Tiu,u. Hence, u,u,u is a unique tripled common fixed point of Tii∈ℕ and g.Lettingg=IdX in Theorem 5, we have the following.Corollary 6.
LetX,A,℧ be a complete generalized C∗-algebra-valued metric space. Suppose that Tii≥0 is a sequence of mappings from X3 into X so that
(23)℧Tiℓ1,ℓ2,ℓ3,Tju1,u2,u3≤A℧ℓ1,Tiℓ1,ℓ2,ℓ3+℧u1,Tju1,u2,u3A∗+B℧u1,ℓ1B∗,where A,B∈A with I≠A=aij, I≠B=bij∈Mm,mA+, A+B2I−A2−1∈ZM. Then, Tii≥0 has a unique tripled fixed point.Example 4.
TakeX=0,1. Given
(24)℧ℓ1,ℓ2=ℓ1−ℓ200ℓ1−ℓ2.
Then,X,A,℧ is a complete generalized C∗-algebra-valued metric space.
ConsiderTi:X3→X and g:X→X as
(25)Tiℓ1,ℓ2,ℓ3=ℓ1,ℓ2,ℓ33i,gℓ1=9ℓ1.Choose(26)A=130013,B=013130.By induction, (3) holds for all ℓ1,ℓ2,ℓ3∈X. Set x=ℓ1+ℓ2+ℓ3/3 and u=u1,u2,u3/3j. Here, for i=1, we have
(27)x−u00x−u≤1300139ℓ1−x+9u1−u009ℓ1−x+9u1−u130013+0131309u1−ℓ1009u1−ℓ1013130.Also,(28)α≔x−13u00x−13u≤1300139ℓ1−x+3u1−13u009ℓ1−x+3u1−13u130013+9013130u13−ℓ1u13−ℓ1+9013130≔β.So,(29)α≤x−u00x−u+23x00x≤131300139ℓ1−x+9u1−u009ℓ1−x+9u1−u130013+013130u1−ℓ100u1−ℓ1013130+23x00x≤β.Clearly,g and Tii≥0 are w-compatible. Therefore, all conditions in Theorem 5 hold, and 0,0,0 is the unique tripled common fixed point of g and Tii≥0.


## 3. Application

Consider the following sequence of the integral equations:(30)xt=∫Eϒ1r,t,s+ϒ2r,t,sfit,s,xs+git,s,xs+hit,s,xsds+ht,for all r,t,s∈E, where E is a Lebesgue measurable set and mE<∞.Denote byX=L∞E the set of essentially bounded measurable functions on E. We consider the following assumptions:
(i)
fi,gi,hi:E×E×ℝ→ℝ, ϒ1:E×E×E→0,∞, ϒ2:E×E×E→−∞,0 are integrable, and h∈L∞E(ii)
There isk∈0,1/2 so that for all ℓ1,ℓ2∈ℝ(31)0≤fit,s,ℓ1s−fit,s,ℓ2s≤kℓ1−ℓ2,−kℓ1−ℓ2≤git,s,ℓ1s−git,s,ℓ2s≤0,0≤hit,s,ℓ1s−hit,s,ℓ2s≤kℓ1−ℓ2,for all s,t∈E with
(32)k≤A2=190019,k≤B2=019190(iii)
sups,t∈E∫Eϒ1r,s,t−ϒ2r,s,tds≤1Theorem 7.
Suppose that assumptions (i)–(iii) hold. Then, (30) has a unique solution in L∞E.Proof.
LetX=L∞E and BL2E be the set of bounded linear operators on the Hilbert space L2E. We endow X with the cone metric ℧:X×X→BL2E defined by ℧f,g=Mf−g, where Mf−g is the multiplication operator on L2E. It is clear that X,BL2E,℧ is a complete C∗-algebra-valued metric space. Define the self-mapping Ti:X3→X by
(33)Tℓ1,ℓ2,ℓ3t=Tiℓ1,ℓ2,ℓ3t=∫Eϒ1r,s,t+ϒ2r,s,tfit,s,ℓ1s+git,s,ℓ2s+hit,s,ℓ3sds+ht,for all ℓ1,ℓ2,ℓ3∈X and r,s,t∈E.
Now, we have(34)℧Tiℓ1,ℓ2,ℓ3,Tju1,u2,u3=MTiℓ1,ℓ2,ℓ3,Tju1,u2,u3.
Using (31), we have
(35)Tiℓ1,ℓ2,ℓ3t−Tju1,u2,u3t=∫Eϒ1r,s,t+ϒ2r,s,tfit,s,ℓ1s+git,s,ℓ2s+hit,s,ℓ3sds−∫Eϒ1r,s,t+ϒ2r,s,tfit,s,u1s+git,s,u2s+hit,s,u3sds=∫Eϒ1r,s,t+ϒ2r,s,tfit,s,ℓ1s−fit,s,u1s+git,s,ℓ2s−git,s,u2s+hit,s,ℓ3s−hit,s,u3sds≤∫Eϒ1r,s,t+ϒ2r,s,tfit,s,ℓ1s−fit,s,u1s+git,s,ℓ2s−git,s,u2s+hit,s,ℓ3s−hit,s,u3sds≤sups,t∈E∫Eϒ1r,s,t+ϒ2r,s,tds.kℓ1−u1+ℓ2−u2+ℓ3−u3≤kℓ1−u1∞+ℓ2−u2∞+ℓ3−u3∞,for all r,s,t∈E.
Therefore, for anyφ∈L2E, we have
(36)Tiℓ1,ℓ2,ℓ3−Tju1,u2,u3=MTiℓ1,ℓ2,ℓ3−Tju1,u2,u3=supφ=1MTiℓ1,ℓ2,ℓ3−Tju1,u2,u3=supφ=1∫ETiℓ1,ℓ2,ℓ3−Tju1,u2,u3tφtφt¯dt≤supφ=1∫Eφt2dtkℓ1−u1∞+ℓ2−u2∞+ℓ3−u3∞≤kℓ1−u1∞+ℓ2−u2∞+ℓ3−u3∞≤B2ℓ1−u1∞+ℓ2−u2∞+ℓ3−u3∞=Bℓ1−u1∞+ℓ2−u2∞+ℓ3−u3∞B∗.
Consequently,(37)℧Tiℓ1,ℓ2,ℓ3,Tju1,u2,u3≤A℧ℓ1,ℓ2,ℓ3+℧u1,Tj,u1,u2,u3A∗+B℧u1,ℓ1B∗.
Hence, all hypotheses of Corollary6 hold. Hence, (30) possesses a unique solution in L∞E.



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