Quasi Controlled K -Metric Spaces over C*-Algebras with an Application to Stochastic Integral Equations

Ouafaa Bouftouh,Samir Kabbaj,Thabet Abdeljawadand Aziz Khan

1Department of Mathematics,Laboratory of Partial Differential Equations,Algebra and Spectral Geometry,Faculty of Sciences,Ibn Tofail University,Kenitra,BP 133,Morocco

2Department of Mathematics and Sciences,Prince Sultan University,P.O.Box 66833,Riyadh,11586,Saudi Arabia

3Department of Medical Research,China Medical University,Taichung,40402,Taiwan

ABSTRACT Generally,the field of fixed point theory has attracted the attention of researchers in different fields of science and engineering due to its use in proving the existence and uniqueness of solutions of real-world dynamic models.C*-algebra is being continually used to explain a physical system in quantum field theory and statistical mechanics and has subsequently become an important area of research.The concept of a C*-algebra-valued metric space was introduced in 2014 to generalize the concept of metric space.In fact,It is a generalization by replacing the set of real numbers with a C*-algebra.After that,this line of research continued,where several fixed point results have been obtained in the framework of C*-algebra valued metric,as well as(more general)C*-algebra-valued b-metric spaces and C*-algebra-valued extended b-metric spaces.Very recently,based on the concept and properties of C*-algebras,we have studied the quasi-case of such spaces to give a more general notion of relaxing the triangular inequality in the asymmetric case.In this paper,we first introduce the concept of C*-algebra-valued quasi-controlled K -metric spaces and prove some fixed point theorems that remain valid in this setting.To support our main results,we also furnish some examples which demonstrate the utility of our main result.Finally,as an application,we use our results to prove the existence and uniqueness of the solution to a nonlinear stochastic integral equation.

KEYWORDS C*-algebra-valued quasi controlled K -metric spaces;leftconvergence;right convergence;fixed point;contraction

1 Introduction

One of the most relevant theories marking the passage from classical to modern analysis is the fixed point theory which was implemented by Banach[1].Several mathematicians have created diverse generalizations of Banach fixed point theory.Wilson,on the other hand,introduced the quasi-metric space that is one of the abstractions of the metric spaces[2].This theory,however,does not include the commutative condition.Numerous mathematicians have adopted this concept to demonstrate some fixed point outcomes,see[3].

Theb-metric spaces concept was first set up by Bakhtin[4]and Czerwik[5].Besides,numerous authors obtained a lot of fixed point results.For example, see [6-10].The extendedb-metric spaces idea was elaborated by Kamran et al.[11]and generalized by Abdeljawad et al.[12]by imposing the control or the double control of thes-relaxed inequality by one or two functions.Mudasir et al.[13]stated new results in the context of dislocatedb-metric spaces and presented an application related to electrical engineering and extended the notion of Kannan maps in view of theF-contraction in this framework,see[14].

In [15,16], Ma et al.introducedC*-algebra valuedb-metric spaces by considering metrics that take values in the set of positive elements of a unitaryC*-algebra.Lately, Asim et al.[17] enlarged this class by definingC*-algebra-valued extendedb-metric spaces.Very recently, Kabbaj et al.[18]have investigated the quasi case of such a metric and they give a more general notion of relaxing the triangular inequality in the asymmetric case[19].Recently,for some work on fixed point theory in the mentioned area,we refer to some published work as[20-34].

In this work, we introduce the notion ofC*-algebra-valued quasi controlledK-metric spaces.We give basic definitions and then employ them to demonstrate fixed point results in such spaces.Examples are also provided to verify the usefulness of our main results.Finally,as an application,we verify the existence of the solution for a nonlinear stochastic integral equation in this setting.

2 Preliminaries

Throughout this paper,A will be a unitaryC*-algebra withIAandσ(δ)is the spectrum ofδ∈A.We set

Note that A+is a cone[20],which induces a partial orderon Ahby

To prove our main results,it will be useful to introduce the following lemma.

Lemma 2.1.[20]Suppose that A is a unitalC*-algebra with a unitIA.

1.ifγ,δ∈Ahandγδ,then for eachξ∈A,ξ*γ ξξ*δξ;

2.ifγ,δ∈Ah,γ,δ0Aandγ δ=δγ,thenγ δ0A;

3.for allγ,δ∈Ah,0Aγδ⇔‖γ‖≤‖δ‖;

Definition 2.1.[17]LetΩ/=∅andΛ:Ω×Ω→AI.AC*-algebra-valued extendedb-metric is a mappingΔ:Ω×Ω→A such that

1.Δ(ω,ϖ)=0Aif and only ifω=ϖ;

2.Δ(ω,ϖ)=Δ(ϖ,ω);

3.Δ(ω,ϖ)Λ(ω,ϖ)[Δ(ω,ν)+Δ(ν,ϖ)].

The triplet(Ω,A,Δ)is called aC*-algebra valued extendedb-metric space.

3 Main Results

In this section,by omitting the symmetry condition,we introduce the notion ofC*-algebra-valued quasi controlledK-metric spaces,whereKis a control function.

Definition 3.1.AC*-algebra-valued quasi controlledK-metric space is the triplet(Ω,A,Δ)whereΩis a non empty set,K:Ω×Ω→AIis aC*-control function andΔ:Ω×Ω→A is a mapping that

1.Δ(ω,ϖ)=0Aif and only ifω=ϖ;

2.Δ(ω,ϖ)K (ω,ϖ)[Δ(ω,ν)+Δ(ν,ϖ)]for all ω,ϖ,ϑ∈Ω.

Remark 3.1.In particular,by takingK (ω,ϖ)=δIA,(Ω,A,Δ)is aC*-algebra-valued quasib-metric space[19].

Example 3.1.LetΩ= [0,1] and A = M2(R).We know that A is aC*-algebra where partial ordering on M2(R)is given as

Define aC*-algebra-valued quasi controlledK-metricΔ:Ω×Ω→R2by:

Given theC*-control functionK:Ω×Ω→AIas

Then,(Ω,A,Δ)is aC*-algebra-valued quasi controlledK-metric space.

Example 3.2.LetΩ=[0,1]and A=M2(C).Define a mappingΔ:Ω×Ω→A as

Let theC*-control functionK:Ω×Ω→A be defined by(for allρ,ϖ∈Ω)

Example 3.3.ConsiderΩ=C(S,C)the space of all continuous functions whereSis compact.Let A =L∞(S)the usual unitalC*-algebra with the sup norm and givenΔ:Ω×Ω→A+for eachφ,ψ∈Ωas

We take

Thus,(Ω,Δ,L∞(S))is aC*-algebra-valued quasi controlledK-metric space.

Next,we introduce some topological concepts onC*-algebra-valued quasi controlledK-metric spaces.

Definition 3.2.Let(Ω,A,Δ)be aC*-algebra-valued quasi controlledK-metric space.The open ballB(ω,r)of centerω∈Ωand radiusr0Ais given by

Example 3.4.Let us define aC*-algebra-valued quasi controlledK-metricΔ:C*×C*→R2+as

with theC*-controlled functionK:C*×C*→]1,+∞[×]1,+∞[given by

Then,it is evident that

The open ballBis given by

ifr|z0|≤1,then

B(z0,r.1A)={z0}

ifr|z0|＞1,then

Remark 3.2.We can also define the closed ball by

B(ω,r)={ϖ∈Ω:Δ(ω,ϖ)r}.

Definition 3.3.Let(Ω,A,Δ)be aC*-algebra-valued quasi controlledK-metric space and let{ϖn}be a sequence inΩ.

1.{ϖn}is called left-converges toϖ∈Ωwith respect to A,if and only if ∀ε≻0A∃k∈N such that

n ＞k⇒Δ(ϖn,ϖ)≺ε.

2.{ϖn}is called right-converges toϖ∈Ωwith respect to A,if and only if ∀ε≻0A∃k∈N such that

n ＞k⇒Δ(ϖ,ϖn)≺ε.

3.{ϖn}is called converges toϖ∈Ωwith respect to A,if and only if

Definition 3.4.Let(X,A,Δ)be aC*-algebra-valued quasi controlledK-metric space.Then

1.{ϖn}is called right-Cauchy with respect to A,if for eachε≻0Athere existsk∈N such that∀p∈N,

2.{ϖn} is called left-Cauchy with respect to A, if for eachε≻0Athere existsk∈N such that∀p∈N,

n ＞k⇒Δ(ϖn+p,ϖn)≺ε.

3.{ϖn}is called Cauchy sequence with respect to A if and only if ∀p∈N,

4.If every Cauchy sequence{ϖn}inΩconverges to some pointϖinΩ,then,the triplet(Ω,A,Δ)is said to be a completeC*-algebra-valued quasi controlledK-metric space.

Example 3.5.TakeΩ=R+and A=R2

LetK:Ω×Ω→AIbe the mapping defined by

K (η,ν)=(2η+2ν+2,2η+2ν+2).

Then,(Ω,A,Δ)is a completeC*-algebra-valued quasi controlledK-metric space.

Example 3.6.LetSbe a compact Hausdorff space and A=C(S)be the set of complex valued continuous functions onS.Note thatC(S)is a unitary commutativeC*-algebra with the usual sup norm such that the involution is defined byψ*(x)=ψ(x)for all x∈S.SettingΩ=L∞(E)whereEis a Lebesgue mensurable set and let us define aC*-algebra-valued quasi controlledK-metricΔ:Ω×Ω→A by

Δ(φ,ψ)(t)=(1+‖φ‖∞+2‖ψ‖∞)‖φ-ψ‖∞et for all φ,ψ∈Ω;t∈[0,1].

Let us define theC*-control operator by

K (φ,ψ)=(1+‖φ‖∞+2‖ψ‖∞)IA.

The condition(i)of Definition 3.1 is clearly satisfied byΔ.Now we check the condition(ii).We takeφ,ψ∈Ωas arbitrary.Then

Therefore,

Δ(φ,ψ)K (φ,ψ)(Δ(φ,φ)+Δ(φ,ψ))for all φ,ψ φ∈Ω.This prove thatΔis aC*-algebra-valued quasi controlledK-metric.Now we want to verify that(X,A,Δ)is a completeC*-algebra-valued quasi controlledK-metric space.Letbe a Cauchy sequence inΩwith respect to A.Then

and

We conclude that the sequenceconverges to the functioninΩrespecting A.

We will fix the notion of a continuous metric in the context presented in this paper since in the literature during the proof of the results in fixed point certain problems arise due to the possible discontinuity of theb-metric with respect to the topology it generates.

Definition 3.5.LetΔbe aC*-algebra-valued quasi controlledK-metric.Δis said to be continuous at(ϖ,ω)if the sequenceconverges toωandconverges toϖthen

Δ(ωn,ϖn)→Δ(ω,ϖ)and Δ(ϖn,ωn)→Δ(ϖ,ω).

Lemma 3.1.Let(Ω,A,Δ)be aC*-algebra-valued quasi controlledK-metric space.SuchΔis continuous in each variable.If a sequence{ωn}∞n=1has a limit,then this limit is unique.

Proof.Fixε≻0A.By assumption,ϖnconverges toωso there existsK1∈N such thatd(ω,ϖn)for alln≥K1.We also assume thatϖnconverges toϖ,so there existsK2∈N such thatd(ϖn,ϖ)for alln≥k2.Then for alln≥K:=max{K1,K2}

Δ(ϖ,ω)K (ϖ,ω)[Δ(ϖ,ϖn)+Δ(ϖn,ω)]K (ϖ,ω)ε.

Asεwas arbitrary,we deduce thatΔ(ϖ,ω)=0,which impliesϖ=ω.

Our main result runs as follows.

Theorem 3.1.Let(Ω,A,Δ)be completeC*-algebra-valued quasi controlledK-metric space such thatΔis a continuous andΓ:Ω→Ωsatisfies the following:

whereθ∈A with‖θ‖A＜1 andsuch thatωn=Γϖn-1=Γnϖ0for an arbitraryϖ0.ThenΓhas a unique fixed point∈Ω.

Proof.Let the sequence {ϖn} be defined byϖn=Γϖn-1=Γnϖ0.From Eq.(1), we obtain by induction

Now we prove that{ϖn}is a right-Cauchy sequence.For anyn,p∈N,we have

Thus,the above inequality implies

Lettingn→∞,we conclude that{ϖn}is a right-Cauchy sequence.Similarly,we prove that{ϖn}is a left-Cauchy sequence.The fact thatΩis complete involves∈Ωsuch that

so that

Then,we get a contradiction,as a resultω=ω*.

Dynamic programming is a powerful technique for solving some complex problems in computer sciences.We illustrate Theorem 3.2 by studying the existence and uniqueness of the solutions of the functional equation presented in the following example.

Example 3.7.LetXandYbe Banach spaces.S⊂Xis the state space andD⊂Yis the decision space.Letη:S×D→S,τ:S×D→R andT:S×D×R →R.Denote byB(S)the set of all real-valued bounded functions onS.Let A =L∞(S)the usual unitalC*-algebra with the sup norm and givenΔ:B(S)×B(S)→A+for eachφ,ψ∈Ωas

(B(S),Δ,L∞(S))is a completeC*-algebra-valued quasi controlledK-metric space.We consider the functional equation

such thatτandTare bounded and

for all(x,y,z1),(x,y,z2)inS×D× R, where 0 ≤α ＜1 and m = ‖T‖.We define a mappingΓ:B(S)→B(S)byΓϖ=h,where

It is easy to getΔ(Γρ,Γϖ)θ*Δ(ρ,ϖ)θsatisfies with

Therefore,the Eq.(1)possesses unique bounded solution onS.

Example 3.8.LetΩ= R and A =M2(C).For anyA∈A, we define its norm as ‖A‖A=max1≤i≤4|ai|.Define a mappingΔ:Ω×Ω→A such that for allρandϖ∈Ω,

Let theC*-control functionK:Ω×Ω→A by:

We define a mappingΓ:Ω→Ωby

It is easy to getΔ(Γρ,Γϖ)θ*Δ(ρ,ϖ)θ

Definition 3.6.LetΩ/= ∅andOΓ(ϖ0)= {Γnϖ0|n∈N} for an arbitraryϖ0∈Ω.A functionΦ:Ω→A is said to beΓ-orbitally lower semi continuous atϖwith respect to A if the sequence{ϖn}inOΓ(ϖ0)is such thatwith respect to A implies

||Φ(ϖ)||A≤lim inf||Φ(ϖn)||A.

Definition 3.7.Let(Ω,A,Δ)be aC*-algebra valued quasi controlledK-metric space.Γ:Ω→Ωis aC*-left-contractive(respectivelyC*-right-contractive mapping)if there existsρ∈Ωand anδ∈A such that

with‖δ‖＜1 for everyϖ∈OΓ(ρ).

Theorem 3.2.Let(Ω,A,Δ)be a completeC*-algebra valued quasi controlledK-metric space such thatΔis continuous.Suppose thatΓ:Ω→ΩisC*-left-contractive for someδ∈A,ϖ0∈Ωandexists for every {ωn} ∈OΓ(ϖ0)such thatThenΓnϖ0→∈Ωasn→∞.Besidesis a fixed point ofΓif and only ifϖ→Δ(ϖ,Γϖ)isΓ-orbitally l.s.c at.

Proof.Similar to Theorem 3.1,we prove that{ϖn}is a Cauchy sequence.SinceΩis complete then

ϖn→ω∈Ω.Assume thatϖ→Δ(ϖ,Γϖ)isΓ-orbitally l.s.c at,we obtain

and this completes the proof.

4 Application

By applying the previous results and involving theC*-algebra valued quasi controlledK-metric space,we prove the existence and uniqueness of a solution of a nonlinear stochastic integral equation given by

where

1.Σis the support of a complete probability space;

2.(Σ,A,P),Λ(τ,ω)is the continuous stochastic free where‖Λ(τ;.)‖L2(Σ,A,P) ＜∞;

3.Θ(τ,ξ,ω)is the stochastic kernel whereΘ(τ,s;.)belongs toL∞(Σ,A,P)such that

4.,ω)is the unknown continuous real-valued stochastic process such that

‖;.)‖L2(Σ,β,P) ＜∞.

LetEbe the space of all continuous functions from R into the spaceL2(Σ,A,P)such thatg(τ,.)∈L2(Σ,A,P),‖g(τ;.)‖L2(Σ,A,P) ＜∞andτ→g(τ,.)is continuous from R intoL2(Σ,A,P)for everyg∈E.

Now,we define the integral operatorΨonEBby

We now claim;ω))is bounded and continuous in mean-square.Indeed

Assume now the functionΛ(τ;ω)is a bounded continuous function from R intoL2(Σ,A,P)and the functionϑ(ξ,;ω))is in theC(R,L2(Ω,β,P))satisfying the condition

whereρandβare constants withandEρis defined as

Define the operatorΓfromEρintoEby

Moreover,under the conditions‖Λ(τ;ω)‖L2(Σ,A,P)+M‖ϑ(τ,0)‖L2(Σ,A,P)≤ρ(1-βM),we get

Hence,;ω)∈EρsoΓis self mapping onEρ.

We prove the existence of solutions to problem 4 utilising our deduced fixed point theorems.Now,letΩ=EρandH=L2(R).We denote the set of all bounded linear operators on Hilbert spaceHby A = B(H).Note that B(H)is a unitaryC*-algebra.We define aC*-algebra quasi controlledK-metricΔ:Ω×Ω→A by:

Similar to the Example 6, one can easily verify the completeness of(Ω,A,Δ).Then, we get by using our assumptions

SinceβM(1+3ρ)＜1,Γsatisfies the inequality(1).Therefore,the integral Eq.(4)has a unique solution by Theorem 3.1.

Example 4.1.LetΣ=]0,1[andα∈]0,[.We consider

Note that for allξ∈R,the functionτ-→Ψ(τ,ξ;.)is continuous from R intoL∞(Σ,β,P).

Assume thatΛ(τ,ω)= 0 and we take.Then,we can check that condition 5 is satisfied with

Now let

5 Conclusion

The results obtained are supported by non-trivial examples and complement and extend some of the most recent results from the literature.We have made a contribution by establishing some basic fixed-point problems considering aC*-algebra valued quasi controlledK-metric.We have proved some existence results for maps satisfying a new class of contractive conditions.The fixed point theorems are essential notions in the theory of integral equations.We have proved that the solution of a nonlinear stochastic integral equation of the Hammerstein type of a more general context using aC*-algebra quasi controlledK-metric spaces.

Future study is to investigate the sufficient conditions to guarantee the existence of a unique positive definite solution of the nonlinear matrix equations in the setting ofC*-algebra-valued quasi controlledK-metric spaces.The conditions of Theorem 3.1 will be verified numerically by giving various values for the given matrices,and the convergence analysis of nonlinear matrix equations will be shown through graphical representations.

Acknowledgement:The authors Thabet Abdeljawad and Aziz Khan would like to thank Prince Sultan University for the support through the TAS research lab.

Funding Statement:The article is financially supported by Prince Sultan University.

Conflicts of Interest:The authors declare that they have no conflicts of interest to report regarding the present study.