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Quasi Controlled -Metric Spaces over -Algebras with an Application to Stochastic Integral Equations
1 Department of Mathematics, Laboratory of Partial Differential Equations, Algebra and Spectral Geometry, Faculty of Sciences, Ibn Tofail University, Kenitra, BP 133, Morocco
2 Department of Mathematics and Sciences, Prince Sultan University, P.O. Box 66833, Riyadh, 11586, Saudi Arabia
3 Department of Medical Research, China Medical University, Taichung, 40402, Taiwan
* Corresponding Author: Thabet Abdeljawad. Email:
(This article belongs to the Special Issue: Computational Aspects of Nonlinear Operator and Fixed Point Theory with Applications)
Computer Modeling in Engineering & Sciences 2023, 135(3), 2649-2663. https://doi.org/10.32604/cmes.2023.023496
Received 28 April 2022; Accepted 16 August 2022; Issue published 23 November 2022
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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 -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
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].
The b-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 extended b-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 the s-relaxed inequality by one or two functions. Mudasir et al. [13] stated new results in the context of dislocated b-metric spaces and presented an application related to electrical engineering and extended the notion of Kannan maps in view of the F-contraction in this framework, see [14].
In [15,16], Ma et al. introduced C∗-algebra valued b-metric spaces by considering metrics that take values in the set of positive elements of a unitary C∗-algebra. Lately, Asim et al. [17] enlarged this class by defining C∗-algebra-valued extended b-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 of C∗-algebra-valued quasi controlled K-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.
Throughout this paper, A will be a unitary C∗-algebra with IA and σ(δ) is the spectrum of δ∈A. We set
Ah={δ∈A:δ∗=δ},A+={δ∈Ah:σ(δ)⊂[0,+∞[};
AI={δ∈ZI:δ⪰IA},ZI={δ∈A:δγ=γδ;∀γ∈A}.
Note that A+ is a cone [20], which induces a partial order ⪯ on Ah by
γ⪯δ⇔δ−γ∈A+.
To prove our main results, it will be useful to introduce the following lemma.
Lemma 2.1. [20] Suppose that A is a unital C∗-algebra with a unit IA.
1. if γ,δ∈Ah and γ⪯δ, then for each ξ∈A,ξ∗γξ⪯ξ∗δξ;
2. if γ,δ∈Ah, γ,δ⪰0A and γδ=δγ, then γδ⪰0A;
3. for all γ,δ∈Ah, 0A⪯γ⪯δ⇔‖γ‖≤‖δ‖;
4. 0⪯γ⪯IA⇔‖γ‖≤1.
Definition 2.1. [17] Let Ω≠∅ and Λ:Ω×Ω→AI. A C∗-algebra-valued extended b-metric is a mapping Δ: Ω×Ω→A such that
1. Δ(ω,ϖ)=0A if and only if ω=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 a C∗-algebra valued extended b-metric space.
In this section, by omitting the symmetry condition, we introduce the notion of C∗-algebra-valued quasi controlled K-metric spaces, where K is a control function.
Definition 3.1. A C∗-algebra-valued quasi controlled K-metric space is the triplet (Ω,A,Δ) where Ω is a non empty set, K:Ω×Ω→AI is a C∗-control function and Δ:Ω×Ω→A is a mapping that
1. Δ(ω,ϖ)=0A if and only if ω=ϖ;
2. Δ(ω,ϖ)≼K(ω,ϖ)[Δ(ω,ν)+Δ(ν,ϖ)]forallω,ϖ,ϑ∈Ω.
Remark 3.1. In particular, by taking K(ω,ϖ)=δ≽IA, (Ω,A,Δ) is a C∗-algebra-valued quasi b-metric space [19].
Example 3.1. Let Ω=[0,1] and A=M2(R). We know that A is a C∗-algebra where partial ordering on M2(R) is given as
(αji)1⩽i,j≤2⪰(βji)1⩽i,j≤2⇔αji≥βji for i=1,2.
Define a C∗-algebra-valued quasi controlled K-metric Δ:Ω×Ω→R2 by:
{Δ(ρ,ϖ)=[0000],iffρ=ϖΔ(ρ,0)=Δ(0,ρ)=[1ρ001ρ],if ρ≠0Δ(ρ,ϖ)=[1+ϖρϖ001+ρρϖ],if ϖρ≠0.
Given the C∗-control function K:Ω×Ω→AI as
K(ρ,ϖ)=[1+1ρ+ϖ+1001+1ρ+ϖ+1].
Then, (Ω,A,Δ) is a C∗-algebra-valued quasi controlled K-metric space.
Example 3.2. Let Ω=[0,1] and A=M2(C). Define a mapping Δ:Ω×Ω→A as
Δ(ρ,ϖ)=[(1+2|ρ|+|ϖ|)|ρ−ϖ|200(1+2|ρ|+|ϖ|)|ρ−ϖ|2].
Let the C∗-control function K:Ω×Ω→A be defined by (for allρ,ϖ∈Ω)
K(ρ,ϖ)=2[1+2|ρ|+|ϖ|001+2|ρ|+|ϖ|].
Example 3.3. Consider Ω=C(S,C) the space of all continuous functions where S is compact. Let A=L∞(S) the usual unital C∗-algebra with the sup norm and given Δ:Ω×Ω→A+ for each φ,ψ∈Ω as
Δ(φ,ψ)(t)={0, if φ=ψ1−11+|φ(t)|, if φ≠0,ψ=01−11+|ψ(t)|, if φ=0,ψ≠0|φ(t)|+2|ψ(t)|, if φψ≠0
We take
K(ψ,φ)(t)=|ψ(t)|+2|φ(t)|+2.
Thus, (Ω,Δ,L∞(S)) is a C∗-algebra-valued quasi controlled K-metric space.
Next, we introduce some topological concepts on C∗-algebra-valued quasi controlled K-metric spaces.
Definition 3.2. Let (Ω,A,Δ) be a C∗-algebra-valued quasi controlled K-metric space. The open ball B(ω,r) of center ω∈Ω and radius r⪰0A is given by
B(ω,r)={ϖ∈Ω:Δ(ω,ϖ)≺r}.
Example 3.4. Let us define a C∗-algebra-valued quasi controlled K-metric Δ:C∗×C∗→R2+ as
Δ(z,z′)={(0,0),if z=z1|zz′|+1|z|,1|zz′|+11|z′|,if z≠z′
with the C∗-controlled function K:C∗×C∗→]1,+∞[×]1,+∞[ given by
K(z,z′)=(1+|z′||z′|,1+|z||z|).
Then, it is evident that
Δ(z,z′)⪯Λ(z,z′)[Δ(z,z′′)+Δ(z′′,z′)],∀z,z′,z′′∈C∗.
The open ball B is given by
B(z0,r.1A)=B(z0,(r,r))={z∈C∗:Δ(z0,z)≺(r,r)}={z0}∪{z∈C∗:z≠z0 and (1|z0z|+1|z0|,1|z0z|+1|z|)≺(r,r)}(|z|+1|z0z|,1+|z0||z0z|)≺(r,r)⇒{|z|+1|z|<r|z0||z|>1+|z0|r|z0|
if r|z0|⩽1, then
B(z0,r.1A)={z0}
if r|z0|>1, then
B(z0,r.1A)={z0}∪{z∈C∗:|z|∈]max(1r|z0|−1+1+|z0||z0|),+∞[}.
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 a C∗-algebra-valued quasi controlled K-metric space and let {ϖn} be a sequence in Ω.
1. {ϖn} is called left-converges to ϖ∈Ω with 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 to ϖ∈Ω with respect to A, if and only if ∀ε≻0A∃k∈N such that
n>k⇒Δ(ϖ,ϖn)≺ε.
3. {ϖn} is called converges to ϖ∈Ω with respect to A, if and only if
limn→∞Δ(ϖ,ϖn)=limn→∞Δ(ϖn,ϖ)=0A.
Definition 3.4. Let (X,A,Δ) be a C∗-algebra-valued quasi controlled K-metric space. Then
1. {ϖn} is called right-Cauchy with respect to A, if for each ε≻0A there exists k∈N such that ∀p∈N,
n>k⇒Δ(ϖn,ϖn+p)≺ε.
2. {ϖn} is called left-Cauchy with respect to A, if for each ε≻0A there exists k∈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,
limn→∞Δ(ϖn,ϖn+p)=limn→∞Δ(ϖn+p,ωn)=0A.
4. If every Cauchy sequence {ϖn} in Ω converges to some point ϖ in Ω, then, the triplet (Ω,A,Δ) is said to be a complete C∗-algebra-valued quasi controlled K-metric space.
Example 3.5. Take Ω=R+ and A=R2
Δ(η,ν)={(0,0), if η=ν(η1+η,η1+η), if η≠0,ν=0(ν1+ν,ν1+ν), if η=0,ν≠0(η+2ν,η+2ν), if ην≠0,η≠ν
Let K:Ω×Ω→AI be the mapping defined by
K(η,ν)=(2η+2ν+2,2η+2ν+2).
Then, (Ω,A,Δ) is a complete C∗-algebra-valued quasi controlled K-metric space.
Example 3.6. Let S be a compact Hausdorff space and A=C(S) be the set of complex valued continuous functions on S. Note that C(S) is a unitary commutative C∗-algebra with the usual sup norm such that the involution is defined by ψ∗(x)=¯ψ(x)forallx∈S. Setting Ω=L∞(E) where E is a Lebesgue mensurable set and let us define a C∗-algebra-valued quasi controlled K-metric Δ:Ω×Ω→A by
Δ(ϕ,ψ)(t)=(1+‖ϕ‖∞+2‖ψ‖∞)‖ϕ−ψ‖∞etforallϕ,ψ∈Ω;t∈[0,1].
Let us define the C∗-control operator by
K(ϕ,ψ)=(1+‖ϕ‖∞+2‖ψ‖∞)IA.
The condition (i) of Definition 3.1 is clearly satisfied by Δ. Now we check the condition (ii). We take ϕ,ψ∈Ω as arbitrary. Then
Δ(ϕ,ψ)(t)=(1+‖ϕ‖∞+2‖ψ‖∞)‖ϕ−ψ‖∞et≤(1+‖ϕ‖∞+2‖ψ‖∞)(‖ϕ−φ‖∞+‖φ−ψ‖∞)et≤K(ϕ,ψ)[Δ(ϕ,ψ)(t)+Δ(φ,ψ)(t)]for all t∈[0,1].
Therefore,
Δ(ϕ,ψ)⪯K(ϕ,ψ)(Δ(ϕ,φ)+Δ(φ,ψ))forallϕ,ψφ∈Ω.
This prove that Δ is a C∗-algebra-valued quasi controlled K-metric. Now we want to verify that (X,A,Δ) is a complete C∗-algebra-valued quasi controlled K-metric space. Let {ϕn}∞n=1 be a Cauchy sequence in Ω with respect to A. Then
limn→∞Δ(ϕn,ϕn+p)=limn→∞Δ(ϕn+p,ϕn)=0A.
We deduce limn→∞‖ϕn+p−ϕn‖∞=0, so {ϕn}∞n=1 is a Cauchy sequence in the space Ω. Since Ω is complete, {ϕn} has a limit ˜ϕ that is also in Ω. Hence it follows that
Δ(ϕn,˜ϕ)⪯e(1+‖ϕn‖∞+2‖˜ϕ‖∞)‖ϕn+p−ϕn‖∞IA
and
Δ(˜ϕ,ϕn)⪯e(1+‖˜ϕ‖∞+2‖ϕn‖∞)‖ϕn+p−ϕn‖∞IA.
We conclude that the sequence {ϕn}∞n=1 converges to the function ˜ϕ in Ω respecting 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 the b-metric with respect to the topology it generates.
Definition 3.5. Let Δ be a C∗-algebra-valued quasi controlled K-metric. Δ is said to be continuous at (ϖ,ω) if the sequence {ωn}∞n=0 converges to ω and {ϖn}∞n=0 converges to ϖ then
Δ(ωn,ϖn)→Δ(ω,ϖ)andΔ(ϖn,ωn)→Δ(ϖ,ω).
Lemma 3.1. Let (Ω,A,Δ) be a C∗-algebra-valued quasi controlled K-metric space. Such Δ is continuous in each variable. If a sequence {ωn}∞n=1 has a limit, then this limit is unique.
Proof. Fix ε≻0A. By assumption, ϖn converges to ω so there exists K1∈N such that d(ω,ϖn)⪯ε2 for all n⩾K1. We also assume that ϖn converges to ϖ, so there exists K2∈N such that d(ϖn,ϖ)⪯ε2 for all n⩾k2. Then for all n⩾K:=max{K1,K2}
Δ(ϖ,ω)⪯K(ϖ,ω)[Δ(ϖ,ϖn)+Δ(ϖn,ω)]⪯K(ϖ,ω)ε.
As ε was arbitrary, we deduce that Δ(ϖ,ω)=0, which implies ϖ=ω.
Our main result runs as follows.
Theorem 3.1. Let (Ω,A,Δ) be complete C∗-algebra-valued quasi controlled K-metric space such that Δ is a continuous and Γ:Ω→Ω satisfies the following:
Δ(Γϖ,Γρ)≼θ∗Δ(ϖ,ρ)θ,∀ϖ,ρ∈Ω(1)
where θ∈A with ‖θ‖A<1 and limn,m→∞‖K(ϖn,ϖm)‖A‖θ‖A≺IA such that ωn=Γϖn−1=Γnϖ0 for an arbitrary ϖ0. Then Γ has a unique fixed point ˜ω∈Ω.
Proof. Let the sequence {ϖn} be defined by ϖn=Γϖn−1=Γnϖ0. From Eq. (1), we obtain by induction
Δ(ϖn,ϖn+1)=Δ(Γϖn−1,Γϖn)⪯θ∗Δ(ϖn−1,ϖn)θ≼(θ∗)2Δ(ϖn−2,ϖn−1)θ2⋮≼(θ∗)nΔ(ϖ0,ϖ1)θn.
Now we prove that {ϖn} is a right-Cauchy sequence. For any n,p∈N, we have
Δ(ϖn,ϖn+p)≼K(ϖn,ϖn+p)[Δ(ϖn,ϖn+1)+Δ(ϖn+1,ϖn+p)]≼K(ϖn,ϖn+p)Δ(ϖn,ϖn+1)+K(ϖn,ϖn+p)K(ϖn+1,ϖn+p)Δ(ϖn+1,ϖn+2)⋮K(ϖn,ϖn+p)K(ϖn+1,ϖn+p)…K(ϖn+p−2,ϖn+p)K(ϖn+p−1,ϖn+p)Δ(ϖn+p−1,ϖn+p)≼K(ϖn,ϖn+p)(θ∗)nK(ϖ0,ϖ1)θn+K(ϖn,ϖn+p)K(ϖn+1,ϖn+p)(θ∗)n+1K(ϖ0,ϖ1)θn+1⋮K(ϖn,ϖn+p)K(ϖn+1,ϖn+p)…K(ϖn+p−2,ϖn+p)+K(ϖn+p−1,ϖn+p)(θ∗)n+p−1Δ(ϖ0,ϖ1)θn+p−1=K(ϖn,ϖn+p)(θ∗)n(Δ(ϖ0,ϖ1)12)2θn+K(ϖn,ϖn+p)K(ϖn+1,ϖn+p)(θ∗)n+1(Δ(ϖ0,ϖ1)12)2θn+1⋮+K(ϖn,ϖn+p)K(ϖn+1,ϖn+p)…K(ϖn+p−2,ϖn+p)K(ϖn+p−1,ϖn+p)(θ∗)n+p−1(K(ϖ0,ϖ1)12)2θn+p−1=K(ϖn,ϖn+p)(Δ(ϖ0,ϖ1)12θn)∗(Δ(ϖ0,ϖ1)12θn)+⋮K(ϖn,ϖn+p)K(ϖn+1,ϖn+p)…K(ϖn+p−1,ϖn+p)(Δ(ϖ0,ϖ1)12θn+p−1)∗(Δ(ϖ0,ϖ1)12θn+p−1)=K(ϖn,ϖn+p)|Δ(ϖ0,ϖ1)12θn|2+K(ϖn,ϖn+p)K(ϖn+1,ϖn+p)|Δ(ϖ0,ϖ1)12θn+1|2⋮K(ϖn,ϖn+p)K(ϖn+1,ϖn+p)…K(ϖn+p−2,ϖn+p)K(ϖn+p−1,ϖn+p)|Δ(ϖ0,ϖ1)12θn+p−1|2=n+p−1∑i=0|Δ(ϖ0,ϖ1)12θn+i|2i∏j=0K(ϖn+p+j,ϖn+p)≼||n+p−1∑i=0|Δ(ϖ0,ϖ1)12θn+i|2||Ai∏j=0‖K(ϖn+j,ϖn+p)‖AIA≼n+p−1∑i=0‖Δ(ϖ0,ϖ1)‖A||θn+i||2Ai∏j=0‖K(ϖn+j,ϖn+p)‖AIA≼‖Δ(ϖ0,ϖ1)‖An+p−1∑i=0||θn+i||2Ai∏j=0‖K(ϖn+j,ϖn+p)‖AIA
Since limn,m→∞‖K(ϖn,ϖm)‖A‖θ‖A<1 so that the series ∑∞n=1‖θn‖A∏ni=1‖K(ϖi,ϖm)‖A converges by ratio test for each m∈N. Let
Vn=n∑i=0||θi||2A∏ij=0‖K(ϖj,ϖm)‖A and V=∞∑i=0||θi||2A∏ij=0‖K(ϖj,ϖm)‖A
Thus, the above inequality implies
Δ(ϖn,ϖn+p)≼‖Δ(ϖ0,ϖ1)‖A||θ2n||A[Vn+p−1−Vn].
Letting n→∞, we conclude that {ϖn} is a right-Cauchy sequence. Similarly, we prove that {ϖn} is a left-Cauchy sequence. The fact that Ω is complete involves ∃˜ω∈Ω such that
limn→∞Δ(˜ω,ωn)=limn→∞Δ(ωn,˜ω)=0A.
Remains to see that ˜ω is a fixed point of Γ. Indeed for any n∈N, we have
Δ(Γ˜ω,˜ω)≼K(Γ˜ω,˜ω)[Δ(Γ˜ω,ϖn+1)+Δ(ϖn+1,˜ω)]=K(Γ˜ω,˜ω)[Δ(Γ˜ω,Γϖn)+Δ(ϖn+1,˜ω)]≼K(T˜ω,˜ω)[θ∗Δ(˜ω,ϖn)θ+Δ(ϖn+1,˜ω)]→0A as n→∞.
Therefore, ˜ω is a fixed point of Γ. To prove uniqueness, we can assume Γ˜ω=˜ω and Γω∗=ω∗ such that ˜ω,ω∗∈Ω. Then by employing Eq. (1), we have
Δ(˜ω,ω∗)=Δ(Γ˜ω,Γω∗)≼θ∗Δ(˜ω,ω∗)θ,
so that
||Δ(˜ω,ω∗)||A=||Δ(Γ˜ω,Γω∗)||A≤||θ∗Δ(˜ω,ω∗)θ||A≤||θ∗||||Δ(˜ω,ω∗)||‖θ‖A=‖θ‖2A||Δ(˜ω,ω∗)||A<||Δ(˜ω,ω∗)||A.
Then, we get a contradiction, as a result ω=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. Let X and Y be Banach spaces. S⊂X is the state space and D⊂Y is the decision space. Let η:S×D→S, τ:S×D→R and T:S×D×R→R. Denote by B(S) the set of all real-valued bounded functions on S. Let A=L∞(S) the usual unital C∗-algebra with the sup norm and given Δ:B(S)×B(S)→A+ for each φ,ψ∈Ω as
Δ(φ,ψ)=1+‖φ‖+‖ψ‖1+‖φ‖‖φ−ψ‖.IA
(B(S),Δ,L∞(S)) is a complete C∗-algebra-valued quasi controlled K-metric space. We consider the functional equation
ϖ(x)=supy∈D[τ(x,y)+T(x,y,ϖ(η(x,y)))](x∈S)(2)
such that τ and T are bounded and
|T(x,y,z1)−T(x,y,z2)|⩽α1+2m|z1−z2|
for all (x,y,z1),(x,y,z2) in S×D×R, where 0≤α<1 and m=‖T‖. We define a mapping Γ:B(S)→B(S) by Γϖ=h, where
h(x)=supy∈D[τ(x,y)+T(x,y,ϖ(η(x,y)))](x∈S).
It is easy to get Δ(Γρ,Γϖ)≼θ∗Δ(ρ,ϖ)θ satisfies with θ=√αIA.
Therefore, the Eq. (1) possesses unique bounded solution on S.
Example 3.8. Let Ω=R and A=M2(C). For any A∈A, we define its norm as ‖A‖A=max1≤i≤4|ai|. Define a mapping Δ:Ω×Ω→A such that for all ρ and ϖ∈Ω,
Δ(ρ,ϖ)=[(1+2|ρ|+|ϖ|)|ρ−ϖ|200(1+2|ρ|+|ϖ|)|ρ−ϖ|2].
Let the C∗-control function K:Ω×Ω→A by:
K(ρ,ϖ)=2[1+2|ρ|+|ϖ|001+2|ρ|+|ϖ|].
We define a mapping Γ:Ω→Ω by
Γρ=ρ3,for allρ∈Ω.
It is easy to get Δ(Γρ,Γϖ)≼θ∗Δ(ρ,ϖ)θ
where θ=[√3300√33]∈A and ‖θ‖A=√33=1√3<1.
Definition 3.6. Let Ω≠∅ and OΓ(ϖ0)={Γnϖ0|n∈N} for an arbitrary ϖ0∈Ω. A function Φ:Ω→A is said to be Γ-orbitally lower semi continuous at ϖ with respect to A if the sequence {ϖn} is such that limn→∞ϖn=ϖ with respect to A implies
||Φ(ϖ)||A⩽lim inf||Φ(ϖn)||A.
Definition 3.7. Let (Ω,A,Δ) be a C∗-algebra valued quasi controlled K-metric space. Γ:Ω→Ω is a C∗-left-contractive (respectively C∗-right-contractive mapping) if there exists ρ∈Ω and an δ∈A such that
Δ(Γϖ,Γ2ϖ)⪯δ∗Δ(ϖ,Γϖ)δ(respectivelyΔ(Γϖ,Γ2ϖ)⪯δ∗Δ(Γ,ϖ)δ)(3)
with ‖δ‖<1 for every ϖ∈OΓ(ρ).
Theorem 3.2. Let (Ω,A,Δ) be a complete C∗-algebra valued quasi controlled K-metric space such that Δ is continuous. Suppose that Γ:Ω→Ω is C∗-left-contractive for some δ∈A, ϖ0∈Ω and limn,m→∞K(ϖn,ϖm) exists for every {ωn}∈OΓ(ϖ0) such that limn,m→∞‖K(ϖn,ϖm)‖A<1‖δ‖A. Then Γnϖ0→˜ω∈Ω as n→∞. Besides ˜ω is a fixed point of Γ if and only if ϖ→Δ(ϖ,Γϖ) is Γ-orbitally l.s.c at ˜ω.
Proof. Similar to Theorem 3.1, we prove that {ϖn} is a Cauchy sequence. Since Ω is complete then ϖn→˜ω∈Ω. Assume that ϖ→Δ(ϖ,Γϖ) is Γ-orbitally l.s.c at ˜ω, we obtain
‖Δ(˜ω,Γ˜ω)‖A≤lim infn→∞‖Δ(Γnϖ0,Γn+1ϖ0)‖A≤lim infn→∞‖δ∗‖A‖Δ(Γn−1ϖ0,Γnϖ0)‖A‖δ‖A≤lim infn→∞‖δ‖2nA‖Δ(ϖ0,ϖ1)‖A→0
We find Δ(˜ω,Γ˜ω)=0. It follows that Γ˜ω=˜ω. Conversely, let ˜ω=Γ˜ω and {ϖn} a sequence in OΓ(ϖ0) with ϖn→˜ω. Then
‖Δ(˜ω,Γ˜ω‖A)=0≤lim infn→∞‖Δ(ϖn,Γϖn‖A),
and this completes the proof.
By applying the previous results and involving the C∗-algebra valued quasi controlled K-metric space, we prove the existence and uniqueness of a solution of a nonlinear stochastic integral equation given by
ϰ(τ;ω)=Λ(τ;ω)+∫RΘ(τ;ξ;ω)ϑ(ξ;ϰ(ξ;ω))dξτ∈R,ω∈Σ,(4)
where
1. Σ is 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 where Θ(τ,s;.) belongs to L∞(Σ,A,P) such that
supτ∈R∫R‖Θ(τ;ξ;ω)‖L∞(Σ,A,P)dξ<∞;
4. ϰ(τ,ω) is the unknown continuous real-valued stochastic process such that
‖ϰ(τ;.)‖L2(Σ,β,P)<∞.
Let E be the space of all continuous functions from R into the space L2(Σ,A,P) such that g(τ,.)∈L2(Σ,A,P), ‖g(τ;.)‖L2(Σ,A,P)<∞ and τ→g(τ,.) is continuous from R into L2(Σ,A,P) for every g∈E.
We consider EB={ϰ∈C(R,L2(Σ,β,P)):‖ϰ(τ,Σ)‖EB=supτ∈R‖ϰ(τ,Σ)‖L2(Σ,A,P)<∞}. Now, we define the integral operator Ψ on EB by
(Ψϰ)(τ;ω)=∫RΘ(τ;s;ω)ϰ(s;ω)d(s)
We now claim (Ψϰ)(τ;ω)) is bounded and continuous in mean-square. Indeed
‖(Ψϰ)(τ;ω)‖L2(Σ,A,P)≤∫R‖Θ(τ;ξ;ω)ϰ(ξ,ω)‖L2(Σ,A,P)dξ≤supξ∈R‖ϰ(ξ,ω)‖L2(Σ,A,P)∫R‖Θ(τ;ξ;ω)‖L∞(Ω,A,P)dξ≤‖ϰ(ξ,ω)‖EB∫R‖Θ(τ;ξ;ω)‖L∞(Σ,A,P)dξ≤M‖ϰ(ξ,ω)‖EB,
where M=supτ∈R∫R‖Θ(τ;s;ω)‖L∞(Σ,A,P)ds. This proves (Ψϰ)(τ;ω))∈EB, that means Ψ is an operator from EB into EB.
Assume now the function Λ(τ;ω) is a bounded continuous function from R into L2(Σ,A,P) and the function ϑ(ξ,ϰ(ξ;ω)) is in the C(R,L2(Ω,β,P)) satisfying the condition
‖ϑ(ξ,ϰ(ξ;ω))−ϑ(ξ,η(ξ;ω))‖L2(Ω,A,P)≤β‖ϰ(ξ;ω)−η(ξ;ω)‖L2(Σ,A,P),∀ϰ,η∈Eρ(5)
where ρ and β are constants with βM<11+3ρ and Eρ is defined as
Eρ={x∈C(R,L2(Σ,A,P)):‖ϰ(ξ,ω)‖Eρ=supξ∈R‖ϰ(ξ,ω)‖L2(Σ,A,P)<ρ}.
Define the operator Γ from Eρ into E by
(Γϰ)(τ;ω)=Λ(τ;ω)+∫RΘ(τ;ξ;ω)ϑ(ξ,ϰ(ξ;ω))dξ.
Moreover, under the conditions ‖Λ(τ;ω)‖L2(Σ,A,P)+M‖ϑ(τ,0)‖L2(Σ,A,P)≤ρ(1−βM), we get
‖(Γϰ)(τ;ω)‖L2(Σ,A,P)≤‖Λ(τ;ω)‖L2(Σ,A,P)+M‖ϑ(t,ϰ(τ;ω))‖L2(Σ,A,P)≤‖Λ(τ;ω)‖L2(Σ,A,P)+M‖ϑ(τ,0)‖L2(Σ,A,P)+Mβ‖ϰ(τ;ω)‖L2(Σ,A,P)≤ρ.
Hence, (Γϰ)(τ;ω)∈Eρ so Γ is self mapping on Eρ.
We prove the existence of solutions to problem 4 utilising our deduced fixed point theorems. Now, let Ω=Eρ and H=L2(R). We denote the set of all bounded linear operators on Hilbert space H by A=B(H). Note that B(H) is a unitary C∗-algebra. We define a C∗-algebra quasi controlled K-metric Δ:Ω×Ω→A by:
Δ(ϰ,η)=π(1+‖ϰ‖L2(Σ,A,P)+2‖η‖L2(Σ,A,P))‖ϰ−η‖L2(Σ,A,P).
Similar to the Example 6, one can easily verify the completeness of (Ω,A,Δ). Then, we get by using our assumptions
||Δ(Γϰ,Γη)||B(H)=sup‖ϕ‖=1⟨π(1+‖Γϰ‖L2(Σ,A,P)+2‖Γη‖L2(Σ,A,P))‖Γϰ−Γη‖L2(Σ,A,P)ϕ,ϕ⟩⩽(1+3ρ)sup‖ϕ‖=1∫R||∫RΘ(τ,ξ;ω)[ϑ(ξ,ϰ(ξ;ω))−ϑ(ξ,η(ξ;ω))]dξ||L2(Σ,A,P)|ϕ(τ)|2dτ≤βM(1+3ρ)sup‖ϕ‖=1∫R|ϕ(τ)|2‖ϰ(ξ;ω)−η(ξ;ω)‖EBdτ≤βM(1+3ρ)supτ∈R‖ϰ(ξ;ω)−η(ξ;ω)‖L2(Σ,A,P)≤βM(1+3ρ)sup‖ϕ‖=1⟨π(1+‖ϰ‖L2(Ω,A,P)+2‖η‖L2(Σ,A,P))‖ϰ−η‖L2(Σ,A,P)ϕ,ϕ⟩≤βM(1+3ρ)||Δ(ϰ,η)||B(H).
Since βM(1+3ρ)<1, Γ satisfies the inequality (1). Therefore, the integral Eq. (4) has a unique solution by Theorem 3.1.
Example 4.1. Let Σ=]0,1[ and α∈]0,16[. We consider
Θ:R×R×Σ→R(τ,ξ,ω)→|τ|αω(τ2+τ+1)(ξ2τ2+1)
Note that for all ξ∈R, the function τ⟼Ψ(τ,ξ;.) is continuous from R into L∞(Σ,β,P).
‖(Ψϰ)(τ;ω)‖L2(Σ,A,P)≤∫R‖Θ(τ;ξ;ω)ϰ(ξ,ω)‖L2(Σ,A,P)dξ≤supξ∈R‖ϰ(ξ,ω)‖L2(Ω,A,P)∫R‖ταω(τ2+τ+1)(ξ2τ2+1)‖L∞(Ω,A,P)dξ≤α‖ϰ(ξ,ω)‖EB∫R|τ|(τ2+τ+1)(ξ2τ2+1)dξ≤απ‖ϰ(ξ,ω)‖EB≤M‖ϰ(ξ,ω)‖EB.
Assume that Λ(τ,ω)=0 and we take ϑ(ξ,ϰ(ξ;ω))=eξ8(eξ−ξ)(1+ξ2+|ϰ(ξ;ω)|). Then, we can check that condition 5 is satisfied with β=e8(e−1).
Now let
E3={x∈C(R,L2(Σ,A,P)):‖ϰ(ξ,ω)‖E2=supξ∈R‖ϰ(ξ,ω)‖L2(Σ,A,P)<2}.
We see that βM(1+3ρ)=10πe48(e−1)<1, so all the assumptions mentioned in the application section are well insured. Hence, there exists unique solution of the nonlinear integral equation given by
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 a
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 of
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.
Bibliography
1. Banach, S. (1922). Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fundamenta Mathematicae, 3(1), 133–181. [Google Scholar]
2. Wallace Alvin, W. (1931). On quasi-metric spaces. American Journal of Mathematics, 53(3), 675–684. DOI 10.2307/2371174. [Google Scholar] [CrossRef]
3. Azam, A., Waseem, M., Rashid, M. (2013). Fixed point theorems for fuzzy contractive mappings in quasi-pseudo-metric spaces. Journal of Fixed Point Theory and Applications, 2013(1), 1–14. [Google Scholar]
4. Bakthin, I. (1989). The contraction mapping principle in almost metric spaces. Functional Analysis, 30, 26–37. [Google Scholar]
5. Czerwik, S. (1993). Contraction mappings in b-metric spaces. Acta Mathematica Universitatis Ostraviensis, 1(1), 5–11. [Google Scholar]
6. Bota, M. F., Guran, L., Petruşel, A. (2020). New fixed point theorems on b-metric spaces with applications to coupled fixed point theory. Journal of Fixed Point Theory and Applications, 22(3), 1–14. DOI 10.1007/s11784-020-00808-2. [Google Scholar] [CrossRef]
7. Goswami, N., Haokip, N., Mishra, V. N. (2019). F-contractive type mappings in b-metric spaces and some related fixed point results. Journal of Fixed Point Theory and Applications, 2019(1), 1–17. [Google Scholar]
8. Khan, A. R., Oyetunbi, D. M. (2020). On some mappings with a unique common fixed point. Journal of Fixed Point Theory and Applications, 22(2), 1–7. DOI 10.1007/s11784-020-00781-w. [Google Scholar] [CrossRef]
9. Ansari, A. H., Saleem, N., Fisher, B., Khan, M. (2017). C-class function on khan type fixed point theorems in generalized metric space. Filomat, 31(11), 3483–3494. DOI 10.2298/FIL1711483A. [Google Scholar] [CrossRef]
10. Saleem, N., Abbas, M., Bin-Mohsin, B., Radenovic, S. (2020). Pata type best proximity point results in metric spaces. Miskolc Mathematical Notes, 21(1), 367–386. DOI 10.18514/MMN.2020.2764. [Google Scholar] [CrossRef]
11. Samreen, M., Kamran, T., Postolache, M. (2018). Extended b-metric space, extended b-comparison function and nonlinear contractions. UPB Scientific Bulletin, Series A, 80(4), 21–28. [Google Scholar]
12. Abdeljawad, T., Mlaiki, N., Aydi, H., Souayah, N. (2018). Double controlled metric type spaces and some fixed point results. Mathematics, 6(12), 320. DOI 10.3390/math6120320. [Google Scholar] [CrossRef]
13. Younis, M., Singh, D., Abdou, A. A. (2022). A fixed point approach for tuning circuit problem in dislocated b-metric spaces. Mathematical Methods in the Applied Sciences, 45(4), 2234–2253. DOI 10.1002/mma.7922. [Google Scholar] [CrossRef]
14. Younis, M., Singh, D. (2022). On the existence of the solution of hammerstein integral equations and fractional differential equations. Journal of Applied Mathematics and Computing, 68(2), 1087–1105. DOI 10.1007/s12190-021-01558-1. [Google Scholar] [CrossRef]
15. Ma, Z., Jiang, L., Sun, H. (2014). C*-algebra valued metric spaces and related fixed point theorems. Journal of Fixed Point Theory and Applicationss, 2014(1), 1–11. [Google Scholar]
16. Ma, Z., Jiang, L. (2015). C*-algebra valued b-metric spaces and related fixed point theorems. Journal of Fixed Point Theory and Applicationss, 2015(1), 1–12. DOI 10.1186/s13663-015-0471-6. [Google Scholar] [CrossRef]
17. Asim, M., Imdad, M. (2020). C*-algebra valued extended b-metric spaces and fixed point results with an application. UPB Scientific Bulletin, Series A, 82(1), 207–218. [Google Scholar]
18. Bouftouh, O., kabbaj, S. (2021). Fixed point theorems in C*-algebra valued asymmetric spaces. arXiv preprint arXiv:2106.11126. [Google Scholar]
19. Bouftouh, O., Kabbaj, S., Abdeljawad, T., Mukheimer, A. (2022). On fixed point theorems in C*-algebra valued b-asymmetric metric spaces. AIMS Mathematics, 7(7), 11851–11861. DOI 10.3934/math.2022661. [Google Scholar] [CrossRef]
20. Murphy, G. J. (2014). C*-algebra and operator theory. London, UK: Academic Press. [Google Scholar]
21. Mlaiki, N., Aydi, H., Souayah, N., Abdeljawad, T. (2018). Controlled metric type spaces and the related contraction principle. Mathematics, 6(10), 194. DOI 10.3390/math6100194. [Google Scholar] [CrossRef]
22. Aleksic, S., Huang, H., Mitrovic, Z., Radenovic, S. (2018). Remarks on some fixed point results in b-metric spaces. Journal of Fixed Point Theory and Applications, 20. DOI 10.1007/s11784-018-0626-2. [Google Scholar] [CrossRef]
23. Collins, J., Zimmer, J. (2007). An asymmetric arzela-ascoli theorem. Topology and its Applications, 154, 2312–2322. DOI 10.1016/j.topol.2007.03.006. [Google Scholar] [CrossRef]
24. Aminpour, A., Khorshidvandpour, S., Mousavi, M. (2012). Some results in asymmetric metric spaces. Mathematica Aeterna, 2, 533–540. [Google Scholar]
25. Mainik, A., Mielke, A. (2005). Existence results for energetic models for rate-independent systems. Calculus of Variations and Partial Differential Equations, 22(1), 73–100. DOI 10.1007/s00526-004-0267-8. [Google Scholar] [CrossRef]
26. Mielke, A., Roubícek, T. (2003). A rate-independent model for inelastic behavior of shape-memory alloys. Multiscale Modeling & Simulation, 1(4), 571–597. DOI 10.1137/S1540345903422860. [Google Scholar] [CrossRef]
27. Rieger, M. O., Zimmer, J. (2005). Young measure flow as a model for damage. Zeitschrift für angewandte Mathematik und Physik, 60(1), 1–32. [Google Scholar]
28. Mennucci, A. (2004). On asymmetric distances. Mathematica Japonica, 24, 327–330. DOI 10.2478/agms-2013-0004. [Google Scholar] [CrossRef]
29. Hicks, T. (1979). A banach type fixed point theorem. Mathematica Japonica, 24, 327–330. [Google Scholar]
30. Batul, S., Kamran, T. (2015). C*-algebra valued contractive type mappings. Journal of Fixed Point Theory and Applicationss, 2015(1), 1–9. [Google Scholar]
31. Mlaiki, N., Asim, M., Imdad, M. (2020). C*-algebra valued partial b-metric spaces and fixed point results with an application. Mathematics, 8(8), 1381. DOI 10.3390/math8081381. [Google Scholar] [CrossRef]
32. Lee, A. C., Padgett, W. (1977). On random nonlinear contractions. Mathematical Systems Theory, 11(1), 77–84. DOI 10.1007/BF01768469. [Google Scholar] [CrossRef]
33. Padgett, W. J. (1973). On a nonlinear stochastic integral equation of the hammerstein type. Proceedings of the American Mathematical Society, 38(3), 625–631. DOI 10.1090/S0002-9939-1973-0320663-2. [Google Scholar] [CrossRef]
34. van An, T., Tuyen, L. Q., van Dung, N. (2015). Stone-type theorem on b-metric spaces and applications. Topology and its Applications, 185, 50–64. DOI 10.1016/j.topol.2015.02.005. [Google Scholar] [CrossRef]
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