C ∗ -Algebra Valued Modular G-Metric Spaces with Applications in Fixed Point Theory

Article

C ^∗ -Algebra Valued Modular G-Metric Spaces with Applications in Fixed Point Theory

Dipankar Das¹ , Lakshmi Narayan Mishra^2,* , Vishnu Narayan Mishra^3,* , Hamurabi Gamboa Rosales^4,* , Arvind Dhaka^5,*, Francisco Eneldo López Monteagudo⁶, Edgar González Fernández⁴

and Tania A. Ramirez-delReal⁷

Citation: Das, D.; Mishra, L.N.;

Mishra, V.N.; Rosales, H.G.; Dhaka, A.; Monteagudo, F.E.L.; Fernández, E.G.; Ramirez-delReal, T.A.

C^∗-Algebra Valued ModularG-Metric Spaces with Applications in Fixed Point Theory.Symmetry2021,13, 2003.

https://doi.org/10.3390/

sym13112003

Academic Editors: Abraham A.

Ungar and Alexei Kanel-Belov

Received: 5 September 2021 Accepted: 9 October 2021 Published: 22 October 2021

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Licensee MDPI, Basel, Switzerland.

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creativecommons.org/licenses/by/

4.0/).

1 Department of Mathematical Sciences, Bodoland University, Kokrajhar 783370, India;

[email protected]

2 Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology (VIT) University, Vellore 632014, India

3 Department of Mathematics, Indira Gandhi National Tribal University, Lalpur, Amarkantak 484887, India

4 Center for Research and Innovation in Information and Communication (INFOTEC), Ciudad de México 14050, Mexico; [email protected]

5 Department of Computer and Communication Engineering, Manipal University Jaipur, Jaipur 303007, India

6 Academic Unit of Electrical Engineering, Autonomous University of Zacatecas, Zacatecas 98000, Mexico;

[email protected]

7 CONACyT—CentroGeo Centro de Investigación en Ciencias de Información Geoespacial, Aguascalientes 20313, Mexico; [email protected]

* Correspondence: [email protected] (L.N.M.); [email protected] (V.N.M.);

[email protected] or [email protected] (H.G.R.); [email protected] (A.D.)

Abstract:This article introduces a new type ofC^∗-algebra valued modularG-metric spaces that is more general than bothC^∗-algebra valued modular metric spaces and modularG-metric spaces.

Some properties are also discussed with examples. A few common fixed point results inC^∗-algebra valued modularG-metric spaces are discussed using the “C∗-class function”, along with some suitable examples to validate the results. Ulam–Hyers stability is used to check the stability of some fixed point results. As applications, the existence and uniqueness of solutions for a particular problem in dynamical programming and a system of nonlinear integral equations are provided.

Keywords:C∗-class function;mGMS;C^∗-avMS;C^∗-avb-MS;C^∗-avGMS; Ulam–Hyers stability

1. Introduction

In recent years,C^∗-algebra has attracted a lot of interest due to its prospective appli- cations in modern mathematics, entropy analysis, fixed point theory, noncommutative geometry, string theory, quantum mechanics, and other fields. LetAbe a Banach algebra and ‘∗’ be involution self-mapping onA. ThenAis said to be aC^∗-algebra if it satisfies for anyz₁,z₂∈A^andα,µ∈C^{: ([1,2])z∗∗}₁ =z₁,(z₁z₂)^∗ =z^∗₂z^∗₁,(αz₁+µz₂)^∗ =αz¯ ^∗₁+µz¯ ^∗₂ andkz^∗₁z₁k=kz₁k², (which easily shows thatkz₁^∗k=kz₁k). LetAbe an unitalC^∗-algebra with the identity 1_Aand zero elementθ. Every element of the setA+={x ∈A:xθ}is called positive, any element ofA+isx∈Awithx=x^∗and spectrumσ(x)⊂R+, where σ(x) ={α∈R:|α1_A−x|=0}.

A partial ordering “⁰⁰ onA^{behaves asx} y ⇔ x−y ∈ A+. For every element θx∈Ahas a unique positive square root, i.e.,|x|= (x^∗x)¹²_.

Ma et al. [3] initiatedC^∗-avMSby replacing real numbers with positive elements of unitalC^∗-algebra, which generalizes metric spaces, and studied some fixed point results.

Ma et al. [4] also generalized this concept and pioneeredC^∗-avb-MS. According to Alsulami et al. [5] and Kadelburg et al. [6], the fixed point results inC^∗-avMSandC^∗-avb-MScan be found as the implications of their classic metric spaces andb-metric spaces, respectively.

Despite this, Mustafa et al. [7] described the significance and obstacles of studying fixed

Symmetry2021,13, 2003. https://doi.org/10.3390/sym13112003 https://www.mdpi.com/journal/symmetry

point theory inC^∗-algebra, and how research on such spaces has become more popular among researchers. Recently, Kumar et al. [8] explained some fixed point results via

“C∗-class function” inC^∗-avMS, which are more general than metric spaces. Due to the importance of the study of fixed point results in the setting ofC^∗-algebra, researchers have initiated more new generalized spaces than metric spaces, such asC^∗-avGMS[9], C^∗-avG_bMS[10],C^∗-avSMS[11],C^∗-algebra valued partial metric spaces [12], etc., which enriches this field (see also [13–28]).

Chistyakov [29,30] initiated modular metric spaces. Since then researchers developed fixed point theory in these spaces. For instance, Zhu et al. [31] studied some fixed point results on asymptotic pointwise contractions in modular metric spaces that generalize metric spaces; Okeke et al. [32] introduced a few fixed point results for rational contractive mappings in modular metric spaces with applications in integrodifferential equations;

Shateri [33] initiatedC^∗-algebra valued modular spaces that generalize modular spaces.

Ege et al. [34] initiated modularb-metric spaces and applied these concepts in fixed point theory. Based on these papers, Moeini et al. [2,35,36] initiatedC^∗-algebra valued modular metric spaces and Das and Mishra [37] introducedC^∗-algebra valued modularb-metric spaces.

Hyers [38] answered Ulam’s [39] question about the stability of functional equations for Banach spaces, and the stability used for the answer is known as Ulam–Hyers stability.

Studies addressing Ulam–Hyers stability results and different stability results in fixed point theory can be seen in [40–50].

Mustafa and Sims [51] initiatedG-metric spaces as a metric space generalization and studied some fixed point results. Researchers developed the study densely forG-metric spaces in fixed point theory, some of which can be seen in [52–58].

Jleli et al. [59] and Samet et al. [60] showed that fixed point results inG-metric spaces can be generated from existence results in the context of quasi metric spaces.

Asadi et al. [61,62] proved some results inG-metric spaces which cannot be obtained from the existence result in the environment of metric space. Agarwal et al. [63] showed that if the contractivity condition of the fixed point result on a G-metric space can be simplified to two variables then an analogous fixed point result in the context of classic metric spaces can be established easily. They also constructed some new fixed point results that cannot be reduced to quasi metric spaces, with new contractive conditions.

Sedghi et al. [64] introducedS-metric spaces and claimed that space is a generalization of G-metric spaces, but Dung et al. [65] explained that this was incorrect. As a result, studying in such environments is both exciting and demanding.

Due to the demand for research in modular metric spaces, Azadifar et al. [66] initi- ated modularG-metric spaces, which generalize modular metric spaces as well as metric spaces. Azadifar et al. [67] also studied common fixed point results in modularG-metric spaces. Okeke et al. [68] studied some fixed point results in modularG-metric spaces and Okeke et al. [69] studied in preordered modularG-metric spaces, which were applied to solve nonlinear integral equations.

From the above study, it can be observed that, along with different generalized metric spaces,G-metric spaces and modularG-metric spaces have numerous applications in fixed point theory. Furthermore, studying the fixed point theorem in the context ofC^∗-algebra has a wide range of applications. Researchers present many applications from the obtained results by expressing multiple situations that exemplify the application domains in distinct generalizedC^∗-avMSas well as generalized modular metric spaces. The foregoing research leads us to investigate modularG-metric spaces inC^∗-algebra in order to generalize the existing spaces. The study of such spaces resulted in the generalization of modularG- metric spaces as well asC^∗-avGMS. Hence, all the results inC^∗-avGMSare automatically generalized compared to the existing spaces mentioned in the above literature.

In this paper, we introduceC^∗-algebra valued modularG-metric spaces via “C∗-class function” to generalize the fixed point results and to offer possible improvements on the structures of some types of metric spaces in algebraic topology. Some fixed point results

are discussed with suitable examples, and the stability of these results is checked by using Ulam–Hyers stability. Applications for existence and uniqueness results for a system of nonlinear integral equations and functional equations in dynamic programming are also discussed.

2. Preliminaries

Following the structure ofC^∗-avGMS[9] andmGMS[66], a new space is introduced calledC^∗-algebra valued modularG-metric space (abbreviatedC^∗-avmGMS).

Definition 1. Let Z be a nonempty set, and S₃be the permutation group on{1, 2, 3}. A mapping Ω : (0,∞)×Z×Z×Z → A+ is called a C^∗-avmGM on Z, if for any z₁,b₂,c₃,d ∈ Z and α>0it satisfies:

(i) Ωα(z₁,b₂,c₃) =θif z₁=b₂=c₃,

(ii) Ωα(z₁,z₁,b2)θ, for all z₁,b2∈Z with z₁6=b2, (iii) Ωα(z_σ1,b_σ2,c_σ3) =_Ωα(z1,b2,c3),σ∈S3,

(iv) Ωα(z1,z1,b2)Ωα(z1,b2,c3)for all z1,b2,c3∈Z with b26=c3, (v) Ωα+µ(z1,b2,c3)Ωα(z1,d,d) +Ωµ(d,b2,c3).

Then(Z,A^,Ω)is said to be C^∗-avmGMS.

Here we discuss some properties and definitions ofC^∗-avmGMS, as follows:

(a) The essential property on a set Zof aC^∗-avmGM,Ω is that for anyz₁,b2,c3 ∈ Z the function 0 < α → _Ωα(z1,b2,c3) ∈ Ais nonincreasing on (0,∞). Moreover, if 0<µ<α, then

Ωα(z₁,b₂,c₃)_Ωα−µ(z₁,z₁,z₁) +_Ωµ(z₁,b₂,c₃)] =_Ωµ(z₁,b₂,c₃). (b) It can be easily checked as that, ifa0∈Zthe set

Z_Ω ={a∈Z: lim

α→∞Ωα(a,a₀,z) =θ}for somez∈Z,

is aC^∗-avmGMSwith the generalized metricG⁰_Ω :Z_Ω×Z_Ω×Z_Ω →Ais given by G⁰_Ω =inf{α>0 :kΩα(z1,b2,c3)k6^α}for allz1,b2,c3∈Z_Ω,

called aC^∗-avmGMS.

(c) If(Z,A^,ω)is aC^∗-avmMSthen(Z,A^,ω)can defineC^∗-avmGMonZby (A^s)_Ω^s_α(z₁,b₂,c₃) =¹

3(ω_α(z₁,b₂) +ω_α(b₂,c₃) +ω_α(c₁,z₃)),

(B^m)_Ω^m_α(z1,b2,c3) =max(ω_α(z1,b2),ω_α(b2,c3),ω_α(c1,z3)), for allα>0.

(d) AnyC^∗-avmGMS,(Z,A,Ω)induces aC^∗−avmM,ω_αby

ω^Ω_α(z1,b2) =Ωα(z1,z1,b2) +Ωα(z1,b2,b2), for allα>0, and satisfies;

Ωα(z₁,b2,c3) 6 Ω^s_α(z₁,b2,c3) 62Ωα(z₁,b2,c3), for allα> 0, and ¹₂Ωα(z₁,b2,c3) 6 Ω^m_α(z1,b2,c3)62Ωα(z1,b2,c3), for allα>0. Further, starting from aC^∗−avmM,ω onZ, we haveω^Ω_α^s(z1,b2) =⁴₃ω_α(z1,b2)andω^Ω_α^m(z1,b2) =2ωα(z1,b2).

Definition 2. Let Z_Ωbe a C^∗-avmGMS. Then for eachα>_0,

(1) Any sequence{an}_n∈Nin Z_Ωis convergent to a∈Z_Ω with respect toAif, for anye>0 there exists N ∈Nsuch that for all n,m>N,Ωα(a,an,am)≺e.

Moreover, it isΩ-G-Cauchy if for all n,m,l>N,Ωα(an,am,a_l)≺e.

(2) A mapping T isΩ-G-continuous with respect toA^{in B}⊆Z_Ωif for every sequence{an}_n∈N⊆ B such that for all n>N,Ωα(an,z,z)≺e, then for all n>N,Ωα(Tan,Tz,Tz)≺e.

(3) Z_ΩisΩ-complete if anyΩ-G-Cauchy sequence with respect toA^isΩ-G-convergent.

(4) A subset B of Z_ΩisΩ-G-bounded with respect toAif for eachα>0and a0∈Z_Ω δ_Ω(B) =sup{kΩα(a0,a,b)k;a,b∈B}<∞,

where,δ_Ω(B)denotes the diameter of B in the C^∗-avmGMS.

Proposition 1. Let Z_Ωbe a C^∗-avmGMS, for eachα>0. Then (1) {an}_n∈Nis aΩ-G-convergent to a with respect toA^; (2) Ωα(an,an,a)→θas n→_∞;

(3) Ωα(an,a,a)→θas n→_{∞; and} (4) Ωα(an,am,a)→θas n,m→∞.

are equivalent.

Proposition 2. Let Z_Ωbe a C^∗-avmGMS, for eachα>0. Then (1) {an}_n∈Nis aΩ-G-Cauchy with respect toA^{; and}

(2) Ωα(am,an,an)→θas n,m→_∞ are equivalent.

Proposition 3. Let(Z,A^,Ω)be a C^∗-avmGMS. For any z₁,b2,c3,a∈Z,andα>0it follows:

(i) ifΩ2α(z₁,b₂,c₃) =θthen z₁=b₂=c₃;

(ii) Ω2α(z₁,b₂,c₃)_Ωα(z₁,z₁,b₂) +_Ωα(z₁,z₁,c₃); (iii) Ω2α(z₁,b2,b2)_2Ωα(b2,z₁,z₁);

(iv) Ω2α(z1,b2,c3)_Ωα(z1,a,c3) +_Ωα(a,b2,c3);

(v) Ω2α(z1,b2,c3) ²₃(Ωα(z1,b2,a) +Ωα(z1,a,c3) +Ωα(a,b2,c3)); (vi) Ω4α(z1,b2,c3)Ωα(z1,a,a) +Ωα(b2,a,a) +Ωα(c3,a,a).

Definition 3. Let Z_Ω be a C^∗-avmGMS, Ωα is said to be symmetric if Ωα(z₁,z₁,b₂) = Ωα(z₁,b2,b2)for all z₁,b2∈Z_Ωandα>0.

Example 1. Let Z= {0, 1, 2}and consider,A = M2(R). Let C ∈ Aand∗, be the involution map such that C^∗=C, definekCk=^q_∑²_i,j=1|c_ij|². Clearly,A^{is a C∗}-algebra. For A= [a_ij]_2×2, B= [bij]2×2∈M2(R); we denote AB if and only if aij6bij.

DefineΩ:(0,∞)×Z×Z×Z→AbyΩα(z1,b2,c3) =diag(^g(z1,b_α^2,c3),

g(z₁,b₂,c₃) α

), for all z1,b2,c3∈ Z andα>0, where

g(z₁,b₂,c₃) =







0i f z1=b2=c3,

1i f (z₁,b₂,c₃)∈ {(0, 0, 1),(0, 0, 2),(1, 1, 2)},

2i f (z1,b2,c3)∈ {(0, 1, 1),(0, 2, 2),(1, 2, 2),(0, 1, 2)}. Then it can be easily checked thatΩαis a C^∗-avmGMS. Since,

Ωα(0, 0, 1)6=_Ωα(0, 1, 1); Ωα(0, 0, 2)6=_Ωα(0, 2, 2); Ωα(1, 1, 2)6=_Ωα(1, 2, 2), soΩαis not symmetric (see [52]).

Now if we takeg(_z1,b2,c3) =|z1−b2|+|b2−c3|+|c3−z1|, then it can be checked thatΩαis aC^∗-avmGMSand symmetric.

Example 2. For the Lebesgue measurable set E and Hilbert space H let, Z=L^∞(E), H=L²(E) andA = B(H), the set of bounded linear operator on H. DefineΩ : (0,∞)×Z×Z×Z → B(H)₊by

Ωα(z1,b2,c3) =β

z1−b2 α

+

b2−c3 α

+

c3−z1 α

∀z1,b2,c3∈Z, α>0,

where β_θ : H → H is the multiplication operator defined by β_θ(_Φ) = θ. Φ,Φ ∈ H. Then (Z_Ω,B(H),Ω)is aΩ-G-complete C^∗-avmGMS.

Example 3. Let Z= l^∞(S)and H=l²(S), S6=φ, andA= B(H). DefineΩ :(0,∞)×Z× Z×Z→B(H)₊by

Ωα({fn},{gn},{ln}) =β

{_fn}−{_gn}

α

+

{_gn}−{_ln} α

+

{_ln}−{_fn} α

∀ {fn},{gn},{ln} ∈Z,α>0, whereβ_θis described as in Example 2. Then(_ZΩ,B(_H)_,Ω)_{is a}Ω-G-complete C^∗-avmGMS.

Example 4(see [9]). Let Z=R^andA=B(H). DefineΩ:(0,∞)×Z×Z×Z→B(H)₊by Ωα(z₁,b₂,c₃) =

z₁−b2

α

+

b2−c3

α

+

c3−z₁ α

1_A∀z₁,b₂,c₃∈Z,α>0.

Then(Z_Ω,B(H),Ω)is aΩ-G-complete C^∗-avmGMS.

Definition 4([2,12]). Define a continuous function H:A+×A+ →A^{for C∗-algebra}A^{. If for} any A,B∈A+,satisfies:

(i) H(A,B)A; and

(ii) H(A,B) =A⇒A=θor B=θ.

Then the function is called “C∗-class function”.

Definition 5 ([2]). A tripled(ψ,ϕ,H∗) where ψ : A+ → A+ in Ψ (set of all continuous functions), ϕ : A+ → A+ inΦ(the class of functions) and H∗ : A+×A+ → A. If for any A,B∈A+satisfies:

AB→H∗(ψ(A),ϕ(A)) H∗(ψ(B),ϕ(B)). Then it is monotone.

Definition 6 ([63]). Let (Z,G) be a G-metric space and T : Z → Z. Then T is said to be G−β−ψ-contractive mapping of type A if there exist two functionsβ: Z×Z×Z→[_0,∞) andψ∈ F_com^(c), family of(c)comparision function such that

β(z1,z2,Tz1)G(Tz1,Tz2,T²z1)≤ψ(G(z1,z2,Tz1)), ∀z1,z2∈Z.

According to Agarwal et al. [63] this type of contractive condition cannot be reduced to a quasi metric.

Asadi and Salimi [62] established the following theorem, which cannot be reduced to quasi metric space as well.

Theorem 1([62]). Let(Z,G)be a G-metric space and T and S be two self-mappings on Z. For all z₁,z2∈Z, nondecreasing and continuous functionψand lower semicontinuous functionφ; if it satisfiesψ(G(Tz1,STz1,Tz2))≤ψ(G(z1,Sz1,z2))−φ(G(z1,Sz1,z2)). Then S and T have a common unique fixed point.

3. Main Results

Let(Z_Ω,A,Ω)be aΩ-completeC^∗-avmGMS, and(T1,T2)be two self-mappings on Z_Ω, satisfying the conditions:

ψ(k_Ωα(T1a,T1b,T1b)k1_A)F∗(ψ(M(a,b,b)),ϕ(M(a,b,b)),

for which,F∗∈C∗,ψ∈Ψandϕ∈Φwith strictly monotone(ψ,ϕ,F∗). M(a,b,b) =kqk²k_Ω2α(T2a,T2b,T2b)k1_A

+kLk²min{kΩα(T₁a,T₂a,T₂a)k,kΩα(T₁b,T₂a,T₂a)k, kΩα(T1a,T2b,T2b)k,kΩα(T1b,T2b,T2b)k}1_A.

In the settingΩα(z₁,b₂,b₂) =ω_α(z₁,b₂)the contractive condition reduces to ψ(kω_λ(T₁a,T₁b)k1_A)F∗(ψ(M(a,b)),ϕ(M(a,b)), where

M(a,b,b) =M(a,b) =kqk²kω_2α(T2a,T2b)k1_A

+kLk²min{kω_α(T₁a,T2a)k,kω_α(T₁b,T2a)k, kω_α(_T1a,T2b)k,kω_α(_T1b,T2b)k}1_A.

Proceeding as in ([59,60]) one can construct the same result in quasiC^∗-avmMSinstead ofC^∗-avmGMS.

Motivated by Theorem 1 [62], the following theorem’s contractive condition cannot be reduced to quasiC^∗-avmMS.

Theorem 2. Let(Z_Ω,A^,Ω)be aΩ-complete C^∗-avmGMS, and T₁and T₂be two self-mappings on Z_Ω, satisfying the condition: for each a,b ∈ Z_Ω, q ∈ A ^{with 0} < kqk < 1, and k_Ωα(T1a,T2T1a,T1b)k<_∞;

ψ(kΩα(T₁a,T₂T₁a,T₁b)k1_A)F∗(ψ(kqk²kΩ2α(a,T₂a,b)k1_A)_, ϕ(kqk²k_Ω2α(a,T2a,b)k1_A)),

for which, F∗ ∈C∗,ψ∈Ψandϕ∈Φwith strictly monotone(ψ,ϕ,F∗). Then T1and T2have a common unique fixed point in Z_Ω.

Proof. Leta0∈Z_Ω. Hence inductivelyan+1=T₁an∀n=0, 1, 2, ... .

ψ(kΩα(an,T2an,an)k1_A)F∗(ψ(kqk²kΩ2α(an−1,T2an−1,an−1)k1_A), ϕ(kqk²k_Ω2α(an−1,T2an−1,an−1)k1_A)), ψ(kqk²kΩ2α(an−1,T2an−1,an−1)k1_A. Sinceψis nondecreasing,

kΩα(an,T2an,an)k1_A kqk²kΩ2α(an−1,T2an−1,an−1)k1_A, kqk²k_Ωα(an−1,T₂an−1,an−1)k1_A, ...,

kqk²ⁿk_Ωα(a₀,T₂a₀,a₀)k1_A,

→0 as n→∞(sincekqk<1).

Now we show that{an}^∞_n=0and{T2an}^∞_n=0areΩ-G-Cauchy sequence. Suppose there existe>0 and subsequence{a_m(k)}and{a_n(k)}withn(k)>m(k)>k>0 such that k_Ωα(a_n(k),a_m(k),a_m(k))k1_A k_Ωα

2(a_n(k),T₂a_m(k),a_m(k))k1_A+2k_Ωα

4(T₂a_m(k),a_m(k),a_m(k))k1_A.

Hence, limk→∞Ωα(a_n(k),a_m(k),a_m(k)) = θ. Therefore{an}^∞_n=0is aΩ-G-Cauchy se- quence and limn→∞Ωα(an,l,l) =θ, for allα>0 and for somel∈Z_Ω. Suppose there exist e>0 and subsequence{T2a_m(k)}and{T2a_n(k)}withn(k)>m(k)>k>0 such that

k_Ωα(T2a_n(k),T2a_m(k),T2a_m(k))k1_A k_Ωα

2(a_n(k),T2a_n(k),a_n(k))k1_A +2k_Ωα

4(T2a_m(k),a_n(k),a_n(k))k1_A.

Hence, limk→∞Ωα(T2a_n(k),T2a_m(k),T2a_m(k)) = θ. Therefore {T2an}^∞_n=0 is a Ω-G- Cauchy sequence and limn→∞Ωα(T₂an,T₂p,T₂p) =_{θ, for all}α>0 and for somep∈Z_Ω.

Sincek_Ωα(an,T₂an,an)k1_A→0 asn→_{∞, so}l=T₂p.

Now,

ψ(kΩα(an,T2an,T1l)k1_A) =ψ(kΩα(T1an−1,T2T1an−1,T1l)k1_A), F∗(ψ(kqk²k_Ωα(an−1,T₂an−1,l)k1_A),

ϕ(kqk²kΩα(an−1,T2an−1,l)k1_A)), ψ(kqk²k_Ωα(an−1,T₂an−1,l)k1_A. Sinceψis non decreasing,

k_Ωα(an,T₂an,T₁l)k1_A kqk²ⁿk_Ωα(a₀,T₂a₀,l)k1_A,

→0 as n→∞(sincekqk<1). Hencel=T₁l=T2p.

ψ(kΩα(l,T2l,l)k1_A) =ψ(kΩα(T1l,T2T1l,T1l)k1_A),

F∗(ψ(kqk²k_Ωα(l,T₂l,l)k1_A),ϕ(kqk²k_Ωα(l,T₂l,l)k1_A)), ψ(kqk²kΩα(l,T2l,l)k1_A.

Sinceψis non decreasing andkqk<1, soΩα(l,T₂l,l) =θ. Hencel=T₁l=T₂l.

Uniqueness:

If possible, letr∈Z_Ωsuch thatr=T1r=T2r.

ψ(kΩα(l,T2l,r)k1_A) =ψ(kΩα(T1l,T2T1l,T1r)k1_A),

F∗(ψ(kqk²k_Ωα(l,T2l,r)k1_A),ϕ(kqk²k_Ωα(l,T2l,r)k1_A)), ψ(kqk²kΩα(l,T2l,r)k1_A.

Clearly,l=r. HenceT₁andT2have a common unique fixed pointr.

Corollary 1. Let (Z_Ω,A^,Ω) be a Ω-complete C^∗-avmGMS, and T be a self-mapping on Z_Ω, satisfying the condition: for each a,b∈ Z_Ω, q∈Awith0<kqk<1, andkΩα(Ta,T²a,Tb)k<∞;

ψ(k_Ωα(Ta,T²a,Tb)k1_A)F∗(ψ(kqk²k_Ω2α(a,Ta,b)k1_A), ϕ(kqk²kΩ2α(a,Ta,b)k1_A)),

for which, F∗ ∈ C∗,ψ∈ _Ψandϕ∈ _Φwith strictly monotone(ψ,ϕ,F∗). Then T has a unique fixed point in Z_Ω.

Example 5. As in Example 2, let Z=l^∞(S), H=l²(S), andA=B(H). DefineΩ:(0,∞)× Z×Z×Z→B(H)₊by

Ωα({fm},{gn},{h_l}) =β

{_fm}−{_gn} α

+

{gn}−{_hl} α

+

{_hl}−{fm} α

∀ {fm},{gn},{h_l} ∈Z;

α>_0,m,n,l∈N.Then(Z_Ω,B(H)_,Ω)is aΩ-G-complete C^∗-avmGMS.

Proof. Define T1 : Z_Ω → Z_Ω by T1({fm}) = {^m+1_m ², 0, 0, . . .} where {fm} = {^m+1_m , 0, 0, . . .} ∈ Z_Ω, m ∈ N. (see [70]) Again define T2 : Z_Ω → Z_Ω by T2({gm}) = {^m+1_m , 0, 0, . . .}where{gm}={(^m+1_m )², 0, 0, . . .} ∈Z_Ω, m∈N.

Supposeψandϕare two self-mappings onA+such thatψ(A) =2Aandϕ(A) =A for allA∈A+, and

(F∗:A+×A+ →A, F∗(A,B) =A−B.

Then(ψ,ϕ,F∗)is strictly monotone. For all{fm},{fn} ∈Z_Ω=l^∞(S), andα>0 we have clearly,

k_Ωα(T₁({fm}),T2T₁({fm}),T₁({fn})k<_∞; n,m∈N(m>n). For everyq,L∈Awith 0<kqk<1,kLk 6=0, we have

ψ(k_Ωα(T₁{fm},T2T₁{fm},T₁{fn})k1_A)F∗(ψ(kqk²k{fm},T2{fm},{fn}k), ϕ(kqk²k{fm},T2{fm},{fn}k).

Hence it satisfies all the conditions of Theorem2. So,T₁andT2have common unique fixed points{1, 0, 0, . . .}.

4. Ulam–Hyers Stability Results inC^∗-avmGMS

Let(Z,A^,Ω)be aC^∗-avmGMSandT : Z_Ω →Z_Ω be a mapping. Let set ofnfixed point equations be

Tⁿz=z,n∈N ⁽¹⁾

called a generalized Ulam–Hyers stability if

(i) there exists a increasing mappingξ:R+ →R+with continuity at 0 andξ(0) =0;

(ii) for anye>0 and for eachz^∗ ∈Z_Ωanesolution of the Equation (1), which satisfies Ωα(z^∗,Tⁿz^∗,Tz^∗)e, n∈N. (2) There exists a solutionx^∗ ∈ Z_Ωof (1) such thatΩα(z^∗,x^∗,x^∗)ξ(e). Then for any s∈R+andk>0;ξ(s) =k.simplies Ulam–Hyers stability of (1).

In the following theorem, two fixed point equations,Tz=zandT²z=zare consid- ered.

Theorem 3. Let(Z_Ω,A^,Ω)be aΩ-complete C^∗-avmGMS satisfying all the condition of Corol- lary1. Moreover,

(a) ψ(kΩα(Ta,T²a,Tb)k1_A + K) F∗(ψ(kqk²kΩ2α(a,Ta,b)k1_A + K),φ(kqk²kΩ2α

(a,Ta,b)k1_A+K),K∈A+.

(b) k_Ωα(Tz^∗,Tz^∗,z^∗)k1_A(1− kqk²)γ1_A, ∀α>0 ;γ>0.

Then Equation (1) is Ulam–Hyers stable.

Proof. From Corollary 1 Fix(T) = {x^∗}. Let e > 0 and z^∗ ∈ Z_Ω be a solution of Equation (2)

ψ(kΩα(x^∗,x^∗,z^∗)k1_A) =ψ(kΩα(x^∗,Tx^∗,z^∗)k1_A) =ψ(kΩα(Tx^∗,T²x^∗,z^∗)k1_A), ψ(k_Ωα

2(Tx^∗,T²x^∗,Tz^∗)k1_A+k_Ωα

2(Tz^∗,Tz^∗,z^∗)k1_A), ψ(kΩ^α

2(Tx^∗,T²x^∗,Tz^∗)k1_A+ (1− kqk²)γ1_A), F∗(ψ(kqk²kΩα(x^∗,Tx^∗,z^∗)k1_A+ (₁− kqk²)_γ1A)_,

ϕ(kqk²k_Ωα(x^∗,Tx^∗,z^∗)k1_A+ (1− kqk²)γ1_A)), ψ(kqk²kΩα(x^∗,Tx^∗,z^∗)k1_A+ (1− kqk²)γ1_A),

this implies thatk_Ωα(x^∗,x^∗,z^∗)k1_Aγ1_A. Hence Equation (1) is Ulam–Hyers stable.

Theorem 4. Let(Z_Ω,A,Ω)be aΩ-complete C^∗-avmGMS satisfying all the conditions of Theo- rem3with (a). Moreover, the onto functionπ:[0,∞)→[0,∞)such thatπ(r) =λr is strictly increasing. Then

(a) Equation (1) is generalized Ulam–Hyers stable.

(b) Fix(T) = {x^∗} and if {zn} ∈ Z_Ω,n ∈ N are such that kΩα(zn,Tzn,Tzn)k1_A → 0 as n→∞, then zn→x^∗as n→∞. (well-posed)

(c) If S : Z_Ω → Z_Ω such that k_Ωα(Tz,Tz,Sz)k1_A η1_A, ∀z ∈ Z_Ω η ∈ [0,∞) then p^∗∈ Fix(S)⇒ kΩα(x^∗,p^∗)k1_Aπ⁻¹(_1−kqk^α _2kη1_A).

Proof. (a) LetFix(T) ={x^∗},e>0 andz^∗∈Z_Ω.

ψ(k_Ωα(x^∗,x^∗,z^∗)k1_A) =ψ(k_Ωα(x^∗,Tx^∗,z^∗)k1_A) =ψ(k_Ωα(Tx^∗,T²x^∗,z^∗)k1_A), ψ(kΩ^α

2(Tx^∗,T²x^∗,Tz^∗)k1_A+kΩ^α

2(Tz^∗,Tz^∗,z^∗)k1_A), ψ(k_Ωα

2(Tx^∗,T²x^∗,Tz^∗)k1_A+e1_A), F∗(ψ(kqk²k_Ωα(x^∗,Tx^∗,z^∗)k1_A+e1_A),

ϕ(kqk²kΩα(x^∗,Tx^∗,z^∗)k1_A+_e1A))_, ψ(kqk²k_Ωα(x^∗,Tx^∗,z^∗)k1_A+e1_A), this implies thatkΩα(x^∗,x^∗,z^∗)k1_{A 1−kqk}¹ _2ke1_A. So,

π(k_Ωα(x^∗,x^∗,z^∗)k1_A) =λk_Ωα(x^∗,x^∗,z^∗)k1_A ^λ

1− kqk²k^e1A.

Therefore we have,k_Ωα(x^∗,x^∗,z^∗)k1_A π⁻¹(_1−kqk^λ _2ke1_A). Hence, Equation (1) is generalized Ulam–Hyers stable.

(b) LetFix(T) ={x^∗},e>0 and{zn} ∈Z_Ω.

ψ(kΩα(x^∗,Tx^∗,zn)k1_A)ψ(kΩ^α₂(zn,Tzn,Tzn)k1_A+kΩ^α₂(Tx^∗,T²x^∗,Tzn)k1_A), F∗(ψ(kqk²k_Ωα(x^∗,Tx^∗,zn)k1_A+k_Ωα

2(zn,Tzn,Tzn)k1_A), ϕ(kqk²kΩα(x^∗,Tx^∗,zn)k1_A+kΩ^α

2(zn,Tzn,Tzn)k1_A)), ψ(kqk²k_Ωα(x^∗,Tx^∗,zn)k1_A+k_Ωα

2(zn,Tzn,Tzn)k1_A), this implies that

kΩα(x^∗,x^∗,zn)k1_A ¹

1− kqk²kkΩ^α

2(zn,Tzn,Tzn)k1_A,

→0as n→_∞.

Hence, we havezn →x^∗ as n→∞.

(c) LetFix(T) ={x^∗}andp^∗∈Fix(S).

ψ(kΩα(x^∗,Tx^∗,p^∗)k1_A) =ψ(kΩα(Tx^∗,T²x^∗,p^∗)k1_A), ψ(k_Ωα

2(Tx^∗,T²x^∗,T p^∗)k1_A+k_Ωα

2(T p^∗,T p^∗,p^∗)k1_A), ψ(kΩ^α

2(Tx^∗,T²x^∗,T p^∗)k1_A+kΩ^α

2(T p^∗,T p^∗,Sp^∗)k1_A), F∗(ψ(kqk²kΩα(x^∗,Tx^∗,p^∗)k1_A+_η1A)_,

ϕ(kqk²k_Ωα(x^∗,Tx^∗,p^∗)k1_A+η1_A)), ψ(kqk²kΩα(x^∗,Tx^∗,p^∗)k1_A+η1_A), this implies thatk_Ωα(x^∗,x^∗,p^∗)k1_{A 1−kqk}¹ _2kη1_A. So,

π(kΩα(x^∗,x^∗,p^∗)k1_A) =λkΩα(x^∗,x^∗,p^∗)k1_A ^α

1− kqk²k^η1A. Therefore we have,k_Ωα(x^∗,x^∗,p^∗)k1_Aπ⁻¹(_1−kqk^λ _2kη1_A).

5. Applications

Shen et al. [9] provided an application for a type of differential equation inC^∗-avGMS.

Pathak et al. [71] for common fixed point and Moeini et al. [2] forC^∗-avmGMSprovided applications to nonlinear integral equations. All of the above inspired the following application (see also [13,14,23,37,56,68,69,72–74]).

First remind, as in Example 2, for allα>0 and f,g,h ∈ L^∞(E), defineΩ:(0,∞)× L^∞(E)×L^∞(E)×L^∞(E)→B(L²(E))₊by

Ωα(f,g,h) =β

f−g α

+

g−h α

+

h−f α

. Then(L^∞(E)_Ω,B(L²(E)),Ω)is aΩ-G-completeC^∗-avmGMS.

Theorem 5. Consider the following system of nonlinear integral equations:

z(r) =I(r) +v(r,z(r)) +µ Z

Et(r,s)q(s,z(s))ds, (3) where r,s∈E;µ∈F; z,I∈L^∞(E)_Ωand v(r,z(r), t(r,s), q(s,z(s)are all real or complex valued functions, which are measurable in r and s on E. Suppose

(i) sups∈ER

E|t(r,s)|dr= M<+_∞;

(ii) For all s∈ E; z1,z2∈L^∞(E)_Ωand v(s,z1(s)),v(s,z2(s))∈ L^∞(E)_Ω, there exists N>1 such that

|v(s,z₁(s)−v(s,z₂(s)|

2√

2 ≥N|z₁(s)−z₂(s)|;

(iii) For all s ∈ E; z1,z2 ∈ L^∞(E)_Ω and q(s,z1(s),q(s,z2(s) ∈ L^∞(E)_Ω, there exists L > 0 such that|q(s,z₁(s)−q(s,z₂(s)| ≤L|z₁(s)−z₂(s)|;

(iv) For a nonempty setBconsists of f,g∈ L^∞(E)_Ωand K:[0, 1]×[0, 1]×R⁺→R^{+, such} that v(r,g(r)) = g(r)−I(r)−αR

Et(r,s)q(s,g(s))ds = f(r), and |I(r) +αR

Et(r,s) q(s,f(r)−I(r)−µR

Et(r,s)q(s,f(s))ds)ds| ≤ ^1+|µ|ML

2√

2N |f(r)−v(r,f(r))|. Then Equation (3) has a unique solution for eachµ∈Fwith ^1+|µ|ML_N <1.

Proof. DefineT₁,T₂:Z_Ω→Z_Ωby

T₁z(r) =z(r)−I(r)−µ Z

Et(r,s)q(s,z(s))ds, T₂z(r) =v(r,z(r))_.

Setkqk²= ^1+|µ|ML_N (<1). Letψandϕbe two self-mappings onB(L²(E))₊such that ψ(_A1) = ¹_2A1andϕ(_A2) = ¹_{4A2for allA1,A2}∈ B(_L²(_E))_{+, and}

(F∗:B(L²(E))₊×B(L²(E))₊→B(L²(E)), F∗(A₁,A₂) = ^√1

2A₁.

Clearly,(ψ,ϕ,F∗)is strictly monotonic. Let f,g ∈ B. ψ(kΩα(T1f,T2T1f,T1g)k1_A) = ¹

2kΩα(T1f,T2T1f,T1g)k1_A,

= ¹ 2kβ

|^T1f−_α^T2T1f|+|^T2T1f_α^−T1g|+|^T1g−_α^T1f|k1_A,

= ¹

2k^T1f −T2T1f

α k_∞1_A+k^T2T1f −T1g

α k_∞1_A+k^T1f−T1g α k_∞1_A,

=k^T1f−T2T1f

α k_∞1_A+ ¹ αsup

s∈E

(f −g) +µ Z

Et(r,s)(q(s,g(s))−q(s,f(s)))ds 1_A, k^T1f−T2T1f

α k_∞1_A+¹+|µ|ML

α kf −gk_∞1_A, k^T1f−T2T1f

α k_∞1_A+¹+|µ|ML 2√

2N k^v(r,f(r))−v(r,g(r)) α k_∞1_A, k^I(r) +αR

Et(r,s)q(s,f(r)−I(r)−µR

Et(r,s)q(s,f(s))ds)ds

α k_∞1_A

+¹+|µ|ML 2√

2N k^T2f −T₂g α k_∞1_A, ¹

2√

2kqk²(k^f −T2f

α k_∞1_A+k^T2f −g

α k_∞1_A+k^g−f α k_∞1_A), ¹

2√

2kqk²k_Ωα(f,T2f,g)k1_A,

F∗(ψ(kqk²kΩα(f,T2f,g)k1_A),φ(kqk²kΩα(f,T2f,g)k1_A)).

By Theorem2, we have a unique solution to the nonlinear integral Equation (3).

LetZ_Ωbe aC^∗-avmGMSandQbe a Banach space andP⊆Q. DefineΘ:V×P→V andD:V×P×R→Rbe two functions, whereV⊆Z_Ω. LetB(V)be the set of Banach spaces consisting all real functional onV. Define a norm,kak=supr∈V|a(r)|and consider the functional equation arising in dynamic programming ([75,76])

a(r) =sups∈P{D(r,s,a(Θ(r,s)))} (4) wherea∈B(V). Define aC^∗-avmGMSonB(V)as in Example 4 by

Ωα(f,g,h) =

f(r)−g(r) α

+

g(r)−h(r) α

+

h(r)−f(r) α

1_A, for allα>0 andf,g,h∈B(V).