Research article

On generalized b-contractions and related applications

  • Received: 02 January 2023 Revised: 27 April 2023 Accepted: 03 May 2023 Published: 30 June 2023
  • MSC : 34A08, 54H25, 47H10

  • In this manuscript, using the idea of b-simulation functions, certain common fixed point results via generalized b-contractions are investigated in the context of b-metric spaces. These findings generalize and supplement various numbers of established results from the existing literature. Examples and applications are also provided for the authenticity of the presented work. Besides, we ensure the existence of a unique common solution for systems of Volterra-Hammerstein integral and Urysohn integral equations, respectively, by applying the established results.

    Citation: Muhammad Rashid, Muhammad Sarwar, Muhammad Fawad, Saber Mansour, Hassen Aydi. On generalized b-contractions and related applications[J]. AIMS Mathematics, 2023, 8(9): 20892-20913. doi: 10.3934/math.20231064

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  • In this manuscript, using the idea of b-simulation functions, certain common fixed point results via generalized b-contractions are investigated in the context of b-metric spaces. These findings generalize and supplement various numbers of established results from the existing literature. Examples and applications are also provided for the authenticity of the presented work. Besides, we ensure the existence of a unique common solution for systems of Volterra-Hammerstein integral and Urysohn integral equations, respectively, by applying the established results.



    Many physical phenomenon can be described by integral equations. Non-linear integral equations play a vital role to solve many mathematical problems arising in engineering and applied science. There are various techniques available in the literature for the existence of solutions of these equations. Many researchers used fixed points techniques to the existence of a unique solution to non-linear integral equations. For instance, refer to [1,2,3,4,5,6,7]. Especially, Shoaib et al. [2] studied fixed point results and its applications to the systems of non-linear integral and differential equations of arbitrary order, while Rashwan and Saleh [8] established fixed point results to find the existence of a unique common solution to a system of Urysohn integral equations. As opposed to that, Pathak et al. [9] and Rashwan and saleh [8] ensured the existence of a unique common solution to a system of Volterra-Hammerstein non-linear integral equations. Additionally, Baklouti and his co-authors have made significant contributions to related areas, including optimal preventive maintenance policies for solar photovoltaic systems (c.f, [15]) and quadratic Hom-Lie triple systems (c.f, [16,17]).

    In the current article, existence of solutions for the following systems of Volterra-Hammerstein integral and the Urysohn integral equations are investigated, respectively:

    D(x)=ξi(x)+baWi(x,y,D(y))dy,

    where x(a,b)R; D,ξiC((a,b),Rn) and Wi:(a,b)×(a,b)×RnRn, for i=1,2;

    and

    D(x)=τi(x)+λt0m(x,y)gi(y,D(y))dy+μ0n(x,y)hi(y,D(y))dy,

    where x(0,),λ,μR,D,Ti,m(x,y),n(x,y),gi(y,D(y)) and hi(y,D(y)) for i=1,2 are measurable functions with real values both in x and y on (0,). The most elaborated result in this area, known as the Banach fixed point theorem, ensures that a solution exists. The extensive usage of the fixed point theory, particularly in metric spaces [11], has then had an impact on the study of its evolution over the past few decades.

    Due to important applications of the Banach contraction principle, many researchers generalized this principle by elaborating the underlying spaces or changing the contractive conditions. See [12,13,14] for details. In recent decades, scholars concentrated on applying such an aforementioned theorem to various generalized metric spaces, see [18,19]. Among these generalized spaces, there is the b-metric space, where the coefficient of the triangle inequality is s1. It was introduced by Bakhtin [13]. Moreover, in [26] Czerwik provided the Banach contraction principle on this space. Recently, Salmi and Noorani [21] presented several properties in these spaces and established some common fixed point theorems in ordered cone b-metric spaces. Khojasteh et al. [34] introduced the concept of a simulation function. This concept has been refined in [22] in order to guarantee the presence of a unique coincidence point for two non-linear mappings.

    Later, the concept of a b-simulation function was introduced to ensure the existence and uniqueness of a fixed point. In [23], Olgun et al. presented the concept of a generalized -contraction. In [24], Jawaher et al. utilized the idea of a b-simulation function and investigated some common fixed points for two contractive mappings. In the same direction, using the concept of a generalized b-contraction with a b-simulation function, Rodjanadid et al. [23] proved some fixed point results in complete b-metric spaces.

    Motivated by the above contributions, using b-simulation functions and b-contractions, some fixed point theorems are constructed. As applications of these findings, some examples and existence results for systems of integral equations are also discussed. We note that by using the presented work, some well known results can be deduced from the existence literature.

    This section includes all those concepts (definitions, theorems, lemmas, etc.) which will help us to prove the main results of this manuscript. These concepts are taken from different papers, like [25,26,27] etc. Throughout the manuscript, the following notions and symbols will be utilized. ϝ, Ω, and represent a non-empty set, a metric, a simulation function and a family of simulation functions, respectively. Also, the initials R and N in the sequel stand for the sets of all real and natural numbers, respectively.

    Definition 2.1. [25] (Metric space) Let ϝ. A function Ω:ϝ×ϝ[0,) is known as a metric on ϝ, if for all a,b,cϝ the following conditions hold:

    m1) Ω(a,b)=0 if and only if a=b;

    m2) Ω(a,b)=Ω(b,a);

    m3) Ω(a,b)Ω(a,c)+Ω(c,b).

    The pair (ϝ,Ω) is a metric space.

    Definition 2.2. [26] (b-metric space) Let ϝ and assume b1. A function Ωb:ϝ×ϝ[0,) is called a b-metric on ϝ if for all a,e,cϝ, the following requirements are satisfied:

    d1) Ωb(a,e)=0 if and only if a=e;

    d2) Ωb(a,e)=Ωb(e,a);

    d3) Ωb(a,e)b[Ωb(a,c)+Ωb(c,e)].

    The pair (ϝ,Ωb) is known as a b-metric space, in short (bMS).

    Definition 2.3. [27] (Convergence, Cauchyness and Completeness) Let {fn} be a sequence in a b-metric space (ϝ,Ωb,b).

    a) {fn} is called b-convergent if and only if there is fϝ such that Ωb(fn,f)0 as n.

    b) {fn} is a b-Cauchy sequence if and only if Ωb(fn,fm)0 as n,m.

    c) The b-metric space is complete if every b-Cauchy sequence is b-convergent.

    Proposition 2.4. [27] The following assertions hold in a b-metric space (ϝ,Ωb,b):

    i) The limit of a b-convergent sequence is unique;

    ii) Each b-convergent sequence is b-Cauchy;

    iii) A b-metric is not continuous generally.

    Definition 2.5. [28] (Simulation function) Let :[0,)×[0,)R be a function. If satisfies the criteria below:

    (1) (0,0) = 0;

    (2) (t,s)<st for all t,s>0;

    (3) if {tn}, {sn} are sequences in (0,) such that

    limntn=limnsn>0,

    then

    limnsup(tn,sn)<0,

    then it is referred as a simulation function. The set of all simulation functions is denoted by the symbol .

    Example 2.6. [28] Let :[0,)×[0,)ϝ be defined by (t,s)=λst for all t,s[0,), where λ[0,1). Then is a simulation function.

    Example 2.7. [28] Let :[0,)×[0,)ϝ be defined by

    (t,s)=Ψ(s)Φ(t) for all t,s[0,), where Ψ,Φ:[0,)[0,)

    are two continuous functions such that Ψ(t)=Φ(t)=0 iff t=0

    and Ψ(t)<tΦ(t) for all t>0. Here, is a simulation function.

    Definition 2.8. [28] (-contraction) Let (ϝ,Ω) be a metric space, T:ϝϝ be a mapping and . T is called a -contraction with regard to if the following condition holds

    (Ω(Tx,Ty),Ω(x,y))0   for  all   x,yϝ.

    If T is a -contraction with respect to , then Ω(Tx,Ty)<Ω(x,y) for all distinct x,yϝ.

    Theorem 2.9. [28] Suppose (ϝ,Ω) is a complete metric space and T:ϝϝ is a - contraction with respect to . Then T has a unique fixed point u in ϝ and for every x0ϝ, the Picard sequence {xn} (where xn=Txn1 for all nN) converges to the fixed point of T.

    Definition 2.10. [29] (Generalized -contraction) Suppose (ϝ,Ω) is a metric space, T:ϝϝ is a mapping and . Then T is referred to as a generalized -contraction with regard to If the following condition is satisfied:

    (Ω(Tx,Ty),M(x,y))0      x,yϝ,

    where

    M(x,y)=max{Ω(x,y),Ω(x,Tx),Ω(y,Ty),12(Ω(x,Ty)+Ω(y,Tx))}.

    Theorem 2.11. [29] Assume (ϝ,Ω) is a complete metric space and T:ϝϝ is a generalized -contraction with respect to , then T has a fixed point in ϝ. Moreover, for every x0ϝ, the Picard sequence {Tnx0} converges to this fixed point.

    Definition 2.12. [33] (b-simulation function) Let (ϝ,Ωb) be a b-metric space with a constant b1. A b-simulation function is a function :[0,)×[0,)R satisfying the following conditions: (1) (t,s)<st for all t,s>0;

    (2) If {tn}, {sn} are two sequences in (0,) such that

    0<limntnlimninfsnlimnsupsnblimntn<,

    then

    limnsup(btn,sn)<0.

    We represent the set of all b-simulation functions by the symbol b.

    Some examples of b-simulation functions are as follows.

    Example 2.13. [33] Let t,s[0,).

    (1) (t,s)=Ψ(s)Φ(t), where \ Φ,Ψ:[0,)[0,) are two continuous functions such that Ψ(t)=Φ(t)=0t=0 and Ψ(t)<tΦ(t) for all t>0;

    (2) (t,s)=sy(t,s)z(t,s)t, where y,z:[0,)×[0,)[0,) are two functions which are continuous respect to each variable, i.e., y(t,s)>z(t,s) for all t,s>0;

    (3) (t,s) = sΦ(s)t, where \ Φ:[0,)[0,) is a lower semi-continuous function such that Φ(t)=0t=0;

    (4) (t,s)=sΦ(s)t, where Φ:[0,)[0,) is such that limtp+Φ(t)<1  p>0;

    (5) (t,s)=λst, where λ[0,).

    Definition 2.14. [31] Let (ϝ,Ωb,b) be a b-metric and y,z be two self mappings on ϝ. Then the pair {y,z} is said to be compatible if

    limnΩb(yzxn,zyxn)=0,

    whenever {xn} is a sequence in ϝ such that

    limnyxn=limnzxn=p   forsome pϝ.

    Lemma 2.15. [24] Let (ϝ,Ωb,b) be a b-metric space. If there exist two sequences {fn} and {rn} such that

    limnΩb(fn,rn)=0,

    whenever {fn} is a sequence in ϝ such that

    limnfn=p   forsome pϝ

    then

    limnrn=p.

    Theorem 2.16. [30] Let T:ϝϝ be a mapping and (ϝ,Ωb) be a complete b-metric space with a constant b1. Assume there is a b-simulation function \ \ such that (bΩb(Tf,Tr),Ωb(f,r))0 for all f,rϝ, then T has a unique fixed point.

    In this work, we introduce generalized b-contraction pairs of self-mappings on a b-metric space. We will show that such mappings have a common fixed point. Some examples and applications are presented making effective the new concepts and obtained results. Well known results in literature are investigated and compared.

    This section includes the main work of this article. We initiate this section with the definition of generalized b-contractions and a related example. Before the proof of the main theorem, we prove some basic lemmas. For the support of the main theorem, some examples are presented. In the last of this section, some remarks are presented.

    Definition 3.1. Consider a b-metric space (ϝ,Ωb,b) with b1, f1,f2:ϝϝ are two mappings and b.

    Then f1,f2 are called generalized b-contractions with respect to if the circumstance listed below is true

    (bΩb(f1s,f1t),Mb(s,t))0 for all s,tϝ,

    where

    Mb(s,t)=max[Ωb(f2s,f2t),Ωb(f2t,f1t),12bΩb(f2s,f1t)].

    Example 3.2. Consider the be metric space (ϝ,Ωb,b) with $ \digamma = [1,2] and \Omega_{b} = (s-t)^2 forall s, t\in \digamma (Here,b=2).Thentheselfmappings f_1, f_2:\digamma \rightarrow \digamma $ defined by

    f1(s)=(s8)2   and
    f2(s)=s8

    are generalized b-contractions with respect to (s,t)=12st. Indeed,

    (bΩb(f1,f1),Mb(s,t))0.

    Lemma 3.3. Suppose (ϝ,Ωb,b) is a b-metric space, and f1,f2:ϝϝ are two generalized b-contractions. Suppose f1(ϝ)f2(ϝ) and there is a b-simulation function such that

    [bΩb(f1t,f1s),Mb(t,s)]0   t,sϝ, (3.1)

    then there is a sequence {an} in ϝ such that

    limnΩb(an1,an)=0.

    Proof. Assume t0 is an arbitrary point. Since f1(ϝ)f2(ϝ), we can construct two sequences {tn} and {sn} such that sn=f1(tn)=f2(tn+1) for every nN. If there is n0N such that tn0=tn0+1, then it follows from the given inequality (3.1) and from (1) that for all nN,

    0(bΩb(f1tn0+1,f1tn0+2),Mb(f2tn0+1,f2tn0+2))0Mb(f2tn0+1,f2tn0+2)bΩb(f1tn0+1,f1tn0+2)=Mb(sn0,sn0+1)bΩb(sn0+1,sn0+2).

    That is,

    Mb(f2t,f2s)=max[Ωb(f2t,f2s),Ωb(f2s,f1s),12bΩb(f2t,f1s)]Mb(f2tn0+1,f2tn0+2)=max[Ωb(f2tn0+1,f2tn0+2),Ωb(f2tn0+2,f1tn0+2),12bΩb(f2tn0+1,f1tn0+2)]=max[Ωb(sn0,sn0+1),Ωb(sn0+1,sn0+2),12bΩb(sn0,sn0+2)].

    Therefore, by triangle inequality,

    Mb(f2tn0+1,f2tn0+2)=max[Ωb(sn0,sn0+1),Ωb(sn0+1,sn0+2)]0max[Ωb(sn0,sn0+1),Ωb(sn0+1,sn0+2)]bΩb(sn0+1,sn0+2).

    Since sn0=sn0+1 implies that Ωb(sn0+1,sn0)=0, consider

    0max[0,Ωb(sn0+1,sn0+2)]bΩb(sn0+1,sn0+2)0<Ωb(sn0+1,sn0+2)bΩb(sn0+1,sn0+2)0<[1b]Ωb(sn0+1,sn0+2)0Ωb(sn0+1,sn0+2)=0
    sn0=sn0+1=sn0+2=sn0+3=,

    which implies that

    limnΩb(sn1,sn)=0.

    Now, suppose that snsn+1 for every nN. Then, it follows from (3.1) and (1) that for every nN, we have

    0[bΩb(f1tn,f1tn+1),Mb(f2tn,f2tn+1)]=[bΩb(sn,sn+1),Mb(sn1,sn)]<Mb(sn1,sn)bΩb(sn,sn+1)=max[Ωb(sn1,sn),Ωb(sn,sn+1)]bΩb(sn,sn+1).

    If Ωb(sn,sn+1)Ωb(sn1,sn), then 0<Ωb(sn,sn+1)bΩb(sn,sn+1). That is,

    bΩb(sn,sn+1)<Ωb(sn,sn+1),

    which is a contradiction. So we have

    Ωb(sn1,sn)Ωb(sn,sn+1)0<Ωb(sn1,sn)bΩb(sn,sn+1)bΩb(sn,sn+1)=Ωb(sn1,sn)  nN.

    This implies that {Ωb(sn1,sn)} is a decreasing sequence of positive real numbers. Thus, there is some Γ0, so that

    limnΩb(sn1,sn)=Γ.

    Assume Γ>0, so from the condition (2), with an=Ωb(sn,sn+1) and bn=Ωb(sn1,sn), one writes

    0limnsup[bΩb(sn,sn+1),Mb(sn1,sn)]<0,

    which is a contradiction. Hence, we get that Γ=0. It ends the proof.

    Remark 3.4. Let (ϝ,Ωb,b) be a b-metric space and assume f1,f2:ϝϝ are two generalized b-contractions. Assume that f1(ϝ)f2(ϝ) and there is a b-simulation function such that

    0[bΩb(f1t,f1s),Mb(f2t,f2s)]  t,sϝ.

    Then there is a sequence {sn} in ϝ such that

    bΩb(sm,sn)Mb(sm1,sn1)  m,nN.

    Proof. By a similar argument of Lemma 3.3 for every nN, we have sn=f1(tn)=f2(tn+1). Hence, from (3.1) and (1), we have for all m,nN,

    0[bΩb(f1tm,f1tn),Mb(f2tm,f2tn)]0[bΩb(sm,sn),Mb(sm1,sn1)]0<Mb(sm1,sn1)bΩb(sm,sn)bΩb(sm,sn)<Mb(sm1,sn1).

    Lemma 3.5. Let (ϝ,Ωb,b) be a b-metric space and assume f1,f2:ϝϝ be two generalized b- contractions. Assume that f1(ϝ)f2(ϝ) and there is a b-simulation function such that the inequality (3.1) holds. Then there exists a sequence {sn} in ϝ, such that {sn} is a bounded sequence.

    Proof. By a similar argument of Lemma 3.3, when for some n0sn0=sn0+1 we have Ωb(si,sj)M for all i,j=0,1,2,, where

    M=max{Ωb(si,sj):i,jn0}.

    Let us assume that snsn+1 for each nN and suppose {sn} is a sequence which is not bounded. Then there is a subsequence {s}nk of {sn} such that for n1=1 and for each kN, nk+1 is the minimum integer such that Ωb(snk+1,snk)>1 and Ωb(sm,snk)1 for nkmnk+11. By triangular inequality, we obtain

    1<Ωb(snk+1,snk)b[Ωb(snk+1,snk+11)+Ωb(snk+11,snk)]b[Ωb(snk+1,snk+11)+1)]=bΩb(snk+1,snk+11)+b).

    Letting k in the above inequality and using Lemma 3.3, we get

    1limkinfΩb(snk+1,snk)limksupΩb(snk+1,snk)b.

    Again from Remark 3.4 we have

    bΩb(snk+1,snk)Mb(snk+11,snk1)=max[Ωb(snk+11,snk1),Ωb(snk1,snk),12bΩb(snk+11,snk)]max[b[Ωb(snk+11,snk)+Ωb(snk,snk1)],Ωb(snk1,snk),12bΩb(snk+11,snk)]max[b[1+Ωb(snk,snk1)],Ωb(snk1,snk),12b(1)]max[b[1+Ωb(snk,snk1)],Ωb(snk1,snk),12b(b)]=max[b(1+Ωb(snk,snk1),Ωb(snk1,snk),12)].

    Now, as

    1<Ωb(snk+1,snk)bbΩb(snk+1,snk)<Mb(snk+11,snk1)max[b[1+Ωb(snk,snk1)],Ωb(snk1,snk),12],

    then taking k, one gets

    blimkMb(snk+11,snk)limkmax[b[1+Ωb(snk,snk1)],Ωb(snk1,snk),12]=max[b(1+0),0,12]=max[b]=b,

    i.e.,

    limkΩb(snk1,snk)=0.

    That is,

    limkMb(snk+11,snk1)=b.

    Using the inequality (3.1) and (2) with ak=Ωb(snk+1,snk) and ck=Mb(snk+11,snk1), we have

    0limksup[bΩb(snk+1,snk),Mb(snk+11,snk1)]<0,

    which is a contradiction. Hence, {sn} is a bounded sequence.

    Lemma 3.6. Suppose (ϝ,Ωb,b) is a b-metric space and assume f1,f2:ϝϝ are two generalized b-contractions. Assume that f1(ϝ)f2(ϝ) and there is a b-simulation function such that the inequality (3.1) holds. Then there is a sequence {sn} in ϝ, such that {sn} is a Cauchy sequence.

    Proof. Using a similar argument as in Lemma 3.3, we have for every nN, sn=f1(tn)=f2(tn+1). If there is noN such that sn0=sn0+1, then we have {sn} is a cauchy sequence. Let us snsn+1 for every nN and let

    Cn=sup{Ωb(si,sj):i,jn}.

    Now, from Lemma 3.3, Cn< for every nN. Since Cn is a positive decreasing sequence, there is some c0 such that

    limnCn=c.

    Let us consider that c>0. Then by the definition of Cn, for every kN there are nk,mkN such that mk>nkk and

    Ck1k<Ωb(smk,snk)Ck.

    Letting k in the inequality above, we have

    limkΩb(smk,snk)=c

    and

    limkΩb(smk1,snk1)=c.

    By the inequality 3.1 and property (1), we have

    0[bΩb(smk,snk),Mb(smk1,snk1)]<Mb(smk1,snk1)bΩb(smk,snk)bΩb(smk,snk)<Mb(smk1,snk1)=max[Ωb(f2tmk,f2tnk),Ωb(f2tnk,f1tnk),12bΩb(f2tmk,f1tnk)]=max[Ωb(smk1,snk1),Ωb(snk1,snk),12bΩb(smk1,snk)]max[Ωb(smk1,snk1),Ωb(snk1,snk),12b[b(Ωb(smk1,smk)+Ωb(smk,snk))]]=max[Ωb(smk1,snk1),Ωb(snk1,snk),12(Ωb(smk1,smk)+Ωb(smk,snk))].

    Letting limk in the above inequality using Lemma 3.3,

    limkΩb(smk,snk)=c

    and

    limkΩb(smk1,snk1)=c.

    We have

    bc=limkbΩb(smk,snk)Mb(smk1,snk1)limkmax[Ωb(smk1,snk1),Ωb(snk1,snk),12(Ωb(smk1,smk)+Ωb(smk,snk))]=max[c,0,12(0+c)]=c.

    Then

    bclimkinfMb(smk1,snk1)limksupMb(smk1,snk1)c.

    From the above inequality and since c>0 that is b=1, then by the property (2) with

    ak=Ωb(smk,snk)

    and

    qk=Mb(smk1,snk1)

    we get

    0limksup[bΩb(smk,snk),Mb(smk1,snk1)]<0

    which is a contradiction, thus c=0, i.e.,

    limncn=0      b1.

    This proves that {sn} is a Cauchy sequence.

    We are now going to present our main result.

    Theorem 3.7. Consider (ϝ,Ωb,b) be a complete b-metric space, and f1,f2:ϝϝ be two generalized b-contractions with f1(ϝ)f2(ϝ) and the pair (f1,f2) is compatible. Assume that a b-simulation function such that 3.1 holds, that is,

    [bΩb(f1t,f1s),Mb(f2t,f2s)]0    t,sϝ.

    If f2 is continuous, then there is a coincidence point of f1 and f2, that is, there exists tϝ such that f1(t)=f2(t). Moreover, if f2 is one to one, then f1 and f2 have a unique common fixed point.

    Proof. Consider x0ϝ. Since f1(ϝ)f2(ϝ), we have for every nN, sn=f1(tn)=f2(tn+1). Now, from Lemma 3.6, the sequence {sn} is Cauchy and since (ϝ,Ωb,b) is a complete b-metric space, there is some sϝ such that

    limnsn=s,

    that is,

    s=limnf1(tn)=limnf2(tn).

    We claim that s is a coincidence point of f1 and f2. Since f2 is continuous, we have

    limnf2f1(tn)=f2f2(tn)=f2(s).

    Also, since {f1,f2} is compatible, we have

    limnΩb(f1f2(tn),f2f1(tn))=0.

    Hence, by Lemma 2.15 we deduce

    limnf1f2(tn)=f2(s).

    Using (3.1), we have

    0[bΩb(f1s,f1f2(tn)),Mb(f2s,f2f2(tn))].

    That is,

    <Mb(f2s,f2f2(tn))bΩb(f1s,f1f2(tn)).

    Letting n,

    0<limninfMb(f2s,f2f2(tn))blimnsupΩb(f1s,f1f2(tn)).

    But

    Mb(f2s,f2f2(tn))=max[Ωb(f2s,f2f2(tn)),Ωb(f2s,f1f2(tn)),12bΩb(f2s,f1f2(tn))]limninfMb(f2s,f2f2(tn))=limninfmax[Ωb(f2s,f2f2(tn)),Ωb(f2s,f1f2(tn)),12bΩb(f2s,f1f2(tn))]=max[Ωb(f2s,f2s),Ωb(f2s,f2s),12bΩb(f2s,f2s)]=0

    which implies that

    0<Mb(f2s,f2f2(tn))bΩb(f1s,f1f2(tn))0<limninfMb(f2s,f2f2(tn))blimnsupΩb(f1s,f1f2(tn))=blimnsupΩb(f1s,f1f2(tn))0.

    Thus,

    limnsupΩb(f1s,f1f2(tn))=0

    that is

    limnf1f2(tn)=f1(s)

    therefore f1(s)=f2(s).

    Now, assume there is pϝ such that f1(p)=f2(p) then the inequality (3.1) and (2) imply that

    0[bΩb(f1s,f1p),Mb(f2s,f2p)]
    =Mb(f2s,f2p)bΩb(f1s,f1p),

    where

    Mb(f2s,f2p)=max[Ωb(f2s,f2p),Ωb(f2p,f1p),12bΩb(f2s,f1p)]=max[Ωb(f2s,f1p),12bΩb(f2s,f1p)]Mb(f2s,f2p)=Ωb(f2s,f1p)0<Ωb(f2s,f1p)bΩb(f1s,f1p)=Ωb(f2s,f1p)bΩb(f2s,f1p)=[1b]Ωb(f2s,f1p)0Ωb(f2s,f1p)=0.

    Hence,

    bΩb(f1s,f1p)Ωb(f1s,f1p).

    If b>1 then f1(s)=f1(p). If b=1, by the condition (2) with

    ak=Ωb(f1s,f1p)

    and

    vk=Mb(f2s,f2p)

    we get

    0limksup[bΩb(f1s,f1p),Mb(f2s,f2p)]<0

    which is a contradiction. Therefore,

    f1(p)=f1(s)=f2(p)=f2(s).

    Now, suppose that f2 is one to one. If s,p are two coincidence points of f1 and f2, In this case, by the above argument we have

    f1(s)=f2(s)=f1(p)=f2(p).

    Since f2 is one to one, it follows that p=s. Also, since f2(s)=f1(s) and the pair {f1,f2} is compatible we have

    f1f2(s)=f2f1(s).

    Therefore,

    f2f1(s)=f1f2(s)=f1f1(s).

    That is, f1(s) is a coincidence point of f1 and f2. Therefore, f1(s)=s and hence

    f1(s)=f2(s)=s.

    That is, f1 and f2 have a unique common fixed point sϝ.

    Corollary 3.8. Let (ϝ,Ωb,b) be a complete b-metric space and f1,f2:ϝϝ be two generalized b-contractions with f1(ϝ)f2(ϝ) and the pair (f1,f2) is compatible. Suppose that there is λ(0,1) such that

    bΩb(f1s,f1t)λMb(s,t)    s,tϝ.

    If f2 is continuous, then there is a coincidence point of f1 and f2, that is, there is tϝ such that f1(t)=f2(t). Moreover, if f2 is one to one, then f1 and f2 have a unique common fixed point.

    Proof. By taking the b-simulation function

    (x,y)=λyx    x,y0.

    The result follows from Theorem 3.7.

    Example 3.9. Let X=[0,) and d:X×XX be defined by

    d(s,t)={0ifs=t;8ifs,t[0,1);3+1s+tifs,t[1,);3325otherwise.

    Then, clearly d is a b-metric on X with b=54.

    Here, we observe that when s=32 and u=2 (both belong to [1,) and t[0,)), we have

    d(s,u)=3+132+2=237

    and

    d(s,t)+d(t,u)=3325+3325=6625.

    Hence, d(s,u)d(s,t)+d(t,u). Hence, d is a b-metric with b=54(>1), but it is not a metric. We now define f,g:XX by

    f(s)={s4+2ifs[0,1);3s2ifs[1,)

    and

    g(s)={sifs[0,1);1sifs[1,).

    Clearly, f and g are b-continuous functions. Now, we define :[0,)×[0,)[0,) by (x,y)=45yx. We have the following possible cases:

    Case 1: s,t[0,1).

    In this case, d(fs,ft)=3+1s+t and Mb(s,t)=8

    (bd(fs,ft),Mb(s,t))=45(8)54(3+1s+t)>0.

    Case 2: s,t[1,).

    Here, d(fs,ft)=3+1s+t and Mb=8. We have

    (bd(fs,ft),Mb(s,t))=45(8)54(3+1s+t)>0.

    Case 3: s[0,1) and t[1,).

    Here, d(fs,ft)=3+1s+t and Mb=8. Also,

    (bd(fs,ft),Mb(s,t))=45(8)54(3+1s+t)>0.

    Case 4: s[1,) and t[0,1).

    In this case d(fs,ft)=3+1s+t and Mb=8. Also,

    (bd(fs,ft),Mb(s,t))=45(8)54(3+1s+t)>0.

    So the pair {f,g} is a generalized b-contraction. It satisfies all the conditions of Theorem 3.7. Hence, f and g have a common unique fixed point.

    Example 3.10. Let ϝ=[0,1] be endowed with the b-metric Ωb(s,t)=(st)2, where b=2. Define f1 and f2 on ϝ by

    f1(s)=(s4)2

    and

    f2(s)=(s4).

    Obviously f1(ϝ)f2(ϝ) and furthermore the pair {f1,f2} is compatible. Consider the b-simulation function given as

    (r,q)=12qr     r,q0.

    For all s,tϝ we have

    0[2Ωb(f1s,f1t),Mb(s,t)]0<12Mb(s,t)2Ωb(f1s,f1t)Ωb(f1s,f1t)<14Mb(s,t).

    Now,

    Ωb(f1s,f1t)=(f1sf1t)2=((s4)2(t4)2)2=(s4+t4)2(s4t4)2(14+14)2(s4t4)2=(24)2Ωb(f2s,f2t)14Mb(s,t)=14max[Ωb(f2s,f2t),Ωb(f2t,f1t),12bΩb(f2s,f1t)].

    As all the requirements of Theorem 3.7 are satisfied, so f1 and f2 have a unique common fixed point, which is 0.

    Example 3.11. Take ϝ=[0,1]. Define Ωb:ϝ×ϝR by Ωb(s,t)=(st)2. Clearly, (ϝ,Ωb) is a complete b-metric with b=2.

    Now, we define the functions f1,f2:[0,1][0,1] by

    f1s=as1+s      sϝ,a(0,12]

    and

    f2t=t1+t      tϝ.

    Clearly,

    f1(s)f2(t)

    and furthermore the pair is [f1,f2] is compatible. Now, consider the b-simulation function :[0,)×[0,)R defined by

    (x,y)=f1f1+1x.

    We have

    (2Ωb(f1s,f1t),Mb(s,t))=Mb(s,t)Mb(s,t)+12Ωb(f1s,f1t)
    Ωb(s,t)Ωb(s,t)+12Ωb(f1s,f1t)
    =(st)2(st)2+12[as1+sat1+t]2
    =(st)2(st)2+12a2(st)2[(1+s)(1+t)]2
    (st)2(st)2+12a2(st)2(st)2+1
    =(st)22a2(st)2(st)2+1
    =(12a2)(st)2(st)2+1
    0    s,tϝ.

    Thus, all the assumption are satisfied of Theorem 3.7, and hence f1 and f2 have a unique common fixed point, which is 0.

    Remark 3.12. If in Lemma 3.3, Mb=Ωb(f2t,f2s) in inequality (3.1), then we will get Lemma 3.1 of [24].

    Remark 3.13. If in Remark 3.4, Mb=Ωb(f2t,f2s) in inequality (3.1), then we will get Remark 3.2 of [24].

    Remark 3.14. If in Lemma 3.5, Mb=Ωb(f2t,f2s) in inequality (3.1), then we will get Lemma 3.3 of [24].

    Remark 3.15. If in Lemma 3.6, Mb=Ωb(f2t,f2s) in inequality (3.1), then we will get Lemma 3.4 of [24].

    Remark 3.16. If in Theorem 3.7, Mb=Ωb(f2t,f2s) in inequality (3.1), then we will get Theorem 3.5 of [24].

    In this section, we present an application of our result to integral equations. Namely, we study the existence of the unique common solution of a system of non linear Urysohn integral equations.

    Let us consider the integral equations

    f(x)=r1(x)+bak1(x,t,f(t))dt (4.1)

    and

    g(x)=r2(x)+bak2(x,t,g(t))dt (4.2)

    where (i) f,r1, r2 and g are unknown functions for each x[a,b].

    (ii) k1 and k2 are kernels defined for x,t[a,b].

    Let us denote

    ϑ1f(x)=bak1(x,t,f(t))dt

    and

    ϑ2g(x)=bak2(x,t,g(t))dt.

    Assume that

    ● (A1) ϑ1f(x)+r1(x)+r2(x)ϑ2(ϑ1f(x)+r1(x))+r2(x)=0

    ● (A2) r1(x)r2(x)+ϑ1f(x)ϑ1g(x)=0.

    We will ensure the existence of a unique common solution of (4.1) and (4.2) that belong to G=(C[a,b],Rn) (the set of continuous mappings defined on [a,b]). For this, define the continuous mappings T1,T2:GG by

    T1f(x)=r1(x)+ϑ1f(x)

    and

    T2g(x)=2f(x)ϑ2f(x)r2(x)

    where f,g,r1,r2G. We claim T1T2.

    Proof. If we show that T2(T1f(x)+r2(x))=T1f(x), then it is conformed that T1T2. Hence,

    T2(T1f(x)+r2(x))=2[T1f(x)+r2(x)]ϑ2[T1f(x)+r2(x)]r2(x)=2T1f(x)+2r2(x)ϑ2[T1f(x)+r2(x)]r2(x)=T1f(x)+r1(x)+r2(x)+ϑ1f(x)ϑ2[T1f(x)+r2(x)]=T1f(x)+[r1(x)+r2(x)+ϑ1f(x)ϑ2[T1f(x)+r2(x)]].

    Using (A1),

    T2[T1f(x)+r2(x)]=T1f(x).

    It shows that

    T1f(x)T2f(x).

    We endow on G the b-metric (with b=2) given as Ωb(x,y)=|xy|2. Here, (G,Ωb) is complete. Further, let us suppose that k1,k2:[a,b]×[a,b]×RnRn are continuous functions satisfying

    |k1(x,t,f(t))k2(x,t,f(t))|Mb(f,g)2(ba), (4.3)

    where

    Mb(f,g)=max{Ωb(T2(f),T2(g)),Ωb(T2(g),T1(g)),14Ωb(T2(f),T1(g))}.

    Theorem 4.1. Under the conditions (A1), (A2) and (4.3), the Eqs (4.1) and (4.2) have a unique common solution.

    Proof. For f,g(G,Rn) and x[a,b], we define the continuous mappings T1,T2:GG by

    T1f(x)=r1(x)+ϑ1f(x)

    and

    T2f(x)=2f(x)ϑ2f(x)r2(x).

    Then we have

    2Ωb(T1(f),T1(g))=2|T1(f)T2(g)|22|T2f(x)T1g(x)|2=2|2f(x)ϑ2f(x)r2(x)r1(x)ϑ1g(x)|2=2|[r1(x)r2(x)+ϑ1f(x)ϑ1g(x)]+ϑ1f(x)ϑ2f(x)|2=2|ϑ1f(x)ϑ2f(x)|22(ba|k1k2|)2dt2(baMb(f,g)2(ba)dt)=Mb(f,g).

    This shows that all the requirements of our main theorem are satisfied, i.e., 2Ωb(T1(f),T1(g))Mb(f,g). Therefore, the integral equations (4.1) and (4.2) have a unique common solution.

    In this section, we give a second application and we ensure the existence of the unique common solution of a system of non linear Volterra-Hammerstein integral equations.

    Let us take ϝ=(L(0,),R) the space of real-valued measurable functions on (0,). Consider

    D(x)=τ1(x)+λt0m(x,y)g1(y,D(y))dy+μ0n(x,y)h1(y,D(y))dy (5.1)

    and

    D(x)=τ2(x)+λt0m(x,y)g2(y,D(y))dy+μ0n(x,y)h2(y,D(y))dy (5.2)

    for all x,y(0,), where λ,μR, and D,τ1,τ2,m(x,y),n(x,y),g1,g2,h1,and h2 are measurable functions with real values in S and r on (0,),

    ϖi=t0m(x,y)gi(y,D(y))dy
    ψi=0n(x,y)hi(y,D(y))dy

    for i=1,2

    f1D(x)=ϖ1D(x)+ψ1D(x)+τ1(x)
    f2D(x)=2D(x)ϖ2D(x)ψ2D(x)τ2(x)

    with

    f1(D(x))f2(v(x)). (5.3)

    Assume that

    (c1):ϖ1D(x)+ψ1D(x)+τ1(x)+τ2(x)ϖ2(ϖ1D(x)+ψ1D(x)+τ1(x)+τ2(x))
    ψ2(ϖ1D(x)+ψ1D(x)+τ1(x)+τ2(x))=0.

    We consider the b-metric space Ωb(x,y)=|xy|2.

    Theorem 5.1. Under the assumption (c1) and the condition (5.3), the system of non linear Volterra-Hammerstein integral equations has a unique common solution.

    Proof. Note that the system of non linear Volterra-Hammerstein integral equations (5.1) and (5.2) has a unique common solution if and only if the system of operator f1 and f2 has a unique common fixed point.

    Now,

    Ωb(f1D(x),f1v(x))=|(f1D(x)f1v(x))|2
    |(f2v(x)f1v(x))|2
    =Ωb(f2(v(x)),f1(v(x)))
    Mb(D(x),v(x))=maxΩb(f2D(x),f2v(x)),Ωb(f2v(x),f1v(x)),14Ωb(f2D(x),f1v(x))
    Ωb(f1D(x),f1v(x))Mb(D(x),v(x)).

    This shows that all the requirements of our main theorem are satisfied, and therefore the integral equations (5.1) and (5.2) have a unique common solution.

    Rodjanadid et al.[23] used the idea of generalized b-contractions with b-simulation functions and proved some fixed point results in complete b-metric spaces. Jawaher et al.[24] utilized the idea of b-simulation functions and investigated some common fixed points for two contractive mappings. In this manuscript, we combined these two ideas and proved some common fixed points for two contractive mappings using the idea of generalized b-contractions with b-simulation functions in b-metric spaces. Different examples and applications are given to demonstrate the validity of the concept and the degree of applicability of our findings. Many applied problems can be described by systems of Fredholm and Volterra integral equations. The presented results can be utilized to study the existence of unique common solutions of these systems.

    The authors would like to thank the Deanship of Scientific Research at Umm Al-Qura University for supporting this work grant code: 23UQU4331214DSR002.

    The authors declare that they have no conflict of interest.



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