# Functional-Type Caristi–Kirk Theorem on Menger Space and Application

Authors: Soumia Chaira; Mohammed Dahmouni; Abderrahim Eladraoui; Mustapha Kabil
Journal: Journal of Mathematics
 (2020)

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## Abstract

In this paper, we extend Caristi’s fixed point theorem in metric spaces to probabilistic metric spaces, and also, we prove some common fixed point theorems for a pair of mappings satisfying a system of Caristi-type contractions in the setting of a Menger space. Two examples are given to support the main results. Furthermore, we have functional equations as an application for the main theorem.


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## Body

## 1. Introduction and Preliminaries

The theory of probabilistic metric spaces is of fundamental importance in random functional analysis especially due to its extensive applications in random differential and random integral equations. In 1942, Menger introduced in [1] a generalization of metric space, called a statistical metric space, by using distribution functions instead of nonnegative real numbers as values of the metric. The connection between probabilistic and geometric concepts has been established in 1956 by Špaček [2]. However, the main influence upon the development of this theory is owed to Schweizer and Sklar and their coworkers (see [3–6]). Since then, a large number of fixed point theorems for single-valued and multivalued mappings in probabilistic metric spaces have been proved by many authors (see, for examples, [7–13]). Since every metric space is a probabilistic metric space, we can use many results in probabilistic metric spaces to prove some fixed point theorems in metric spaces and Banach spaces.In this paper, first, we extend Caristi’s fixed point theorem in metric spaces to probabilistic metric spaces. Secondly, we prove, in the setting of a Menger space, some common fixed point theorems for a pair of mappings satisfying a system of Caristi-type contractions. Two examples are given to support our main results.Letℝ+ denote the set of positive real numbers and ℕ denote the set of all natural numbers. For brevity, Tx and Sx will be denoted by Tx and Sx, respectively. Let c:ℝ+⟶ℝ+ be some function. Let, for α∈ℝ+,(1)liminft⟶α+ct=supinfε>0cα,α+ε,limsupt⟶α+ct=infsupε>0cα,α+ε.Definition 1.
We say that a functionc is right lower (upper) semicontinuous at α if liminft⟶α+ct=cαlimsupt⟶α+ct=cα.
From the definition, it is easy to prove that the following proposition is needed in the sequel.Proposition 1.
Ifc is a right lower (upper) semicontinuous function at α, then it is right locally bounded below (above) at α, that is, there exists λ>0 such that infcα,α+λ>−∞supcα,α+λ<+∞.Definition 2.
A mappingg:ℝ+⟶ℝ+ is said to be locally bounded above if it is bounded above on each 0,a, a>0.
In order to set the framework needed to state our main results, we recall the following notions.Definition 3.
(distribution function). A mappingF:ℝ⟶ℝ+ is called a distribution function if it is nondecreasing, left continuous with infF=0, and supF=1.
In what follows, we always denote byD the set of all distribution functions.Example 1.
A simple example of distribution function is a Heaviside function:(2)Ht=0,ift≤0,1,ift>0.Definition 4.
(probabilistic metric space). A probabilistic metric space (PM-space, for short) is an ordered pairX,ℱ, where X is an abstract set of elements and ℱ is a mapping of X×X⟶D (we shall denote the distribution function ℱp,q by Fp,q, and Fp,qt will represent the value of Fp,q at t∈ℝ). The functions Fp,q, p,q∈X, are assumed to satisfy the following conditions:(1)
∀t>0, Fp,qt=1 if and only if p=q(2)
Fp,q0=0(3)
Fp,q=Fq,p(4)
IfFp,qt1=1 and Fq,rt2=1, then Fp,rt1+t2=1Definition 5.
(see [14]). Let X,ℱ be a probabilistic metric space. For p∈X and t>0, the strong t-neighborhood of p is the set(3)Npt=q∈X:Fp,qt>1−t.
The strong neighborhood system atp is the collection Np=Npt:t>0, and the strong neighborhood system for X is the union N=∪p∈XNp.Definition 6.
(see [14]). The ε,λ−topology on X,ℱ is the topology introduced on X by the family of the neighborhoods Upε,λp∈X,ε>0,λ>0, where(4)Upε,λ=q∈X:Fp,qε>1−λ.Remark 1.
Upt,t=Npt for all t>0.Definition 7.
(triangular norm). A triangular norm (briefly at-norm) is a binary operation τ on the unit interval 0,1 which is associative, commutative, and nondecreasing in each of its variables and such that τx,1=x for every x∈0,1.Example 2.
The following are the three basict-norms:(i)
Minimum:τMx,y=minx,y(ii)
Product:τpx,y=xy(iii)
Lukasiewiczt-norm: τLx,y=maxx+y−1,0Definition 8.
(Menger space). A Menger space is a triplet X,ℱ,τ where X,ℱ is a PM-space and τ is a t-norm satisfying the following triangle inequality:(5)Fp,rt1+t2≥τFp,qt1,Fq,rt2,∀p,q,r∈X,t1,t2≥0.
The following result is due to Schweizer et al. [6].Theorem 1.
IfX,ℱ,τ is a Menger space such that supt<1τt,t=1, then X,ℱ,τ is a Hausdorff space in the topology induced by the family Upε,λp∈X,ε>0,λ>0 of neighborhoods.Definition 9.
LetX,ℱ,τ be a Menger space. A sequence pnn∈ℕ in X is said to be(i)
τ-convergent to p∈X if for any ε>0 and any λ>0, there exists a positive integer N=Nε,λ such that Fpn,pε>1−λ, whenever n≥N(ii)
τ-Cauchy sequence if for any ε>0 and any λ>0, there exists a positive integer N=Nε,λ such that Fpn,pmε>1−λ, whenever n,m≥NDefinition 10.
A Menger spaceX,ℱ,τ is said to be τ-complete if each τ-Cauchy sequence in X is τ-convergent to some point in X.Definition 11.
LetX,ℱ,τ be a Menger space such that supt<1τt,t=1, and let S be a self-mapping on X,ℱ,τ. S is said to be a τ-continuous mapping if for each sequence pnn∈ℕ in X which is τ-convergent to a point p∈X, the sequence Spnn∈ℕ is τ-convergent to Sp.
The following result was established by Schweizer and Sklar in [5].Theorem 2.
LetX,ℱ,τ be a Menger space such that supt<1τt,t=1. Then, a sequence pnn∈ℕ in X is τ-convergent to p∈X if and only if for each t∈ℝ,(6)limn⟶∞Fpn,pt=Ht.The following theorem establishes a connection between metric spaces and Menger spaces.Theorem 3 (see [13]).
LetX,d be a metric space. Let ℱ:X×X⟶D be the mapping defined by(7)Fp,qt=Ht−dp,q,p,q∈X,t∈ℝ.
Then,X,ℱ,τM is a Menger space. It is complete if the metric d is complete.
We say that at-norm τ satisfies the condition P if for all y∈0,1,(8)limx⟶1−τx,y=y.Remark 2.
It is easy to see that if at-norm τ satisfies the condition P, then(9)supt<1τt,t=1.Theorem 4 (see [15]).
LetX,ℱ,τ be a Menger space with τ satisfying the condition P. Let pnn∈ℕ and qnn∈ℕ be two sequences in X such that pnn∈ℕ is τ-convergent to p∈X and qnn∈ℕ is τ-convergent to q∈X. Then,(1)
For any givent∈ℝ, we have(10)liminfn⟶∞Fpn,qnt≥Fp,qt.(2)
Ift∈ℝ is any continuous point of Fp,q, then(11)limn⟶∞Fpn,qnt=Fp,qt.Lemma 1.
(see [15], Proposition 2). If X,ℱ,τ is a Menger space with τ satisfying the condition P, then it is metrizable. In addition, if X,ℱ,τ is sequentially complete, then it must be net-complete.


## 2. Main Results

Theorem 5.
LetX,ℱ,τ be a complete Menger space with τ satisfying the condition P. Let T,S:X⟶X be two mappings such that, for all p∈X and for all t>0,(12)Fp,Spt≥Ht−hcϕp,cϕTpϕp−ϕTp.Fp,Tpt≥Ht−hcϕp,cϕSpϕp−ϕSp,where ϕ:X⟶ℝ+ is a lower semicontinuous function, c:ℝ+⟶ℝ+ is a right locally bounded from above, and h is a locally bounded function from ℝ+×ℝ+ to ℝ+.
Then, there exists an elementp∗∈X such that Tp∗=p∗=Sp∗.Proof.
Step 1: letα=infϕX. Since c is locally bounded from the above, there exists λ>0 such that μ=supcα,α+λ<∞. For this μ, there exists ν>0 such that ht,s≤ν for all s,t∈0,μ.For somep0∈X such that α≤ϕp0≤α+λ, we consider the following set:(13)X0=p∈X:ϕp≤ϕp0.One can see thatX0 is nonempty and closed since ϕ is lower semicontinuous. For all p∈X0, ϕTp≤ϕp. If not, then there exists some p1∈X0 such that ϕp1<ϕTp1. By applying (12), we obtain Fp1,Sp1t=1, for all t>0; this implies that Sp1=p1 and consequently Tp1=p1, which is contradiction. Hence, ϕTp≤ϕp, for all p∈X0. So,(14)ϕTp≤ϕp≤ϕx0,ϕSp≤ϕp≤ϕp0,for allp∈X0.Thus,TX0⊂X0 and SX0⊂X0. And since ϕp, ϕTp, ϕSp∈α,α+λ, for all p∈X0, we obtain(15)maxcϕp,cϕTp,cϕSp≤μ,maxhcϕp,cϕTp,hcϕp,cϕSp≤ν.Step 2: we define a partial order “⪯” onX0 as follows:(16)p⪯q⟺Fp,qt≥Ht−νϕp−ϕq,∀t>0.Ifp,q∈X0 such that p⪯q, then ϕq≤ϕp. On the contrary, the reflexivity and The symmetry of “⪯” are obvious. Let us prove the transitivity.Ifp,q,r∈X0 such that p⪯q, q⪯r, then ϕr≤ϕq≤ϕp≤ϕp0Moreover, lett be a positive real number.Ift≤νϕp−ϕr, we have(17)Fp,rt≥0=Ht−νϕp−ϕr.Ift>νϕp−ϕr, then, by considering t1,t2>0 such that t1>νϕp−ϕq, t2>νϕq−ϕr, and t=t1+t2, we have(18)Fp,rt≥τFp,qt1,Fq,rt2≥τHt1−νϕp−ϕq,Ht2−νϕq−ϕr=τ1,1=1=Ht−νϕp−ϕr.Again following (16), we have p⪯r. This shows that “≺¯” is a partial order on X0.Step 3: now, letpαα∈I be a totally ordered subset of X0, where I is an index set. We stipulate that pα⪯pβ⟺α≤β. Hence, I,≤ is a directed set, and ϕpαα∈I is a decreasing monotone net. Let s≥0 such that ϕpα⟶s. So, for any given ε>0, λ>0, and ε>λ, there exists α0∈I such that when α0≤α, we have s≤ϕpα<s+λ/ν. Hence, for any α,β∈I such that α0≤α≤β, we have ϕpα−ϕpβ≤λ/ν, and then(19)Fpα,pβε≥Hε−νϕpα−ϕpβ≥Hε−λ=1>1−λ.
This implies thatϕpαα∈I is a Cauchy net on X0. By the completeness of X0 and according to Proposition 2 in [15], there exists some p¯∈X0 such that pα⟶p¯. In view of the lower semicontinuity of ϕ, it follows that(20)ϕp¯≤liminfαϕpα=limαϕpα=s≤ϕpα,∀α∈I.
Now, we prove thatp¯ is an upper bound of pαα∈I. In fact, for any given α∈I, when t>0 is a continuous point of Fpα,p¯, then, by applying Theorem 1 in [15], we have(21)Fpα,p¯t=limβFpα,pβt≥liminfβHt−νϕpα−ϕpβ≥Ht−νϕpα−ϕp¯.
Using the same idea as in [15], Theorem 3, we prove that(22)Fpα,p¯t≥Ht−νϕpα−ϕp¯,∀t>0,∀α∈I.
This means that, for allα∈I, pα⪯p¯, i.e., p¯ is an upper bound of pαα∈I. By Zorn’s lemma, X0,⪯ has a maximal element p∗. If ϕSp∗≤ϕTp∗, then, according to (1), we have(23)Fp∗,Sp∗t≥Ht−νϕp∗−ϕSp∗,∀t>0.
Hence,p∗⪯Sp∗, which implies that p∗=Sp∗ and p∗=Tp∗. The same conclusion holds if ϕTp∗≤ϕSp∗.
If we takeT=S and hx,y=1, for all x,y∈ℝ+×ℝ+, we obtain the following result.Corollary 1 (see [15]).
LetX,ℱ,τ be a complete Menger space with τ satisfying the condition P. Let T:X⟶X be a mapping such that(24)Fp,Tpt≥Ht−ϕp−ϕTp,∀p∈X,∀t>0,where ϕ:X⟶ℝ+ is a lower semicontinuous function. Then, there exists an element p∗∈X such that Tp∗=p∗.The following corollary is the metric version of Theorem5.Corollary 2 (see [16]).
LetX,d be a complete metric space. Let T,S:X⟶X be two mappings such that(25)dp,Sp≤hcϕp,cϕTpϕp−ϕTp,dp,Tp≤hcϕp,cϕSpϕp−ϕSp,∀p∈X,where ϕ:X⟶ℝ+ is a lower semicontinuous function, c:ℝ+⟶ℝ+ is a right locally bounded from the above, and h is a locally bounded function from ℝ+×ℝ+ to ℝ+.Then, there exists an elementp∗∈X such that Tp∗=p∗=Sp∗.Proof.
Consider the mappingℱ:X×X⟶D defined as follows:(26)ℱp,q=Fp,qt=Ht−dp,q,t∈ℝ,p,q∈X.
By Theorem3, the space X,ℱ,τM is a complete Menger space. It is easy to see that the t-norm τM satisfies the condition P. Therefore, all conditions of Theorem 5 are satisfied. Then, the conclusion holds.Example 3.
Consider the setX=ℝ+ endowed with the metric d defined by(27)dp,q=p−q,for allp,q∈X2,and define the function F:X×X⟶D as follows:(28)Fp,qt=Fp,qt=Ht−p−q,for allp,q∈X2,t∈ℝ.
Consider the two mappingsT,S:X⟶X such that(29)Sp=p4,ifp∈0,1;p4+34,ifp∈1,+∞,Tp=p2,ifp∈0,1;p2+12,ifp∈1,+∞,and the lower semicontinuous function ϕ:X⟶0,+∞ defined by(30)ϕp=exp2pifp∈0,1;exp2p+1ifp∈1,+∞.
Let us show that, for allp∈X and t>0,(31)Fp,Spt≥Ht−ϕp−ϕTp;Fp,Tpt≥Ht−ϕp−ϕSp.
Since the functionH is nondecreasing, it suffices to show that, for all p∈X,(32)S:p−Sp≤ϕp−ϕTp;p−Tp≤ϕp−ϕSp.(i)
Ifp∈0,1, we have p−Sp=3/4p, p−Tp=1/2p, and(33)cϕp−ϕTp=e2p−ep=p+∑n=2+∞1n!2pn−pn,ϕp−ϕSp=e2p−ep/2=32p+∑n=2+∞1n!2pn−p2n.
Then, we obtainp−Sp≤ϕp−ϕTp and p−Tp≤ϕp−ϕSp.(ii)
Ifp=1, system (S) is obvious.(iii)
Ifp>1, we have p−Sp=3/4p−1, p−Tp=1/2p−1, and(34)ϕp−ϕTp=e2p+1−ep+2=p−1+∑n=2+∞1n!2p+1n−p+2n,ϕp−ϕSp=e2p−ep/2=32p−32+∑n=2+∞1n!2p+1n−p2+52n.
Then, we obtainp−Sp≤ϕp−ϕTp and p−Tp≤ϕp−ϕSp.
Hence, we have(35)Fp,Spt≥Ht−ϕp−ϕTp,Fp,Tpt≥Ht−ϕp−ϕSp,∀p∈X,∀t>0,and 0, 1 are two common fixed points of T and S.Theorem 6.
LetX,ℱ,τ be a complete Menger space with τ satisfying the condition P. Let ϕ,ψ:X⟶ℝ+ be two lower semicontinuous functions and T,S:X⟶X be two mappings such that, for all p∈X and for all t>0,(36)Fp,Spt≥Ht−hcϕp+ψp,cϕTp+ψTpψp−ψTp,Fp,Tpt≥Ht−hcϕp+ψp,cϕSp+ψSpϕp−ϕSp,where c:ℝ+⟶ℝ+ is a right locally bounded function from the above and h is a locally bounded function from ℝ+×ℝ+ to ℝ+.Assume that(i)
ϕ⚬T, ϕ⚬S, ψ⚬T, and ψ⚬S are lower semicontinuous functions(ii)
There existsp0∈X such that ψSp0≤ψTp0 and ϕTp0≤ϕSp0Then, there exists an elementp∗∈X such that Tp∗=p∗=Sp∗.Proof.
The setX0=p∈X:ψSp≤ψTpandϕTp≤ϕSp is nonempty p0∈X0 and closed because the functions ϕ⚬T, ϕ⚬S, ψ⚬T, and ψ⚬S are lower semicontinuous.Step 1: letα=infϕ+ψX0. Since the function c is locally bounded from the above, there exists λ>0 such that μ=supcα,α+λ<∞. Also, there exists ν>0 with ht,s≤ν, whenever t,s∈0,μ.Letp1∈X0 such that α≤ϕ+ψp1≤α+λ. Consider the set X1 defined by(37)X1=p∈X0:ϕ+ψp≤ϕ+ψp1.Then, it is nonemptyp1∈X1 and closed since ϕ+ψ is lower semicontinuous.Letp∈X1. Since p∈X0, ψSp≤ψTp and ϕTp≤ϕSp. So,(38)ϕ+ψTp≤ϕTp+ψp≤ϕSp+ψp≤ϕ+ψp≤ϕ+ψp1.Then,Tp∈X1. Similarly, we can show that Sp∈X1, for all p∈X1.Sinceϕ+ψp, ϕ+ψTp, ϕ+ψSp∈α,α+λ,(39)maxcϕ+ψp,cϕ+ψTp,cϕ+ψSp≤μ.And consequently, for allp∈X1,(40)hcϕ+ψp,cϕ+ψTp≤ν,hcϕ+ψp,cϕ+ψSp≤ν.Step 2: we introduce the partial order “⪯” defined onX1 by(41)p⪯q⟺Fp,qt≥Ht−νϕ+ψp+ϕ+ψq,∀t>0.
By the same arguments as in the proof of Theorem5, one can show that the partial ordered set X1,⪯ has a maximal element p∗.
For allt>0,(42)Fp∗,Sp∗t≥Ht−hcϕp∗+ψp∗,cϕTp∗+ψTp∗ψp∗−ψTp∗≥Ht−νψp∗−ψTp∗≥Ht−νψp∗−ψSp∗.
SinceH is nondecreasing and ϕp∗−ϕSp∗≥0,(43)Fp∗,Sp∗t≥Ht−νψp∗+ϕp∗−ϕSp∗−ψSp∗≥Ht−νψ+ϕp∗−ψ+ϕSp∗.
Hence,p∗⪯Sp∗, which implies that Sp∗=p∗ and Tp∗=p∗.Example 4.
Consider the spaceX=ℝ+ with the metric d defined by(44)dp,q=p−q,for allp,q∈X2.
Consider the two mappingsT,S:X⟶X such that(45)Sp=p3,ifp∈0,1;p3+23,ifp∈1,+∞,Tp=p5,ifp∈0,1;p5+45,ifp∈1,+∞.
Consider the two lower semicontinuous functionsϕ,ψ:X⟶0,+∞ such that(46)ϕp=e5p,ifp∈0,1;e5p+1,ifp∈1,+∞,ψp=e3p,ifp∈0,1;e3p+1,ifp∈1,+∞.
One can see that(47)ϕSp=e5p/3,ifp∈0,1;e5p/3+13/3,ifp∈1,+∞.ψTp=e3p/5,ifp∈0,1;e3p/5+17/5,ifp∈1,+∞.
Consider the functionF:X×X⟶D defined by(48)Fp,qt=Fp,qt=Ht−p−q,for allp,q∈X2,t∈ℝ.
Letp∈X. Let us show that(49)p−Sp≤ψp−ψTp;p−Tp≤ϕp−ϕSp.(i)
Ifp∈0,1, we have p−Sp=2/3p, p−Tp=4/5p, and(50)ψp−ψTp=e3p−e3p/5=12p5+∑n=2+∞1n!3pn−3p5n,ϕp−ϕSp=e5p−e5p/3=103p+∑n=2+∞1n!5pn−5p3n.Then,p−Sp≤ψp−ψTp and p−Tp≤ϕp−ϕSp.(ii)
Ifp=1, the system is obvious.(iii)
Ifp>1, we have p−Sp=2/3p−1, p−Tp=4/5p−1, and(51)ψp−ψTp=e3p+1−e3p+17/5=125p−1+∑n=2+∞1n!3p+1n−3p+175n,ϕp−ϕSp=e5p+1−e5p+13/3=103p−1+∑n=2+∞1n!5p+1n−5p+133n.
Then,p−Sp≤ψp−ψTp and p−Tp≤ϕp−ϕSp.
In all the cases, we have(52)p−Sp≤ψp−ψTp;p−Tp≤ϕp−ϕSp.
SinceH is nondecreasing,(53)Fp,Spt≥Ht−ψp−ψTp;Fp,Tpt≥Ht−ϕp−ϕSp,∀p∈X,∀t>0.
Moreover, the functionsψ⚬T, ψ⚬S, ϕ⚬T, and ϕ⚬S are lower semicontinuous, ϕT0≤ϕS0, and ψS0≤ψT0.
Therefore, all the assumptions of Theorem5 are satisfied, and 0 is a common fixed point of S and T.


## 3. Application

LetC1 be the space of the functions f:ℕ∗⟶0,+∞ such that the series ∑n≥12−n/fn is convergent and C2 be the space of the functions f:ℕ∗⟶0,+∞ such that the series ∑n≥12−nfn is convergent. We consider two metrics d1 and d2 defined, respectively, on C1 and C2 by(54)d1f,g=∑n=1+∞12n1fn−1gn,d2f,g=∑n=1+∞12n=fn−gn.Since the functionm:C1,d1⟶C2,d2 defined by mf=1/f, for all f∈C1, is a homeomorphism and C2,d2 is complete, then also C1,d1 is complete.Letℱ be the family of distribution functions defined on C1 by(55)Ff,gt=Ht−d1f,g,for  allt∈0,+∞.Then, the spaceC1,ℱ,τM is a complete Menger space.Now, letc≥1, and three sequences g:ℕ∗⟶1,+∞, h:ℕ∗⟶0,+∞, and r:ℕ∗⟶1,+∞.Theorem 7.
The equationfn=gnfn−1rn+hn, n∈ℕ∗, of unknown f, has a solution in C1.Proof.
LetCg,h,c be the set of elements f of C1 such that f1=c and(56)fn≥gnfn−1rn+hn,for alln∈ℕ∗.
Cg,h,c is nonempty since the sequence f, defined by f1=c and fn=+∞ for all n≥2, is an element of Cg,h,c.
Cg,h,c is a closed subset of C1. Indeed, let f∈C1 and fpp∈Cg,h,cℕ∗ such that fpτM−convergent to f. Then, for ε>0 and 0<λ<1, there exists N∈ℕ such that(57)Ffp,fε=Hε−d1fp,f>1−λ,for allp≥N,so ε−d1fp,f>0, for all p≥N, which shows that d1fp,f⟶0; according to [17], Lemma 1, we have fpn⟶fn, for all n∈ℕ∗, when p⟶+∞. We know that, for each p∈ℕ∗,(58)fpn≥gnfpn−1rn+hn,for  alln∈ℕ∗.
Let us fixn≥2. By letting p⟶+∞ in the above inequality, we obtain(59)fn≥gnfn−1rn+hn.
Consider the mappingT defined on Cg,h,c by(60)Tf1=c,Tfn=gnfn−1rn+hn,for  alln≥2.
Then, we havef∈Cg,h,c.(i)
Sincern≥1, hn≥0, and gn≥1, for all n∈ℕ∗, then fn≥fn−1, for all n∈ℕ∖0,1. Hence, fnn∈ℕ∗ is an increasing sequence.(ii)
Letn∈ℕ∖0,1. Then, we have(61)Tfn≥gnfn−1rn+hn≥gngn−1fn−2rn−1+hn−1rn+hn≥gnTfn−1rn+hn,andTf1=c, which shows that TCg,h,c⊂Cg,h,c.(iii)
∑n=1+∞1/2nfn=1/2c+∑n=2+∞1/2nfn≤1/2c+1/c∑n=2+∞1/2n=1/c.The functionφ:Cg,h,c⟶[0,+∞ defined by(62)φf=1c−∑n=1+∞12nfnis lower semicontinuous inCg,h,c.(iv)
Letf∈Cg,h,c. Since fn≥Tfn, for all n∈ℕ∗,(63)d1f,φTf=∑n=1+∞12n1Tfn−∑n=1+∞12n1fn=φf−φTf.
And sinceH is nondecreasing, we have, for all t≥0,(64)Ff,Tft=Ht−df,Tf≥Ht−φf−φTf.
By applying Corollary2, there exists f∗∈C1 such that(65)Tf∗n=f∗n,for  alln∈ℕ∗.Example 5.
The following functional equation(66)f1=e,fn=n2fn−1+en−12e2n−1−n2,for  alln≥2,admits a solution in C1. In fact, one can apply Theorem 7 for gn=n2≥1, hn=en−12e2n−1−n2≥0, and rn=1, for all n∈ℕ∗.


## 4. Conclusion

The results in this paper(1)
Extend the main results of [16] from a standard metric space to a Menger space(2)
Generalize and improve the main result of [15]



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*Source: 3931426-2020-06-11.xml*