# A New Coupled Fixed Point Result in Extended Metric Spaces with an Application to Study the Stability of Set-Valued Functional Equations

Authors: Ahmed H. Soliman; A. M. Zidan
Journal: Journal of Function Spaces
 (2019)

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## Abstract

In this paper, we introduce a new coupled fixed point theorem in a generalized metric space and utilize the same to study the stability for a system of set-valued functional equations.


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## Body

## 1. Introduction and Preliminaries

Discussing the stability of functional equations, we pursue the inquiry put forward in 1940 by Ulam [1] which states that the solution of an equation varying marginally from a given solution, should of need be near the solution for the given equation. Three popular techniques to establish the stability from a perspective Hyers–Ulam of functional equations are the direct technique [2], the technique of invariant means [3, 4] and the fixed point technique (see [5]). In the last technique, different definitely known fixed point theorems are utilized, just as some new have been demonstrated and utilized in a specific circumstance. In 1991, Baker [6] studied the stability of functional equations via the banach fixed point theorem. Since fixed point technique of Baker, Radu [7] gave the stability of an equation of functional by the option of fixed point technique which was presented by Diaz and Margolis [8]. The fixed point technique has given a great deal of impact in the advancement of the stability of functional equations. We allude to numerous papers of stability of equations of functional utilizing the fixed point technique in references on the stability of equations of functional (see [8–10]).In 2008, Park and An [11] used the fixed point technique to study the stability of functional equations due to Cauchy–Jensen. In 2009, Gao et al. [12] defined the generalized Cauchy–Jensen equation as follows:LetS be an abelian group and n-divisible, where n∈N, the set of all natural numbers, and X be a normed space with the norm ‖,‖X. For any function f:S→X, the equation(1)nfx+yn+z=fx+fy+nfz,for eachx,y,z∈S and n∈N is said to be the generalized equation of Cauchy–Jensen. In special case, when n=2, the equation is called the Cauchy–Jensen equation.Recently, in 2018, Kaskasem et al. [13] introduced the stability by Hyers–Ulam–Rassias of the generalized set-valued functional equations of Cauchy-Jensen given by(2)αfx+uα+z=fx⊕fy⊕αfz,for eachx,y,z∈Xα≥2.The objective of our paper is basically two fold. The first goal is introduce a new fixed point technique dealing with coupled fixed point results for nonlinear contractive mappings on the generalized metric space due to Diaz et al. [8]. The second goal is to apply our new coupled fixed point results to study the stability for the following coupled system of the generalized set-valued functional equations of Cauchy–Jensen:(3)αfx+uα+z,y+vα+w=fx,y⊕fu,v⊕αfz,w,(4)αgx+uα+z,y+vα+w=gx,y⊕gu,v⊕αgz,w,for allx,y,z,u,v,w∈X,α≥2 and f,g:G→X.Next, we recall some preliminaries that will be used in the main results of this paper.LetY be a Banach space. We defined the following:(i)
2Y = the set of power sets of Y;(ii)
CbY = all bounded and closed subsets of Y;(iii)
CcY = all convex and closed subsets of Y;(iv)
CcbY = all convex closed and bounded subsets of Y.Definition 1 [12].
On2Y, they consider the addition and the scalar multiplication as follows:(5)C+C′=x+x′x∈C,x′∈C′andλC=λxx∈C,
whereC,C′∈2Y and λ∈R, the set of all real numbers. Also, we define the following:(6)C⊕C′=C+C′.¯
Then(7)λC+λC′=λC+C′andλ+μC⊆λC+μC.
Also, whenC is convex, we obtain(8)λ+μC=λC+μC,
for allλ,μ∈R. For any set C∈2Y, the distance function d.,C and the support function s.,C are defined by(9)dx,C=infx∈Yx−yy∈C,(10)dx∗,C=supx∗∈Y∗x−yx∈C.
For all setsC,C′∈CbY, the Hausdorff distance between C and C′ is defined by(11)hC,C′=infλ>0C⊆C′+λBY,C′⊆C+λBY,
whereBY is the closed unit ball in Y.Proposition 1 [13].
For anyC,C′,K,K′∈CcbY and λ>0, the following properties hold:(1)
hC⊕C′,K⊕K′≤hC⊕K+C′⊕K′;(2)
hλC,λK=λhC,K.Definition 2 [14].
LetX be a set. A distance mapping d:X×X→0,∞ is said to be a generalized metric on X if the following conditions are hold:1da,b=0 for all a,b∈X if and only if a=b;2da,b=db,a for all a,b∈X;3da,u≤da,b+db,u for all a,b,u∈X;4 every d-Cauchy sequence in X is d−convergent, i.e., limn,m→∞dxn,xm=0 for a sequence xn∈X, n=1,2,… implies the existence of an element x∈X with limn→∞dx,xn=0, (x is unique by 1 and 3).
By the fact that not every two points inX have necessarily a finite distance. One might call such a space a generalized complete metric space.Example 1.
LetX=a,b,c. Define d on X×X as follows:(12)da,b=db,a=1,db,c=dc,b=dc,a=da,c=∞,dx,x=0,∀x∈X.
ThenX,d is a generalized metric space.Definition 3 [15].
LetX,≺_ be a partially ordered space and let F:X×X→X. The function F is said to have the mixed monotone property if Fa,b is nondecreasing monotone in a and is nonincreasing monotone in b, that is, for each a,b∈X,(13)a1,a2∈X,a1≺_a2⇒Fa1,b≺_Fa2,b,(14)b1,b2∈Y,b1≺_b2⇒Fa,b1≺_Fa,b2.Definition 4 [15].
A paira,b∈X×X is called a coupled fixed point of the function F:X×X→X if a=Fa,b and b=Fb,a.


## 2. Main Coupled Fixed Point Results

Theorem 1.
Suppose thatX,d is a complete generalized metric space and the function F:X×X→X be a continuous mapping having the mixed monotone property on X. Assume that there exists a k∈0,1 such that for a,b,u,v∈X, the following holds:(15)dFa,b,Fu,v≤k2da,u+db,v,
for alla≤u,b≥v and da,b<∞. If there exist a0,b0∈X such that a0≤Fa0,b0 and b0≥Fb0,a0. Then the following alternative holds: either.
(i)
for alln≥0, we have(16)dFna,b,Fn+1a,b=dFnb,a,Fn+1b,a=∞,or
(ii)
F has a coupled fixed point in X, that is, there exist a,b∈X such that a=Fa,b and b=Fb,a.Proof.
By the given assumptions, there existsa0,b0∈X×X such that a0≤Fu0,v0 and b0≥Fb0,a0. Then, we can define a1,b1∈X×X such that a1=Fa0,b0 and b1=Fb0,a0, then a0≤Fa0,b0=a1 and b0≥Fb0,a0=b1. Also there exists a2,b2∈X×X such that a2=Fa1,b1 and b2=Fb1,a1. Since F has the mixed monotone property, we have,(17)a1=Fa0,b0≤Fa1,b1=FFa0,b0=FFa0,b0,Fb0,a0=a2,(18)b1=Fb0,a0≤Fb1,a1=FFb0,a0=FFb0,a0,Fa0,b0=b2.
Continuing in this way, we construct two sequencesan and bn in X such that(19)an+1=Fn+1a0,b0=FFna0,b0=FFna0,b0,Fnb0,a0(20)bn+1=Fnn+1b0,a0n=FFnb0,a0=FFnb0,a0,Fna0,b0
for alln=0,1,2,....
There are two mutually exclusive possibilities: either
(a)
for every integeri=0,1,2...., one has(21)dFna0,b0,Fn+1a0,b0=∞anddFnb0,a0,Fn+1b0,a0=∞.which is exactly the alternative i of the conclusion of the theorem, or else
(b)
some integeri=0,1,2...., one has(22)dFna0,b0,Fn+1a0,b0<∞anddFnb0,a0,Fn+1b0,a0<∞.
Now, we need to show that (b) implies alternativeii of the conclusion of the theorem.
If case (b) holds, letN=Na0,b0 denote a particular one. For definiteness, one could choose the smallest of all integer n≥0, such that(23)dFna0,b0,Fn+1a0,b0<∞anddFnb0,a0,Fn+1b0,a0<∞.
Then, by (15), since dFNa0,b0,FN+1a0,b0<∞ and dFNb0,a0,FN+1b0,a0<∞, we get(24)dFN+1a0,b0,FN+2a0,b0=dFFNa0,b0,FFN+1a0,b0,=dFFNa0,b0,FNb0,a0,FN+1a0,b0,FN+1b0,a0,≤k2dFNa0,b0,FN+1a0,b0+dFNb0,a0,FN+1b0,a0,<∞.
Also,(25)dFN+1b0,a0,FN+2b0,a0≤k2dFNb0,a0,FN+1b0,a0+dFNa0,b0,FN+1a0,b0<∞.
However at this point, the triangle property (12  in Definition 2 infers that, at whatever point n>N, one has for each L=1,2,⋯, that(26)dFna0,b0,Fn+1a0,b0≤∑i=1LdFn+i−1a0,b0,Fn+ia0,b0,≤∑i=1Lkn+1−i−N2n+1−i−NdFNa0,b0,FN+1a0,b0,≤k2n−N1−k/2L1−k/2dFNa0,b0,FN+1a0,b0.
Since0<L<1, then the sequence Fna0,b0n=0∞ and similarly the sequence Fnb0,a0n=0∞ are d−Cauchy sequences and by (15) in Definition 2 they are d−convergent. In other words, there exist a,b∈X such that(27)limn→∞dFna0,b0,a=0andlimn→∞dFnb0,a0,b=0.
At last, we guaranteeFa,b=a and Fb,a=b, since F is continuous at a,b then we have(28)Fa,b=limn→∞FFna0,b0,Fnb0,a0,=limn→∞Fn+1a0,b0,=a,(29)Fb,a=limn→∞FFnb0,a0,Fna0,b0,=limn→∞Fn+1b0,a0,=b.Remark 1.
Letf:X→X be a mapping from X into itself. If we put fa=Fa,b and fb=Fb,a in Theorem 1, then one can deduce the following theorem.Theorem 2.
Suppose thatX,d is a partially ordered complete generalized metric space and the function f:X→X be a continuous strictly contractive mapping, that is, there exists a number k<1 such that(30)dfa,fb≤kda,b,∀a≥b.
If there existsa0∈X with a0≤fa0.
Then the following alternative holds: either
(I)
for alln≥0, we have(31)dfna,fn+1a=∞,or
(II)
f has a coupled fixed point in X, that is, there exist a∈X such that a=fa.Proof 2.
By the given assumptions, there existsa0∈X such that a0≤fu0. Then, we can define a1∈X such that a1=fa0, then a0≤fa0=a1. Also there exists a2∈X such that a2=fa1. Since f has the mixed monotone property, we have,(32)a1=fa0≤fa1=ffa0=ffa0=a2.
Continuing in this way, we construct two sequencesan in X such that(33)an+1=fn+1a0=ffna0=ffna0
for alln=0,1,2,....
There are two mutually exclusive possibilities: either
(A)
for every integeri=0,1,2...., one has(34)dfna0,fn+1a0=∞, which is exactly the alternative I of the conclusion of the theorem, or else
(B)
some integeri=0,1,2...., one has(35)dfna0,fn+1a0<∞.
Now, we need to show that (B) implies alternative (II) of the conclusion of the theorem.
If case (B)  holds, letN=Na0 denote a particular one. For definiteness, one could choose the smallest of all integer n≥0, such that(36)dfna0,fn+1a0<∞.
Then, by (30), since dfNa0,fN+1a0<∞, we get(37)dfN+1a0,fN+2a0=dffNa0,ffN+1a0=dffNa0,fNb0,fN+1a0,fN+1b0≤kdfNa0,fN+1a0<∞.
However at this point, the triangle property (12) in Definition 3 infers that, at whatever point n>N, one has for each L=1,2,…, that(38)dfna0,fn+1a0≤∑i=1Ldfn+i−1a0,fn+ia0≤∑i=1Lkn+1−i−NdfNa0,fN+1a0≤kn−N1−kLdfNa0,fN+1a0.
Since0<L<1, then the sequence fna0n=0∞ is a d−Cauchy sequence and by (15) in Definition 2 it is d−convergent. In other words, there exists a point a∈X such that(39)limn→∞dfna0,a=0.
At last, we guaranteelimn→∞fna=x, since f is continuous at a then we have(40)fa=limn→∞ffna0=limn→∞fn+1a0=a.Remark 2.
We note that the contractive condition in [8, Theorem 1.6] is slightly stronger than the condition (30) of Theorem 2.


## 3. Stability of the Cauchy–Jensen Functional Equations

LetX be a real normed space and Y be a real banach space.Definition 5.
Letf,g:X×X→CcbY be two set-valued mappings.
(1)
The coupled generalized Cauchy-Jensen set-valued functional equation is defined by(41)αfx+uα+z,y+vα+w=fx,y⊕fu,v⊕αfz,w,(42)αgx+uα+z,y+vα+w=gx,y⊕gu,v⊕αgz,w,for all x,y,z,u,v,w∈X and α≥2
(2)
Every solution of the generalized Cauchy-Jensen set-valued functional equation is called a Cauchy–Jensen set-valued mapping.Theorem 3.
Letf,g be two set-valued mappings defined on X×X into CcbY,⊕,h such that there exists a function
ψ
:
X
×
X
×
X
→
0
,
∞ satisfying(43)hαfx+uα+z,y+vα+w,fx,y⊕fu,v⊕αfz,w≤ψx,u,z+ψy,v,w,
for allx,y,z,u,v,w∈X and α≥2. If there exists L<1 such that(44)ψx,y,z≤L2αψαx,αy,αz,(45)ψu,v,w≤L2αψαu,αv,αw,
for allx,y,z,u,v,w∈X, then there exists unique generalized Cauchy–Jensen set-valued mappings F,G:X×X→CcbY,⊕,h such that(46)hfx,y,Fx,y≤11−L2+αψx,x,x+ψy,y,y,(47)hgy,x,Gy,x≤11−L2+αψx,x,x+ψy,y,y,
for allx,y∈X. Moreover, if there exist positive real number r and M with r<1 such that diam fx,y≤M‖x,y‖Xr,gx,y≤M‖x,y‖Xr for all x,y∈X, then Fx,y,Gy,x are singleton sets.Proof 3.
First, we consider the setS=g:X×X→CcbY|g0,0=0 and introduce the generalized metric on X as follows:(48)dgx,y,fx,y=infM∈0,∞hgx,y,fx,y≤Mη,
whereη=ψx,x,x+ψy,y,y and infψ=+∞. Then S,d is a complete generalized metric space (see[[16], Theorem (3)]). Now, we consider a linear mapping T:S×S→S such that(49)Tfx,y,gx,y=1βfβx,βy,∀x,y∈X,(50)Tgx,y,fx,y=1βgβx,βy,∀x,y∈X.
whereβ=2/α+1.
Next, we show thatT is a strictly contractive mapping with Lipschitz constant L. Let g,f∈S with dfx,y,ux,y=K and dgx,y,vx,y=K′ for some K,K′∈R+. It follows from (48) that(51)hfx,y,ux,y≤Kψx,x,x+ψy,y,y,(52)hgx,y,vx,y≤K′ψx,x,x+ψy,y,y,
for allx,y∈X. From Proposition 1, (44), (45) and (52) we obtain that(53)hTf,g,Tu,v=h1βfβx,βy,1βuβx,βy=1βhfβx,βy,uβx,βy≤L2K+K′ψx,x,x+ψy,y,y,
for allx∈X. Hence, dTf,g,Tu,v≤L/2K+K′, that is, dTf,g,Tu,v≤L/2df,u+dg,v. Therefore, we suppose that x=u,z=β−2/βx,y=v,w=β−2/βy and in (43) since fx,y is convex, we have(54)hβfβx,βy,β2fx,y≤ψx,x,x+ψy,y,y,
for allx,y∈X. Then, we have(55)β2h1βfβx,βy,fx,y≤ψx,x,x+ψy,y,y,
for allx,y∈X. Thus, by (2), we have(56)hTf,g,f≤1β2ψx,x,x+ψy,y,y,
for allx,y∈X.
Similarly, one can deduce that(57)hTg,f,g≤1β2ψx,x,x+ψy,y,y,
and so(58)dTf,g,f≤1β2<∞,(59)dTg,f,g≤1β2<∞.
for allx,y∈X.
By Theorem1, there exist two mappings F,G:X×X→CcbY,h such that the following conditions hold;
(a)
F
,
G is a coupled fixed point of T, that is, Fx,y=TFx,y,Gx,y and Gx,y=TGx,y,Fx,y, for all x,y∈X. Then we have(60)Fx,y=TFx,y,Gx,y=1βFβx,βy⇒Fβx,βy=βFx,y,(61)Gx,y=TGx,y,Fx,y=1βGβx,βy⇒Gβx,βy=βGx,y.
(b)
The sequencesTnf,g and Tng,f converge to F,G respectively. This implies the following equality:(62)Fx,y=limn→∞1βnfβnx,βny∀x,y∈X,(63)Gx,y=limn→∞1βngβnx,βny∀x,y∈X.
(c)
We obtain thatdf,F≤1/1−Ldf,Tf,g and dg,G≤1/1−Ldg,Tg,f which implies to the following inequality:(64)df,F≤11−Lβanddg,G≤11−Lβ.
Thus the inequalities (46) hold.
It follows from (41) and (42) that(65)hαFx+uα+z,y+vα+w,x,u⊕y,v⊕αz,w=limn→∞1βnhαfβnx+βnuα+βnz,βny+βnuα+βnw,,fβnx,βny⊕fβnu,βnv⊕fβnz,βnw≤limn→∞1βnψβnx,βnu,βnz+ψβny,βnv,βnw=0,
for allx,y,z,u,v,w∈X. Thus, we have(66)hαFx+uα+z,y+vα+w,Fx,y⊕Fu,v⊕αFz,w=0.
So, we have(67)αFx+uα+z,y+vα+w=Fx,u⊕Fy,v⊕αFz,w,
and similarly, one can get that(68)αGx+uα+z,y+vα+w=Gx,u⊕Gy,v⊕αGz,w,
for allx,y,z,u,v,w∈X. Moreover, let r and M be positive real numbers with r<1 and diamfx,y≤MM‖x,y‖r for all x,y∈X. Then, we have(69)diam1βnfβnx,βny≤1βnfβnx,βny
for allx,y∈X. Since 1/βn−rn<1, we have limn→∞1/βn‖βnx,βny‖Xr=0.
This implies thatFx=limn→∞1/βnβnx,βnyXr is a singleton set. This completes the proof.Open Problem 13.
Can our results in this paper be extended in generalizedb−metric spaces as in Aydi and Czerwik [17] and Karapinar et al. [18].



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