# Common Fixed Points viaλ-Sequences in G-Metric Spaces

Authors: Yaé Ulrich Gaba
Journal: Journal of Mathematics
 (2017)

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## Abstract

We useλ-sequences in this article to derive common fixed points for a family of self-mappings defined on a complete G-metric space. We imitate some existing techniques in our proofs and show that the tools employed can be used at a larger scale. These results generalize well known results in the literature.


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## Body

## 1. Introduction and Preliminaries

The generalization of the Banach contraction mapping principle has been a heavily investigated branch of research. In recent years, several authors have obtainedfixed point andcommon fixed point results for various classes of mappings in the setting of many generalized metric spaces. One of them, the G-metric space, is our focus in this paper, and fixed point results, in this setting, presented by authors like Abbas et al. [1], Gaba [2, 3], Mustafa and Sims [4], Sihag et al. [5], and many more, are enlightening on the subject. Moreover, in [6], we introduced the concept of λ-sequence which extended the idea of α-series proposed by Sihag et al. in [5]. The present article exclusively presents natural extensions of some results already given by Abbas et al. [1] and Sihag et al. [5] and therefore generalizes some recent results regarding fixed point theory in G-metric spaces. We also show how the idea of λ-sequence is used in proving some of these results. The method builds on the convergence of an appropriate series of coefficients. We also make use of a special class of homogeneous functions. Recent and similar work can also be read in [2, 3, 6, 7].We recall here some key results that will be useful in the rest of this manuscript. The basic concepts and notations attached to the idea ofG-metric spaces can be read extensively in [4], but for the convenience of the reader, we discuss the most important ones.Definition 1 (compare [4, Definition 3]).
LetX be a nonempty set, and let the function G:X×X×X→[0,∞) satisfy the following properties: (G1)
G(x,y,z)=0 if x=y=z whenever x,y,z∈X.(G2)
G(x,x,y)>0 whenever x,y∈X with x≠y.(G3)
G(x,x,y)≤G(x,y,z) whenever x,y,z∈X with z≠y.(G4)
G(x,y,z)=G(x,z,y)=G(y,z,x)=… (symmetry in all three variables).(G5)
(1)Gx,y,z≤Gx,a,a+Ga,y,zfor any pointsx,y,z,a∈X.Then,(X,G) is called a G-metric space.The property (G3) is crucial and shall play a key role in our proofs.Proposition 2 (compare [4, Proposition 6]).
Let(X,G) be a G-metric space.
Then, for a sequence(xn)⊆X, the following are equivalent: (i)
(xn) is G-convergent to x∈X.(ii)
limn,m→∞⁡G(x,xn,xm)=0.(iii)
limn→∞⁡G(x,xn,xn)=0.(iv)
limn→∞⁡G(xn,x,x)=0.Proposition 3 (compare [4, Proposition 9]).
In aG-metric space (X,G), the following are equivalent: (i)
The sequence(xn)⊆X is G-Cauchy.(ii)
For eachε>0, there exists N∈N such that G(xn,xm,xm)<ε for all m,n≥N.Definition 4 (compare [4, Definition 9]).
AG-metric space (X,G) is said to be complete if every G-Cauchy sequence in (X,G) is G-convergent in (X,G).Definition 5 (compare [4, Definition 4]).
AG-metric space (X,G) is said to be symmetric if (2)Gx,y,y=Gx,x,y,∀x,y∈X.Definition 6 (compare [3, Definition 2.1]).
A sequencexnn≥1 in a metric space (X,d) is a λ-sequence if there exist 0<λ<1 and n(λ)∈N such that (3)∑i=1L-1dxi,xi+1≤λLfor  eachL≥nλ+1.Definition 7 (compare [2, Definition 6]).
A sequencexnn≥1 in a G-metric space (X,G) is a λ-sequence if there exist 0<λ<1 and n(λ)∈N such that (4)∑i=1L-1Gxi,xi+1,xi+1≤λLfor  eachL≥nλ+1.Definition 8 (compare [5, Definition 2.1]).
For a sequenceann≥1 of nonnegative real numbers, the series ∑n=1∞an is an α-series if there exist 0<λ<1 and n(λ)∈N such that (5)∑i=1Lai≤λLfor  eachL≥nλ.Remark 9.
For a givenλ-sequence xnn≥1 in a G-metric space (X,d), the sequence βnn≥1 of nonnegative real numbers, defined by (6)βi=dxi,xi+1,xi+1,is an α-series.Moreover, any nonincreasingλ-sequence of elements of R+ endowed with the max (the max metric m refers to mx,y=max⁡x,y) metric is also an α-series. Therefore, λ-sequences generalize α-series, but to ease computations, we shall consider, throughout the paper, α-series (however, the reader can convince himself that using λ-sequences does not add to the complexity of the problem).


## 2. First Generalizations Results

We begin with the following generalization of [5, Theorem 2.1], the main result of Sihag et al.LetΦ be the class of continuous, nondecreasing, subadditive, and homogeneous functions F:[0,∞)→[0,∞) such that F-1(0)={0}.Theorem 10.
Let(X,G) be a complete G-metric space and {Tn} be a family of self-mappings on X such that(7)FGTix,Tjy,Tkz≤FΘki,jGx,Tix,Tix+12Gy,Tjy,Tjy+Gz,Tkz,Tkz+FΔki,jGx,y,z,for all x,y,z∈X with x≠y, 0≤Θki,j,Δki,j<1;i,j,k=1,2,…, and some F∈Φ, homogeneous with degree s. If(8)∑i=1∞Θi+2i,i+1s+Δi+2i,i+1s1-Θi+2i,i+1sis an α-series, then {Tn} have a unique common fixed point in X.Proof.
We will proceed in two main steps.
Claim  1.
{
T
n
}
n
≥
1 have a common fixed point in X.
For anyx0∈X, we construct the sequence (xn) by setting xn=Tnxn-1,n=1,2,….
We assume without loss of generality thatxm≠xn for all n≠m∈N. Using (7), we obtain, for the triplet (x0,x1,x2),(9)FGx1,x2,x3=FGT1x0,T2x1,T3x2≤Θ31,2sFGx0,x1,x1+12Gx1,x2,x2+12Gx2,x3,x3+Δ31,2sFGx0,x1,x2.
By property (G3) ofG, one knows that(10)Gxi,xi+1,xi+1≤Gxi-1,xi,xi+1,Gxi,xi,xi+1≤Gxi,xi+1,xi+2.
Hence,(11)FGx1,x2,x3=FGT1x0,T2x1,T3x2≤Θ31,2sFGx0,x1,x2+12Gx1,x2,x3+12Gx1,x2,x3+Δ31,2sFGx0,x1,x2=Θ31,2sFGx1,x2,x3+Θ31,2s+Δ31,2sFGx1,x2,x0;that is,(12)FGx1,x2,x3≤Θ31,2s+Δ31,2s1-Θ31,2sFGx0,x1,x2.
Also, we get(13)FGx2,x3,x4≤Θ42,3s+Δ42,3s1-Θ42,3sFGx1,x2,x3≤Θ42,3s+Δ42,3s1-Θ42,3sΘ31,2s+Δ31,2s1-Θ31,2sFGx0,x1,x2.
Repeating the above reasoning, we obtain(14)FGxn,xn+1,xn+2≤∏i=1nΘi+2i,i+1s+Δi+2i,i+1s1-Θi+2i,i+1sFGx0,x1,x2.
If we set(15)ri=Θi+2i,i+1s+Δi+2i,i+1s1-Θi+2i,i+1s,we have that(16)FGxn,xn+1,xn+2≤∏i=1nriFGx0,x1,x2.
Therefore, for alll>m>n>2,(17)Gxn,xm,xl≤Gxn,xn+1,xn+1+Gxn+1,xn+2,xn+2+⋯+Gxl-1,xl-1,xl≤Gxn,xn+1,xn+2+Gxn+1,xn+2,xn+3+⋯+Gxl-2,xl-1,xl.
Using the fact thatF is subadditive, we write(18)FGxn,xm,xl≤∏i=1nri+∏i=1n+1ri+⋯+∏i=1l-2riFGx0,x1,x2=∑k=0l-n-2∏i=1n+kriFGx0,x1,x2=∑k=nl-2∏i=1kriFGx0,x1,x2.
Now, letλ and n(λ) be as in Definition 8; then, for n≥n(λ) and using the fact that the geometric mean of nonnegative real numbers is at most their arithmetic mean, it follows that(19)FGxn,xm,xl≤∑k=nl-21k∑i=1krikFGx0,x1,x2=∑k=nl-2αkFGx0,x1,x2≤αn1-αFGx0,x1,x2.
Asn→∞, we deduce that G(xn,xm,xl)→0. Thus, (xn) is a G-Cauchy sequence.
And sinceX is complete, there exists u∈X such that (xn)G-converges to u.
Moreover, for any positive integersk,l, we have (20)FGxn,Tku,Tlu=FGTnxn-1,Tku,Tlu≤FΘln,kGxn-1,Tnxn-1,Tnxn-1+12Gu,Tku,Tku+Gu,Tlu,Tlu+FΔln,kGxn-1,u,u.
Lettingn→∞ and using property (G3), we obtain(21)FGu,Tku,Tlu≤Θln,ksFGu,Tku,Tlu,and this is a contradiction, unless u=Tku=Tlu, since Θln,k<1. Then, u is a common fixed point of {Tn}.
Claim  2. u is the unique common fixed point of {Tm}.
Finally, we prove the uniqueness of the common fixed pointu. To this aim, let us suppose that v is another common fixed point of {Tm}; that is, Tm(v)=v,∀m≥1. Then, using (7) again, we have(22)FGu,v,v=FGTmu,Tmv,Tmv≤Δmm,msFGu,v,v,which yields u=v, since Δmm,m<1. So, u is the unique common fixed point of {Tm}.Theorem 11.
Let(X,G) be a complete G-metric space and {Tn} be a family of self-mappings on X such that(23)FGTipx,Tjpy,Tkpz≤FΘki,jGx,Tipx,Tipx+12Gy,Tjpy,Tjpy+Gz,Tkpz,Tkpz+FΔki,jGx,y,z,for all x,y,z∈X with x≠y, 0≤Θki,j,Δki,j<1;i,j,k=1,2,…, some positive integer p, and some F∈Φ, homogeneous with degree s. If(24)∑i=1∞Θi+2i,i+1s+Δi+2i,i+1s1-Θi+2i,i+1sis an α-series, then {Tn} have a unique common fixed point in X.Proof.
It follows from Theorem10 that the family {Tnp} have a unique common fixed point x∗. Now, for any positive integers i,j,i≠j, (25)Tix∗=TiTipx∗=TipTix∗,Tjx∗=TjTjpx∗=TjpTjx∗;that is, Ti(x∗) and Tj(x∗) are also fixed points for Tip and Tjp (remember that any fixed point of Tip is a fixed point of Tjp for i≠j; cf. Theorem 10). Since the common fixed point of {Tnp} is unique, we deduce that (26)x∗=Tix∗=Tjx∗∀i.The next result, corollary of Theorem10, corresponds to the result presented by Sihag et al. [5, Theorem 2.1].Corollary 12 (compare [5, Theorem 2.1]).
Let(X,G) be a complete G-metric space and {Tn} be a family of self-mappings on X such that(27)GTix,Tjy,Tkz≤Θki,jGx,Tix,Tix+Gy,Tjy,Tjz+Δki,jGx,y,z,for all x,y,z∈X with x≠y, 0≤Θki,j,Δki,j<1,i,j,k=1,2,…. If(28)∑i=1∞Θi+2i,i+1+Δi+2i,i+11-Θi+2i,i+1is an α-series, then {Tn} have a unique common fixed point in X.Proof.
In Theorem10, take F=Id[0,∞) (the identity map on [0,∞)), j=k and y=z.Example 13.
LetX=[0,1] and G(x,y,z)=max⁡{x,y,z} whenever x,y∈[0,1] and G(x,x,x)=0 for x∈[0,1]. Clearly, (X,G) is a complete G-metric space.Following the notation in the definition, we set the following.Θ
k
i
,
j
=
1
/
(
1
+
2
η
)
2, where η=min⁡{i,j,k} and Δki,j=0. We also define Ti(x)=x/16i for all x∈X and i=1,2,… and F:[0,∞)→[0,∞),x↦x. Then, F is continuous, nondecreasing, subadditive, and homogeneous of degree s=1/2 and F-1(0)={0}. Assume i<j≤k and x>y≥z. Hence, we have(29)FDTix,Tjy,Tjz=x16i,Fδi,jDx,Tix,Tix+Dy,Tjy,Tjz=11+2i2x+y.Therefore, condition (27) is satisfied for all x,y∈X with x≠y. Moreover, since F is homogeneous of degree s=1/2, the sequence (30)si=δi,i+1s1-δi,i+1s=12isatisfies the condition on (8); that is, it is an α-series.Then, by Corollary12, {Tn} has a common fixed point, which is this case x∗=0.


## 3. Second Generalizations Results

The next generalization is that of [1, Theorem 2.1], the main result of Abbas et al. Instead of considering three maps, we consider a family of maps like in the previous case. Moreover, to show the reader that λ-sequences do not add to the complexity of the problem, we shall use them in the next statement.Theorem 14.
LetX be a complete G-metric space (X,G) and {Tn} be a sequence of self-mappings on X. Assume that there exist three sequences (an), (bn), and (cn) of elements of X such that(31)GTix,Tjy,Tkz≤Δki,jGx,y,z+Θki,jGTix,x,x+Gy,Tjy,y+Gz,z,Tkz+Λki,jGTix,y,z+Gx,Tjy,z+Gx,y,Tkz,for all x,y,z∈X with 0≤Δki,j+3Θki,j+4Λki,j<1/2,i,j,k=1,2,…, where Δki,j=G(ai,aj,ak), Θki,j=G(bi,bj,bk), and Λki,j=G(ci,cj,ck). If the sequence (ri), where(32)ri=Δi+2i,i+1+2Θi+2i,i+1+3Λi+2i,i+11-Θi+2i,i+1-Λi+2i,i+1,is a nonincreasing λ-sequence of R+ endowed with the max (the max metric m refers to mx,y=max⁡{x,y}) metric, then {Tn} have a unique common fixed point in X. Moreover, any fixed point of Ti is a fixed point of Tj for i≠j.Proof.
We will proceed in two main steps.
Claim  1. Any fixed point of Ti is also a fixed point of Tj and Tk for i≠j≠k≠i.
Assume thatx∗ is a fixed point of Ti and suppose that Tjx∗≠x∗ and Tkx∗≠x∗. Then,(33)Gx∗,Tjx∗,Tkx∗=GTix∗,Tjx∗,Tkx∗≤Δki,jGx∗,x∗,x∗+Θki,jGTix∗,x∗,x∗+Gx∗,Tjx∗,x∗+Gx∗,x∗,Tkx∗+Λki,jGTix∗,x∗,x∗+Gx∗,Tjx∗,x∗+Gx∗,x∗,Tkx∗≤Θki,j+Λki,jGx∗,Tjx∗,Tkx∗+Gx∗,Tjx∗,Tkx∗≤2Θki,j+2Λki,jGx∗,Tjx∗,Tkx∗,which is a contradiction unless Tix∗=x∗=Tjx∗=Tkx∗.
Claim  2. For any x0∈X, we construct the sequence (xn) by setting xn=Tnxn-1,n=1,2,…. We assume without loss of generality that xn≠xm for all n≠m. Using (31), we obtain(34)Gx1,x2,x3=GT1x0,T2x1,T3x2≤Δ31,2Gx0,x1,x2+Θ31,2Gx1,x0,x0+Gx1,x2,x1+Gx2,x2,x3+Λ31,2Gx1,x1,x2+Gx0,x2,x2+Gx0,x1,x3.
By property (G3), one can write(35)Gx1,x2,x3=GT1x0,T2x1,T3x2≤Δ31,2Gx0,x1,x2+Θ31,2Gx1,x0,x2+Gx1,x2,x0+Gx1,x2,x3+Λ31,2Gx1,x0,x2+Gx0,x1,x2+Gx0,x1,x3.
Again, since(36)Gx0,x1,x3≤Gx0,x2,x2+Gx2,x1,x3≤Gx0,x1,x2+Gx2,x1,x3,we obtain(37)Gx1,x2,x3=GT1x0,T2x1,T3x2≤Δ31,2Gx0,x1,x2+Θ31,2Gx1,x0,x2+Gx1,x2,x0+Gx1,x2,x3+Λ31,2Gx1,x0,x2+Gx0,x1,x2+Gx0,x1,x2+Gx2,x1,x3;that is, (38)1-Θ31,2-Λ31,2Gx1,x2,x3≤Δ31,2+2Θ31,2+3Λ31,2Gx0,x1,x2.
Hence,(39)Gx1,x2,x3≤Δ31,2+2Θ31,2+3Λ31,21-Θ31,2-Λ31,2Gx0,x1,x2.
Also, we get(40)Gx2,x3,x4≤Δ42,3+2Θ42,3+3Λ42,31-Θ42,3-Λ42,3Gx1,x2,x3≤Δ42,3+2Θ42,3+3Λ42,31-Θ42,3-Λ42,3Δ31,2+2Θ31,2+3Λ31,21-Θ31,2-Λ31,2Gx0,x1,x2.
Repeating the above reasoning, we obtain(41)Gxn,xn+1,xn+2≤∏i=1nΔi+2i,i+1+2Θi+2i,i+1+3Λi+2i,i+11-Θi+2i,i+1-Λi+2i,i+1Gx0,x1,x2.
If we set(42)ri=Δi+2i,i+1+2Θi+2i,i+1+3Λi+2i,i+11-Θi+2i,i+1-Λi+2i,i+1,we have that(43)Gxn,xn+1,xn+2≤∏i=1nriGx0,x1,x2.
Therefore, for alll>m>n>2,(44)Gxn,xm,xl≤Gxn,xn+1,xn+1+Gxn+1,xn+2,xn+2+⋯+Gxl-1,xl-1,xl≤Gxn,xn+1,xn+2+Gxn+1,xn+2,xn+3+⋯+Gxl-2,xl-1,xl,(45)Gxn,xm,xl≤∏i=1nri+∏i=1n+1ri+⋯+∏i=1l-2riGx0,x1,x2=∑k=0l-n-2∏i=1n+kriGx0,x1,x2=∑k=nl-2∏i=1kriGx0,x1,x2.
Now, letλ and n(λ) be as in Definition 8; then, for n≥n(λ) and using the fact that the geometric mean of nonnegative real numbers is at most their arithmetic mean, it follows that(46)Gxn,xm,xl≤∑k=nl-21k∑i=1krikGx0,x1,x2=∑k=nl-2λkGx0,x1,x2≤λn1-λGx0,x1,x2.
Asn→∞, we deduce that G(xn,xm,xl)→0. Thus, (xn) is a G-Cauchy sequence.
Moreover, sinceX is complete, there exists u∈X such that (xn)G-converges to u.
If there existsn0 such that Tn0u=u, then, by Claim  1, the proof of existence is complete.
Otherwise, for any positive integersk,l, we have (47)Gxn,Tku,Tlu=GTnxn-1,Tku,Tlu≤Δln,kGxn-1,u,u+Θln,kGTnxn-1,xn-1,xn-1+Gu,Tku,u+Gu,u,Tlu+Λln,kGTnxn-1,u,u+Gxn-1,Tku,u+Gxn-1,u,Tlu.
Lettingn→∞ and using property (G3), we obtain(48)Gu,Tku,Tlu≤Θln,kGu,Tku,u+Gu,u,Tlu+Λln,kGu,Tku,u+Gu,u,Tlu≤2Θki,j+2Λki,jGu,Tku,Tlu+Gu,Tku,Tlu,and this is a contradiction, unless u=Tku=Tlu.
Finally, we prove the uniqueness of the common fixed pointu. To this aim, let us suppose that v is another common fixed point of Tm; that is, Tm(v)=v,∀m≥1. Then, using (31), we have(49)Gu,v,v=GTnu,Tkv,Tlv≤Δln,kGu,v,v+3Λln,kGu,v,v,which yields u=v. So, u is the unique common fixed point of {Tm}.Following the same lines of the proof of Theorem11, one can prove the next theorem.Theorem 15.
LetX be a complete G-metric space (X,G) and {Tn} be a sequence of self-mappings on X. Assume that there exist three sequences (an), (bn), and (cn) of elements of X such that(50)GTipx,Tjpy,Tkpz≤Δki,jGx,y,z+Θki,jGTipx,x,x+Gy,Tjpy,y+Gz,z,Tkpz+Λki,jGTipx,y,z+Gx,Tjpy,z+Gx,y,Tkpz,for all x,y,z∈X with 0≤Δki,j+3Θki,j+4Λki,j<1/2,i,j,k=1,2,…, some positive integer p, where Δki,j=G(ai,aj,ak), Θki,j=G(bi,bj,bk), and Λki,j=G(ci,cj,ck). If the sequence (ri), where(51)ri=Δi+2i,i+1+2Θi+2i,i+1+3Λi+2i,i+11-Θi+2i,i+1-Λi+2i,i+1,is a nonincreasing λ-sequence of R+ endowed with the max (the max metric m refers to mx,y=max⁡{x,y}) metric, then {Tn} have a unique common fixed point in X. Moreover, any fixed point of Ti is a fixed point of Tj for i≠j.The next result, corollary of Theorem14, corresponds to the result presented by Abbas et al. [1, Theorem 2.1].Corollary 16.
LetX be a complete G-metric space (X,G), f,g,h mappings on X. Assume that there exist three nonnegative real numbers a, b, and c such that(52)Gfx,gy,hz≤aGx,y,z+bGfx,x,x+Gy,gy,y+Gz,z,hz+cGfx,y,z+Gx,gy,z+Gx,y,hz,for all x,y,z∈X with 0≤a+3b+4c<1.
Then,f,g,h have a unique common fixed point in X. Moreover, any fixed point of f is a fixed point of g and h and vice versa.Proof.
In Theorem14, take T1=f, T2=g, T3=h. Also, set (53)Δ31,2=a,Θ31,2=b,Λ31,2=c.
Hence, we have(54)0≤a+3b+4c<12⟹0≤a+3b+4c<1⟺0≤ri=r=a+2b+3c1-b-c<1.
The sequenceri=r is constant, so, in Definition 8, if we choose λ=1/2 and n(λ)=1, it is clear that ∑i=1∞ri is an α-series. Indeed, since(55)a+2b+3c1-b-c<a+3b+4c<12,therefore, for any L≥n(λ)+1=1+1=2,(56)∑i=1L-1ri=∑i=1L-1r<12L-1≤12L.We conclude this manuscript with the following result, whose proof is straightforward, following the steps of the proofs of the earliest results.Theorem 17.
LetX be a complete G-metric space (X,G) and {Tn} be a sequence of self-mappings on X. Assume that there exist three sequences (an), (bn), and (cn) of elements of X such that(57)FGTipx,Tjpy,Tkpz≤FΔki,jGx,y,z+Θki,jGTipx,x,x+Gy,Tjpy,y+Gz,z,Tkpz+Λki,jGTipx,y,z+Gx,Tjpy,z+Gx,y,Tkpz,for all x,y,z∈X with 0≤Δki,js+3Θki,js+4Λki,js<1/2,i,j,k=1,2,…, some positive integer p, and some F∈Φ, homogeneous with degree s, where Δki,j=G(ai,aj,ak), Θki,j=G(bi,bj,bk), and Λki,j=G(ci,cj,ck). If the sequence (ri) where(58)ri=Δi+2i,i+1s+2Θi+2i,i+1s+3Λi+2i,i+1s1-Θi+2i,i+1s-Λi+2i,i+1sis a nonincreasing λ-sequence of R+ endowed with the max (the max metric m refers to mx,y=max⁡{x,y}) metric, then {Tn} have a unique common fixed point in X. Moreover, any fixed point of Ti is a fixed point of Tj for i≠j.In addition to the examples provided by Abbas et al. and Sihag et al., illustrations of all the above results can be read in [2, Example 2.5] and [6, Example 2.8].



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*Source: 6018054-2017-11-19.xml*