Volume 2010, Article ID 458086,16pages doi:10.1155/2010/458086
Research Article
On Mappings with Contractive Iterate at a Point in Generalized Metric Spaces
Ljiljana Gaji ´c and Zagorka Lozanov-Crvenkovi ´c
Department of Mathematics and Informatics, University of Novi Sad, Trg Dositeja Obradov´ca 4, 21000 Novi Sad, Serbia
Correspondence should be addressed to Ljiljana Gaji´c,[email protected] Received 7 September 2010; Revised 1 December 2010; Accepted 29 December 2010 Academic Editor: B. Rhoades
Copyrightq2010 L. Gaji´c and Z. Lozanov-Crvenkovi´c. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Using the setting of generalized metric space, the so-called G-metric space, fixed point theorems for mappings with a contractive and a generalized contractive iterate at a point are proved. These results generalize some comparable results in the literature. A common fixed point result is also proved.
1. Introduction
Sehgal in1proved fixed point theorem for mappings with a contractive iterate at a point and therefore generalized a well-known Banach theorem.
Theorem 1.1. LetX, dbe a complete metric space and letT :X → Xbe a continuous mapping with property that for everyx∈Xthere existsnx∈Nso that for everyy∈X
d
Tnxx, Tnxy
≤q·d x, y
, whereq∈0,1. 1.1
ThenT has a unique fixed pointuinXand limkTkx0 u, for eachx0∈X.
Guseman2extended Sehgal’s result by removing the condition of continuity ofT and weakening 1.1to hold on some subset Bof X such thatTB ⊆ B, where, for some x0 ∈B,Bcontains the closure of the iterates ofx0. Further extensions appear in3,4. Our aim in this study is to show that these results are valid in more general class of spaces.
In 1963, S. G¨ahler introduced the notion of 2-metric spaces but different authors proved that there is no relation between these two function and there is no easy relationship between
results obtained in the two settings. Because of that, Dhage5introduced a new concept of the measure of nearness between three or more object. But topological structure of so called D-metric spaces was incorrect. Finally, Mustafa and Sims6introduced correct definition of generalized metric space as follows.
Definition 1.2see6. LetXbe a nonempty set, and letG :X×X×X → Rbe a function satisfying the following properties:
G1Gx, y, z 0 ifxyz;
G20< Gx, x, y; for allx, y∈X, withx /y;
G3Gx, x, y≤Gx, y, z, for allx, y, z∈X, withz /y;
G4Gx, y, z Gx, z, y Gy, z, x · · ·,symmetry in all three variables;
G5Gx, y, z≤Gx, a, a Ga, y, z, for allx, y, z, a∈X.
Then the functionGis called a generalized metric, or, more specifically, aG-metric onX, and the pairX, Gis called aG-metric space.
Clearly these properties are satisfied whenGx, y, zis the perimeter of triangle with vertices atx,y, andz∈R2, moreover takingain the interior of the triangle shows thatG5 is the best possible.
Example 1.3. LetX, dbe an ordinary metric apace, thenX, dcan defineG-metrics onXby EsGsx, y, z dx, y dy, z dx, z,
EmGmx, y, z max{dx, y, dy, z, dx, z}.
Example 1.4see6. Letx{a, b}. DefineGonX×X×Xby
Ga, a, a Gb, b, b 0, Ga, a, b 1, Ga, b, b 2, 1.2
and extendGtoX×X×Xby using the symmetry in the variables. Then it is clear theX, G is aG-metric space.
Definition 1.5see6. LetX, Gbe aG-metric space, and let{xn}be sequence of points of X, a pointx∈Xis said to be the limit of the sequence{xn}, if limn,m→ ∞Gx, xn, xm 0, and one says that the sequence{xn}isG-convergent tox
Thus, ifxn → xin aG-metric spaceX, G, then for any >0, there existsN∈Nsuch thatGx, xn, xm< , for alln, m≥N.
Definition 1.6see6. LetX, Gbe aG-metric space, a sequence{xn}is calledG-Cauchy if for every > 0, there is N ∈ Nsuch thatGxn, xm, xl < , for alln, m, l ≥ N; that is, if Gxn, xm, xl → 0 asn, m, l → ∞.
AG-metric spaceX, Gis said to beG-completeor completeG-metricif everyG- Cauchy sequence inX, GisG-convergent inX, G.
Proposition 1.7 see6. LetX, Gbe a G-metric space, then the function Gx, y, zis jointly continuous in all three of its variables.
Definition 1.8 see 6. A G-metric space X, G is called symmetric G-metric space if Gx, y, y Gy, x, x, for allx, y∈X.
Proposition 1.9see6. EveryG-metric spaceX, Gwill define a metric spaceX, dGby dG
x, y G
x, y, y G
y, x, x
, ∀x, y∈X. 1.3
Note that ifX, Gis a symmetricG-metric space, then dG
x, y 2G
x, y, y
, ∀x, y∈X. 1.4
However, ifX, Gis nonsymmetric, then by theG-metric properties it follows that 3
2G x, y, y
≤dG
x, y
≤3G x, y, y
, ∀x, y∈X, 1.5
and that in general these inequalities cannot be improved.
Proposition 1.10 see 6. A G-metric space X, G is G-complete if and only ifX, dG is a complete metric space.
In recent years a lot of interesting papers were published with fixed point results inG- metric spaces, see7–18. This paper is our contribution to the fixed point theory inG-metric spaces.
2. Fixed Point Results
LetX, Gbe aG-metric space,f : X → Xa mapping,B⊆ Xsuch that for someq∈ 0,1 and each forx∈Bthere exists a positive integernnxsuch that
G
fnxz, fnxx, fnxx
≤q·max
Gz, x, x, G
z, fnxx, fnxx , G
fnxz, x, x 2.1
for allz∈B. Then we writef∈ 1. If G
fnxz, fnxx, fnxx
≤q·max
Gz, x, x,1 2 G
z, fnxz, fnxz G
x, fnxx, fnxx , 1
2 G
z, fnxx, fnxx G
fnxz, x, x
2.2
for allz∈B, we writef ∈ 2.
Theorem 2.1. Letf ∈ 1orf∈ 2. LetB⊆X, withfB⊆B. If there existsu∈Bsuch that for nnu,fnu u, thenuis the unique fixed point offinB. Moreover,fky0 → u,k → ∞, for anyy0∈Bforf ∈ 1, and forf ∈ 2ifq <2/3.
Proof. IfX, Gis a symmetric space thandGz, x 2Gz, x, xand2.1becomes
dG
fnxz, fnxx
≤qmax
dGz, x, dG
z, fnxx , dG
x, fnxz
, 2.3
and2.2becomes
dG
fnxz, fnxx
≤qmax
dGz, x,1 2 dG
z, fnxz dG
x, fnxx , 1
2 dG
z, fnxx dG
x, fnxz ,
2.4
thus the result follows from Theorem 12 in3and it is valid for anyq <1. Suppose now that X, Gis nonsymmetric space. Then by inequality1.5we have that2.1becomes
G
fnxz, fnxx, fnxx
≤2q·max
Gz, x, x, G
z, fnxx, fnxx , G
fnxz, x, x ,
2.5
and2.2becomes
G
fnxz, fnxx, fnxx
≤2q·max
Gz, x, x,1 2 G
z, fnxz, fnxz G
x, fnxx, fnxx , 1
2 G
z, fnxx, fnxx G
fnxz, x, x .
2.6
Since 2qneed not be less then 1 we can use metric fixed point results only forq <1/2.
On the other side, using the concept ofG-metric space, we are going to prove the result, if the first case for any 0< q <1, and in the second one for 0< q <2/3. This means that our results are real generalization in the case of nonsymmetricG-metric spaces.
Let f ∈ 1. Uniqueness follows from 2.1, since for fnz z, it follows that Gz, u, u Gfnz, fnu, fnu≤qGz, u, u. Nowfnfu fuimplies thatfu u.
Lety0∈B, and assumefmy0/ufor eachm. Formsufficiently large writemknr, k≥1, and 1≤r < n. Then
G fm
y0
, u, u G
fknr y0
, fnu, fnu
≤qmax G
fk−1nr y0
, u, u , G
fk−1nr y0
, fnu, fnu , G
fm y0
, u, u,
qG
fk−1nr y0
, u, u
≤ · · · ≤qkG fr
y0
, u, u
≤qkmax G
fp y0
, u, u
: 1≤p < n ,
2.7
soGfmy0, u, u → 0,m → ∞.
Iff ∈ 2, uniqueness follows from2.2since forfnz z, it follows thatGz, u, u Gfnz, fnu, fnu≤qmax{Gz, u, u,0}and furtherfu u. Now for anyy0∈B
G fm
y0
, u, u G
fknr y0
, fnu, fnu
≤qM
y0, m, u
, 2.8
where
M
y0, m, u
⎧⎪
⎪⎪
⎪⎪
⎪⎨
⎪⎪
⎪⎪
⎪⎪
⎩ G
fk−1nr y0
, u, u , 1
2 G
fk−1nr y0
, fm y0
, fm y0
0 , 1
2 G
fk−1nr y0
, u, u G
fm y0
, u, u .
2.9
ForMy0, m, u 1/2Gfk−1nry0, u, u Gfmy0, u, uwe have
1 2G
fk−1nr y0
, u, u
< 1 2G
fm y0
, u, u , 1
2G fm
y0
, u, u
< 1 2G
fk−1nr y0
, u, u 2.10
which is a contradiction, and therefore
M
y0, m, u max
G
fk−1nr y0
, u, u ,1
2G
fk−1nr y0
, fm y0
, fm y0
. 2.11
IfMy0, m, u 1/2Gfk−1nry0, fmy0, fmy0then
G fm
y0
, u, u
≤ q 2G
fk−1nr y0
, u, u qG
u, u, fm y0
. 2.12
So
2 1−q
G fm
y0
, u, u
≤qG
fk−1nr y0
, u, u
. 2.13
Therefore,Gfmy0, u, u≤hGfk−1nry0, u, u, wherehmax{q, q/21−q}. Forh <1, Gfmy0, u, u → ∞,m → ∞.
Forf:X → Xthe setOf;x0 {fnx0:n∈N}is called the orbit forx0∈X.
Theorem 2.2. LetX, Gbe a completeG-metric space and letf :X → X be a mapping. Suppose that for somex0∈Xthe orbitOf;x0is complete, and that: for someq∈0,1and eachx∈ Of;x0 there is an integernx≥1 such that
G
fnxz, fnxx, fnxx
≤q·Gz, x, x 2.14
for allz∈ Of;x0.
Thenxkfnxk−1xk−1,k∈N, converges to someu∈Xand for allm, k∈N,m > k
Gxk, xk, xm≤ qk
1−q2max G
fpx0, x0, x0
: 1≤p≤nx0
2.15
If inequality in2.14holds for allx∈ Of;x0, thenfnuu uandfkx0 → u,k → ∞.
Moreover, iffOf;x0⊆ Of;x0, thenuis the fixed point off.
Proof. IfX, G is a symmetric G-metric space the statement easily follows from Guseman fixed point result2. LetX, Gbe nonsymmetricG-metric space. Then by inequality1.5
dG
fnxz, fnxx
≤2qdGz, x. 2.16
Thus, one can use the fixed point result in metric space only forq <1/2. But here, using the concept ofG-metric, we prove the result for any 0< q <1. At first let us show that
sup
m G
fmx0, x0, x0
M <∞. 2.17
For anym ∈ N, sufficiently large, there existk, r ∈ N, 1 ≤ r ≤ nx0−1 such that mk·nx0 r. Then
G
fmx0, x0, x0
≤G
fknx0rx0, fnx0x0, fnx0x0 G
fnx0x0, x0, x0
≤qG
fk−1nx0rx0, x0, x0
G
fnx0x0, x0, x0
≤qG
fk−1nx0rx0, fnx0x0, fnx0x0
1q G
fnx0x0, x0, x0
≤q2G
fk−2nx0rx0, x0, x0
1q
G
fnx0x0, x0, x0
≤ · · ·
≤qkG
frx0, x0, x0
1q· · ·qk−1 G
fnx0x0, x0, x0
≤ 1
1−qmax G
fpx0, x0, x0
: 1≤p≤nx0
M <∞.
2.18
Now, for eachk∈N Gxk, xk, xk1 G
fnxk−1xk−1, fnxk−1xk−1, fnxkfnxk−1xk−1
≤qG
xk−1, xk−1, fnxkxk−1
≤ · · ·
≤qkG
x0, x0, fnxkx0
≤qkM.
2.19
For allm, k∈N,m > k, it follows that
Gxk, xk, xm≤Gxk, xk, xk1 Gxk1, xk1, xk2 · · ·Gxm−1, xm−1, xm≤ qk 1−qM,
2.20
so{xk}is Cauchy sequence and there existsulimkxk, for someu∈X, and inequality2.15 is proved.
If we suppose that inequality in2.14is satisfied for all x ∈ Of;x0, then, for all k∈N,
G
fnuu, fnuu, fnuxk
≤qGu, u, xk 2.21
so limkfnuxk fnuu.
On the other hand, G
fnuxk, xk, xk
G
fnufnxk−1xk−1, fnxk−1xk−1, fnxk−1xk−1
≤qG
fnuxk−1, xk−1, xk−1
≤ · · · ≤qkG
fnux0, x0, x0
2.22
implies that limkGfnuxk, xk, xk 0.
SinceGis continuous it means that G
fnuu, u, u
0. 2.23
Hencefnuu u.
Now, let us suppose thatfOf;x0⊆ Of;x0. Sincef ∈ 1byTheorem 2.1uis the fixed point offinXand limkfkx0 u.
Fornx 1, in inequality2.14independently onx, we are going to simplify the proof and to relax the condition in2.14.
Corollary 2.3. LetX, Gbe a completeG-metric apace and letf:X → X. Suppose that there exist a pointx0∈Xandq∈0,1withOf;x0complete and
G
fz, fx, fx
≤qGz, x, x 2.24
for eachx, zfx∈ Of;x0. Then{fkx0}converges to some pointu∈Xand for allk, m∈N, m > k,
Gxk, xk, xm≤ qk 1−qG
x0, x0, fx0
. 2.25
If 2.24holds, for allx∈ Of;x0orfis orbitally continuous atu, thenuis a fixed point off.
Proof. IfX, Gis a symmetric space thandGx, z 2Gz, x, xso2.24becomes dG
fz, fx
≤qdGz, x, 2.26
and result follows from Theorem 2 in19.
Now, letX, Gbe a nonsymmetricG-metric space. Then sincexkfxk−1,k∈N, Gxk, xk, xk1≤qGxk−1, xk−1, xk ≤ · · · ≤qkG
x0, x0, fx0
, 2.27
so for allm, k∈N,m > k,
Gxk, xk, xm≤ qk 1−qG
x0, x0, fx0
, 2.28
and there existsu limkxk. If2.24holds for allx ∈ Of;x0, then byTheorem 2.2, since nu 1, it follows thatfu u.
The fact thatfis orbitally continuous atx u, and that limkfkx0 u, implies that limkfk1x0 fu, and thereforeufu.
Remark 2.4. Let us note that this result is very close to Theorem 2.1 in8.
Remark 2.5. In the statements abovefdoes not have to be continuous.
The next theorems are generalizations of ´Ciri´c fixed point results in4.
Theorem 2.6. LetX, Gbe a complete metric space andT :X → X a mapping. Suppose that for eachx∈Xthere exists a positive integernnxsuch that
G
Tnx, Tnx, Tny
≤qmax
G x, x, y
, G
x, x, Ty , . . . , G
x, x, Tny ,1
2Gx, x, Tnx Gx, Tnx, Tnx
2.29
holds for someq < 2/3 and ally∈ X. ThenT has a unique fixed pointu∈X. Moreover, for every x∈X, limmTmx u.
Proof. IfX, Gis a symmetric space thendGx, y 2Gx, x, yand inequality2.29becomes dG
Tnx, Tny
≤qmax dG
x, y , dG
x, Ty
, . . . , dGx, Tnx
, 2.30
for ally∈X. Then the result follows from Theorem 2.1 in4and it is true for allq <1.
Now suppose thatX, Gis nonsymmetric space. Then, by definition of the metricdG
and inequality1.5we have dG
Tnx, Tny
≤2qmax dG
x, y , dG
x, Ty , . . . , dG
x, Tny
, dGx, Tnx
. 2.31
But 2qneed not to be less than 1, so we will prove the statement by usingG-metric.
First, let us prove prove that
Gx, x, Tmx≤ 1
1−qbx, m1,2, . . . , 2.32
where bx max
Gx, x, Tx, G
x, x, T2x
, . . . , Gx, x, Tnx,1
2Gx, x, Tnx Gx, Tnx, Tnx
. 2.33
Clearly2.32is true form1,2, . . . , n. Suppose thatm > n, and that2.32is true fori≤m and let us prove it forim1. Letm1−nr. Now
G
x, x, Tm1x,
≤Gx, x, Tnx G
Tnx, Tnx, Tm1x , G
Tnx, Tnx, Tm1x
≤qbx, 2.34
where
bx max
Gx, x, Trx, G
x, x, Tr1x
, . . . , Gx, x, Trnx, 1
2Gx, x, Tnx Gx, Tnx, Tnx
.
2.35
Ifbx Gx, x, Tnr, then2.34imply
G
x, x, Tm1x
≤ 1
1−qGx, x, Tnx≤ 1
1−qbx. 2.36
Ifbx/Gx, x, Tnr, then2.34imply
G
x, x, Tm1x
≤Gx, x, Tnx q
1−qbx≤ 1
1−qbx. 2.37
Thus by induction we obtain2.32.
Let us prove that{Tmx}mis a Cauchy sequence. Letx0x,n0 nx0,x1Tn0x0, and we define inductively a sequence of integers and a sequence of points{xk}kinXas follows:
nk nxk, andxk1 Tnkxk,k 0,1, . . .. Evidently, {xk}k is a subsequence of the orbit {Tmx0}m. Using this sequence we will prove that{Tmx0}mis a Cauchy sequence.
Letxk be any fixed member of {xk}k and letxp Tpx0 and xq Tqx0 be any two members of the orbit which follow afterxk. Thenxp Trxkandxq Tsxkfor somerands, respectively. Now, using2.29we get
G
xk, xk, xp
G
Tnk−1xk−1, Tnk−1xk−1, Tnk−1Trxk−1
≤ 3
2qGxk−1, xk−1, Tr1xk−1, 2.38 where
Gxk−1, xk−1, Tr1xk−1 max
Gxk−1, xk−1, Trxk−1, G
xk−1, xk−1, Tr1xk−1
, . . . , Gxk−1, xk−1, Trnk−1xk−1, Gxk−1, xk−1, Tnk−1xk−1
.
2.39
Similarly,Gxk−1, xk−1, Tr1xk−1≤3/2qGxk−2, xk−2, Tr2xk−2, where
Gxk−2, xk−2, Tr2xk−2 max{Gxk−2, xk−2, Tr1xk−2, . . . , Gxk−2, xk−2, Tnk−2xk−2} 2.40
Repeating this argumentktimes we get
G
xk, xk, xp
≤ 3
2q k
Gx0, x0, Trkx0. 2.41
HenceGxk, xk, xp≤3/2qkbx0. SimilarlyGxk, xk, xq≤3/2qkbx0, so
G
xp, xp, xq
≤2G
xk, xk, xp
G
xk, xk, xq
≤ 3
2q k
·3bx0. 2.42
Sinceq <2/3, it follows that{Tmx0}mis a Cauchy sequence. Let limmTmx0 u∈X.
We show thatuis a fixed point ofT. First, let us prove thatTnu u, wheren nu. For m≥nnu, we now have
GTnu, Tnu, TnTmx0
≤qmax
Gu, u, Tmx0, G
u, u, Tm1x0
, . . . , Gu, u, Tmnx0,
1
2Gu, u, Tnu Gu, Tnu, Tnu
,
2.43
and on lettingmtend to infinity it follows that
GTnu, Tnu, u≤qmax
0,1
2Gu, u, Tnu Gu, Tnu, Tnu
. 2.44
Forq <2/3 we haveTnuu.
To show thatuis a fixed point ofT, let us suppose thatTu /uand letGu, u, Tku max{Gu, u, Tru: 1≤r≤nnu}. Then
G
u, u, Tku G
Tnu, Tnu, TnTku
≤qmax
G
u, u, Tku , G
u, u, Tk1u
, . . . , G
u, u, Tknu , . . . , 1
2Gu, u, Tnu Gu, Tnu, Tnu
≤ 3 2qG
u, u, Tku .
2.45
Sinceq <2/3, it follows thatGu, u, Tku 0, which implies thatuis a fixed point ofT. Let us suppose that for somez∈X,Tzz. Then,
Gu, u, z GTnu, Tnu, Tnz≤qmax{Gu, u, z,0} 2.46
implies thatzuand thusuis the unique fixed point inX.
If we suppose thatT is continuous, then we may prove the following theorem.
Theorem 2.7. LetX, Gbe a completeG-metric space and letT :X → Xbe a continuous mapping which satisfies the condition: for eachx∈Xthere is a positive integernnxsuch that
G
Tnx, Tnx, Tny
≤max G
x, x, y , G
x, x, Ty , . . . , G
x, x, Tny
, Gx, x, Tx, . . . , Gx, x, Tnx 2.47
for someq <1 and ally∈X. ThenThas a unique fixed pointu∈Xand limmTmx u∈X, for everyx∈X.
Proof. Letxbe an arbitrary point inX. Then, as in the proof ofTheorem 2.6, the orbit{Tmx}m is bounded and is a Cauchy sequence in the completeG-metric spaceXand so it has a limit uinX. Since by the hypothesisTis continuous,
TnuuTnulim
m Tmxlim
m Tmnuu. 2.48
Therefore,uis a fixed point ofTnu. By the same argument as in the proof ofTheorem 2.6, it follows thatuis a unique fixed point ofT.
Remark 2.8. The condition thatT is a continuous mapping can be relaxed by the following condition:Tnxis continuous at a pointx∈X.
3. A Common Fixed Point Result
Now, we are going to prove Hadˇzi´c20fixed point theorem in 2-metric space, in a manner ofG-metric spaces.
Theorem 3.1. LetX, Gbe a completeG-metric space,SandT : X → X one to one continuous mappings,A:X → SX∩TXcontinuous mapping commutative withSandT. Suppose that there exists a pointx0∈Xsuch thatOA;x0is complete and that the following conditions are satisfied:
iFor everyx∈ OA;x0there existsnx∈Nso that for allz∈Xand someq∈0,1
G
Anxz, Anxx, Anxx
≤qmin{GTx, Tx, Sz, GTx, Sx, Tz, GTx, Sx, Sz, GSx, Sx, Tz}.
3.1
iiThere existsM >0 such that for allz∈ OA;x0
GSx0, Sx0, z≤M <∞. 3.2
Then there exists one and only one elementu∈Xsuch that
AuSuTuu. 3.3
e.g., there exists a unique common fixed point forA, S, andT
Proof. SinceAX⊆SX∩TXstarting withx0we can define the sequence{xn} ⊆Xsuch that
Tx2k−1Anx2k−2x2k−2, Sx2kAnx2k−1x2k−1.
3.4
Let
yn
⎧⎨
⎩
Tx2k−1, n2k−1,
Sx2k, n2k, k∈N. 3.5
We are going to prove that{yn}is Cauchy sequence
G
y2k−1, y2k−1, y2k
G
Anx2k−2x2k−2, Anx2k−2x2k−2, Anx2k−1x2k−1
G
Anx2k−2x2k−2, Anx2k−2x2k−2, Anx2k−1T−1Anx2k−2x2k−2
≤qG
Sx2k−2, Sx2k−2, Anx2k−1x2k−2 qG
Anx2k−3x2k−3, Anx2k−3x2k−3, Anx2k−1S−1Anx2k−3x2k−3
≤ · · · ≤q2k−2G
Tx1, Tx1, Anx2k−1x1
q2k−2G
Anx0x0, Anx0x0, Anx2k−1T−1Anx0x0
≤q2k−1G
Sx0, Sx0, Anx2k−1x0
≤q2k−1M.
3.6
Similarly one can prove thatGy2k, y2k, y2k1≤q2kM,k∈N, for allm, k∈N,m > k, G
yk, yk, ym
≤G
yk, yk, yk1 G
yk1, yk1, ym
≤ · · ·
≤m−1
jk
G
yj, yj, yj1
≤ qk
1−qM. 3.7
Thus we proved that{yn}is a Cauchy sequence, so there existsu∈Xsuch that
limn ynu. 3.8
It obvious that limkTx2k−1limkSx2ku.
At first we will prove thatAuu G
y2k, y2k, Ay2k1
GSx2k, Sx2k, ATx2k1 G
Anx2k−1x2k−1, Anx2k−1x2k−1, AAnx2kx2k
G
Anx2k−1x2k−1, Anx2k−1x2k−1, AAnx2kS−1Anx2k−1x2k−1
≤qG
Tx2k−1, Tx2k−1, AAnx2kx2k−1
qG
Anx2k−2x2k−2, Anx2k−2x2k−2, AAnx2kT−1Anx2k−2x2k−2
≤q2G
Sx2k−2, Sx2k−2, AAnx2kx2k−2
≤ · · ·
≤q2kG
Sx0, Sx0, AAnx2kx0
≤q2k·M,
3.9
so limkGy2k, y2k, Ay2k1 0.
Now, since thatGandAare continuous we have thatGu, u, Au 0 soAuu.
Further, let us prove thatTuu.
G
y2k, y2k, Ty2k
G
Anx2k−1x2k−1, Anx2k−1x2k−1, TAnx2k−1x2k−1
≤qGTx2k−1, Tx2k−1, STx2k−1 qG
Anx2k−2x2k−2, Anx2k−2x2k−2, SAnx2k−2x2k−2
≤q2GSx2k−2, Sx2k−2, TSx2k−2 q2G
Anx2k−3x2k−3, Anx2k−3x2k−3, TAnx2k−3x2k−3
≤ · · ·
≤q2kGSx0, Sx0, TSx0
3.10
implies that
limk G
y2k, y2k, Ty2k
Gu, u, Tu 0, 3.11
andTuu. Similarly one can see thatSuu, so we prove that
AuSuTuu. 3.12
If we suppose thatωis some other common fixed point forA, S, andT then we have that
Gu, u, ω G
Anuu, Anuu, Anuω
≤qGSu, Su, Tω qGu, u, ω< Gu, u, ω 3.13
which is contradiction!
So, the common fixed point forA, S, andTis unique, and proof is completed.
Remark 3.2. ForS T IdX condition2.14is satisfied but theTheorem 2.2is not just a consequence ofTheorem 3.1since inTheorem 2.2we do not suppose thatfis continuous.
Acknowledgments
The authors are thankful to professor B. E. Rhoades, for his advice which helped in improving the results. This work was supported by grants approved by the Ministry of Science and Technological Development, Republic of Serbia, for the first author by Grant no. 144016, and for the second author by Grant no. 144025.
References
1 V. M. Sehgal, “A fixed point theorem for mappings with a contractive iterate,” Proceedings of the American Mathematical Society, vol. 23, pp. 631–634, 1969.
2 L. F. Guseman Jr., “Fixed point theorems for mappings with a contractive iterate at a point,”
Proceedings of the American Mathematical Society, vol. 26, pp. 615–618, 1970.
3 B. E. Rhoades, “A comparison of various definitions of contractive mappings,” Transactions of the American Mathematical Society, vol. 226, pp. 257–290, 1977.
4 L. ´Ciri´c, “On Sehgal’s maps with a contractive iterate at a point,” Publications de l’Institut Math´ematique, vol. 33, no. 47, pp. 59–62, 1983.
5 B. C. Dhage, “Generalised metric spaces and mappings with fixed point,” Bulletin of the Calcutta Mathematical Society, vol. 84, no. 4, pp. 329–336, 1992.
6 Z. Mustafa and B. Sims, “A new approach to generalized metric spaces,” Journal of Nonlinear and Convex Analysis, vol. 7, no. 2, pp. 289–297, 2006.
7 Z. Mustafa, H. Obiedat, and F. Awawdeh, “Some fixed point theorem for mapping on completeG- metric spaces,” Fixed Point Theory and Applications, vol. 2008, Article ID 189870, 12 pages, 2008.
8 M. Abbas and B. E. Rhoades, “Common fixed point results for noncommuting mappings without continuity in generalized metric spaces,” Applied Mathematics and Computation, vol. 215, no. 1, pp.
262–269, 2009.
9 Z. Mustafa, W. Shatanawi, and M. Bataineh, “Fixed point theorem on uncompleteG-metric spaces,”
Journal of Matehematics and Statistics, vol. 4, no. 4, pp. 196–210, 2008.
10 W. Shatanawi, “Fixed point theory for contractive mappings satisfyingΦ-maps inG-metric spaces,”
Fixed Point Theory and Applications, vol. 2010, Article ID 181650, 9 pages, 2010.
11 Z. Mustafa and B. Sims, “Fixed point theorems for contractive mappings in complete G-metric spaces,” Fixed Point Theory and Applications, vol. 2009, Article ID 917175, 10 pages, 2009.
12 A. Dehghan Nezhad and H. Mazaheri, “New results inG-best approximation inG-metric spaces,”
Ukrainian Mathematical Journal, vol. 62, no. 4, pp. 648–654, 2010.
13 R. Saadati, S. M. Vaezpour, P. Vetro, and B. E. Rhoades, “Fixed point theorems in generalized partially orderedG-metric spaces,” Mathematical and Computer Modelling, vol. 52, no. 5-6, pp. 797–801, 2010.
14 Z. Mustafa, W. Shatanawi, and M. Bataineh, “Existence of fixed point results inG-metric spaces,”
International Journal of Mathematics and Mathematical Sciences, vol. 2009, Article ID 283028, 10 pages, 2009.
15 S. Manro, S. S. Bhatia, and S. Kumar, “Expansion mapping theorems inG-metric spaces,” International Journal of Contemporary Mathematical Sciences, vol. 5, no. 51, pp. 2529–2535, 2010.
16 R. Chugh, T. Kadian, A. Rani, and B. E. Rhoades, “Property P inG-metric spaces,” Fixed Point Theory and Applications, vol. 2010, Article ID 401684, 12 pages, 2010.
17 Z. Mustafa, F. Awawdeh, and W. Shatanawi, “Fixed point theorem for expansive mappings inG- metric spaces,” International Journal of Contemporary Mathematical Sciences, vol. 5, no. 50, pp. 2463–2472, 2010.
18 M. Abbas, T. Nazir, and S. Radenovi´c, “Some periodic point results in generalized metric spaces,”
Applied Mathematics and Computation, vol. 217, no. 8, pp. 4094–4099, 2010.
19 S. Park, “A unified approach to fixed points of contractive maps,” Journal of the Korean Mathematical Society, vol. 16, no. 2, pp. 95–105, 1980.
20 O. Hadˇzi´c, “On common fixed point theorems in 2-metric spaces,” Zbornik Radova Prirodno- Matematiˇckog Fakulteta. Serija za Matemati, vol. 12, pp. 7–18, 1982.
