Volume 2011, Article ID 297360,14pages doi:10.1155/2011/297360
Research Article
Impact of Common Property (E.A.) on Fixed Point Theorems in Fuzzy Metric Spaces
D. Gopal,
1M. Imdad,
2and C. Vetro
31Department of Mathematics and Humanities, National Institute of Technology, Surat, Gujarat 395007, India
2Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India
3Dipartimento di Matematica e Informatica, Universit`a degli Studi di Palermo, Via Archirafi 34, 90123 Palermo, Italy
Correspondence should be addressed to D. Gopal,gopal.dhananjay@rediffmail.com Received 7 November 2010; Accepted 9 March 2011
Academic Editor: Jerzy Jezierski
Copyrightq2011 D. Gopal et al. 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.
We observe that the notion of common propertyE.A.relaxes the required containment of range of one mapping into the range of other which is utilized to construct the sequence of joint iterates. As a consequence, a multitude of recent fixed point theorems of the existing literature are sharpened and enriched.
1. Introduction and Preliminaries
The evolution of fuzzy mathematics solely rests on the notion of fuzzy sets which was introduced by Zadeh1in 1965 with a view to represent the vagueness in everyday life.
In mathematical programming, the problems are often expressed as optimizing some goal functions equipped with specific constraints suggested by some concrete practical situations.
There exist many real-life problems that consider multiple objectives, and generally, it is very difficult to get a feasible solution that brings us to the optimum of all the objective functions. Thus, a feasible method of resolving such problems is the use of fuzzy sets 2. In fact, the richness of applications has engineered the all round development of fuzzy mathematics. Then, the study of fuzzy metric spaces has been carried out in several wayse.g.,3,4. George and Veeramani5modified the concept of fuzzy metric space introduced by Kramosil and Mich´alek 6 with a view to obtain a Hausdorfftopology on fuzzy metric spaces, and this has recently found very fruitful applications in quantum particle physics, particularly in connection with both string and ε∞ theorysee 7and references cited therein. In recent years, many authors have proved fixed point and common fixed point theorems in fuzzy metric spaces. To mention a few, we cite 2,8–15. As patterned
in Jungck 16, a metrical common fixed point theorem generally involves conditions on commutatively, continuity, completeness together with a suitable condition on containment of ranges of involved mappings by an appropriate contraction condition. Thus, research in this domain is aimed at weakening one or more of these conditions. In this paper, we observe that the notion of common propertyE.A.relatively relaxes the required containment of the range of one mapping into the range of other which is utilized to construct the sequence of joint iterates. Consequently, we obtain some common fixed point theorems in fuzzy metric spaces which improve many known earlier resultse.g.,11,15,17.
Before presenting our results, we collect relevant background material as follows.
Definition 1.1see18. LetX be any set. A fuzzy set inXis a function with domainXand values in0,1.
Definition 1.2see6. A binary operation∗:0,1×0,1 → 0,1is a continuoust-norm if it satisfies the following conditions:
i∗is associative and commutative, ii∗is continuous,
iiia∗1afor everya∈0,1,
iva∗b≤c∗difa≤candb≤dfor alla, b, c, d∈0,1.
Definition 1.3see5. A tripletX, M,∗is a fuzzy metric space wheneverXis an arbitrary set,∗is a continuoust-norm, andMis a fuzzy set onX×X×0, ∞satisfying, for every x, y, z∈Xands, t >0, the following conditions:
iMx, y, t>0,
iiMx, y, t 1 if and only ifxy, iiiMx, y, t My, x, t,
ivMx, y, t∗My, z, s≤Mx, z, t s, vMx, y,·:0, ∞ → 0,1is continuous.
Note thatMx, y, tcan be realized as the measure of nearness betweenxandywith respect tot. It is known thatMx, y,·is nondecreasing for allx, y ∈ X. LetX, M,∗be a fuzzy metric space. Fort > 0, the open ballBx, r, twith centerx ∈X and radius 0< r <1 is defined byBx, r, t {y∈X :Mx, y, t >1−r}. Now, the collection{Bx, r, t:x∈X, 0< r <1,t >0}is a neighborhood system for a topologyτonXinduced by the fuzzy metric M. This topology is Hausdorffand first countable.
Definition 1.4see5. A sequence{xn}inXconverges toxif and only if for eachε >0 and eacht >0, there existsn0∈Nsuch thatMxn, x, t>1−εfor alln≥n0.
Remark 1.5see5. LetX, dbe a metric space. We definea∗babfor alla, b∈0,1and Mdx, y, t t/t dx, yfor everyx, y, t∈ X×X×0, ∞, thenX, Md,∗is a fuzzy metric space. The fuzzy metric spaceX, Md,∗is complete if and only if the metric space X, dis complete.
With a view to accommodate a wider class of mappings in the context of common fixed point theorems, Sessa19introduced the notion of weakly commuting mappings which was
further enlarged by Jungck20by defining compatible mappings. After this, there came a host of such definitions which are scattered throughout the recent literature whose survey and illustrationup to 2001is available in Murthy21. Here, we enlist the only those weak commutatively conditions which are relevant to presentation.
Definition 1.6 see 20. A pair of self-mappings f, g defined on a fuzzy metric space X, M,∗is said to be compatibleor asymptotically commutingif for allt >0,
nlim→ ∞M
fgxn, gfxn, t
1, 1.1
whenever{xn}is a sequence inXsuch that limn→ ∞fxnlimn→ ∞gxnz, for somez∈X.
Also, the pairf, gis called noncompatible, if there exists a sequence {xn}inX such that limn→ ∞fxn limn→ ∞gxn z, but either limn→ ∞Mfgxn, gfxn, t/1 or the limit does not exist.
Definition 1.7 see 10. A pair of self-mappings f, g defined on a fuzzy metric space X, M,∗is said to satisfy the propertyE.A.if there exists a sequence{xn}inXsuch that limn→ ∞fxnlimn→ ∞gxnzfor somez∈X.
Clearly, compatible as well as noncompatible pairs satisfy the propertyE.A..
Definition 1.8 see10. Two pairs of self mappings A, S and B, Tdefined on a fuzzy metric spaceX, M,∗are said to share common propertyE.A.if there exist sequences{xn} and{yn}inX such that limn→ ∞Axn limn→ ∞Sxn limn→ ∞Byn limn→ ∞Tyn zfor somez∈X.
For more on properties E.A. and common E.A., one can consult 22 and 10, respectively.
Definition 1.9. Two self mappingsfandgon a fuzzy metric spaceX, M,∗are called weakly compatible if they commute at their point of coincidence; that is,fxgximpliesfgxgfx.
Definition 1.10see23. Two finite families of self mappings{Ai}and{Bj}are said to be pairwise commuting if
iAiAjAjAi, i, j∈ {1,2, . . . , m}, iiBiBj BjBi, i, j∈ {1,2, . . . , n},
iiiAiBjBjAi, i∈ {1,2, . . . , m}andj∈ {1,2, . . . , n},
The following definitions will be utilized to state various results inSection 3.
Definition 1.11see15. LetX, M,∗be a fuzzy metric space andf, g :X → X a pair of mappings. The mappingf is called a fuzzy contraction with respect tog if there exists an upper semicontinuous functionr:0, ∞ → 0, ∞withrτ< τ for everyτ >0 such that
1 M
fx, fy, t −1≤r
1 m
f, g, x, y, t−1
, 1.2
for everyx, y∈Xand eacht >0, where m
f, g, x, y, t
min M
gx, gy, t , M
fx, gx, t , M
fy, gy, t
. 1.3
Definition 1.12see15. LetX, M,∗be a fuzzy metric space and f, g : X → X a pair of mappings. The mappingf is called a fuzzyk-contraction with respect togif there exists k∈0,1, such that
1 M
fx, fy, t−1≤k
1 m
f, g, x, y, t−1
, 1.4
for everyx, y∈Xand eacht >0, where m
f, g, x, y, t
min M
gx, gy, t , M
fx, gx, t , M
fy, gy, t
. 1.5
Definition 1.13. Let A, B, Sand T be four self mappings of a fuzzy metric space X, M,∗.
Then, the mappingsAandBare called a generalized fuzzy contraction with respect toSand T if there exists an upper semicontinuous functionr :0, ∞ → 0, ∞, withrτ< τ for everyτ >0 such that for eachx, y∈Xandt >0,
1 M
Ax, By, t−1≤r
1 min
M
Sx, Ty, t
, MAx, Sx, t, M
By, Ty, t−1
. 1.6
2. Main Results
Now, we state and prove our main theorem as follows.
Theorem 2.1. LetA, B,S andT be self mappings of a fuzzy metric spaceX, M,∗such that the mappingsAandBare a generalized fuzzy contraction with respect to mappingsSand T. Suppose that the pairsA, Sand B, Tshare the common property (E.A.) and SXand TX are closed subsets ofX. Then, the pairA, Sas well asB, Thave a point of coincidence each. Further,A, B, S andT have a unique common fixed point provided that both the pairsA, SandB, Tare weakly compatible.
Proof. Since the pairs A, S and B, T share the common property E.A., there exist sequences{xn}and{yn}inXsuch that for somez∈X,
nlim→ ∞Axn lim
n→ ∞Sxn lim
n→ ∞Byn lim
n→ ∞Tynz. 2.1
SinceSXis a closed subset ofX, therefore limn→ ∞Sxn z∈SX, and henceforth, there exists a pointu∈Xsuch thatSuz.
Now, we assert thatAuSu. If not, then by1.6, we have 1
M
Au, Byn, t −1≤r
1 min
M
Su, Tyn, t
, MAu, Su, t, M
Byn, Tyn, t−1
, 2.2
which on makingn → ∞, for everyt >0, reduces to
1
MAu, z, t−1≤r
1
min{MAu, z, t} −1 2.3
that is a contradiction yielding therebyAu Su. Therefore,uis a coincidence point of the pairA, S.
IfTXis a closed subset ofX, then limn→ ∞Tynz∈TX. Therefore, there exists a pointw∈Xsuch thatTwz.
Now, we assert thatBwTw. If not, then according to1.6, we have
1
MAxn, Bw, t−1≤r
1
min{MSxn, Tw, t, MAxn, Sxn, t, MBw, Tw, t}−1 , 2.4
which on makingn → ∞, for everyt >0, reduces to
1
Mz, Bw, t−1≤r
1
min{Mz, Bw, t}−1 , 2.5
which is a contradiction as earlier. It follows thatBw Twwhich shows thatwis a point of coincidence of the pairB, T. Since the pairA, Sis weakly compatible andAuSu, hence AzASuSAuSz.
Now, we assert thatzis a common fixed point of the pairA, S. Suppose thatAz /z, then using again1.6, we have for allt >0,
1
MAz, Bw, t−1≤r
1
min{MAz, Bw, t}−1 , 2.6
implying thereby thatAzBwz.
Finally, using the notion of weak compatibility of the pairB, Ttogether with1.6, we get Bz z Tz. Hence, z is a common fixed point of both the pairs A, S and B, T.
Uniqueness of the common fixed pointzis an easy consequence of condition1.6.
The following example is utilized to highlight the utility ofTheorem 2.1over earlier relevant results.
Example 2.2. LetX 2,20andX, M,∗be a fuzzy metric space defined as
M x, y, t
t
t x−y if t >0, x, y∈X. 2.7
DefineA, B, S, T :X → Xby
Ax
⎧⎨
⎩
2 ifx2, 3 ifx >2,
Sx
⎧⎨
⎩
2 ifx2, 6 ifx >2,
Bx
⎧⎪
⎪⎪
⎨
⎪⎪
⎪⎩
2 ifx2, 6 if 2< x≤5, 3 ifx >5,
Tx
⎧⎪
⎪⎪
⎨
⎪⎪
⎪⎩
2 ifx2, 18 if 2< x≤5, 12 ifx >5.
2.8
Then, A, B, S and T satisfy all the conditions of the Theorem 2.1 with rτ kτ, where k ∈ 4/9,1and have a unique common fixed pointx 2 which also remains a point of discontinuity.
Moreover, it can be seen that AX {2,3}/⊂{2,12,18} TX and BX {2,3,6}/⊂{2,6} SX. Here, it is worth noting that none of the earlier theoremswith rare possible exceptionscan be used in the context of this example as most of earlier theorems require conditions on the containment of range of one mapping into the range of other.
In the foregoing theorem, if we setrτ kτ,k∈0,1, andMx, y, t t/t |x−y|, then we get the following result which improves and generalizes the result of Jungck16, Corollary 3.2in metric space.
Corollary 2.3. LetA, B, Sand T be self mappings of a metric spaceX, dsuch that d
Ax, By
≤kmax d
Sx, Ty
, dAx, Sx, d
By, Ty
, 2.9
for everyx, y ∈X,k ∈ 0,1. Suppose that the pairsA, SandB, Tshare the common property (E.A.) andSXandTXare closed subsets ofX. Then, the pairA, Sas well asB, Thave a point of coincidence each. Further,A, B, SandT have a unique common fixed point provided that both the pairsA, SandB, Tare weakly compatible.
By choosingA, B, SandT suitably, one can deduce corollaries for a pair as well as for two different trios of mappings. For the sake of brevity, we deduce, by settingA Band ST, a corollary for a pair of mappings which is an improvement over the result of C. Vetro and P. Vetro15, Theorem 2.
Corollary 2.4. LetA, Sbe a pair of self mappings of a fuzzy metric spaceX, M,∗such thatA, S satisfies the property (E.A.),Ais a fuzzy contraction with respect toSandSXis a closed subset ofX. Then, the pairA, Shas a point of coincidence, whereas the pairA, Shas a unique common fixed point provided that it is weakly compatible.
Now, we know that A fuzzy k-contraction with respect to S implies A fuzzy contraction with respect toS. Thus, we get the following corollary which sharpen of15, Theorem 4.
Corollary 2.5. LetAand Sbe self mappings of a fuzzy metric space X, M,∗such that the pair A, Senjoys the property (E.A.),Ais a fuzzyk-contraction with respect toS, andSXis a closed
subset ofX. Then, the pairA, Shas a point of coincidence. Further,AandShave a unique common fixed point provided that the pairA, Sis weakly compatible.
3. Implicit Functions and Common Fixed Point
We recall the following two implicit functions defined and studied in 14 and 23, respectively.
Firstly, following Singh and Jain14, letΦbe the set of all real continuous functions φ:0,14 → R, non decreasing in first argument, and satisfying the following conditions:
iforu, v≥0,φu, v, u, v≥0, orφu, v, v, u≥0 implies thatu≥v, iiφu, u,1,1≥0 implies thatu≥1.
Example 3.1. Defineφt1, t2, t3, t4 15t1−13t2 5t3−7t4. Then,φ∈Φ.
Secondly, following Imdad and Ali 23, let Ψ denote the family of all continuous functionsF :0,14 → Rsatisfying the following conditions:
iF1: for everyu >0,v≥0 withFu, v, u, v≥0 orFu, v, v, u≥0, we haveu > v, iiF2:Fu, u,1,1<0, for each 0< u <1.
The following examples of functionsF∈Ψare essentially contained in23.
Example 3.2. DefineF :0,14 → RasFt1, t2, t3, t4 t1−φmin{t2, t3, t4}, whereφ:0,1 → 0,1is a continuous function such thatφs> sfor 0< s <1.
Example 3.3. DefineF:0,14 → RasFt1, t2, t3, t4 t1−kmin{t2, t3, t4}, wherek >1.
Example 3.4. DefineF:0,14 → RasFt1, t2, t3, t4 t1−kt2−min{t3, t4}, wherek >0.
Example 3.5. DefineF : 0,14 → RasFt1, t2, t3, t4 t1−at2−bt3−ct4, wherea >1 and b, c≥0 b, c /1.
Example 3.6. DefineF :0,14 → RasFt1, t2, t3, t4 t1−at2−bt3 t4, wherea >1 and 0≤b <1.
Example 3.7. DefineF:0,14 → RasFt1, t2, t3, t4 t31−kt2t3t4, wherek >1.
Before proving our results, it may be noted that above-mentioned classes of functions ΦandΨare independent classes as the implicit functionFt1, t2, t3, t4 t1−kmin{t2, t3, t4}, wherek >1belonging toΨ does not belongs toΦasFu, u,1,1<0 for allu >0, whereas implicit functionφt1, t2, t3, t4 15t1−13t2 5t3−7t4belonging toΦdoes not belongs toΨ asFu, v, u, v 0 impliesuvinstead ofu > v.
The following lemma interrelates the property E.A. with the common property E.A..
Lemma 3.8. LetA, B, SandTbe self mappings of a fuzzy metric spaceX, M,∗. Assume that there existsF∈Ψsuch that
F M
Ax, By, t , M
Sx, Ty, t
, MSx, Ax, t, M
By, Ty, t
≥0, 3.1
for all x, y ∈ X and t > 0. Suppose that pair A, S (or B, T) satisfies the property (E.A.), and AX ⊂ TX(or BX ⊂ SX). If for each{xn},{yn} in X such that limn→ ∞Axn limn→ ∞Sxn (or limn→ ∞Byn limn→ ∞Tyn), we have liminfn→ ∞MAxn, Byn, t > 0 for all t >0, then, the pairsA, SandB, Tshare the common property (E.A.).
Proof. If the pairA, Senjoys the propertyE.A., then there exists a sequence{xn}inXsuch that limn→ ∞Axn limn→ ∞Sxn zfor somez∈X. SinceAX⊂TX, hence for eachxn there existsyninXsuch thatAxnTyn, henceforth limn→ ∞Axn limn→ ∞Tyn z. Thus, we haveAxn → z,Sxn → zandTyn → z.
Now, we assert thatByn → z. We note thatByn → zif and only ifMAxn, Byn, t → 1. Assume that there existst0 > 0 such thatMAxn, Byn, t0 1, then by hypothesis there exists a subsequence of{xn}, say{xnk}, such that
M
Axnk, Bynk, t0
→ lim inf
n→ ∞M
Axn, Byn, t0
u >0. 3.2
By3.1, we have F
M
Axnk, Bynk, t , M
Sxnk, Tynk, t
, MSxnk, Axnk, t, M
Bynk, Tynk, t
≥0, 3.3 which on makingk → ∞, reduces to
Fu,1,1, u≥0, 3.4
implying thereby thatu >1, which is a contradiction. Hence limn→ ∞Byn zwhich shows that the pairsA, SandB, Tshare the common propertyE.A..
With a view to generalize some fixed point theorems contained in Imdad and Ali11, 23we prove the following fixed point theorem which in turn generalizes several previously known results due to Chugh and Kumar24, Turkoglu et al.25, Vasuki18, and some others.
Theorem 3.9. LetA, B, SandT be self mappings of a fuzzy metric spaceX, M,∗. Assume that there existsF∈Ψsuch that
F M
Ax, By, t , M
Sx, Ty, t
, MSx, Ax, t, M
By, Ty, t
≥0, 3.5
for allx, y ∈ X andt > 0. Suppose that the pairsA, SandB, Tshare the common property (E.A.) andSXandTXare closed subsets ofX. Then, the pairA, Sas well asB, Thave a point of coincidence each. Further,A, B, SandT have a unique common fixed point provided that both the pairsA, SandB, Tare weakly compatible.
Proof. Since the pairsA, SandB, Tshare the common propertyE.A., then there exist two sequences{xn}and{yn}inXsuch that
nlim→ ∞Axn lim
n→ ∞Sxn lim
n→ ∞Byn lim
n→ ∞Tynz, 3.6
for somez∈X.
SinceSX is a closed subset ofX, then limn→ ∞Sxn z ∈ SX. Therefore, there exists a pointu∈Xsuch thatSuz. Then, by3.5we have
F M
Au, Byn, t , M
Su, Tyn, t
, MSu, Au, t, M
Byn, Tyn, t
≥0, 3.7
which on makingn → ∞reduces to
FMAu, z, t, MSu, z, t, MSu, Au, t, Mz, z, t≥0, 3.8
or, equivalently,
FMAu, z, t,1, MAu, z, t,1≥0, 3.9
which givesMAu, z, t 1 for allt > 0, that is,Au z. Hence,Au Su. Therefore,uis a point of coincidence of the pairA, S.
SinceTXis a closed subset ofX, then limn→ ∞Tyn z ∈ TX. Therefore, there exists a pointw∈Xsuch thatTwz. Now, we assert thatBwz. Indeed, again using3.5, we have
FMAxn, Bw, t, MSxn, Tw, t, MSxn, Axn, t, MBw, z, t≥0. 3.10
On makingn → ∞, this inequality reduces to
FMz, Bw, t, Mz, z, t, Mz, z, t, MBw, z, t≥0, 3.11
that is,
FMz, Bw, t,1,1, Mz, Bw, t≥0, 3.12
implying thereby thatMz, Bw, t>1, for allt >0. HenceTw Bw z, which shows that wis a point of coincidence of the pairB, T. Since the pairA, Sis weakly compatible and AuSu, we deduce thatAzASuSAuSz.
Now, we assert thatzis a common fixed point of the pairA, S. Using3.5, we have FMAz, Bw, t, MSz, Tw, t, MSz, Az, t, MBw, Tw, t≥0, 3.13 that isFMAz, z, t, MAz, z, t,1,1≥ 0. Hence,MAz, z, t 1 for allt >0 and therefore Azz.
Now, using the notion of the weak compatibility of the pairB, Tand 3.5, we get BzzTz. Hence,zis a common fixed point of both the pairsA, SandB, T. Uniqueness ofzis an easy consequence of3.5.
Example 3.10. In the setting of Example 2.2, retain the same mappings A, B, S and T and defineF:0,14 → RasFt1, t2, t3, t4 t1−φmin{t2, t3, t4}withφr √
r.
Then, A, B, S and T satisfy all the conditions of Theorem 3.9 and have a unique common fixed pointx2 which also remains a point of discontinuity.
Further, we remark that Theorem 2 of Imdad and Ali23cannot be used in the context of this example, as the required conditions on containment in respect of ranges of the involved mappings are not satisfied.
Corollary 3.11. The conclusions ofTheorem 3.9remain true if3.5is replaced by one of the following conditions:
iMAx, By, t≥φmin{MSx, Ty, t, MSx, Ax, t, MBy, Ty, t}, whereφ:0,1→ 0,1is a continuous function such thatφs> sfor all 0< s <1.
iiMAx, By, t≥kmin{MSx, Ty, t, MSx, Ax, t, MBy, Ty, t}, wherek >1.
iiiMAx, By, t≥kMSx, Ty, t min{MSx, Ax, t, MBy, Ty, t}, wherek >0.
ivMAx, By, t ≥ aMSx, Ty, t bMSx, Ax, t cMBy, Ty, t, where a > 1 and b, c≥0b, c /1.
vMAx, By, t ≥ aMSx, Ty, t bMSx, Ax, t MBy, Ty, t, wherea > 1 and 0≤b <1.
viMAx, By, t≥kMSx, Ty, tMSx, Ax, tMBy, Ty, t, wherek >1.
Proof. The proof of various corollaries corresponding to contractive conditions i–vi follows fromTheorem 3.9and Examples3.2–3.7.
Remark 3.12. Corollary 3.11corresponding to conditioniis a result due to Imdad and Ali 11, whereasCorollary 3.11corresponding to various conditions presents a sharpened form of Corollary 2 of Imdad and Ali23. Similar to this corollary, one can also deduce generalized versions of certain results contained in17,18,24.
The following theorem generalizes a theorem contained in Singh and Jain14.
Theorem 3.13. LetA, B, SandT be self mappings of a fuzzy metric spaceX, M,∗. Assume that there existsφ∈Φsuch that
φ M
Ax, By, kt , M
Sx, Ty, t
, MAx, Sx, t, M
By, Ty, kt
≥0, φ
M
Ax, By, kt , M
Sx, Ty, t
, MAx, Sx, kt, M
By, Ty, t
≥0,
3.14
for allx, y ∈ X,k ∈0,1andt >0. Suppose that the pairsA, SandB, Tenjoy the common property (E.A.) andSXandTXare closed subsets ofX. Then, the pairsA, SandB, Thave a point of coincidence each. Further,A, B, SandThave a unique common fixed point provided that both the pairsA, SandB, Tare weakly compatible.
Proof. The proof of this theorem can be completed on the lines of the proof ofTheorem 3.9, hence details are omitted.
Example 3.14. In the setting ofExample 2.2, we defineφt1, t2, t3, t4 15t1−13t2 5t3−7t4, besides retaining the rest of the example as it stands.
Then, all the conditions ofTheorem 3.13withk∈1/4,1are satisfied.
Notice that 2 is the unique common fixed point ofA, B, S and T, but this example cannot be covered by Theorem 3.1 due to Singh and Jain14asAX {2,3}/⊂{2,12,18}
TXandBX {2,3,6}/⊂{2,6}SX. This example cannot also be covered byTheorem 3.9 of this paper asφu, u,1,1 2u−1impliesφ1,1,1,1 0 which contradictsF1.
Now, we statewithout proofthe following result.
Theorem 3.15. Let {A1, A2, . . . , Am},{B1, B2, . . . , Bn},{S1, S2, . . . , Sp}, and {T1, T2, . . . , Tq} be four finite families of self mappings of a fuzzy metric spaceX, M,∗such that the mappingsA A1A2· · ·Am, B B1B2· · ·Bn, S S1S2· · ·Spand T T1T2· · ·Tq satisfy3.5. Suppose that the pairsA, SandB, Tshare the common property (E.A.) andSXas well asTXare closed subsets ofX. Then, the pairsA, SandB, Thave a point of coincidence each. Further, provided the pairs of families{Ai},{Sk}and{Br},{Tt}commute pairwise, wherei∈ {1, . . . , m}, k ∈ {1, . . . , n}, r ∈ {1, . . . , p}, andt∈ {1, . . . , q}, thenAi, Sk, BrandTthave a unique common fixed point.
Proof. The proof of this theorem can be completed on the lines of Theorem 3.1 due to Imdad et al.26, hence details are avoided.
By settingA A1 A2 · · · Am, B B1 B2 · · · Bn,S S1 S2 · · · Sp andT T1 T2 · · · Tq inTheorem 3.15, one can deduce the following result for certain iterates of mappings which is a partial generalization ofTheorem 3.9.
Corollary 3.16. LetA, B, SandT be four self mappings of a fuzzy metric spaceX, M,∗such that Am, Bn, Spand Tq satisfy the condition 3.5. Suppose that the pairsAm, Spand Bn, Tqshare the common property (E.A.) and SpX as well asTqX are closed subsets ofX. Then, the pairs Am, SpandBn, Tqhave a point of coincidence each. Further,A, B, SandThave a unique common fixed point provided that the pairsA, SandB, Tcommute pairwise.
Remark 3.17. Results similar to Corollary 3.11 as well asCorollary 3.16 can be outlined in respect ofTheorem 3.13,Theorem 3.15, andCorollary 3.16. But due to the repetition, details are avoided.
Now, we conclude this note by deriving the following results of integral type.
Corollary 3.18. LetA, B, SandT be four self mappings of a fuzzy metric spaceX, M,∗. Assume that there exist a Lebesgue integrable functionϕ:R → Rand a functionφ:0,14 → Rsuch that
φu,1,u,1
0
ϕsds≥0,
φu,1,1,u
0
ϕsds≥0, or
φu,u,1,1
0
ϕsds≥0 3.15
impliesu1. Suppose that the pairsA, SandB, Tshare the common property (E.A.) andSX andTXare closed subsets ofX. If
φMAx,By,t,MSx,Ty,t,MSx,Ax,t,MBy,Ty,t 0
ϕsds≥0 ∀x, y∈X andt >0, 3.16
then the pairsA, SandB, Thave a point of coincidence each. Further,A, B, SandThave a unique common fixed point provided that both the pairsA, SandB, Tare weakly compatible.
Proof. Since the pairsA, SandB, Tshare the common propertyE.A., then there exist two sequences{xn}and{yn}inXsuch that
nlim→ ∞Axn lim
n→ ∞Sxn lim
n→ ∞Byn lim
n→ ∞Tynz, 3.17
for somez∈X. SinceSXis a closed subset ofX, then limn→ ∞Sxn z∈SX. Therefore, there exists a pointu∈Xsuch thatSuz. Now, we assert thatAuSu. Indeed, by3.16, we have
φMAu,Byn,t,MSu,Tyn,t,MSu,Au,t,MByn,Tyn,t 0
ϕsds≥0. 3.18
On makingn → ∞, it reduces to
φMAu,z,t,1,Mz,Au,t,1 0
ϕsds≥0, 3.19
which impliesMAu, z, t 1, and soAuz.
BeingTXa closed subset ofX, repeating the same argument, we deduce that there exists a pointw∈Xsuch thatBwTw.
Since the pairA, Sis weakly compatible andAuSu, we deduce thatAzASu SAuSz.
Now, we assert thatzis a common fixed point of the pairA, S. Using3.16, with xzandyw, we have
φMAz,z,t, MAz,z,t,1,1 0
ϕsds≥0, 3.20
that impliesMAz, z, t 1. HenceAz z. Similarly, we prove thatBz Tz zand so zis a common fixed point ofA, B, SandT. Uniqueness ofz is a consequence of condition 3.16.
Corollary 3.19. LetA, B, SandT be four self mappings of a fuzzy metric spaceX, M,∗. Assume that there exist a Lebesgue integrable functionϕ :R → R and a functionφ :0,14 → R, where φ∈Φ, such that
φMAx,By,t,MSx,Ty,t,MSx,Ax,t,MBy,Ty,t 0
ϕsds≥0, ∀x, y∈X, t >0, φu,u,1,1
0
ϕsds≥0, ∀u∈0,1.
3.21
Suppose that the pairs A, S and B, T enjoy the common property (E.A.) and SX and TX are closed subsets ofX. Then, the pairsA, SandB, Thave a point of coincidence each. Further,
A, B, Sand T have a unique common fixed point provided that both the pairsA, SandB, Tare weakly compatible.
Proof. The proof is the same ofCorollary 3.18, so details are omitted.
Acknowledgment
C. Vetro is supported by University of Palermo, Local University project R. S. ex 60%. The authors are grateful to Professor Dorel Mihet for going through the manuscript and for useful suggestions.
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