Semicompatibility and fixed point theorem in fuzzy metric space using implicit function

IJRET: International Journal of Research in Engineering and Technology ISSN: 2319-1163
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Volume: 01 Issue: 03 | Nov-2012, Available @ http://www.ijret.org 248
SEMICOMPATIBILITY AND FIXED POINT THEOREM IN FUZZY
METRIC SPACE USING IMPLICIT FUNCTION
V. H. BADSHAH1
, YOGITA R. SHARMA2
1,2
Prof., School of Studies in Mathematics, Vikram University, Ujjain, M.P. 456010, INDIA, yogitasharma17@yahoo.com
Abstract
In this paper we proved fixed point theorem of four mapping on fuzzy metric space based on the concept of semi copatibility using
implicit relation. These results generalize several corresponding relations in fuzzy metric space. All the results of this paper are new.
Keywords: fuzzy metric space, compatibility, semi compatibility, implicit relation. 2000 AMS Mathematics Subject
Classification: 47H10, 54H25
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1. INTRODUCTION
Semicompatible maps in d-topological space introduced by
Cho et al.[2].They define a pair of self maps to be compatible
if conditions (i) (,ST SyTy=implies that ; (ii) for sequence
{STyTSy=}nxin Xand xX∈,whenever {}nSxx→,{}nTxx→,then
as ,hold. However, in fuzzy metric space (ii) implies (i), taking
nSTxTx→n→∞nxy=for all andnxTySy==.So we define a
semecompatible pair of self maps in fuzzy metric space by
condition (ii) only. Saliga [9] and Sharma et. al [10] proved
some interesting fixed point results using implicit real
functions and semicompatibility in d-complete topological
spaces. Recently, Popa in [8] used the family of implicit real
functions to find the fixed points of two pairs of
semicompatible maps in a d-complete topological space. Here,
denotes the family of all real continuous functions 4F4F
()4:FRR+→ satisfying the following properties.
(i) There exists such that for every with or we have.
1h≥0,0uv≥≥(),,,0Fuvuv≥(),,,0,Fuvvu≥uhv≥
(ii) , for all . (),,0,00Fuu<0u>
Jungck and Rhoades [6], Dhage [3] termed a pair of self maps
to be coincidentally commuting or equivalently weak
compatible if they commute at their coincidence points. This
concept is most general among all the commutativity concepts
in this field as every pair of commuting self maps is R-weakly
commuting, each pair of R-weakly commuting self maps is
compatible and each pair of compatible self maps is weak
compatible but reverse is not always true. Similarly, every
semicompatible pair of self maps is weak compatible but the
reverse is not always true. The main object of this paper is to
obtain some fixed point theorems in the setting of fuzzy metric
space using weak compatibility, semicompatibility, and an
implicit relation.
2. PRELIMINARIES
Definition 2.1. A binary operation is called a continuous t-
norm if 2*:[0,1][0,1]→
([0,1],∗) is an abelian topological monoid with unit 1 such
that a∗b ≤ c∗d whenever
a ≤ c and b ≤ d for all a,b, c, and [0,1]d∈.
Examples of t-norm are a∗b = ab and a∗b = min{a,b}.
Definition 2.2 (Kramosil andMich´alek [7]). The 3-tuple
(X,M,∗) is called a fuzzy metric
space if X is an arbitrary set, ∗ is a continuous t-norm, andM
is a fuzzy set in 2[0,)X×∞
satisfying the following conditions for all ,,xyzX∈ and s,t > 0:
(FM-1) M(x, y,0) = 0;
(FM-2) M(x, y, t) = 1, for all t > 0 if and only if x = y;
(FM-3) M(x, y, t) =M(y,x, t);
(FM-4) M(x, y, t)∗M(y,z, s) ≥M(x,z, t +s);
(FM-5) M(x, y,·) : [0,∞)→[0,1] is left continuous.
Note that M(x, y, t) can be thought of as the degree of nearness
between x and y with
respect to t. We identify x = y with M(x, y, t) = 1 for all t > 0.
The following example shows that every metric space induces
a fuzzy metric space.

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Example 2.3 (George and Veeramani [4]). Let (X,d) be a
metric space. Define a∗b =
min{a,b} and for all ,,abX∈
()()(),,,0,,,00.,tMxyttMxytdxy=∀>+
Then (X,M,∗) is a fuzzy metric space. It is called the fuzzy
metric space induced by the metric d.
Lemma 2.4 (Grabiec [5]). For all ,xyX∈, M(x, y,·) is a
nondecreasing function.
Definition 2.5 (Grabiec [5]). Let (X,M,∗) be a fuzzy metric
space. A sequence {xn} in
X is said to be convergent to a point x _ X if limn→∞M(xn,x,
t) = 1 for all t > 0. Further,
the sequence {}nx is said to be a Cauchy sequence in X if
()lim,,1nnnpMxxt→∞+= for all
t > 0 and p > 0. The space is said to be complete if every
Cauchy sequence in it convergesto a point of it.
Remark 2.6. Since ∗ is continuous, it follows from (FM-4)
that the limit of a sequence in a fuzzy metric space is unique.
In this paper, (X,M,∗) is considered to be the fuzzy metric
space with condition
(FM-6) limt→∞M(x, y, t) = 1, for all ,xyX∈
Lemma 2.7 (Cho [1]). Let {yn} be a sequence in a fuzzy metric
space (X,M,∗) with the condition (FM-6). If there exists a
numbe ()0,1k∈ such that M(yn+2, yn+1,kt) ≥M(yn+1, yn, t),
for all t > 0, then {yn} is a Cauchy sequence in X.
Lemma 2.8. Let A and B be two self-maps on a complete fuzzy
metric space (X,M,∗) such that for some , for all (0,1k∈ ,xyX∈
and t > 0, M(Ax,By,kt) ≥ Min_M(x, y, t),M(Ax,x, t)_. (2.2)
Then A and B have a unique common fixed point in X.
Proof. Let . Taking pX∈0xp=, define sequence {}nx in X by
22nnAxx +=and 2122nnBxx+= . By taking 22,nnxxyx+==and
22,n xxyx−==, respectively, in the contractive condition, we
obtain that ()(11,,,,nnnn MxxktMxxt+−≥, 0,t∀>∀ (2.3)
Therefore by Lemma 2.7, {}nxis a Cauchy sequence in X,
which is complete. Hence, {}nxconverges to some u in X.
Taking 2nxx= and y = u and letting n→∞ in the contractive
condition, we get Bu = u. Similarly, by putting x = u and
21nyx+=, we get Au = u. Therefore, u is the common fixed
point of the maps A and B. The uniqueness of the common
fixed point follows from the contractive condition. _
Definition 2.9. Let A and S be mappings from a fuzzy metric
space (X,M,∗) into itself.
Themappings are said to be weak compatible if they commute
at their coincidence points, that is, Ax = Sx implies that ASx =
SAx.
space (X,M,∗) into itself.
Then the mappings are said to be compatible if
(lim,,1,nnnMASxSAxt→∞= 0t∀> (2.4) whenever {}nx is a
sequence in X such that limlim.nnnnAxSxxX→∞→∞== (2.5)
Proposition 2.11 [12]. Self-mappings A and S of a fuzzy
metric space (X,M,∗) are compatible, then they are weak
compatible. The converse is not true.
space (X,M,∗) into itself. Then the mappings are said to be
semicompatible if ()lim,,1,nnMASxSxt→∞= 0t∀> (2.6)
whenever {}nx is a sequence in X such that
limlim.nnnnAxSxxX→∞→∞== (2.7)
2.1. A Class of Implicit Relation.
Let +Rbe the set of all non-negative real numbers. Let Φ be
the collection of all function satisfying: ()++→RR3:φ
(a) φ is lower semcontinuous in each coordinate variable.
(b) Let ()vvuu,,φ≥ or (vuvu,, φ≥.Then , for ,where
kvu≥+∈Rvu,()11,1,1>=kφ.
(c) ()uuuu>,,φ, for . {}0−∈+Ru
3. MAIN RESULTS
3.1. Theorem: Let A,B,S and T be self mappings of a
complete Fuzzy metric space satisfying that
(∗,,MX()()XTXA⊆, (1)
()()XSXB⊆ the pair ( is semicompatible and SA,()TB, is weak
compatible; (2)
one of A or is continuous. (3)
S
for some Φ∈φthere exists such that for all (1,0∈k Xyx∈,and
0>t ()()()(){}ktTyByMtSxAxMtTySxMktbyAxM,,,,,,,,min,,≥ (4)
Then A,B,S and T have a unique common fixed point in X.
Proof: Let be any arbitrary point as Xx∈0()()XTXA⊆ and
()(XSXB⊆ there exists Xxx∈21, such that
2110,SxBxTxAx==.Inductively construct sequences { and
}ny{}nx in X such that

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22122212212,+++++====nnnnnnSxBxyTxAxy for
,...2,1,0=nNow using (4) with we get 122,+==nnxyxx
()()()(){}ktTxBxMtSxAxMtTxSxMktBxAxMnnnnnnnn,,,,,,,,min,,
121222122122++++≥
That is
()()()(){}ktyyMtyyMtyyMktyyMnnnnnnnn,,,,,,,,min,,221221212
22212++++++≥
()(tyyMktyyMnnnn,,,,2122212+++≥
Similarly, 322232221222,++++++====nnnnnnSxBxyTxAxy
using 2212,++==nnxyxx
()(tyyMktyyMnnnn,,,,22123222++++≥
Thus for any n and t we have
()(tyyMktyyMnnnn,,,,11−+≥
Hence by Lemma [2.7] ,{ is a Cauchy sequence in X which is
complete.Therefore converges to , its subsequences
}ny{}nyXu∈{}{}{}{}122122,,,++nnnnTxSxBxAx also
converges to that is u
{}uAxn→2 , { }uBxn→+12
{}uSxn→2 , { (5) }uTxn→+12
Case I : S is continuous. In this case , we have
SuSAxn→2 , (6) SuxSn→22
The semicompatibility of the pair ()SA, gives
SuASxnn=∞→2lim (7) Step 1: By putting in (4), we obtain
122,+==nnxySxx
()()()(){}221221222121,,min,,,,,,,,nnnnnnnnMASxBxktMSSxTx
tMASxSSxtMBxTxkt+++≥
Letting using (5),(6) and (7) and the continuity of the norm * ,
we have n→∞
()()()(){},,min,,,,,,,,MSuuktMSuutMSuSutMuukt≥
that is
()(){},,min,,,1,1MSuuktMSuut≥
(),,1MSuukt≥
It is non decreasing
(),,1,0MSuuktt≥>
Therefore
.Suu= (8)
Step 2: By putting 21,nxuyx+== in (4)
We obtain that
()()()(){}21212121,,min,,,,,,,,nnn
MAuBxktMSuTxtMAuSutMBxTxkt+++≥
Taking limit as and using (5) and (8) n→∞
()(){},,min1,,,,1MAuuktMAuut≥
(),,1MAuut≥ for 0t>
Therefore
Auu=
Hence . (9) AuuSu== Step 3:
As ()()AXTX⊆ there exists wX∈such that AuSuuTw===.
By putting 2,nxxyw== in (4)
We obtain
()()()(){}2222,,min,,,,,,,,nnnnMAxBwktMSxTwtMAxSxtMBwT
wkt≥
Taking limit as and using (5) we get n→∞
()(){},,min1,1,,,MuBwktMBwukt≥
We have for all (),,MuBukt≥ 0t>
Hence (),,MuBut=
Thus u Bw=
Therefore BwTwu==
Since is weak compatible. (,BT
We get TBwBTw=
that is (10) .BuTu=
Step 4:
By putting ,xuyu==in (4),(9) and (10)
We obtain that
()()()(){},,min,,,,,,,,MAuBuktMSuTutMAuSutMBuTukt≥
that is ()(){},,min,,,1,1MAuBuktMAuBut≥
(),,MAuBut≥ for all 0t>
Thus (),,MAuBut=
We have .AuBu=
Therefore uA. uSuBuTu====
That is u is a common fixed point of ,,,.ABST Case II : A is
continuous. We have the semicompatibility of the pair
2nASxAu→(),AS gives. 2nASxSu→
By uniqueness of limit in fuzzy metric space we obtain that
.AuSu=
Step 5:
By putting 21,nxuyx+== in (4) we obtain that
()()()(){}21212121,,,,,,,,,,nnn
MAuBxktminMSuTxtMAuSutMBxTxkt+++≥
Taking limit as and using (5) and (8) we get n→∞
()(){},,min1,,,,1MAuuktMAuut≥
We have for all (),,1MAuut≥0t>
Which gives .uAu=
Uniqueness:
Let be another common fixed point of z,,,ABST.
Then zAzBzSzTz==== putting xu= and yz= in (4) we get
()()()(){},,min,,,,,,,,MAuBzktMSuTztMAuSutMBzTzkt≥
that is
()(){},,,,,1,1MuzktMuzt≥
Therefore we have (),,1Muzt≥ for all 0t>
Hence (),,1Muzt=
That isuz. =
Therefore u is the unique common fixed point of the self maps
,,,.ABST
REFERENCES
[1] Y. J. Cho, Fixed points in fuzzy metric spaces, J. Fuzzy
Math. 5 (1997), no. 4, 949–962.
[2] Y. J. Cho, B. K. Sharma, and D. R. Sahu, Semi-
compatibility and fixed points, Math. Japon. 42(1995), no. 1,
91–98.
[3] B. C. Dhage, On common fixed points of pairs of
coincidentally commuting mappings in D-metric spaces,
Indian J. Pure Appl. Math. 30 (1999), no. 4, 395–406.
[4] A. George and P. Veeramani, On some results in fuzzy
metric spaces, Fuzzy Sets and Systems 64(1994), no. 3, 395–
399.
[5] M. Grabiec, Fixed points in fuzzy metric spaces, Fuzzy
Sets and Systems 27 (1988), no. 3, 385–389.

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[6] G. Jungck and B. E. Rhoades, Fixed points for set valued
functions without continuity, Indian J.Pure Appl. Math. 29
(1998), no. 3, 227–238.
[7] I. Kramosil and J.Mich´alek, Fuzzy metrics and statistical
metric spaces, Kybernetika (Prague) 11(1975), no. 5, 336–
344.
[8] V. Popa, Fixed points for non-surjective expansion
mappings satisfying an implicit relation, Bul.S¸ tiint¸.Univ.
Baia Mare Ser. B Fasc. Mat.-Inform. 18 (2002), no. 1, 105–
108.
[9] L.M. Saliga, Fixed point theorems for non-self maps in d-
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(1996), no. 1, 103–110.
[10] B. K. Sharma, D. R. Sahu, M. Bounias, and A. Bonaly,
Fixed points for non-surjective expansion mappings, Int. J.
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[11] B. Singh and S. Jain, Semi-compatibility and fixed point
theorems in Menger space, Journal of the Chungcheong
Mathematical Society 17 (2004), no. 1, 1–17.

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