Some Fixed Point Theorem for Generalized ( , ) - Contractive Mappings in Strong M- Fuzzy Metric Spaces 1 M. Jeyaraman, 2 S. Sowndrarajan and 3 D. Poovaragavan 1 PG and Research Department of Mathematics, Raja Doraisingam Govt. Arts College, Sivaganga, Tamil Nadu, India. [email protected]2 PG and Research Department of Mathematics, Raja Doraisingam Govt. Arts College, Sivaganga, Tamil Nadu, India. [email protected]3 Department of Mathematics, Govt. Arts College for Women, Sivaganga, Tamil Nadu, India. [email protected]Abstract In this paper, we introduce generalized ( , ) - contractive mapping in strong M -fuzzy metric spaces and prove fixed point theorems to this class of maps. Further we introduce a generalized ( , )- contractive mapping f with respect a mapping g and prove common fixed point theorems. We provide examples in support of our results. Mathematics Subject Classification: 47H10, 54H25. Key Words:Strong fuzzy metric space, generalized ( , ) contractive mappings, generalized fuzzy metric spaces. International Journal of Pure and Applied Mathematics Volume 119 No. 12 2018, 3119-3131 ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu Special Issue ijpam.eu 3119
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International Journal of Pure and Applied MathematicsVolume 119 No. 12 2018, 3119-3131ISSN: 1314-3395 (on-line version)url: http://www.ijpam.euSpecial Issue ijpam.eu
3119
1. Introduction
The concept of fuzzy metric space was introduced in different ways by various
authors [3,4] and the fixed point theory in these spaces has been intensively
studied. The notion of fuzzy metric space, introduced by Kramosil and Michalek
[4] was modified by George and Veeramani [2] and that obtained a Hausdorff
topology for this section of fuzzy metric spaces. Gregori and Sapena [3] have
introduced a kind of contractive mappings in fuzzy metric spaces in the sense of
George and Veeramani and proved a fuzzy Banach contraction theorem using a
strong condition for completeness, which is Completeness in the sense of
Grabiec, or G-completeness. In 2010, Gregori et al introduced Strong fuzzy
metric space and proved a fixed point theorem. Motivated from Azizollah et al
[1] we have developed this paper. In 2006, Sedghi and Shobe [6] defined ℳ- fuzzy
metric spaces and proved a common fixed point theorem for weakly compatible
mapping in this space. In this paper, we first introduce generalized contractive
conditions of maps and also prove some fixed point theorems for generalized
( , )-contractive mapping in strong M -fuzzy metric spaces.
2. Preliminaries
Definition 2.1
A binary operation : [0,1] ×[0,1] → [0,1] is said to be a continuous t-norm if
it satisfies the following conditions :
i) is associative and commutative,
ii) is continuous,
iii) a 1 = a for all a [0, 1],
iv) a b ≤ c d whenever a ≤ c and b ≤ d for all a, b, c, d [0, 1].
Definition 2.2
A 3-tuple (X, ℳ, ) is called ℳ-fuzzy metric space if X is an arbitrary non
empty set, is a continuous t-norm and ℳ is a fuzzy set on X3 × (0,∞),
satisfying the following conditions: for each x, y, z, a X and t, s > 0.
(ℳ1) ℳ(x, y, z, t) > 0,
(ℳ 2) ℳ(x, y, z, t) = 1 if and only if x = y = z,
(ℳ3) ℳ (x, y, z, t) = ℳ(p{x, y, z}, t), when p is the permutation function.
(ℳ4) ℳ(x, y, a, t) ℳ (a, z, z, s) ≤ ℳ (x, y, z, t + s),
(ℳ5) ℳ(x, y, z, . ) : (0, ∞) → [0,1] is continuous,
(ℳ6) ℳ(x, y, z, t) = 1 for all x, y, z X.
Definition 2.3
Let (X, ℳ, ) be a ℳ-fuzzy metric space. The ℳ -fuzzy metric is said to be
strong (non-Archimedean) if it satisfies
(ℳ4') :ℳ(x, y, z, t) ≥ ℳ(x, y, a, t) ℳ(a, z, z, t), for each x, y, z X and each
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t > 0.
Remark 2.4
Axiom (ℳ4' ) cannot replace axiom (ℳ4) in the above definition of fuzzy
metric, since in that case, ℳ could not be a fuzzy metric on X .
Note that it is possible to define a strong fuzzy metric by replacing (ℳ4) by
(ℳ4') and demanding in (ℳ5) that the function ℳ(x, y, z, . ) be an increasing
continuous function on t, for each x, y, z X. (In fact, in such a case we have
that
ℳ(x, y, z, t + s) ≥ ℳ(x, y, a, t + s) ℳ(a, z, z, t + s) ≥ ℳ(x, y, a, t) ℳ(a, z, z,
s)).
Remark 2.5
Not every ℳ-fuzzy metric space is a strong fuzzy metric space.
Definition 2.6
Let (X, ℳ, ) be a ℳ-fuzzy metric space.
1. A sequence {xn} in X is said to be convergent to a point x X if ℳ(xn, x, x,
t)= 1 for all t > 0.
2. A sequence {xn} in X is called a Cauchy sequence if, for each 0 < ϵ < 1 and t >
0, there exits n0 ℕ such that ℳ(xn, xm, xm, t) > 1-ϵ for each n, m ≥ n0.
3. A fuzzy metric space in which every Cauchy sequence is convergent is said to
be complete.
4. A fuzzy metric space in which every sequence has a convergent subsequence is
said to be compact.
3. Main Results
Definition 3.1
Let ψ: (0, 1] → [1, ∞) be a function which satisfies the following conditions.
(i). ψ is continuous and non-increasing, and
(ii). ψ(x) = 1 if and only if x = 1.
We denote by the class of all functions which satisfies the above conditions.
Note that in fact the map ψ : (0, 1] → [1, ∞) defined by ψ(t) = is in .
Definition 3.2
Let φ : (0, 1] × (0, 1] → (0, 1] be a function which satisfies the following
conditions.
(i). φ is upper semi continuous and non-decreasing, and
(ii). φ (s, t) = 1 if and only if s = t = 1.
We denote by the class of all functions which satisfies the above conditions.
Note that in fact the map φ : (0, 1] × (0, 1] → (0, 1] defined by Φ(s, t) =
st is in .
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Now, we introduce generalized ( , ) - contractive mapping in fuzzy metric
space.
Definition 3.3
Let (X, ℳ, ) be a ℳ - fuzzy metric space. We say that a mapping T: X→ X is
generalized ( , ) - contractive mapping if there exists ( , ) × such
that,
(ℳ(Tx, Ty, Ty, t)) ≤ ψ (N(x, y, y, t) φ ( N' (x, y, y, t) , N'' (x, y, y, t)), (3.3.1)
for all x, y X and for all t > 0, where
N(x, y, y, t) = min {ℳ(x, y, y, t), ℳ(x, Tx, Tx , t), ℳ(y, Ty ,Ty ,t)},
N' (x, y, y, t) = min {ℳ (x, y, y, t), ℳ(x, Tx, Tx, t), ℳ(x, Ty, Ty, t)},
N'' (x, y, y, t) = min {ℳ (x, y, y, t),ℳ(y, Ty ,Ty, t), ℳ(y, Tx, Tx, t)}.
Definition 3.4
Let (X, ℳ, ) be a ℳ- fuzzy metric space and f, g be two self mappings on X. A
mapping f is said to be generalised ( , )- contractive with respect to g if
there exists ( , ) × such that,
(ℳ(fx, gy, gy, t)) ≤ ψ (N(x, y, y, t) φ( N' (x, y, y, t) , N'' (x, y, y, t)), (3.4.1)
for all x, y X and for all t > 0, where
N(x, y, y, t) = min {ℳ(x, y, y, t), ℳ(x, fx, fx, t), ℳ(y, gy, gy, t)},
N' (x, y, y, t) = min {ℳ(x, y, y, t), M(x, fx, fx ,t), ℳ(x, gy, gy, t)},
N'' (x, y, y, t) = min {ℳ(x, y, y, t), ℳ(y, Ty ,Ty, t), ℳ(y, Tx, Tx, t)}.
The following propositions are useful to prove our main results.
Proposition 3.5
Let (X, ℳ, ) be a strong ℳ-fuzzy metric space. Let T : X → X be a
generalized ( , ) - contractive mapping. Fix x0 X. Define a sequence {xn} in
X by xn+1 = Txn for n = 0, 1, 2,… If (xn, xn+1, xn+1, t) = 1 for all t > 0 then
{xn} is a Cauchy sequence.
Proof
Since the mapping T is generalized ( , ) - contractive there exist ( , ) ×
such that (ℳ(Tx, Ty, Ty, t) ≤ ψ(N(x, y, y, t) φ( N' (x, y, y, t) , N'' (x, y, y, t))
for all x, y X.
Suppose that sequence {xn} is not a Cauchy sequence. Then there exist ϵ (0,
1) and t0 > 0 such that for all k ≥1, there are positive integers m(k), n(k) ℕ with
m(k) > n(k) ≥ k and
ℳ( xn(k), xm(k), xm(k), t0) ≤ 1- ϵ . (3.5.1)
We assume that m(k) is the least integer exceeding n(k) and satisfying the above
inequality, that is equivalently,
ℳ(xn(k), xm(k)-1,xm(k)-1 , t0) > 1- ϵ and ℳ(xn(k), xm(k), xm(k), t0) ≤ 1- ϵ . Now, we have
Since is continuous and is upper semi continuous with respect to both
variables on taking limit superior in (3.9.11), we get, (lt) ≤ (lt) (lt, lt). (3.9.12)
Which implies (lt, lt) = 1. By property of , lt = 1. Hence by {xn} is a Cauchy
sequence. Since (X, ℳ, ) is a complete strong fuzzy metric space there exist u
X such that xn → u. Without loss of generality we assume that f is continuous.
As x2n-1 → u as n →∞, the continuity of f implies that fx2n-1 = x2n → fu as n
→∞, by uniqueness of the limit, we obtain fu = u. Therefore u F (f) = F (g).
We will show u is unique.
Suppose that v F(f, g) = F (f ) = F (g). For each t > 0, we have
ψ (M( u, v, v, t)) = ψ(ℳ(fu, gv, gv, t)) (3.9.13)
≤ ψ (N( u, v, v, t)) φ (N'( u, v, v, t) , (N''( u, v, v, t)),
where N( u, v, v, t) = min {ℳ ( u, v, v, t), ℳ(u, fu, fu, t), ℳ( v, gv, gv, t)}
= min {ℳ( u, v, v, t), 1, 1 } = ℳ( u, v, v, t), (3.9.14)
N'( u, v, v, t) = min {ℳ ( u, v, v, t), ℳ(u, fu, fu, t), ℳ( u, gv, gv, t)
= min {ℳ( u, v, v, t), 1, ℳ ( u, v, v, t) } = ℳ( u, v, v, t}, (3.9.15)
N''( u, v, v, t) = min {ℳ ( u, v, v, t), ℳ( v, gv, gv, t), ℳ(v, fu, fu, t)}
= min {ℳ( u, v, v, t), 1, ℳ( u, v, v, t)} = ℳ( u, v, v, t). (3.9.16)
From (3.9.13), (3.9.14), (3.9.15) and (3.9.16), we have
ψ (ℳ( u, v, v, t)) = ψ(ℳ( fu, gv, gv, t)) ≤ ψ (ℳ( u, v, v, t) φ (ℳ( u, v, v, t) , ℳ( u, v, v, t)).
Thus, φ (ℳ( u, v, v, t) , ℳ( u, v, v, t)) = 1, which implies ℳ ( u, v, v, t) = 1,
Therefore u = v.
Example 3.10
Let X= [0, ∞) and f, g :X →X defined by fx = , and g(x) = . Let M be a strong
fuzzy metric space define by ℳ(x, y, y, t) = ( ) D(x, y, z), where D(x, y, z) = |x-y| + |y-z|
+ |x-z|.
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Let : (0, 1]→ [1, ∞) and : (0, 1]× (0, 1]→(0, 1] define by (t) = and (s, t) = st.
We prove that f is ( , ) - generalized contractive mapping with respect to g.
Thus by Theorem 3.9 we conclude that f and g have a unique common fixed
point in X, in fact 0 is a common fixed point for f and g in X.
References
[1] Azizollah Aziz, Mohammed Moosaei and Gita Zare, Fixed point theorems for almost generalized C-contractive mappings in ordered complete metric spaces, fixed point theory and application, springer.
[2] A. George and P. Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets and Systems, 64(1994), 395-399.
[3] V. Gregori and A. Sapena, On fixed point theorems in fuzzy metric spaces, Fuzzy Sets andSystems, 125(2002), 245-252.
[4] I. Kramosil and J. Michalek, Fuzzy metric and statistical metric spaces, Kyber-netica, 11(1975), 326-334.
[5] R. Muthuraj and R.Pandiselvi, Common fixed point theorem for compatible mapping of type (P-1) and type (P-2) in ℳ-fuzzy metric spaces, Arab journal of mathematics and Mathematical sciences, 3(1) (2013), 1-10.
[6] S.Sedghi and N. Shobe, Fixed point theorem in ℳ-fuzzy metric spaces with property (E), Advances in Fuzzy Mathematics, 1(1) (2006), 55-65.
[7] L.A. Zadeh, Fuzzy sets, Information and Control, 8, (1965), 338-353.
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