Our main purpose in this paper is to introduce the notion of φ-contractive mapping in fuzzy metric spaces and to present two new results on the existence and the approximation of fixed point of nonlinear contractions mappings and semigroups in fuzzy metric spaces. These results are of interest in view of analogous results in metric spaces (see for exampl

Abdelhak Benbrik

Abderrahim Mbarki

Abedelmalek Ouahab

Tahiri Ismail

Wafa Ahid

CiteSeerX

Applied Mathematical Sciences, Vol. 7, 2013, no. 7, 307 - 313
Nonlinear Contractions and Semigroups
in Complete Fuzzy Metric Spaces
Abderrahim Mbarki
National school of Applied Sciences
P.O. Box 669, Oujda University Morocco
MATSI Laboratory
ambarki@ensa.univ-oujda.ac.ma
Abdelhak Benbrik
Department of Mathematics
Oujda university, 60000 Oujda Morocco
MATSI Laboratory
benbrik@sciences.univ-oujda.ac.ma
Abedelmalek Ouahab
Department of Mathematics
Oujda university, 60000 Oujda Morocco
MATSI Laboratory
ouahab05@yahoo.fr
Wafa Ahid
Department of Mathematics
Oujda university, 60000 Oujda Morocco
MATSI Laboratory
ahid.wafa@gmail.com
Tahiri Ismail
Department of Mathematics
Oujda university, 60000 Oujda Morocco
MATSI Laboratory
tahiri.ismail@mail.com
308 A. Mbarki, A. Benbrik, A. Ouahab, A. Wafa and T. Ismail
Abstract
Our main purpose in this paper is to introduce the notion of φ-
contractive mapping in fuzzy metric spaces and to present two new re-
sults on the existence and the approximation of fixed point of nonlinear
contractions mappings and semigroups in fuzzy metric spaces. These
results are of interest in view of analogous results in metric spaces (see
for example [1], [7])
Mathematics Subject Classification: Primary 47H10, 54H25
Keywords: Fixed point, fuzzy metric, left reversible semigroup, nonlinear
contractive conditions
1 Introduction
Our terminology and notation for fuzzy metric spaces conform of that George
et al. [5, 6]. Recently Gregori et al. [4] have showed that the study of the
intuitionistic fuzzy metric space (IFMS) (X, M, N, ∗,♦) can be reduced to the
study of the fuzzy metric space (FMS) (X, M, ∗). More exactly, the topology
τ(MN) of an IFMS (X, M, N, ∗,♦) coincides with the topology τ(M) generated
by the FMS (X, M, ∗), which has as a base the family of open sets. So, our
study is limited to FMSs.
Definition 1.1 A subset B of FMS (X, M, ∗) is called fuzzy bounded if for
each t > 0 there exists λ ∈ (0, 1) such that M(x, y, t) ≥ λ for all x, y ∈ B.
Remark 1.2 Let (X, d) be a metric space. Denote a ∗ b = min{a, b} for all
a, b ∈ [0, 1] and let Md be a fuzzy set on X2 × (0,∞) defined as follows:
Md(x, y, t) =
t
t + d(x, y)
.
It easy to check is that (X, Md, ∗) is a fuzzy metric space, and
B(x, r, t) = {y ∈: d(x, y) < rt
1 − r}
So (X, Md, ∗) is a complete fuzzy metric space if and only if (X, d) is a complete
metric space. Moreover, a nonempty subset A of (X, Md, ∗) is fuzzy bounded
if and only if A is bounded in (X, d)
Lemma 1.3 Every convergence sequence of a FMS is fuzzy bounded.
Nonlinear contractions in complete fuzzy metric spaces 309
Proof. Let {xn} be a convergence sequence of a FMS (X, M, ∗) then there
exists x ∈ X such that xn → x as n → ∞, which implies, for each t > 0 that
M(xn, x,
t
2
) → 1, then there exists N ∈ IN such that M(xn, x, t2) > 12 , ∀n ≥ N .
It follows
M(xn, x,
t
2
) ≥ R, ∀n ∈ IN
where R = min
{
1
2
, min{n≤N} M(xn, x, t2)
}
. Then
M(xn, xp, t) ≥ M(xn, x, t
2
) ∗ M(xp, x, t
2
) ≥ R ∗ R.
For all p, n ∈ IN. This completes the proof of the Lemma
Definition 1.4 Let (X, M, ∗) be a FMS. Let φ : [0,∞) → [0,∞) is an
upper semi-continuous from the right function such that φ(0) = 0 and φ(t) < t
for t > 0.
We will say the sequence {xn} in X is fuzzy φ-contractive if for each t > 0,
t
M(xn+2, xn+1, t)
− t ≤ φ
( t
M(xn+1, xn, t)
− t
)
for all n ∈ IN.
A selfmap f on (X, M, ∗) is called fuzzy φ-contractive if for each t > 0,
t
M(f(x), f(y), t)
− t ≤ φ
( t
M(x, y, t)
− t
)
for all x, y ∈ X.
Remark 1.5 It is not hard to prove that every φ-contractive selfmap f on
a metric (X, d) is fuzzy φ-contractive on (X, Md, ∗) ( ∗ a t-norm such that
(X, Md, ∗) is a fuzzy metric space). As it very easy to check that a fuzzy
contractive mapping of contractive constant k is a φ-contractive with r →
φ(r) = rk
Lemma 1.6 Every fuzzy bounded φ-contractive sequence {xn} of a FMS
(X, M, ∗) is a fuzzy Cauchy sequence.
Proof. Let {xn} be a fuzzy bounded contractive sequence. Since {xn} is
bounded then for t > 0 there exists λ > 0 such that M(xn, xp, t) ≥ λ. On the
other hand, since {xn} is φ-contractive then for n > p ≥ 0 we have
t
M(xn, xp, t)
− t ≤ φp
( t
M(xn−p, x0, t)
− t
)
310 A. Mbarki, A. Benbrik, A. Ouahab, A. Wafa and T. Ismail
it follows
t
M(xn, xp, t)
− t ≤ φp
( t
λ
− t
)
Letting p → ∞. Then we obtain
t
M(xn, xp, t)
− t → 0
which implies that {xn} is a Cauchy sequence.
2 Main Results
Theorem 2.1 Let (X, M, ∗) be a complete fuzzy metric space and f be a φ-
contractive self-map on X. If there exists x0 such that the sequence {fn(x0)}
is fuzzy bounded. Then f has a unique fixed point in X. Furthermore, the
Picard iterates associate to each point of X converge to the fixed point
Proof. Let x0 ∈ X, such that xn = fn(x0), n ∈ IN is bounded. Since f is
φ-contractive therefore {xn} is fuzzy bounded φ-contractive sequence. Hence
from the Lemma 1.2 {xn} is a Cauchy sequence in a complete fuzzy metric
space. Then there exists a point x ∈ X such that xn → x as n → ∞. Now we
shall show that fx = x.
Since f is φ-contractive then
t
M(f(xn), f(x), t)
− t ≤ φ
( t
M(xn, x, t)
− t
)
→ 0
as n → ∞, then limn→∞ M(f(xn), f(x), t) = 1 for each t > 0. Therefore,
limn→∞ xn = f(x) that is f(x) = x. Hence f has a fixed point x ∈ X. Next,
let z ∈ X then
t
M(fn(z), x, t)
− t = t
M(fn(z), fn(x), t)
− t ≤ φn
( t
M(z, x, t)
− t
)
→ 0
as n → ∞, then limn→∞ M(fn(z), x, t) = 1 for each t > 0. Therefore, limn→∞ fn(z) =
x. We claim that x is the unique fixed point of f . For this suppose that y
(x 
= y ) is another fixed point of f in X. Then
t
M(x, y, t)
− t = t
M(f(x), f(y), t)
− t ≤ φn
( t
M(x, y, t)
− t
)
Letting n → ∞ we obtain M(x, y, t) = 1 for each t > 0, that is x = y, which
is a contradiction. This completes the proof of the theorem.
As a consequence of Remark 1.2, Lemma 1.1 and Theorem 1.1, we can obtain
the following
Nonlinear contractions in complete fuzzy metric spaces 311
Corollary 2.2 [1] Suppose that T is a φ-contractive selfmap on a complete
metric space (X, d). Then T has a unique fixed point z and, moreover, for any
x ∈ X, the sequence of iterates {T n(x)} converges to z.
Corollary 2.3 [3] Let (X, M, ∗) be a complete fuzzy metric space in which
fuzzy contractive sequences are Cauchy. Let f : X → X be an fuzzy contractive
mapping. Then f has a unique fixed point
3 Common Fixed Point Theorem
Let S be a semigroup of selfmaps on (X, M, ∗). For any x ∈ X, the orbit of x
under S starting at x is the set O(x) defined to be {x} ∪ Sx, where Sx is the
set {g(x) : g ∈ S}. We say that S is left reversible if, for any f, g in S, there
are a, b such that fa = gb. It is obvious that left reversibility is equivalent to
the statement that any two right ideals of S have nonempty intersection.
The next theorem can be seen as fuzzy version of a result of Huang et al. [5]
on metric space
Theorem 3.1 Suppose S is a left reversible semigroup of φ-contractive self-
maps on a complete fuzzy metric space (X, M, ∗). If there exists a point x0 in
X such that O(x0) is fuzzy bounded, then S has a unique common fixed point
z and, moreover, for any f ∈ S and x ∈ X, the sequence of iterates {fn(x)}
converges to z.
Proof. It follows from the Theorem 1.1 that each f ∈ S has a unique fixed
point zf in X and for any x ∈ X, the sequence of iterates {fn(x)} converges to
zf . So, to complete the proof it suffices to show that zf = zg for any f, g ∈ S.
Let n be a positive integer, the left reversibility of S shows that are an and bn
in S such that fnan = g
nbn, and so
M(zf , zg, t) ≥ M(zf , fnan(x0), t
2
) ∗ M(zg, gnbn(x0), t
2
). (1)
Also, since f is φ-contractive we then have
t
M(fn(x0), fnan(x0), s)
− t ≤ φn
( t
M(x0, an(x0), s)
− t
)
Letting n → ∞ in the last inequality and using the fact that O(x0) is bounded
we obtain
M(fn(x0), f
nan(x0), s) → 1. (2)
Since
M(zf , f
nan(x0), t) ≥ M(zf , fn(x0), t
2
) ∗ M(fn(x0), fnan(x0), t
2
). (3)
312 A. Mbarki, A. Benbrik, A. Ouahab, A. Wafa and T. Ismail
Letting n → ∞ in (3) and using fn(x0) → zf , we get
fnan(x0) → zf .
Likewise, we also have gnbn(x0) → zg which implies that, as n → ∞ in (1) we
obtain
M(zf , zg, t) → 1
This completes the proof of the Theorem.
Corollary 3.2 [7] Suppose S is a left reversible semigroup of selfmaps on
a complete metric space (X, d) such that the following conditions (i) and (ii)
are satisfied
i. There is x0 in X, its orbit O(x0) is bounded;
ii. There exists a gauge function ϕ : [0,∞) → [0,∞) such that d(fx, fy) ≤
ϕ(d(x, y)) for any f in S and x, y in X.
Then S has a unique common fixed point z and, moreover, for any f ∈ S and
x ∈ X, the sequence of iterates {fn(x)} converges to z.
References
[1] D. W. Boyd and J. S. W. Wong, On nonlinear contractions, Proc. Amer.
Math. Soc. 20, (1969), 458- 464.
[2] T.H. Chang, Common fixed point theorems in Menger spaces, Bull. Inst.
Math. Acad. Sinica, 22 no. 1, (1994), 17-29.
[3] V. Gregori and A. Sapena, On fixed point theorems in fuzzy metric spaces,
Fuzzy Sets and Systemes 125, (2002), 245-253.
[4] V. Gregori, Romaguera S. and Veeramani P., A note on intuitionistic
fuzzy metric spaces, Chaos, Solitons and Fractals 28 no. 4, (2006), 902-
905.
[5] A. George and Veeramani P., On some results in fuzzy metric spaces,
Fuzzy Sets and Systems, 64 no 3, (1994 ), 395- 399.
[6] A. George and Veeramani P., On some results of analysis for fuzzy metric
spaces, Fuzzy Sets and Systems, 90 no. 3, (1997), 365- 368.
[7] Y. Y. Huang and C. C. Hong, A note on left reversible semigroups of
contractions, Far East J. Math. Sci.4 no. 1, (1996), 81-87.
Nonlinear contractions in complete fuzzy metric spaces 313
[8] B. Schweizer and A. Sklar, Probabilistic Metric Spaces, North-Holland
Series in Probability and Applied Mathematics.5 (1983).
Received: September, 2012


Nonlinear Contractions and Semigroups in Complete Fuzzy Metric Spaces

http://citeseerx.ist.psu.edu/viewdoc/download;jsessionid=2FD64B79C24EC8D763652BDA69B05F3C?doi=10.1.1.1047.9908&rep=rep1&type=pdf

Nonlinear Contractions and Semigroups in Complete Fuzzy Metric Spaces

Abstract

Similar works

Full text

Available Versions

CiteSeerX