NEW ZEALAND JOURNAL OF MATHEMATICS
Volume 35 (2006), 109–114
A FIXED POINT THEOREM IN THE MENGER
PROBABILISTIC METRIC SPACE
Abdolrahman Razani
(Received September 2004)
Abstract. In this article, a fixed point theorem in the Menger probabilistic
metric space is proved. In addition, the existence of a periodic point in this
space is shown. Finally, two questions would arise.
1. Introduction
The notion of a probabilistic metric space was introduced by Menger [4] in 1942.
Fixed point theory in this space is studied by many authors, for instance [2], [5],
[7] and so on. It is necessary to mention that fixed point theorems are main tools
for mathematicians to study the problem of existence of a solution for a system
of differential equations in probabilistic metric space, for instance see [2]. The
propose of this paper is to present a fixed point theorem in Menger probabilistic
metric space. Due to do this and for the sake of convenience, first, we recall some
definitions and notations in [1], [2], [3], [6] and [7].
Definition 1.1. A mapping F : R → R+ is called a distribution function if it is
nondecreasing and left-continuous with inft∈R F (t) = 0 and supt∈R F (t) = 1.
Let D+ be the set of all distribution functions F such that F (0) = 0 ( F is a
nondecreasing, left-continuous mapping from R into [0, 1] such that supx∈R F (x) =
1).
Definition 1.2. A probabilistic metric space (briefly, PM-space)
is an order pair
+
(S, F ) where S is a nonempty set and F : S × S → D F (p, q) is denoted by Fp,q
for every (p, q) ∈ S × S satisfies the following conditions:
1. Fu,v (x) = 1 for all x > 0 if and only if u = v (u, v ∈ S).
2. Fu,v (x) = Fv,u (x) for all u, v ∈ S and x ∈ R.
3. If Fu,v (x) = 1 and Fv,w (y) = 1, then Fu,w (x + y) = 1 for u, v, w ∈ S and
x, y ∈ R.
If only 1 and 2 hold, the ordered pair (S, F ) is said to be a probabilistic semimetric
space.
Definition 1.3. A mapping T : [0, 1] × [0, 1] → [0, 1] is called a triangular norm
(abbreviated, a t-norm) if the following conditions are satisfied:
1991 Mathematics Subject Classification 47H10, 54H25.
Key words and phrases: Contractive mapping, Fixed point, Menger space, Triangular norm.
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ABDOLRAHMAN RAZANI
(I) T (a, 1) = a for every a ∈ [0, 1],
(II) T (a, b) = T (b, a) for every a, b ∈ [0, 1],
(III) a ≥ b, c ≥ d → T (a, c) ≥ T (b, d) (a, b, c, d ∈ [0, 1]),
(IV) T a, T (b, c) = T T (a, b), c (a, b, c ∈ [0, 1]).
Example 1.4. The following are the four basic t-norms:
(I) The minimum t-norm, TM , is defined by
TM (x, y) = min(x, y),
(II) The product t-norm, TP , is defined by
TP (x, y) = x.y,
(III) The Lukasiewicz t-norm, TL , is defined by
TL (x, y) = max(x + y − 1, 0),
(IV) The weakest t-norm, the drastic product, TD , is defined by
min(x, y) if max(x,y)=1,
TD (x, y) =
0
otherwise.
As regards the pointwise ordering, we have the inequalities
TD < TL < TP < TM .
Definition 1.5. A Menger probabilistic metric space (briefly, Menger PM-space)
(see [6]) is a triple (S, F, T ), where (S, F ) is a probabilistic metric space, T is a
triangular norm (abbreviated t-norm) and the following inequality holds:
Fu,v (x + y) ≥ T (Fu,w (x), Fw,v (y)),
(1)
for all u, v, w ∈ S and every x > 0, y > 0.
Schweizer, Sklar and Thorp [7] proved that if (S, F, T ) is a Menger PM-space
with sup0<t<1 T (t, t) = 1, then (S, F, T ) is a Hausdorff topological space in the
topology τ induced by the family of (ε, λ)−neighborhoods
{Up (ε, λ) : p ∈ S, ε > 0, λ > 0},
where
Up (ε, λ) = {u ∈ S : Fu,p (ε) > 1 − λ}.
Definition 1.6. Let (S, F, T ) be a Menger PM-space with sup0<t<1 T (t, t) = 1.
(1) A sequence {un } in S is said to be τ −convergent to u ∈ S (we write un → u)
if for any given ε > 0 and λ > 0, there exists a positive integer N = N (ε, λ)
such that Fun ,u (ε) > 1 − λ whenever n ≥ N .
(2) A sequence {un } in S is called a τ −Cauchy sequence if for any ε > 0 and
λ > 0, there exists a positive integer N = N (ε, λ) such that Fun ,um (ε) > 1 − λ,
whenever n, m ≥ N .
(3) A Menger PM-space (S, F, T ) is said to be τ -complete if each τ -Cauchy sequence in S is τ -convergent to some point in S.
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A FIXED POINT THEOREM
The rest of the paper is organized as follows: In Section 2, a fixed point theorem
is given. Finally, the existence of a periodic point is proved in Section 3.
2. Fixed Point Under Contractive Map
In this section, the definition of contractive map is rewritten and an iterative
theorem is proved. This theorem shows that if we have the existence of a convergent
subsequence of an iterate sequence (of a contractive map) then we can prove the
existence of a fixed point. In order to do this, we recall the following definition:
Definition 2.1. Let (S, F, T ) be a Menger PM-space. We say the mapping f :
S → S is contractive, if for all u 6= v ∈ S,
Ff (u),f (v) (x) ≥ Fu,v (x),
(2)
for all every x > 0, but Ff (u),(v) 6= Fu,v .
Lemma 2.2. Let (S, F, T ) be a Menger PM-space, where T is a continuous t−norm.
Let (un , vn , xn ) be a sequence in S × S × R+ and (un , vn , xn ) → (u, v, x0 ), where
u, v ∈ S and x0 > 0. If Fu,v is continuous at x0 , then Fun ,vn (xn ) → Fu,v (x0 ).
Proof. Note that
lim inf Fun ,vn (xn ) ≥ lim inf T T Fun ,u (δ), Fu,v (xn − 2δ) , Fv,vn (δ)
n
n
= T T 1, lim inf Fu,v (xn − 2δ) , 1
n
=
lim inf Fu,v (xn − 2δ)
n
≥ Fu,v (x0 − 3δ)
since xn > x0 − δ eventually,
and so lim inf Fun ,vn (xn ) ≥ Fu,v (x−
0 ) since δ > 0 is arbitrary. By the same
n
+
lim supFun ,vn (xn ) ≤ Fu,v (x0 ). Since Fu,v is continuous at x0 , then
n
argument,
Fun ,vn (xn ) → Fu,v (x0 ).
Now we have the main theorem as follows:
Theorem 2.3. Let (S, F, T ) be a Menger PM-space, where T is a continuous
t−norm. If A is a contractive mapping of S into itself such that there exists a
point u ∈ S whose sequence of iterates (An (u)) contains a convergent subsequence
Ani (u) ; then ξ = limi→∞ Ani (u) ∈ S is a unique fixed point.
Proof. Suppose A(ξ) 6= ξ and consider the sequence Ani +1 (u) which, it can
easily be verified, converges to A(ξ).
Choose x to be such that
FA(ξ), A2 (ξ) (x) > Fξ,A(ξ) (x),
with FA(ξ), A2 (ξ) and Fξ,A(ξ) are both continuous at x. This is possible, because
there must be some value x1 > 0 with
FA(ξ), A2 (ξ) (x1 ) > Fξ,A(ξ) (x1 ),
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ABDOLRAHMAN RAZANI
and since FA(ξ),A2 (ξ) and Fξ,A(ξ) are both left continuous at x1 , we have
FA(ξ), A2 (ξ) (x) > Fξ,A(ξ) (x)
for all x in some interval. Each function has at most countably many discontinuities,
so a point can be found in this interval at which both are continuous.
Now, construct neighborhoods B1 , B2 of ξ, A(ξ), respectively, such that for some
ε > 0 and all u ∈ B1 , v ∈ B2 we have
FA(u), A(v) (x) > Fu,v (x) + ε.
For large enough i, we have
FAni +1 (u), Ani +2 (u) (x) > FAni (u), Ani +1 (u) (x) + ε.
Applying A a few more times, gives
FAni +1 (u), Ani+1 +1 (u) (x) > FAni (u), Ani +1 (u) (x) + ε,
and hence
FAni (u), Anj +1 (u) (x) > FAni (u), Ani +1 (u) (x) + (j − i)ε
for sufficiently large i and j. Letting j → ∞ gives the contradiction.
In order to prove the uniqueness of ξ, suppose there is a η 6= ξ with A(η) = η,
then it follows that
Fξ,η (x) = FA(ξ),A(η) (x) < Fξ,η (x),
which is contradiction. This proves the uniqueness and, thus, accomplishes the
proof of this theorem.
Theorem 2.3 implies some information on the convergence of a sequence of iterates.
Remark 2.4. Let all assumptions of Theorem 2.3 hold. If (An (u)), u ∈ S, contains
a convergent subsequence (Ani (u)), then limn→∞ An (u) exists and coincides with
the fixed point ξ.
Proof. We have limi→∞ Ani (u) = ξ. Given 1 > δ > 0 there exists, then, a positive
integers N0 such that i > N0 implies Fξ,Ani (u) (t) > 1 − δ. If m = ni + l (ni fixed, l
variable) is any positive integer > ni then
Fξ,Am (u) (x) = FAl (ξ),
which proves the above assertion.
Ani +l (u) (x)
> Fξ,Ani (u) (x) > 1 − δ,
3. Periodic Points
In this section, first, we define a periodic point or an eventually fixed point.
Then we prove the existence of a periodic point in the Menger PM-space. Finally,
two questions would arise.
Definition 3.1. Let (S, F, T ) be a Menger PM-space, and f is a self-mapping of
S. Then ξ is a periodic point or an eventually fixed point, if there exists a positive
integer k such that f k (ξ) = ξ.
A FIXED POINT THEOREM
113
Definition 3.2. Let (S, F, T ) be a Menger PM-space, a mapping f : S → S is
called locally contractive if and only if:
∀u∈S ∃0<λ<1 ∀p,q∈{v∈S:Fu,v (x)>1−λ} Fp,q (x) ≤ Ff (p),f (q) (x)
(3)
for all x > 0.
Definition 3.3. Let (S, F, T ) be a Menger PM-space. A mapping f : S → S is
called λ-uniformly locally contractive if and only if λ does not depend on u.
Theorem 3.4. Let (S, F, T ) be a Menger PM-space, with a continuous t−norm T
defined as T (a, b) = TM (a, b) for a, b ∈ [0, 1]. Suppose f is a λ-uniformly locally
contractive self-mapping of S such that
there exists a point u ∈ S whose sequence of iterates
(f n (u)) contains a convergent subsequence f ni (u) ,
(4)
then ξ = limi→∞ f ni (u) is a periodic point of f .
Proof. By the condition (4), there exists a positive integer N1 such that for i > N1
implies
(5)
Ff ni (u), ξ (x) > 1 − λ for all 0 < λ < 1 and x > 0.
Also f is λ-uniformly locally contractive, thus the last inequality implies
Ff ni +1 (u),f (ξ) (x) ≥ Ff ni (u),ξ (x)
and so Ff ni +1 (u),f (v) (x) > 1 − λ. After ni+1 − ni iterations we obtain:
Ff ni+1 (u),f ni+1 −ni (ξ) (x) > 1 − λ.
Note that
Fξ,f ni+1 −ni (ξ) (x) ≥ T Fξ,f ni+1 (u) (x0 ), Ff ni+1 (u),f ni+1 −ni (ξ) (x1 )
> T (1 − λ), (1 − λ) ,
(6)
where x = x0 + x1 , and also the last equality is hold because Fξ,f ni+1 (u) (x0 ) > 1 − λ
by (5) and Ff ni+1 (u), f ni+1 −ni (ξ) (x1 ) > 1 − λ by the same argument as above for
x1 instead of x0 . Now, due to the definition of TM , i.e. TM (a, b) = min{a, b}, we
obtain:
Fξ,f ni+1 −ni (ξ) (x) > 1 − λ.
(7)
Suppose that η = f ni+1 −ni (ξ) 6= ξ. If we call A = f ni+1 −ni , then by the same proof
of Theorem 2.3, we obtain a contradiction. Hence, putting k = ni+1 − ni , we have
f k (ξ) = ξ as asserted.
Corollary 3.5. If, in Theorem 3.4, Fξ,f (ξ) (x) > 1 − λ, then k = 1. Indeed
Ff k (ξ), f k+1 (ξ) (x) = Fξ,f (ξ) (x),
hence, f (ξ) 6= ξ contradicts (4).
Question 1. It is natural to ask whether Theorem 3.4 would remain true if λuniformly locally contractive self-map is substituted by locally contractive self-map.
Question 2. It is natural to ask whether Theorem 3.4 would remain true if T is
defined in general case.
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ABDOLRAHMAN RAZANI
Acknowledgments. I thank the referee, whose helpful comments led to many
improvements in this manuscript. The author would like to thank the Institute for
studies in Theoretical Physics and Mathematics (IPM), Teheran, Iran, for supporting this research (No. 83340116).
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Abdolrahman Razani
Department of Mathematics, Faculty of
Science
Imam Khomeini International University
PO Box: 34194-288
Qazvin
IRAN
[email protected]
Institute for Studies in Theoretical Physics
and Mathematics (IPM)
PO Box 19395-5531
Tehran
IRAN