Abstract
In this paper, we first present a fixed point theorem for set-valued fuzzy contraction type maps in complete fuzzy metric spaces which extends and improves some well-know results in literature. Then by presenting an endpoint result we initiate endpoint theory for fuzzy contraction maps in fuzzy metric spaces.
02000 Mathematics Subject Classification: 47H10, 54H25.
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1. Introduction and preliminaries
Many authors have introduced the concept of fuzzy metric spaces in different ways [1–4]. Kramosil and Michalek [5] introduced the fuzzy metric space by generalizing the concept of the probabilistic metric space to fuzzy situation. George and Veeramani [6, 7] modified the concept of fuzzy metric space introduced by Kramosil and Michalek [5] and obtained a Hausdorff topology for this kind of fuzzy metric spaces. Recently, the fixed point theory in fuzzy metric spaces has been studied by many authors [8–18]. In [11], the following definition is given.
Definition 1.1. A sequence (t n ) of positive real numbers is said to be an s-increasing sequence if there exists m0 ∈ ℕ such that t m + 1 ≤ tm+1, for all m ≥ m0.
Gregori and Sapena [11] proved the following fixed point theorem.
Theorem 1.2. Let (X, M, *) be a complete fuzzy metric space such that for every s-increasing sequence (t n ) and every x, y ∈ X
Suppose f : X → X is a map such that for each x, y ∈ X and t > 0, we have
where 0 < k < 1. Then, f has a unique fixed point.
In this article, we first give a fixed point theorem for set-valued contraction maps which improve and generalize the above-mentioned result of Gregori and Sapena. Then, in Section 2, we initiate endpoint theory in fuzzy metric spaces by presenting an endpoint result for set-valued maps.
To set up our results in the next section we recall some definitions and facts.
Definition 1.3 (3). A binary operation * : [0, 1] × [0, 1] → [0, 1] is called a continuous t-norm if ([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, ∈ [0, 1]. Examples of t-norm are a * b = ab and a * b = min{a, b}.
Definition 1.4 (6). The 3-tuple (X, M, *) is called a fuzzy metric space if X is an arbitrary non-empty set, * is a continuous t-norm, and M is a fuzzy set on X2 × [0, ∞) satisfying the following conditions, for each x, y, z ∈ X and t, s > 0,
-
(1)
M(x, y, t) > 0,
-
(2)
M(x, y, t) = 1 if and only if x = y,
-
(3)
M(x, y, t) = M(y, x, t),
-
(4)
M(x, y, t) * M(y, z, s) ≤ M(x, z, t + s),
-
(5)
M(x, y, t) : (0, ∞) → [0,1] is continuous.
Example 1.5. [6] Let (X, d) be a metric space. Define a * b = ab (or a * b = min{a, b}) and for all x, y ∈ X and t > 0,
Then (X, M, *) is a fuzzy metric space. We call this fuzzy metric M induced by the metric d the standard fuzzy metric.
Definition 1.6. Let (X, M, *) be a fuzzy metric space.
-
(1)
A sequence {x n } is said to be convergent to a point x ∈ X if limn→∞ M(x n , x, t) = 1 for all t > 0.
-
(2)
A sequence {x n } is called a Cauchy sequence if
for all t > 0.
-
(3)
A fuzzy metric space in which every Cauchy sequence is convergent is said to be complete.
-
(4)
A subset A ⊆ X is said to be closed if for each convergent sequence {x n } with x n ∈ A and x n → x, we have x ∈ A.
-
(5)
A subset A ⊆ X is said to be compact if each sequence in A has a convergent subsequence.
Throughout the article, let denote the class of all compact subsets of X.
Lemma 1.7. [10]For all x, y ∈ X, M(x, y,.) is non-decreasing.
Definition 1.8. Let (X, M, *) be a fuzzy metric space. M is said to be continuous on X2 × (0, ∞) if
whenever {(x n , y n , t n )} is a sequence in X2 × (0, ∞) which converges to a point (x, y, t) ∈ X2 × (0, ∞); i.e.,
Lemma 1.9. [10]M is a continuous function on X2 × (0, ∞).
2. Fixed point theory
The following lemma is essential in proving our main result.
Lemma 2.1. Let (X, M, *) be a fuzzy metric space such that for every x, y ∈X, t > 0 and h > 1
Suppose {x n } is a sequence in X such that for all n ∈ ℕ,
where 0 < α < 1. Then {x n } is a Cauchy sequence.
Proof. For each n ∈ ℕ and t > 0, we have
Thus for each n ∈ ℕ, we get
Pick the constants h > 1 and l ∈ ℕ such that
Hence, for m ≥ n, we get
Then, from the above, we have
for each t > 0. Therefore, we get
for each t > 0 and so {x n } is a Cauchy sequence.
In 2004, Rodríguez-López and Romaguera [19] introduced Hausdorff fuzzy metric on the set of the non-empty compact subsets of a given fuzzy metric space.
Definition 2.2. ([19]) Let (X, M, *) be a fuzzy metric space. For each and t > 0, set
Lemma 2.3. [19]Let (X, M, *) be a fuzzy metric space. Then, the 3-tupleis a fuzzy metric space.
Now we are ready to prove our first main result.
Theorem 2.4. Let (X, M, *) be a complete fuzzy metric. Suppose F : X → X is a set-valued map with non-empty compact values such that for each x, y ∈ X and t > 0, we have
where α : [0, ∞) → [0,1) satisfying
and. Furthermore, assume that (X, M, *) satisfies (2.1) for some x0 ∈ X and x1 ∈ Fx0. Then F has a fixed point.
Proof. Let t > 0 be fixed. Notice first that if A and B are non-empty compact subsets of X and x ∈ A then by [19, Lemma 1], there exists a y ∈ B such that
Thus given α ≤ H M (A, B, t) there exists a point y ∈ B such that
Let x0 ∈ X and x1 ∈ Fx0. If Fx0 = Fx1 then x1 ∈ Fx1 and x1 is a fixed point of F and we are finished. So, we may assume that Fx0 ≠ Fx1. From (2.2), we get
Since x1 ∈ Fx0 and F is compact valued then by Rodríguez-López and Romaguera [19, Lemma 1] there exists a x2 ∈ Fx1 satisfying
Continuing this process, we can choose a sequence {x n }n ≥ 0in X such that xn+1∈ Fx n satisfying
Then, the sequence {M(xn+1, xn+2, t)} n is non-decreasing.
Thus {d(xn+1, xn+2, t)} n is a non-negative non-increasing sequence and so is convergent, say to, l ≥ 0. Since by the assumption
then there exists k < 1 and N ∈ ℕ such that
Since M(x, y,.) is non-decreasing then (2.3) together with (2.4) yield
Then from the above, we get
Hence by Lemma 2.1, we get {x n }, which is a Cauchy sequence. Since (X, M, *) is a complete fuzzy metric space, then there exists such that , that means , for each t > 0. Thus, , for each t > 0. Since
then there exists k < l < 1 such that
Now we claim that . To prove the claim notice first that since and then for each t > 0, we get
Since xn+1∈ Fx n then from (2.5), we have
Thus there exists a sequence such that
for each t > 0. For each n ∈ ℕ, we have
Hence, from the above, we get
which means . Since is closed (note that is compact), and then, we get .
Corollary 2.5. Let (X, M, *) be a complete fuzzy metric. Suppose F : X → X is a set-valued map with non-empty compact values such that for each x, y ∈ X and t > 0, we have
where 0 < k < 1. Furthermore, assume that (X, M, *) satisfies (2.1) for some x0 ∈ X and x1 ∈ Fx0. Then F has a fixed point.
From Corollary 2.5, we get the following improvement of the above mentioned result of Gregori and Sapena [11] (note that for each t > 0 and h > 1, the sequence t n = thnis s-increasing).
Theorem 2.6. Let (X, M, *) be a complete fuzzy metric space. Suppose f : X → X is a map such that for each x, y ∈ X and t > 0, we have
where 0 < k < 1. Furthermore, assume that (X, M, *) satisfies (2.1) for some x0 ∈ X, each t > 0 and h > 1. Then f has a fixed point.
Let (X, d) be a metric space and A and B are non-empty closed bounded subsets of X. Now set
Then H is called the Hausdorff metric. Now, we are ready to derive the following version of Mizoguchi-Takahashi fixed point theorem [20].
Corollary 2.7. Let (X, d) be a complete metric space. Suppose F : M → M is a set-valued map with non-empty compact values such that for some k < 1
where α : [0, ∞) → [0,1) satisfying
Then F has a fixed point.
Proof. Let (X, M, *) be standard fuzzy metric space induced by the metric d with a * b = ab. Now we show that the conditions of Theorem 2.4 are satisfied. Since (X, d) is a complete metric space then (X, M, *) is complete. It is easy to see that (X, M, *) satisfies (2.1). For each non-empty closed bounded subsets of X, we have
By the above and our assumption, we have
for each t > 0 and each x, y ∈ X. Therefore, the conclusion follows from Theorem 2.4.
3. Endpoint theory
Let X be a non-empty set and let F : X → 2Xbe a set-valued map. An element x ∈ X is said to be an endpoint (invariant or stationary point) of F, if Fx = {x}. The investigation of the existence and uniqueness of endpoints of set-valued contraction maps in metric spaces have received much attention in recent years [21–26].
Definition 3.1. Let (X, M, *) be a fuzzy metric space and let F : X → X be a multi-valued mapping. We say that F is continuous if for any convergent sequence x n → x0 we have H M (Fx n , Fx0, t) → 1 as n → ∞, for each t > 0.
As far as we know the following is the first endpoint result for set-valued contraction type maps in fuzzy metric spaces.
Theorem 3.2. Let (X, M, *) be a complete fuzzy metric space and letbe a continuous set-valued mapping. Suppose that for each x ∈ X there exists y ∈ Fx satisfying
where k ∈ [0,1). Then, F has an endpoint.
Proof. For each x ∈ X, define the function f : X → [0, ∞) by f(x, t) = H M (x, Fx, t) = infy∈FxM(x, y, t), x ∈ X. Suppose that {x n } converges to x; then for any y ∈ Fx and z ∈ Fx n , we have
Since y ∈ Fx is arbitrary then from the above, we get
It follows from the continuity of F that
Hence,
whenever x n → x. Let x0 ∈ X. Then by (3.1) there exists a x1 ∈ Fx0 such that
Continuing this process, we can choose a sequence {x n }n≥0in X such that xn+1∈ Fx n satisfying
From the definition of H M (x n , Tx n ), we have
From (3.2) and (3.3), we get
which implies that {H M (x n , Fx n , kt)} n is a non-negative non-decreasing sequence of real numbers and so is convergent. To find the limit of {H(x n , Fx n , kt)} n notice that
Since Fx0 is compact then there exists a y0 ∈ Fx0 such that
(3.5) together with (3.6) imply that for each n ∈ ℕ
From (2.1) we have and so
From (3.2), we get
from which and Lemma (2.1), we get {x n } is a Cauchy sequence. Since (X, M, *) is a complete fuzzy metric space then there exists a such that . By assumption the function f(x) = H M (x, Fx, t) is upper semicontinuous, then
Thus
and so .
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Acknowledgements
This research was in part supported by the grant from IPM (90470017). The second author was also partially supported by the Center of Excellence for Mathematics, University of Shahrekord.
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Kiany, F., Amini-Harandi, A. Fixed point and endpoint theorems for set-valued fuzzy contraction maps in fuzzy metric spaces. Fixed Point Theory Appl 2011, 94 (2011). https://doi.org/10.1186/1687-1812-2011-94
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DOI: https://doi.org/10.1186/1687-1812-2011-94