On Fuzzy Contractive Mappings in Fuzzy Metric Spaces

In this paper we introduce a new fuzzy contraction mapping and prove that such mappings have ﬁxed point in τ -complete fuzzy metric spaces. As an application, we shall utilize the results obtained to show the existence and uniqueness of random solution for the following random linear random operator equation. Moreover, we shall show that the existence and uniqueness of the solutions for nonlinear Volterra integral equations on a kind of particular fuzzy metric space.


INTRODUCTION
The concept of fuzzy metric spaces was obtained in different ways [7,13,14,[24][25][26]. To obtain a Hausdorff topology of fuzzy metric spaces, [7,8,13] modified the concept of fuzzy metric space introduced by Kramosil and Michalek [14] and showed that every ordinary metric induces a fuzzy metric in the sense of George and Veeramani [7]. Fixed point theorems play a fundamental role in demonstrating the existence of solutions to a wide variety of problems arising in mathematics, physics, engineering, medicine and social sciences. The study of fixed point theorems in fuzzy mathematics was instigated by Weiss [32], Butnariu [3], Singh and Talwar [30], Estruch and Vidal [5],Wang et al. [33], Mihet [22], Qiu et al. [23] and Beg and Abbas [2]. Heilpern [11] introduced the concept of fuzzy contraction mappings and established the fuzzy Banach contraction principle on a complete metric linear spaces with the d ∞ -metric for fuzzy sets. Azam and Beg [1] proved common fixed point theorems for a pair of fuzzy mappings satisfying Edelstein, Alber and Guerr-Delabriere type contractive conditions in ametric linear space. Rashid [25] give a generalization of Hicks type contractions and SB-type contractions in fuzzy metric spaces. He proved some fixed point theorems for these new type contraction mappings on fuzzy metric spaces.
In this paper we introduce a new fuzzy contraction mapping and prove that such mappings have fixed point in complete fuzzy metric spaces. As an application, we shall utilize the results obtained to show the existence and uniqueness of random solution for the following random linear random operator equation. Moreover, we shall show that the existence and uniqueness of the solutions for nonlinear Volterra integral equations on a kind of particular fuzzy metric space. The structure of this paper is as follows. In Section 2, we recall some definitions and the uniform structure of fuzzy metric spaces. In Section 3, we give some fixed point theorems for several contraction type mappings in FM-spaces and we shall utilize the results obtained to show the existence and uniqueness of random solution for the following random linear random operator equation. Section 4 is devoted to give some common fixed point theorems for Contraction type mappings. Our results generalize and extend many known results in fuzzy metric spaces and probabilistic metric spaces, see [13,18,22,24,25,31,32].

PRELIMINARIES
First we recall some of the basic concepts, which will be used in the sequel.

Definition 3. [14]
A triple (X, M, T ) is called a fuzzy metric space (briefly, a FM-space) if X is an arbitrary (non-empty ) set, T is a continuous t-norm and M is a fuzzy set on X × X × [0, ∞) such that the following axioms hold: (FM-1) M (x, y, 0) = 0 for all x, y ∈ X, (FM-5) M (x, z, t + s) ≥ T (M (x, y, t), M (y, z, s)) for all x, y, z ∈ X and for all t, s ∈ [0, ∞).
We will refer to the fuzzy metric spaces in the sense of Kramosil and Michalek as KM-fuzzy metric spaces. If, in the above definition, the condition (FM-5) is replaced by the condition (FM-5A) M (x, z, max{t, s}) ≥ T (M (x, y, t), M (y, z, s)) for all x, y, z ∈ X and for all t, s ∈ [0, ∞), then (X, M, T ) is called a non-Archimedean fuzzy metric space. It is easy to check that (FM-5A) implies (FM-5), that is, every a non-Archimedean fuzzy metric space is itself a fuzzy metric space.
Example 1. Let (X, d) be an ordinary metric space and let β be a nondecreasing and continuous function from (0, ∞) into (0, 1) such that lim t→∞ β(t) = 1 . Some examples of these functions are for all x, y ∈ X. It is easy to see that is a non-Archimedean fuzzy metric space.

Definition 4. [7]
A triple (X, M, T ) is called a fuzzy metric space (briefly, a FM-space) if X is an arbitrary (non-empty ) set, T is a continuous t-norm and M is a fuzzy set on X × X × [0, ∞) such that the following axioms hold: (FM-1) M (x, y, t) > 0 for all x, y, z ∈ X, (FM-2) M (x, y, t) = 1 for every t > 0 if and only if x = y, (FM-3) M (x, y, t) = M (y, x, t) for all x, y ∈ X and t > 0, , M (y, z, s)) for all x, y, z ∈ X and for all t, s ∈ [0, ∞).
We will refer to the fuzzy metric spaces in the sense of George and Veeramani as GV-fuzzy metric spaces. Let (X, M, ∆) be a fuzzy metric space. The open ball B M (x, r, t) for t > 0 with centre x ∈ X and radius r, 0 < r < 1 is defined by: A subset A of a fuzzy metric space (X, M, ∆) is said to be open if given any point x ∈ A, there exists 0 < r < 1, and t > 0 such that B(x, r, t) ⊆ A. The familly : is a topology on X , τ M is called the topology on X induced by the fuzzy metric M . (X, τ M ) is a Housdorff first countable topological space (see [7]). (ii) A sequence {x n } in X is said to be τ -Cauchy if for any ǫ > 0 and λ ∈ (0, 1), there exists a positive integer N 0 = N 0 (ǫ, λ) such that whenever n, m ≥ N 0 .
(iii) (X, M ) is said to be complete if every τ -Cauchy sequence in X is τ -convergent to some point in X.
Definition 6 ([27]). A mapping T : X → X is said to be τ -continuous if, for any sequence

Fixed point theorems for Contraction type mappings
In this section, we give some fixed point theorems for several contraction type mappings in FM-spaces.
Clearly, it follows that the condition (2) is equivalent to the following condition: In what follows, we shall adopt the following notations: Let (X, M ) be a F M -space and T be a self-mapping on (X, M ). For any x, y ∈ X and positive integer i, we denote Throughout this section, we always assume that (X, M, ∆) is a τ -complete Menger F Mspace, ∆ is a continuous t-norm, and Φ(t) is a function satisfying the condition (Φ).
is fuzzy bounded, and that for any x ∈ X, there exists a positive integer m(x) such that, for all t ≥ 0, Then T has a fixed point in X, and for any x 0 ∈ X, the sequence {x n } defined by x n = T n x 0 , n ∈ N, τ -converges to a fixed point of T .
Proof. For any x 0 ∈ X, by assumptions, there exists a positive integer m(x 0 ) such that Now we choose a sequence {m(j)} of positive integers defined as follows: By the induction, we can prove the following inequality holds: By using Lemma 1 which implies that {x n } is a τ -Cauchy sequence in X. By the τ -completeness of (X, M, ∆), we can suppose x n τ − → x * ∈ X. By the τ -continuity of T , it is easy to see that x * is a fixed point of T . This completes the proof.
is fuzzy bounded, and suppose that there exists a positive integer m such that, for any x ∈ X, the following holds: Then the conclusion of Theorem 1 still holds.
Lemma 2. Let T be a τ -continuous self-mapping on (X, M, ∆). Then the following conditions are equivalent: (i) There exists a positive integer m such that, for any x ∈ X, (ii) There exists a positive integer m such that, for any x ∈ X and any non-negative integer k, Proof. The proof is straightforward and so is omitted.

fuzzy bounded, and that the condition (i) or (ii) in Lemma 2 is satisfied.
Then the conclusion of Theorem 1 still holds.
Theorem 4. Let T be a τ -continuous self-mapping on (X, M, ∆) Suppose that there exist positive integers m, n such that for any x, y ∈ X and any t ≥ 0, Then T has a unique fixed point in X, and for any x 0 ∈ X, the sequence {x n } defined by x n = T n x 0 , n = 1, 2, · · · , τ -converges to this fixed point.
Proof. Without loss of generality, we can assume that m ≥ n. For any x ∈ X and a non-negative integer k, put y = T m−n+k x. It follows from (6) that, for all t ≥ 0, we have By Theorem 4, T has a fixed point in X, and for any x 0 ∈ X, the sequence {x n = T n x 0 } τ -converges to a fixed point x * of T . Suppose that y * is another fixed point of T . Then for all t ≥ 0, we have Repeating the above procedure by the condition (Φ), we obtain which implies that x * = y * . This completes the proof.
, there exists a constant k ∈ (0, 1) such that, for all t ≥ 0 and x, y ∈ X, Then T has a unique fixed point x * , and for each x 0 ∈ X, the sequence {x n } defined by x n = T n x 0 , n = 1, 2, · · · , τ -converges to the point x * in X.
(b) First we prove that x * is a fixed point of T m(x * ) . Letting m(x * ) = m * , by the assumption, there exist t * ∈ R such that M (x * , T m * x * , t * ) = 1. Denote It is obvious that t 0 ≤ t * . Next we prove that t 0 = 0. In fact, if t 0 > 0, by the left-continuity of Φ, there exist t 1 , t 2 ∈ R + , 0 < t 2 < t 1 < t 0 , such that Φ(t 2 ) > t 0 . From (19), it follows that On the other hand, it follows from which contradicts M (x * , T m * x * , t 1 ) < 1. Hence we have t 0 = 0 and so M (x * , T m * x * , t) = 1 for all t > 0, i.e., T m * x * = x * .
Next we prove that x * is the unique fixed point of T m * . Then, for any t > 0, we have i.e., x * = y * . Thus, the fixed point of T m * is unique. Finally, we prove that x * is also the unique fixed point of T and T n x 0 τ − → x * . In fact, since T m * x * = x * , T m * T x * = T x * . Noting that x * is the unique fixed point of T m * , thus we have x * = T x * . The uniqueness of the fixed point x * is obvious. For any n ∈ Z + , n > m * , we may write it by n = km * + s, 0 ≤ s < m * . For any t > 0, by (17), we have, for all t > 0, Letting i → ∞, we obtain, for all t > 0, This implies that T n x 0 → x * as k → ∞. This completes the proof.
Theorem 7. Let (X, M, ∆) be a τ -complete Menger F M -space and the t-norm satisfy the condition that for any t 0 ∈ (0, 1] is continuous at t = 1. Let T : X → X be a mapping satisfying the following conditions: (ii) for each x ∈ X, there exists m(x) ∈ Z + such that for all y ∈ X and t ∈ R + , where k ∈ (0, 1) is a constant.
Then T has a unique fixed point x * ∈ X, and for any x 0 ∈ X, the iterative sequence {T n x 0 } τ -converges to the fixed point x * .
Proof. First we note that, under the assumption on ∆, sup 0<t<1 ∆(t, t) = 1. Letting Φ(t) = t/k, then it satisfies the conditions in Theorem 6. Next, we prove that, for all t > 0, In fact, for any t > 0, we have Taking k 1 ∈ (k, 1), we have 0 < k k1 < 1 and for all t > 0. Since M (x * , x i , t/k 1 ) → 1 and M (x * , T m * x i , t/k 1 ) → 1 for all t > 0 (i → ∞), there exists n ∈ I + such that, for i > N 0 , we have On the other hand, we have Letting i → ∞ in the preceding expression, we have, for all t > 0, Hence M (x * , T m * x * , t) = 1 for all t > 0. By Theorem 6, the conclusion is obtained. This completes the proof.
By the same way as stated above, we can prove the following: is fuzzy bounded, and that for each x ∈ X there exists an integer m(x) ≥ 1 such that for all y ∈ X and t ≥ 0, where k is a constant with k ∈ (0, 1). Then T has a unique fixed point x * in X and T n x 0 → x * for each x 0 ∈ X.
As an application, in the following, we shall utilize the results obtained above to show the existence and uniqueness of random solution for the following random linear random operator equation (20) x(ω) = T (ω, x(ω)) + y(ω).
In the sequel, we shall always assume that (Ω, A, P ) is a complete probability measure space, (X, d) is a separable linear metric space, and (X, B) is a measurable space, where B is the σ-algebra of all Borel subsets of X.
Let T be a mapping from Ω × X into X. For any positive integer i and x, y ∈ X, we denote Let A ⊂ X and denote δ(A) = sup It is known that if (X, d) is a complete metric space, then the induced Menger F M -space (X, M, ∆) is a τ -complete Menger F M -space and that a sequence {x n } in X τ -converges to a point x * ∈ X if and only if {x n } in X converges in the metric d to x * .  : Ω × X → X is a d-continuous random operator, and that for any x ∈ X and any ω ∈ Ω, O s (ω, x; 0, ∞) is d-bounded. Suppose further that there exist positive integers m, n satisfying the following: for all x, y ∈, where the random operator S : Ω × X → X is defined as follows: Then for any given X-valued random variableỹ(ω), there exists a unique random solution x * (ω) of the following nonlinear random operator equation: (22) x(ω) = T (ω, x(ω)) +ỹ(ω), and for any X-valued random variable x 0 (ω), the sequence x n (ω) = T (ω, x n−1 (ω)) +ỹ(ω), n = 1, 2, · · · of X-valued random variable converges almost surely to x * (ω).
Next since Φ(ω, t) satisfies the conditions (Φ 1 ) and (Φ 2 ), we have that Φ −1 (ω, t) satisfies the condition (Φ). From Lemma (4), it follows that S(ω, .) is τ -continuous. Since each d-bounded set in X is fuzzy bounded, it follows that for any x ∈ X, the orbit O s (ω, x; 0, ∞) is fuzzy bounded. Hence all the conditions of Theorem 4 are satisfied. By using Theorem 4, there exist an unique fixed point x * (ω) ∈ X and for any X-valued random variable x 0 (ω), when ω =ω the sequence {x n (ω)} ∞ n=0 converges in τ (hence in d) to x * (ω). By the arbitrariness ofω ∈ G, this implies that x n (ω) → x * (ω) a.s. By Lemma 3, it is easy to know that {x n (ω)} is a sequence of X-valued random variables, so that its strong limit x * (ω) is also an X-valued random variable, and x * (ω) is the unique random fixed point of S(., .). Accordingly, x * (ω) is the unique random solution of the equation (22). This completes the proof.
Theorem 10. Let (X, M, ∆), ∆, and Φ be the same as above. Let {T n } ∞ n=1 be a sequence of self-adjoint on (X, M, ∆). Suppose that there exists a functional sequence {m n (x)} ∞ n=1 : (X, M, ∆) → Z + such that for each n ∈ Z + and each x ∈ X, m n (x)|m b (T n x) and that, for any i, j ∈ Z + , i = j and x, y ∈ X, the following holds: for all t ≥ 0. Suppose further that there exists some x 0 ∈ X such that the set {x n } ∞ n=0 ⊂ X defined by (24) x n = T mn(xn−1) n x n−1 for all n = 1, 2, · · · is fuzzy bounded. Then there exists an unique common fixed point x * ∈ X and x n τ − → x * .
Proof. For given x 0 ∈ X, we prove that the sequence {x n } defined by (24) is a τ -Cauchy sequence in X. In fact, for any i, j ∈ Z + , it follows from (23) that, for all t ≥ 0, Therefore, for any m, n ∈ Z + (m < n), from (25) By the arbitrariness of n ∈ Z + (n > m), we have By the induction, it is easy to prove that for all m = 1, 2, · · · and t ≥ 0. In view of condition (Φ) and the fuzzy boundedness of the set {x n } defined by (24), it follows that for all t ≥ 0, Consequently, for any given ǫ > 0 and 0 < λ < 1, there exists N = N (ǫ, λ) ∈ Z + such that for all m ≥ N , inf Therefore, we have This implies that the sequence {x n } defined by (24) is a τ -Cauchy sequence in X. By the τ -completeness of X, we can suppose that x n τ − → x * ∈ X. Now we prove that x * is a common periodic point of {T n }, i.e., Indeed, for any i ∈ Z + , it follows from (23) that, for any n > i, we have Let G 0 be the set of all discontinuity points of M (x * , T mi(x * ) i x * , t). Since Φ m is strictly increasing, we know that Φ −m (G 0 ) is the set of discontinuity points of M (x n , T mi(x * ) i x * , Φ m (t)), m = 1, 2, · · · . Moreover, G 0 , Φ −m (G 0 ), m = 1, 2, · · · are the countable sets and so is also countable. LetG = R + \ G. When t = 0 or t ∈G (i.e., t is a common continuity point Repeating this procedure, we can prove that Letting n → ∞ and noting the condition (Φ), for all t ∈G or t = 0, we have (31) M When t ∈ G with t > 0, by the density of real numbers, there exist t 1 , t 2 ∈G such that 0 < t 1 < t < t 2 . Since them distribution function is nondecreasing, we have, from (31) Taking S = I X (the identity mapping on X) ,we have As an application, in the sequel we use some result stated above to show the existence and uniqueness of the solutions for nonlinear Volterra integral equations on a kind of particular fuzzy metric space.
In what follows, let [0, a] be a fixed real interval (0 < a < ∞) and (X, . X ) a real Banach space. We denote by C([0, a]; X) the Banach space of all X-valued continuous functions defined on [0, a] with norm defined by (38) x C = sup 0≤t≤a x(t) X , x(t) ∈ C([0, a]; X).
As well as the norm . C , the space C([0, a]; X) can be endowed with another norm . * which is defined as follows: (39) x * = sup for all x, y ∈ C([0, a]; X) and t ∈ R + , where the mappings T and T m are defined as follows: (T x)(t) = y(t) +

Conclusion
In this article we introduced a new fuzzy contraction mapping and prove that such mappings have fixed point in τ -complete fuzzy metric spaces. As an application, we utilized the results obtained to show the existence and uniqueness of random solution for the following random linear random operator equation. Also, we showed that the existence and uniqueness of the solutions for nonlinear Volterra integral equations on a kind of particular fuzzy metric space.