Some results on common best proximity point in fuzzy metric spaces

In this paper, we define the concepts of commute proximally, dominate proximally, weakly dominate proximally and common best proximity point in fuzzy metric space (abbreviated, FM -space). We prove some common best proximity point and common fixed point theorems for dominate proximally and weakly dominate proximally mappings in FM -space under certain conditions. Our results generalize many known results in metric space.


Introduction and preliminaries
The notion of fuzzy sets introduced by Zadeh [25], proved a turning point in the development of mathematics.This notion laid the foundation of fuzzy mathematics.Fuzzy set theory has applications in applied sciences such as neural network theory, stability theory, mathematical programming, modelling theory, engineering sciences, medical sciences (medical genetics, nervous system), image processing, control theory, communication etc. Kramosil and Michalek [15] introduced the notion of a fuzzy metric space by generalizing the concept of the probabilistic metric space to the fuzzy situation.George and Veeramani [12], modified the concept of fuzzy metric spaces introduced by Kramosil and Michalek [15].Fixed point theory in fuzzy metric spaces was initiated by Grabiec [13].Subrahmanyam [22] gave a generalization of Jungck [14] common fixed point theorem for commuting mappings in the setting of fuzzy metric spaces, whereas Vasuki [23] gave a fuzzy version of a result contained in Pant [18].Thereafter, many authors established fuzzy versions of a host of classical metrical common fixed point theorems (e.g.[1,20,23]).
In nonlinear analysis, the theory of fixed points is an essential instrument to solve the equation T x = x for a self mapping T defined on a subset of an abstract space such as a metric space, a normed linear space or a topological vector space.If T is a non-self mapping from A to B, then the aforementioned equation does 178 H. Shayanpour and A. Nematizadeh not necessarily admit a solution.However, in such circumstances, it may be speculated to determine an element x for which the error d(x, T x) is minimum, where d is the distance function, in which case x and T x are in close proximity to each other.In fact, best approximation theorems and best proximity point theorems are applicable for solving such problems.In view of the fact that d(x, T x) is at least d(A, B), a best proximity point theorem guarantees the global minimization of d(x, T x) by the requirement that an approximate solution x satisfies the condition d(x, T x) = d(A, B).Such optimal approximate solutions are called best proximity points of the mapping T .Further, it is interesting to observe that best proximity theorems also emerge as a natural generalization of fixed point theorems.A best proximity point reduces to a fixed point if the mapping under consideration is a self mapping.Investigation of several variants of contractions for the existence of a best proximity point can be found in [2,3,4,5,7,8,9,10,17,19,21,24].Eldred et al. [11] have established a best proximity point theorem for relatively non-expansive mappings.Further, Anuradha and Veeramani have focussed on best proximity point theorems for proximal pointwise contraction mappings [6].
In this paper, we establish some definitions and basic concepts of the common best proximity point in the framework of fuzzy metric spaces.
We first bring notation, definitions and known results, which are related to our work.
(George and Veeramani [12]) The 3-tuple (X, M, * ) is said to be a fuzzy metric space (abbreviated, F M -space) if X is a nonempty set, * is a continuous t-norm and M is a fuzzy set on X × X × [0, ∞) satisfying the following conditions: Some results on common best proximity point in fuzzy metric spaces 179 for any x, y, z ∈ X and t, s > 0.
Example 1.4.[12] Let (X, d) be a metric space.Define then (X, M, * M ) is a fuzzy metric space.
To this end, assume towards a contradiction that for some x, y and z in X.We can choose t > d λ (x, y) and s > d λ (y, z) so that t + s < d λ (x, z).
To see that d λ (x, y) is left continuous, let, λ ∈ (0, 1) and let {λ i } be a non-decreasing sequence converging to λ, Then {d λ i (x, y)} is a non-increasing sequence bounded below by zero and hence converges to a number p; since d λi (x, y) ≥ d λ (x, y), p ≥ d λ (x, y).We complete the proof by showing that p = d λ (x, y).If p > d λ (x, y), then from the definition of d λ (x, y), M (x, y, p) > 1 − λ.Choose an n large enough so that M (x, y, p

we have a contradiction. ✷
The family {d λ : λ ∈ (0, 1)} of all such pseudometrics will be called the family of pseudometrics associated with the fuzzy metric M .
and we have a contradiction.Therefore d λ (x, y) < ǫ. ✷ From this lemma it is clear that the neighborhood U x (ε, λ) of x in the M topology is the d λ -spherical neighborhood of x of radius ǫ.We thus have immediately the following basic result.Theorem 1.12.Let (X, M, * M ) be a F M -space.The topology on X generated by the family of pseudometrics associated with the fuzzy metric M is the same as the topology induced by M .Definition 1.13.A sequence (x n ) in a F M -space (X, M, * ) is said to be convergent to a point x ∈ X if and only if for every ε > 0 and λ ∈ (0, 1), there exists for all t > 0, in this case we say that limit of the sequence (x n ) is x.Definition 1.14.A sequence (x n ) in a F M -space (X, M, * ) is said to be Cauchy sequence if and only if for every ε > 0 and λ ∈ (0, 1), there exists n 0 (ε, λ) ∈ N such that M (x n+p , x n , ε) > 1 − λ for all n ≥ n 0 (ε, λ) and every p ∈ N or for every λ ∈ (0, 1), d λ (x n+p , x n ) → 0 for all p ∈ N, or lim n→∞ M (x n , x n+p , t) = 1, for all t > 0 and p ∈ N.
Also, a F M -space (X, M, * ) is said to be complete if and only if every Cauchy sequence in X, is convergent.
Proposition 1.15.The limit of a convergent sequence in a F M -space (X, M, * ) is unique.16.Let (X, M, * ) be a F M -space and (x n ) be a sequence in X.If sequence (x n ) converges to x ∈ X, then M (x, x, t) = 1 for all t > 0.
Proof: It is obvious.✷ Definition 1.17.Let A and B be nonempty subsets of a F M -space (X, M, * ).Let which is said to be the fuzzy distance of A, B.
Definition 1.18.Let A and B be nonempty subsets of a F M -space (X, M, * ).We define the following sets: Definition 1.19.Let A and B be nonempty subsets of a F M -space (X, M, * ) and T, S : A → B be two mappings.We say that an element x ∈ A is a common best proximity point of the mappings S and T , if for all t ≥ 0.
It is clear that the notion of a common fixed point coincided with the notion of a common best proximity point when the underlying mapping is a self mapping.Also, it can be noticed that common best proximity point is an element at which both function x → M (x, Sx, t) and x → M (x, T x, t) for all t ≥ 0, attain global supremum.
Definition 1.20.Let A and B be nonempty subsets of a F M -space (X, M, * ) and T, S : A → B be two mappings.We say that T, S are commute proximally if for all t ≥ 0, then Sv = T u, where x, y, u, v ∈ A.
Example 1.21.Let (X, M, * ) be a F M -space and T, S : X → X be two mappings such that T S = ST .Clearly M (X, X, t) = 1 for all t ≥ 0 and so if then by the hypothesis, u = Sx and v = T x.Therefore Sv = ST x = T Sx = T u, hence T, S are commute proximally.
Definition 1.22.Let A and B be nonempty subsets of a F M -space (X, M, * ) and T, S : A → B be two mappings.We say that the mapping T is to dominate the mapping S proximally if for all t ≥ 0, then there exists a α ∈ (0, 1) such that for all t ≥ 0, where Some results on common best proximity point in fuzzy metric spaces 183 Definition 1.24.Let A and B be nonempty subsets of a F M -space (X, M, * ) and T, S : A → B be two mappings.We say that the mapping T is to weakly dominate the mapping S proximally if for all t ≥ 0, then there exists a α ∈ (0, 1) such that for all t ≥ 0, where Obviously, if T dominates S proximally, then T weakly dominates S proximally.The following example shows that the converse is not true, in general. .
In this article, we introduce two new classes of mappings, called dominate proximally and weakly dominate proximally in fuzzy metric space.We provide sufficient conditions for the existence and uniqueness of common best proximity points and common fixed points for weakly dominate proximally mappings in F M -space.Our results generalize many known results in metric space.Examples are given to support our main results.

Main Results
Now we state and prove our main theorem about existence and uniqueness of a common best proximity point for dominate proximally and weakly dominate proximally mappings in F M -space under certain conditions.Theorem 2.1.Let A and B be nonempty subsets of a complete fuzzy metric space (X, M, * M ) such that A 0 and B 0 are nonempty and A 0 is closed.If the mappings T, S : A → B satisfy the following conditions: (i) T weakly dominates S proximally, Some results on common best proximity point in fuzzy metric spaces 185 (ii) S and T commute proximally, (iii) S and T are continuous, (iv) S(A 0 ) ⊂ B 0 and S(A 0 ) ⊂ T (A 0 ).
Then, there exists a unique element x ∈ A such that for all t ≥ 0.
Proof: First, suppose that there exists an element u ∈ A 0 such that Su = T u.By the hypothesis, there exists an element x ∈ A 0 such that so, Sx = T x.Once again, by the hypothesis, there exists an element v ∈ A 0 such that Since T weakly dominates S, then from (2.1) and (2.2), for all t ≥ 0, we get which implies x = v, by Lemma 1.5.Therefore, it follows that So, x is a common best proximity point of the mappings S and T .If x ′ is another common best proximity point of the mappings S and T , in other words then by using the same argument as above we can show that x = x ′ .Second, we claim that there exists an element u ∈ A 0 such that Su = T u.To support the claim, let x 0 be a fixed point element in A 0 .By the hypothesis, there exists an element x 1 ∈ A 0 such that Sx 0 = T x 1 .This process can be carried on.Having chosen x n ∈ A 0 , by the hypothesis, we can find an element x n+1 ∈ A 0 such that Sx n = T x n+1 .By the condition (iv), there exists an element u n ∈ A 0 such that M (u n , Sx n , t) = M (A, B, t) for all t ≥ 0 and n ∈ N.
Further, it follows from the choice x n and u n that So, by the condition (i), we have for all t ≥ 0. Thus, we have for all t ≥ 0. In the following we show by induction that for each n ∈ N and for each t ≥ 0, there exists If n = 1, then by (2.3), we have for some 1 ≤ m ≤ 2 and for all t ≥ 0. Thus (2.4) holds for n = 1.Assume towards a contradiction that (2.4) is not true and take n 0 > 1, be the least natural number such that (2.4) does not hold.So there exists t 0 > 0, such that for all 1 ≤ m ≤ n 0 + 1, we have If then by the hypothesis we have for some 1 ≤ m ≤ n 0 , a contradiction.Thus and form (2.3), we have By the condition (i), we get then from (2.6) and the above, we have Some results on common best proximity point in fuzzy metric spaces 187 for some 1 ≤ m ≤ n 0 ≤ n 0 + 1, a contradiction.Therefore from (2.6) and the above, we get for some 1 ≤ m ∈ {n 0 − 1, n 0 , n 0 + 1} ≤ n 0 + 1. Again by the condition (i), we have then from (2.7) and the above, we have then from the above, we have a contradiction, since if the above inequality becomes equality, then we can assume that m = n 0 .Therefore from the above, we get for some 1 ≤ m ′ ≤ n 0 + 1. Therefore by continuing this process, we see that for each 1 ≤ k ≤ n 0 , there exists 1 ≤ m ≤ n 0 + 1 such that If k = n 0 in (2.8), then this is a contradiction by (2.5).So (2.4) holds for all n ∈ N. Suppose that (d λ ) λ∈(0,1) is the family of pseudometrics in Definition 1.8, by Theorem 1.12 the family of pseudometrics (d λ ) λ∈(0,1) generates the topology induced by M on X.We obtain by (2.4) that for u n and every λ ∈ (0, 1), (2.10) We now show that ∞ n=1 a n = lim n→∞ s n < ∞.Assume towards a contradiction that lim n→∞ s n = ∞.By the hypothesis we can assume without loss of generality that s n = 0 for all n ∈ N.So by the hypothesis the series Some results on common best proximity point in fuzzy metric spaces 189 is convergent.From (2.11), we get there exists n ∈ N such that for every m ∈ N, taking the limit as m → ∞, we get 1 ≤ 1 2 , a contradiction.Therefore for every λ ∈ (0, 1) and p ∈ N, we have Then (u n ) is a Cauchy sequence and by the hypothesis there exists some element u ∈ A 0 such that lim n→∞ u n = u.By the hypothesis it is easy to see that Su n = T u n+1 , for all n ∈ N, now by the continuity of the mappings S and T we get Su = T u, so the desired result is obtained.✷ The following corollary, is immediate.Then, there exists a unique element x ∈ A such that M (x, Sx, t) = M (A, B, t) = M (x, T x, t), for all t ≥ 0.
Corollary 2.3.Let (X, M, * M ) be a complete F M -space, S be a self mapping on X and T be a continuous self mapping on X such that commutes with S. If S(X) ⊆ T (X) and there exists a constant α ∈ (0, 1) such that M (Sx, Sy, αt) ≥ min{M (T x, T y, t), M (T x, Sx, t), M (T x, Sy, t), M (T y, Sx, t)}, (2.12) for every x, y ∈ X and t ≥ 0. Then S and T have a unique common fixed point.
Proof: We used the assumption of continuity of S in Theorem 2.1 to show that  Case 5. Let x ∈ [0, 4  5 ) ∪ ( is called a triangular norm (abbreviated, t-norm) if the following conditions are satisfied: (i) a * b = b * a, (ii) a * (b * c) = (a * b) * c, (iii) a * b ≥ c * d, whenever a ≥ c and b ≥ d, (iv) a * 1 = a, for every a, b, c, d ∈ [0, 1].Two typical examples of continuous t-norm are a * M b = min{a, b} and a * P b

Corollary 2 . 2 .
Let A and B be nonempty subsets of a complete F M -space (X, M, * M ) such that A 0 and B 0 are nonempty and A 0 is closed.If the mappings T, S : A → B satisfy the following conditions: (i) T dominates S proximally, (ii) S and T commute proximally, (iii) S and T are continuous, (iv) S(A 0 ) ⊂ B 0 and S(A 0 ) ⊂ T (A 0 ).