Abstract: This study has considered the problem of finding a best approximation pair, i.e., two points which achieve the minimum distance between two nonempty sets in fuzzy normed spaces. we defined F-proximinal set and F-approximately compact relative to any set and study the existence and uniqueness of best proximity points.
INTRODUCTION
The best proximity pair evolves as a generalization of the concept of best approximation. Recently, Bauschke et al. (2004), Kim (2006), Cuenya and Bonifacio (2008) and Mohsenalhosseini et al. (2011) obtained some results on characterization and finding the best proximity points in linear normed spaces. Shams et al. (2009) studied the best approximation pairs in probabilistic normed spaces. This paper attempted to investigate the concept of best approximation pairs in fuzzy normed spaces and get some results on existence and compactness of the best proximity sets.
PRELIMINARIES
Definition 2.1: A binary operation *:[0, 1]x[0, 1]→[0, 1] is said to be a continuous t-nom if ([0, 1], *) is a topological monoid with unit 1 such that a*b≤c*d whenever, a≤c and b≤d (a, b, c, d ε [0, 1]).
If αεR+0, then we define:
Definition 2.2: 3-tuple (X, N, *) is said to be a fuzzy normed space if X is a vector space, * is a continues t-norm and N is a fuzzy set on Xx(0, ∞) satisfying the following conditions for every x, y εX and t, s>0,
| • | N (x, t)>0 |
| • | N (x, t) = 1⇔x = 0 |
| • | N (αx, t) = N(x, t/|α|), for all α≠0 |
| • | N (x, t)* N(y, s)≤N(x+y, t+s) |
| • | N (x,.) is a nondecreasing function on R and limt→∞ N(x, t = 1) |
In addition, if for t>0, x→N(x, t) is a continuous map on X; then (X, N,*) is called a strong fuzzy normed space.
Lemma 2.1: Let N be a fuzzy norm. Then:
| • | N (x, t) is nondecreasing with respect to t for each xεX |
| • | N (x-y, t) = N(y-x, t) |
Definition 2.3: Let (X, N, *) be a fuzzy normed space. The open ball B (x, r, t) and the closed ball B[x, r, t] with the center xεX and radius 0<r<1, t>0 are defined as follows:
Lemma 2.2: If (X, N, *) is a fuzzy normed space. Then:
| • | The function (x, y)→x+y is continuous |
| • | The function (α, x)→αx is continuous |
Example 2.1: Let (X, ||.||) be a normed space. If we define a*b = ab or a*b = min {a, b} and:
Then (X, N, *) is a fuzzy normed space. In particular if k = m = n = 1 we have:
which is called the standard fuzzy norm induced by the norm ||.||.
Definition 2.4: A sequence {xn} in a fuzzy normed space (X, N, *) is called a convergence sequence to xεX, if for each t>0 and each 0<ε<1 there exists NεN such that for all n≥N, we have:
Definition 2.5: Let (X, N, *) be a fuzzy normed space. For each t>0, a subset A⊆X is called F-bounded if there exists 0<r<1 such that N(x, t)>1-r for all xεX.
MAIN RESULTS
Definition 3.l: Let G and H are two nonempty subset of a fuzzy normed space (X, N, *). For t>0, let:
and:
An element (g0, h0)εGxH is said to be a F- best approximation pair relative (G, H) if:
We shall denote by PtG, H, the set of all elements of F-best approximation pair relative (G, H) i.e.:
Also an element g0εA is said to be a F- best approximation to G from H if:
We denote by PtG(H), the set of all elements of F- best approximation to G from H i.e:
Definition 3.2: For a fuzzy normed space X and nonempty subsets G and H, a sequence {xn} is said to converge in distance to H if limn→4 N(xn-H, t) = N(G-H, t).
Definition 3.3: Let (X, N, *) be a fuzzy normed space. For each t>0 and for nonempty subsets G and H of X, GxH is called F- proximinal pair relative to (G, H) if PtG, H is non-void. GxH is called F-quasi Chebyshev pair if PtG, H is a compact set. Also G is called t-proximinal relative to H if PtG (H) is non-void for some H⊂X\G. G is called F- quasi Chebyshev relative to H, if PtG (H) is a compact set for some H⊂X\G.
Example 3.1: Let X =R. For a, b ε{0, 1}, let a*b = ab. Define N:Rx(0, +∞)→[0, 1], by N(x, t) = t/t+|x|. Then (X, N, *) is the standard fuzzy normed space. Let G = [0, 2] and H = [3, 4], then for each gεG and hεH, it is easy too see that N(3-2, t)>N(g-h, t). So N(3-2, t) = N(G-H, t) and N(3-H, t) = N(G-H, t). Hence, for each t>0, (3, 2) is a F-best approximation pair relative to (G, H) and 3 is a F-best approximation to G from H.
Lemma 3.1: Let G and H be nonempty subsets of a strong fuzzy normed
space (X, N, *) and G is compact. Then
Proof: Let for all t>0, N(G-H, t) = 1. As X is first countable, there exists a sequence {gn} in G such that N(gn-H, t)→N(G-H, t). Since, G is compact set, there exists a subsequence {gnk} and g0 in G such that gnk→g0 and then N(gnk-H, t)→N(g0-H, t). Therefore:
for all t>0. Hence, N(g0-H, t) = 1 and so
for all t>0.
Definition 3.4: Let (X, N, *) be a fuzzy normed space and G and H be nonempty subsets of X. We say that the subset G is F-approximately compact relative to H if every sequence gnεG with the property that, for all t>0, N(gn-H, t)→N(G-H, t), has a subsequence convergent to an element of G.
Theorem 3.1: Let G and H are nonempty subsets of a strong fuzzy normed space (X, N, *) and G is F-approximately compact relative to H, then G is a F- proximinal set relative to H.
Proof: By definition, there exists {gn}⊂G such that N(n-H, t)→N(G-H, t). Since, G is F- approximately compact relative to H, so there exists a subsequence gnk and g0εG such that gnk→g0. Since, (X, N, *) is a strong fuzzy normed space, we have, N(gnk-H, t)→N(g0-H, t). Hence, for all t>0, N(g0-H, t) = N(G-H, t).
Theorem 3.2: Let G is a F- approximately compact relative to H then, G is F- quasi Chebyshev relative to H.
Proof: Let {gn} be a sequence in PtG(H). It is obvious that there exists a subsequence {gnk} and g0εG such that gnk→g0 and this complete the proof.
Theorem 3.3: Let G and H be nonempty subsets of a Fuzzy normed space (X, N, *). If G is F- approximately compact and H is compact, then G is F- approximately compact relative to H.
Proof: Let gnεA be any sequence converging in distance
to H and let the sequence hnεH for all t>0 satisfies, limN(gn-hn,
t) = N(G-H, t). Since, H is compact, hn
Then gn' converges in distance to g0 and, since N(G-H,
t) = N(G-h0, t) and G is approximately compact, gn
Theorem 3.4: Let G and H be nonempty subsets of a Fuzzy normed space (X, N, *). If G is F-approximately compact and H is compact, then GxH is F-quasi Chebyshev set relative to (G, H).
Proof: Let (gn, hn)εGxH be any sequence in
PtG, H. Then for every t>0, N(gn-hn,
t)→N(G-H, t). Since, H is compact, hn
Therefore, limN(gn'-h0, t) = N(G-h0, t). Since,
G is F-approximately compact, gn
Lemma 3.2: Let G and H be nonempty subsets of a fuzzy normed space (X, N, *). If G is F- approximately compact and F-bounded and H is F-boundedly compact, then G is F-approximately compact to H.
Proof: Let {gn} be a sequence converges in distance to H and let hnεH satisfies:
As {gn} is F-bounded, so is {hn}. Since, H is F-boundedly
compact, hn
Lemma 3.3: Let G and H be nonempty subsets of a fuzzy normed space (X, N, *). If G is closed and F-boundedly compact and H is F-bounded, then G is F-approximately compact to H.
Proof: Suppose {gn} be a sequence such that N(gn-H, t)→ N(G-H, t) and choose hn in H such that:
As {hn} is F- bounded, so is {hn'}; hence, gn
Theorem 3.5: Let G and H be nonempty subsets of a fuzzy normed space (X, N, *). If G is F -proximinal and H is compact, then GxH is F-proximinal pair relative to (G, H).
Proof: Suppose hnεH satisfies limN(hn-G,
t) = (G-H, t). By compactness of H, hn
Theorem 3.4: Let G and H be nonempty subsets of a fuzzy normed space (X, N, *). If G is F- proximinal and F-bounded and H is closed and F-boundedly compact, then GxH is F- proximinal pair relative to (G, H).
Proof: Suppose hnεH satisfies limN(hn-G,
t) = N(G-H, t). Since, G is F-bounded, {hn} must also be F-bounded
so hn
Lemma 3.5: Let G and H be nonempty subsets of a fuzzy normed space (X, N, *). If G is approximately compact and H is compact, then K = {gεG:∃hεH, N(g-h, t) = N(G-h, t)} is compact.
Proof: Let gn be a sequence in K and for every nεN choose
{hn} in H such that an minimizes the distance from G to {hn}.
Since H is compact, hn
Then limN(gn'-h0, t) = N(h0-G, t). Therefore, {gn'} converges in distance to h0. Since, G is approximately compact, so it is converges subsequentially.