**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 lim_{t→∞} 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 {x_{n}} 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 *N*εN 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 (g_{0}, h_{0})εGxH is said to be a F- best
approximation pair relative (G, H) if:

We shall denote by P^{t}_{G, H}, the set of all elements of
F-best approximation pair relative (G, H) i.e.:

Also an element g_{0}εA is said to be a F- best approximation
to G from H if:

We denote by P^{t}_{G}(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 {x_{n}} is said to converge in distance to H if lim_{n→4}
N(x_{n}-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 P^{t}_{G, H} is non-void. GxH is called F-quasi Chebyshev pair if P^{t}_{G, H} is a compact set. Also G is called t-proximinal relative to H if P^{t}_{G} (H) is non-void for some H⊂X\G. G is called F- quasi Chebyshev relative to H, if P^{t}_{G} (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
is nonempty if and only if N(G-H, t) = 1 for each t>0.

**Proof:** Let for all t>0, N(G-H, t) = 1. As X is first countable, there exists a sequence {g_{n}} in G such that N(g_{n}-H, t)→N(G-H, t). Since, G is compact set, there exists a subsequence {gn_{k}} and g_{0} in G such that gn_{k}→g_{0} and then N(gn_{k}-H, t)→N(g_{0}-H, t). Therefore:

for all t>0. Hence, N(g_{0}-H, t) = 1 and so .
Conversely, suppose there exists a ,
then for all t>0, N(g_{0}-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 g_{n}εG with the property that, for all t>0, N(g_{n}-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 {g_{n}}⊂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 gn_{k} and g_{0}εG such that gn_{k}→g_{0}. Since, (X, N, *) is a strong fuzzy normed space, we have, N(gn_{k}-H, t)→N(g_{0}-H, t). Hence, for all t>0, N(g_{0}-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 {g_{n}} be a sequence in P^{t}_{G}(H). It is obvious that there exists a subsequence {gn_{k}} and g_{0}εG such that gn_{k}→g_{0} 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 g_{n}εA be any sequence converging in distance
to H and let the sequence h_{n}εH for all t>0 satisfies, limN(g_{n}-h_{n},
t) = N(G-H, t). Since, H is compact, h_{n}h_{n'}→h_{0}εH.
Hence:

Then g_{n'} converges in distance to g_{0} and, since N(G-H,
t) = N(G-h_{0}, t) and G is approximately compact, g_{n}g_{n}'→g_{0}εG;
that is, g_{n} converges subsequentially to an element of G.

**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 (g_{n}, h_{n})εGxH be any sequence in
P^{t}_{G, H}. Then for every t>0, N(g_{n}-h_{n},
t)→N(G-H, t). Since, H is compact, h_{n}h_{n'}→h_{0}εH.
Hence:

Therefore, limN(g_{n'}-h_{0}, t) = N(G-h_{0}, t). Since,
G is F-approximately compact, g_{n}g_{n'}→g_{0}εG.
Hence:

**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 {g_{n}} be a sequence converges in distance to H and let h_{n}εH satisfies:

As {g_{n}} is F-bounded, so is {h_{n}}. Since, H is F-boundedly
compact, h_{n}h_{n'}→h_{0}εX.

**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 {g_{n}} be a sequence such that N(g_{n}-H, t)→ N(G-H, t) and choose h_{n} in H such that:

As {h_{n}} is F- bounded, so is {h_{n'}}; hence, g_{n}g_{n'}→g_{0}εG,
which complete the proof.

**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 h_{n}εH satisfies limN(h_{n}-G,
t) = (G-H, t). By compactness of H, h_{n}h_{n'}→h_{0}εH
so N(G-h_{0}, t) = N(G-H, t) Since, G is F-proximinal, there exists
g_{0}εG such for all t>0, N(g_{0}-h_{0}, t)
= N(G-h_{0}, t), so N(g_{0}-h_{0}, t) = N(G-H, t).

**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 h_{n}εH satisfies limN(h_{n}-G,
t) = N(G-H, t). Since, G is F-bounded, {h_{n}} must also be F-bounded
so h_{n}h_{n'}→h_{0}εH.

**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 g_{n} be a sequence in K and for every nεN choose
{h_{n}} in H such that an minimizes the distance from G to {h_{n}}.
Since H is compact, h_{n}h_{n'}→h_{0}εH
Hence:

Then limN(g_{n'}-h_{0}, t) = N(h_{0}-G, t). Therefore, {g_{n'}} converges in distance to h_{0}. Since, G is approximately compact, so it is converges subsequentially.