The main purpose of this paper is to study the approximate best proximity pair of cyclic maps and their diameter in fuzzy normed spaces defined by Bag and Samanta. First, approximate best point proximity points on fuzzy normed linear spaces are defined and four general lemmas are given regarding approximate fixed point and approximate best proximity pair of cyclic maps on fuzzy normed spaces. U...

this paper is concerned with the best proximity pair problem in hilbert spaces. given two subsets $a$ and $b$ of a hilbert space $h$ and the set-valued maps $f:a o 2^ b$ and $g:a_0 o 2^{a_0}$, where $a_0={xin a: |x-y|=d(a,b)~~~mbox{for some}~~~ yin b}$, best proximity pair theorems provide sufficient conditions that ensure the existence of an $x_0in a$ such that $$d(g(x_0),f(x_0))=d(a,b).$$

This paper is concerned with the best proximity pair problem in Hilbert spaces. Given two subsets $A$ and $B$ of a Hilbert space $H$ and the set-valued maps $F:A o 2^ B$ and $G:A_0 o 2^{A_0}$, where $A_0={xin A: |x-y|=d(A,B)~~~mbox{for some}~~~ yin B}$, best proximity pair theorems provide sufficient conditions that ensure the existence of an $x_0in A$ such that $$d(G(x_0),F(x_0))=d(A,B).$$

Notice that best proximity point results have been studied to find necessaryconditions such that the minimization problemminx∈A∪Bd(x,Tx)has at least one solution, where T is a cyclic mapping defined on A∪B.A point p ∈ A∪B is a best proximity point for T if and only if thatis a solution of the minimization problem (2.1). Let (A,B) be a nonemptypair in a normed...

We introduce the notion of quasi-cyclic-noncyclic pair and its relevant new notion of coincidence quasi-best proximity points in a convex metric space. In this way we generalize the notion of coincidence-best proximity point already introduced by M. Gabeleh et al cite{Gabeleh}. It turns out that under some circumstances this new class of mappings contains the class of cyclic-noncyclic mappings ...