p-best approximation on probabilistic normed spaces

American Journal of Applied Sciences, Jan, 2009 by M Shams, S.M. Vaezpour, R. Saadati

INTRODUCTION

An interesting and important generalization of the notion of metric space was introduced by Menger under the name of statistical metric space, which is now called probabilistic metric space. Menger made a contribution to resolving the interpretative issue of quantum mechanics. The idea of Menger was to use distribution functions instead of nonegative real numbers.

It is also of fundamental importance in probabilistic functional analysis and nonlinear analysis and applications (2,3). Studies of such spaces by numerous authors followed (4-7). An important family of probabilistic metric spaces are probabilistic normed spaces (briefly, PN-spaces). Probabilistic normed spaces were introduced by Serstnev in 1963 (8).

In this research, we introduce the concept of best approximation in probabilistic normed spaces and present some results. Best approximation play a key role in many areas.

In the sequel after an introduction to probabilistic normed spaces, we define the concept of best approximation in probabilistic normed space and generalize some definitions such as set of best approximation, proximinal set and approximately compact set (9), (10)

A distribution function (briefly, a d.f.) is a nondecreasing function F defined on R, with F (-[infinity]) = 0 and F (-[infinity]) = 1. The set of all distribution functions that are left continuous on (-[infinity],[infinity]) will be denoted by [DELTA]. The subset of those d.f's such that F (0) = 0 will be denoted by [[DELTA].sup. ] and for every a [member of] R, [[upsilon.sub.a] is the d.f. defined by,

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The set [DELTA], as well as its subsets, can be partially ordered by the usual pointwise order; in this order, 80 is the maximal element in [[epsoilon].sub.o].is the maximal element in [[DELTA].sup. ].

A triangle function is a mapping [tau]: [[DELTA].sup. ]x[[DELTA].sup. ][right arrow][[DELTA].sup. ] that is commutative, associative, nondecreasing in each variable and which has 80 as identity.

A continuous t-norm is a continuous binary operation on [0,1] that is commutative, associative, nondecreasing in each variable and has 1 as identity.

Definition 1.1: A probabilistic normed space (briefly denoted by PN space) is a triple (V, [upsilon], [tau]) where V is a vector space over the field K of real or complex numbers, [upsilon] is a function from V into [[DELTA].sup. ], [tau] is a continuous triangle function and for every choice of x and y in V and any a y in V and any a[not equal to] K the following conditions hold:

* (N1) [tau] (x) = [[upsilon].sub.0] if and only if, x = [theta] ([theta] is the null vector in V)

* (N2) [tau] (ax)(t) = [tau] (x)(t\|a|) for all t in [R.sup. ],

* (N3) [tau] (x y) [greater than or equal to] [tau]([upsilon])(x), [upsilon](y))

[tau] is called a probabilistic norm on V (briefly P-norm) and it is called a strong probabilistic norm if for t> 0, x[right arrow] [upsilon] (x)(t) is a continuous map onV.

In the sequel we shall frequently denote the distribution function [upsilon] (x) by [upsilon] and its value at t by [[upsilon].sup.x] (t).

Definition 1.2: Let G [member of] [[DELTA].sup. ] be different from [upsilon] and [[upsilon].sub.[infinity]] let (V, II.II) be a normed space and define [upsilon]: V [right arrow] [[DELTA].sup. ] by [[upsilon].sub.[theta]] = [[upsilon].sub.[theta]] and, if x [not equal to] [theta], by

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The pair (V, [upsilon]) is called the simple space generated by (V II.II) and G.

Definition 1.3: Let {[P.sub.n]} be a sequence in (V, [upsilon], [tau]) Then

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Definition 1.4: Let (V, [upsilon], [tau]) be a PN space. A subset A of X is said to be p-bounded if and only if, there exists t> 0 such that [upsilon].sub.p-q] (t)>1-t for all p, q [member of] A.

Let (V, [upsilon], [tau]) be a probabilistic normed space. The open ball [N.sub.t] (t) with the center x [member of] Vand radius t>0 is defined as follow:

[N.sub.p](t) = {q:[[upsilon].sub.p - p](t)>1 - t}

Definition 1.5: A probabilistic normed space (V, [upsilon], [tau]) is said to be a strong probabilistic normed space if for x in V, t>0, y[right arrow][[upsilon.sub.x-y] (t) is a continuous map on V.

MAIN RESULTS

Definition 2.1: Let A and C are two nonempty subset of a probabilistic normed space (V, [upsilon], [tau]). For t>0, let

[[upsilon].sub.A - C](t) = sup{[[upsilon].sub.a - c](t):(a,c)[euro]AXC} = sup{[[upsilon].sub.a - c](t):a[euro]A}

An element [a.sub.0] [member of]A is said to be a p-best approximation to C from A if,

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Also an element ([a.sub.0], [c.sub.0])[member of] AxC is said to be a p-best approximation pair relative to (A, C) if,

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We shall denote by [P.sub.A].sup.t]] (C) the set of elements of p-best approximation to C from A i.e.,

[P.sub.A.sup.t](C) = {a[euro]A:[[upsilon].sub.a - c](t) = [[upsilon].sub.A - C](t),[for all]t>0}

And dente by [P.sub.A].sub.t] (C), the set of all elements of p-best approximation pair to (A,C) i.e.,

[P.sub.[A,C].sup.t] = {(a - c)[euro]AXC:[[upsilon].sub.a - c](t) = [[upsilon].sub.A - C](t),[for all]t>0}

Definition 2.2: A sequence converges sub-sequentially if it has a convergent subsequence. In the above notation [x.sub.n] > [x.sub.n], [right arrow] [x.sub.0] identifies the subsequence and the point to which it converges. Recall that a subset C of an PN space is compact if every sequence in C converges sub-sequentially to an element of C.

Definition 2.3: For a probabilistic normed space X and nonempty subsets A and C a sequence [a.sub.n] [member of] A is said to converge in distance to C if [lim.sub.n[right arrow] [infinity] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (t) = [[upsilon].sub.A-C] (t).

Example 2.4: Let V = R, (V,[??]) be a normed space. For p,q[member of] V define, [upsilon]: V [right arrow] [[DELTA].sup. ] as [[upsilon].sup.p] (t) = [[member of].sub.||p||] (t). For a, b [member of] R.sup. ], [tau] ([[upsilon.sub.a], [[[upsilon].sub.b]) = [[upsilon].sub.a b]. Then it is easy to prove that (V, [upsilon], [tau]) is a PN space. Let A = [0,2], C = [3,4], then for a [member of] A and c [member of] C, [[upsilon].sub.||3-2||] (t) > [[upsilon].sub.||a-c|| (t). So [[upsilon].sub.3-2] (t) = [[Upsilon].Sub.C-A] (t) and [[upsilon].sub.3-A] (t) = [[upsilon].sub.C-A] (t). Hence for each t > 0, 3 is a p-best approximation to A from C. Also (3,2) is a p-best approximation pair relative to (A,C).

Definition 2.5: Let (V, [upsilon], [tau]) be a PN space, the nonempty subset A[subset]V is called p-proximinal (p-quasi Chebyshev) set relative to C if [P.sub..sup.t](C) is nonvoid (compact) for some C[supset]V/A. Also, for nonempty subsets A and C of B, AxC is called p-proximinal (p-quasi Chebyshev), pair relative to (A,C) if [P.sub.A].sup.t].sup.c] is nonvoid (compact).