Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2013, Article ID 978738, 7 pages
http://dx.doi.org/10.1155/2013/978738
Research Article
New Generalization of -Best Simultaneous Approximation in
Topological Vector Spaces
Mahmoud Rawashdeh and Sarah Khalil
Department of Mathematics and Statistics, Jordan University of Science and Technology, Irbid 22110, Jordan
Correspondence should be addressed to Mahmoud Rawashdeh; msalrawashdeh@just.edu.jo
Received 20 January 2013; Revised 23 April 2013; Accepted 23 April 2013
Academic Editor: Naseer Shahzad
Copyright © 2013 M. Rawashdeh and S. Khalil. his is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly
cited.
Let be a nonempty subset of a Hausdorf topological vector space , and let be a real-valued continuous function on . If for
each =(
1
,
2
,...,
)∈
, there exists
0
∈ such that
()=∑
=1
(
−
0
)= inf {∑
=1
(
−):∈}, then is called
-simultaneously proximal and
0
is called -best simultaneous approximation for in . In this paper, we study the problem
of -simultaneous approximation for a vector subspace in . Some other results regarding -simultaneous approximation in
quotient space are presented.
1. Introduction
Let be a closed subset of a Hausdorf topological vector
space and a real-valued continuous function on . For
∈, set
() = inf
∈
(−). A point
0
∈ is
called -best approximation to in if
()=(−
0
).
he set
() = { ∈ :
() = ( − )} deno-
tes the set of all -best approximations to in . Note
that this set may be empty. he set is said to be -pro-
ximal (-Chebyshev) if for each ∈,
() is nonempty
(singleton). he notion of -best approximation in a vector
space was given by Breckner and Brosowski [1] and in
a Hausdorf topological space by Narang [2, 3]. For a
Hausdorf locally convex topological vector space and a
continuous sublinear functional on , certain results on
best approximation relative to the functional were proved
in [1, 4]. By using the existence of elements of -best
approximation, certain results on ixed points were proved by
Pai and Veermani in [5]. In addition, for a topological vector
space relative to upper semicontinuous functions, some
results on best approximation were proved by Haddadi and
Hamzenejad [6]. Moreover, Naidu [7] proved some results on
best simultaneous approximation related to -nearest point
and topological vector space .
Analogous to the problem of simultaneous approxima-
tion [8], we introduce the concept of best -simultaneous
approximation as follows.
Deinition 1. Let be a non-empty subset of a Hausdorf
topological vector space , and let be a real-valued
continuous function on . A point
0
∈ is called -
best simultaneous approximation in if there exists =
(
1
,
2
,...,
)∈
such that
()= inf {
∑
=1
(
−):∈}=
∑
=1
(
−
0
).
(1)
he set of all -best simultaneous approximations to =
(
1
,
2
,...,
)∈
in is denoted by
()={∈:
()=
∑
=1
(
−)}. (2)
he set is called -simultaneously proximal (-sim-
ultaneously Chebyshev) if for each =(
1
,
2
,...,
)∈
,
() ̸ = (singleton). If =1, simultaneous -proximal is
precisely -proximal.