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1

On simultaneous strong proximinality

Gupta Sahil, Narang T.D.

Fasciculi Mathematici

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2021

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nr 65

47--55

EN

In this paper, we extend the notions of simultaneous strong proximinality and simultaneous strong Chebyshevity available in Banach spaces to metric spaces and prove that if W is a simultaneously approximatively compact subset of a metric space (X, d) then W is simultaneously strongly proximinal. The converse holds if the set of all best simultaneous approximations to every bounded subset S of X from W is compact. We show that simultaneously strongly Chebyshev sets are precisely the sets which are simultaneously strongly proximinal and simultaneously Chebyshev. How simultaneous strong proximinality is transmitted to and from quotient spaces has also been discussed when the underlying spaces are metric linear spaces.

2

Best approximation in metric spaces

Narang T. D., Gupta S.

Fasciculi Mathematici

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2017

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Nr 58

113--121

EN

The aim of this paper is to prove some results on the existence and uniqueness of elements of best approximation and continuity of the metric projection in metric spaces. For a subset M of a metric space (X, d), the nature of set of those points of X which have at most one best approximation in M has been discussed. Some equivalent conditions under which an M-space is strictly convex have also been given in this paper.

3

Invariant points of best approximation and best simultaneous approximation

Chandok S., Narang T.D.

Fasciculi Mathematici

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2013

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Nr 51

61-70

EN

In this paper we generalize and extend Brosowski-Meinardus type results on invariant points from the set of best approximation to the set of best simultaneous approximation, which is not necessarily starshaped. As a consequence some results on best approximation are deduced. The proved results extend and generalize some of the results of R. N. Mukherjee and V. Verma [Publ. de l’Inst. Math. 49(1991) 111-116], T.D. Narang and S. Chandok [Selcuk J. Appl. Math. 10(2009) 75-80; Indian J. Math. 51(2009) 293-303], and of G. S. Rao and S. A. Mariadoss [Serdica-Bulgaricae Math. Publ. 9(1983) 244-248].

4

Common fixed points of nonexpansive mappings with applications to best and best simultaneous approximation

Chandok S., Narang T.D.

Journal of Applied Analysis

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2012

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Vol. 18, nr 1

33-46

EN

We prove some new results on the existence of common fixed points for nonexpansive and asymptotically nonexpansive mappings in the framework of convex metric spaces. We also obtain some results on common fixed points from the set of best and best simultaneous approximations as applications. The proved results generalize and extend some of the known results in the literature.