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1

Best approximation and fixed points for rational-type contraction mappings

Chandok Sumit

Journal of Applied Analysis

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2019

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Vol. 25, nr 2

205--209

EN

In this paper, we prove a fixed point theorem for a rational type contraction mapping in the frame work of metric spaces. Also, we extend Brosowski-Meinardus type results on invariant approximation for such class of contraction mappings. The results proved extend some of the known results existing in the literature.

2

Approximation of functions and their conjugates in Lp and uniform metric by Euler means

Volosivets S. S., Tyuleneva A. A.

Demonstratio Mathematica

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2018

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Vol. 51, nr 1

141--150

EN

For 2π-periodic functions from Lp (where 1 < p < ∞) we prove an estimate of approximation by Euler means in Lp metric generalizing a result of L. Rempuska and K. Tomczak. Furthermore, we show that this estimate is sharp in a certain sense. We study the uniform approximation of functions by Euler means in terms of their best approximations in p-variational metric and also prove the sharpness of this estimate under some conditions. Similar problems are treated for conjugate functions.

3

Concrete minimal 3 x 3 Hermitian matrices and some general cases

Klobouk A. H., Varela A.

Demonstratio Mathematica

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2017

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Vol. 50, nr 1

330--350

EN

Given a Hermitian matrix M ∈ M₃ (ℂ) we describe explicitly the real diagonal matrices Dм such that ║M + Dм║ ≤ ║M + D║ for all real diagonal matrices D ∈ M3(ℂ), where ║ · ║ denotes the operator norm. Moreover, we generalize our techniques to some n × n cases.

4

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].

5

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.

6

Common fixed point results with applications in convex metric space

Abbas M.

Fasciculi Mathematici

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2008

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Nr 39

6-15

EN

Sufficient conditions for the existence of a common fixed point for uniformly Cq— commuting mappings satisfying a generalized contractive conditions in the framework of a convex metric space are obtained. As an application, related results on best approximation are derived. Our results generalize various known results in the literature.

7

Common fixed point, best approximation and generalized f-weak contraction multivalued mapping in p-normed spaces

Abbas M., Rhoades B.

Demonstratio Mathematica

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2008

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Vol. 41, nr 1

137-144

EN

We establish coincidence point and common fixed point results for multivalued f-weak contraction mappings which assume closed values only. As an application, related common fixed point and invariant approximation are obtained in the setup of certain metrizable topological vector spaces. Our results provide extensions as well as substantial improvements of several well known results in the literature.

8

Invariant approximations noncommuting maps and strongly M-starshaped metric spaces

Naz A.

Demonstratio Mathematica

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2003

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Vol. 36, nr 4

893--898

EN

In this paper, we obtain some results on invariant approximations in strongly M-starshaped metric spaces, which extend some known results.

9

On the structure of the set of best ||.||Φ-approximants

Mazzone F.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

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2002

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[Z] 42(1)

87-102

EN

Certain properties of Fi-approximants and || || (fi)-approximants were studied by Landers and Rogge, where || || (fi) is the Luxemburg norm. In particular, they investigated the existence of best ||.|| (fi)-approximants and the structure of the . set of best ||.||(fi)-approximants. These authors proved that the set of best ||.||(fi)-approximants of f given a Fi-closed lattice C is a lattice. In this paper we show that this result does not hold if we consider the Orlicz norm in place of the Luxemburg norm. Furthermore, we see that for a large class of functions Fi and measurable spaces the following statements are equivalent: 1) the set of all best || || (fi)-approximants to f in C is a lattice, for every Fi-closed lattice C and f L_fi. 2) (L_fi,,||.||fi) = (L_p,m||.||_p), for some m > 0 and 1 < p < oo.

10

Continuity of the metric projection in Orlicz space

Teng Y., Wang T., Li M.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

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1999

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[Z] 39

109-116

EN

Let (f(n)) be a sequence of functions converging in norm to f in some rotund Orlicz function or se-quence space endowed with the Luxemburg norm or the Orlicz norm, and let (C(n)) be a sequence of convex sets satisfying some condition and tending in suitable way to a ser C. Then the best norm approximation of f(n) with respect to C(n) converges in norm to the best approximation of f with respect to C.

11

On some means of double Fourier series

Taberski R.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

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1998

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[Z] 38

139-148

EN

For bivariate periodic functions Lebesgue-integrable on the basic square and corresponding p>0, the rates of Lp - convergence of the means (1), (3) are estimated. The one-dimensional case was considered in Sections 3, 4 of [6].