International Journal of Theoretical and Applied Mathematics
Volume 4, Issue 3, June 2018, Pages: 29-34
Received: Aug. 27, 2018;
Accepted: Sep. 18, 2018;
Published: Nov. 7, 2018
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Gaj Ram Damai, Department of Mathematics, Tribhuvan University, Siddhnath Science Campus, Mahendranagar, Nepal
Prakash Muni Bajracharya, Central Department of Mathematics, Tribhuvan University, Kirtipur, Kathmandu, Nepal
In this paper, we give some new conditions for the existence and uniqueness of mutually nearest points of two sets, i.e., two points which achieve the minimum distance between two sets in a metric or Banach spaces. These conditions are seen in case of compact sets, weak Compact sets, closed and convex sets, weakly sequentially compact sets and boundedly compact sets and their combinations. The study is confined to metric spaces and normed or Banach spaces. Some geometric properties of a Banach spaces like; strictly convexity, uniformly convexity, P-Property and weak P-Property are introduced. Also, we introduce the concept of generalized weak P-Property and give some interesting results. The present work may be briefly outlined as follows: It is the mathematical study that is motivated by the desire to seek answers to the following basic questions, among others. Which subsets are mutually proximinal? How does one recognize when given elements x ∈ A and y ∈ B are the nearest points of A and B? which is called a natural extension of the best approximation problem. Can one describe some useful algorithms for actually computing nearest points between two given sets? And how to find closely related sets to the proximity maps.
Gaj Ram Damai,
Prakash Muni Bajracharya,
Mutually Nearest Points for Two Sets in Metric Spaces, International Journal of Theoretical and Applied Mathematics.
Vol. 4, No. 3,
2018, pp. 29-34.
P. M. Bajracharya, G. R. Damai, “A note on existence and uniqueness of nearest points in a set", International Journal of Research and Review, (2018), 5(1), 5.-54.
P. M. Bajracharya and G. R. Damai, “On Nearest Points of a Set", GIJASH, (2018) Vol.2, Issue 3, 5-8.
V. R. Sankar and A. A. Eldred, “Characterization of Strictly Convex Spaces and Applications," Journal of Optimization Theory and Applications, (2014) Volume 160, No.2, 703-710.
Xiubin Xu, “A Result on Best Proximity Pair of Two Sets", Journal of Approximation Theory, (1988)54, 322-325.
Xianfa Luo, “Characterizations and Uniqueness of Mutually Nearest Pointsfor Two Sets in Normed Spaces, "Numerical Functional Analysis and Optimization, (2014)35(5), 611-622.
C. C. Adams and R. D. Fran-zosa, “Introduction to topology Pure and applied," Upper Saddle River, NJ, Pearson Prentice Hall, (2008).
J. Zhang, Y. Su, Q. Cheng, “A note on a best proximity point theorem for Geraghty contractions," Fixed Point Theory Appl., 2.13, 99 (2013).
V. R. Shankar, “A best proximity point theorem for weakly contractive non-self mappings," Nonlinear Anal. (2011)74, 4804-4808.
R. Shakarchi, “Problems and Solutions for Undergraduate Analysis," Library of Congress Cataloging in-Publication Data (1998), ISBN 0-387-98235-3.
Chong Li, “On Mutually Nearest and Mutually Furthest Points in Reflexive Banach Spaces," Journal of Approximation Theory, (2000)103, 1-17.
D. V. Pai, “Proximal Points of Convex Sets in normed Linear Spaces," Yokohama Math., (1974)9, 53-78.
D. V. Pai and H. G. ter Morsche, “On proximal points of pairs of sets," T. H.-Report 75-WSK-OI January (1975).
D. V. Pai, “A Characterization of Smooth Normed Linear Spaces,” Journal of Approximation Theory, (1976)17, 315-32.
W. K. Gunt, “An alternating projections method for certain linear problems in a Hilbert space," IMA Journal of Numerical Analysis (1995) 15, 291-305.
Lin-Bor-Luh, “Distance-sets in normed vector spaces”. Nieuw Arch. Wiskunde, (1966) (3)14, 23-3.
J. J. Dionisio, ‘‘On the separation of convex sets in R". Univ. Lisboa Revista Fae., Ci. (A 2), 1.(1963\64), 185-94.
J. W. Tukey, “Some notes on the separation of convex sets,’’ Portugaliae Math., (1942)3, 95-1.2.
V. L. Klee, “Separaticn properties of convex sets,”Proe. Am. math. Soc., (1955)6, 313-17.
D. V. Pai, “Proximinal points of convex sets in normed linear spaces,”Research Report, 1.1. T., Bombay. th. Soc., 6, 3(1972), 13-17.
W. Cheney, and A. A. Goldstein, “Proximity maps for convex sets,”Proc. Am. math. Soc.,(1969)1., 448-5.
M. Nicolescu, “Sur la meilleure approximation d'une fonction donnee par les functions d'une famille donnee,”But. Fae. Sfi., Cern; Uli,(1938)12, 120-280.
I. Singer, “Best Approximation in Normed Linear by Elements of Linear Subs paces,” Springer-Verlag, Berlin-Heideiberg, (1970).