Best approximation for nonconvex set in q -normed space

Hemant Kumar Nashine

Archivum Mathematicum (2006)

  • Volume: 042, Issue: 1, page 51-58
  • ISSN: 0044-8753

Abstract

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Some existence results on best approximation are proved without starshaped subset and affine mapping in the set up of q -normed space. First, we consider the closed subset and then weakly compact subsets for said purpose. Our results improve the result of Mukherjee and Som (Mukherjee, R. N., Som, T., A note on an application of a fixed point theorem in approximation theory, Indian J. Pure Appl. Math. 16(3) (1985), 243–244.) and Jungck and Sessa (Jungck, G., Sessa, S., Fixed point theorems in best approximation theory, Math. Japonica 42(2) (1995), 249–252.) and some known results (Dotson,W. G., Jr., On fixed point of nonexpansive mappings in nonconvex sets, Proc. Amer. Math. Soc. 38(1) (1973), 155–156.), (Latif, A., A result on best approximation in p-normed spaces, Arch. Math. (Brno) 37 (2001), 71–75.), (Nashine,H. K., Common fixed point for best approximation for semi-convex structure, Bull. Kerala Math. Assoc. (communicated).) are obtained as consequence. To achieve our goal, we have introduced a property known as “Property(A)”.

How to cite

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Nashine, Hemant Kumar. "Best approximation for nonconvex set in $q$-normed space." Archivum Mathematicum 042.1 (2006): 51-58. <http://eudml.org/doc/249793>.

@article{Nashine2006,
abstract = {Some existence results on best approximation are proved without starshaped subset and affine mapping in the set up of $q$-normed space. First, we consider the closed subset and then weakly compact subsets for said purpose. Our results improve the result of Mukherjee and Som (Mukherjee, R. N., Som, T., A note on an application of a fixed point theorem in approximation theory, Indian J. Pure Appl. Math. 16(3) (1985), 243–244.) and Jungck and Sessa (Jungck, G., Sessa, S., Fixed point theorems in best approximation theory, Math. Japonica 42(2) (1995), 249–252.) and some known results (Dotson,W. G., Jr., On fixed point of nonexpansive mappings in nonconvex sets, Proc. Amer. Math. Soc. 38(1) (1973), 155–156.), (Latif, A., A result on best approximation in p-normed spaces, Arch. Math. (Brno) 37 (2001), 71–75.), (Nashine,H. K., Common fixed point for best approximation for semi-convex structure, Bull. Kerala Math. Assoc. (communicated).) are obtained as consequence. To achieve our goal, we have introduced a property known as “Property(A)”.},
author = {Nashine, Hemant Kumar},
journal = {Archivum Mathematicum},
keywords = {Best approximation; demiclosed mapping; fixed point; $I$-nonexpansive mapping; $q$-normed space; best approximation; demiclosed mapping; fixed point},
language = {eng},
number = {1},
pages = {51-58},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Best approximation for nonconvex set in $q$-normed space},
url = {http://eudml.org/doc/249793},
volume = {042},
year = {2006},
}

TY - JOUR
AU - Nashine, Hemant Kumar
TI - Best approximation for nonconvex set in $q$-normed space
JO - Archivum Mathematicum
PY - 2006
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 042
IS - 1
SP - 51
EP - 58
AB - Some existence results on best approximation are proved without starshaped subset and affine mapping in the set up of $q$-normed space. First, we consider the closed subset and then weakly compact subsets for said purpose. Our results improve the result of Mukherjee and Som (Mukherjee, R. N., Som, T., A note on an application of a fixed point theorem in approximation theory, Indian J. Pure Appl. Math. 16(3) (1985), 243–244.) and Jungck and Sessa (Jungck, G., Sessa, S., Fixed point theorems in best approximation theory, Math. Japonica 42(2) (1995), 249–252.) and some known results (Dotson,W. G., Jr., On fixed point of nonexpansive mappings in nonconvex sets, Proc. Amer. Math. Soc. 38(1) (1973), 155–156.), (Latif, A., A result on best approximation in p-normed spaces, Arch. Math. (Brno) 37 (2001), 71–75.), (Nashine,H. K., Common fixed point for best approximation for semi-convex structure, Bull. Kerala Math. Assoc. (communicated).) are obtained as consequence. To achieve our goal, we have introduced a property known as “Property(A)”.
LA - eng
KW - Best approximation; demiclosed mapping; fixed point; $I$-nonexpansive mapping; $q$-normed space; best approximation; demiclosed mapping; fixed point
UR - http://eudml.org/doc/249793
ER -

References

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  1. Brosowski B., Fixpunktsatze in der Approximationstheorie, Mathematica (Cluj) 11 (1969), 165–220. (1969) MR0277979
  2. Carbone A., Some results on invariant approximation, Internat. J. Math. Math. Soc. 17(3) (1994), 483–488. (1994) Zbl0813.47067MR1277733
  3. Dotson W. G., Fixed point theorems for nonexpasive mappings on starshaped subsets of Banach space, J. London Math. Soc. 4(2) (1972), 408–410. (1972) MR0296778
  4. Dotson W. G., Jr., On fixed point of nonexpansive mappings in nonconvex sets, Proc. Amer. Math. Soc. 38(1) (1973), 155–156. (1973) MR0313894
  5. Hicks T. L., Humpheries M. D., A note on fixed point theorems, J. Approx. Theory 34 (1982), 221–225. (1982) MR0654288
  6. Jungck G., An iff fixed point criterion, Math. Mag. 49(1) (1976), 32–34. (1976) Zbl0314.54054MR0433425
  7. Jungck G., Sessa S., Fixed point theorems in best approximation theory, Math. Japonica 42(2) (1995), 249–252. (1995) Zbl0834.54026MR1356383
  8. Köthe G., Topological vector spaces I, Springer-Verlag, Berlin 1969. (1969) MR0248498
  9. Latif A., A result on best approximation in p-normed spaces, Arch. Math. (Brno) 37 (2001), 71–75. Zbl1068.41055MR1822766
  10. Meinardus G., Invarianze bei linearen Approximationen, Arch. Rational Mech. Anal. 14 (1963), 301–303. (1963) MR0156143
  11. Mukherjee R. N., Som T., A note on an application of a fixed point theorem in approximation theory, Indian J. Pure Appl. Math. 16(3) (1985), 243–244. (1985) Zbl0606.41048MR0785288
  12. Nashine H. K., Common fixed point for best approximation for semi-convex structure, Bull. Kerala Math. Assoc. (communicated). 
  13. Park S., Fixed points of f-contractive maps, Rocky Mountain J. Math. 8(4) (1978), 743–750. (1978) Zbl0398.54030MR0513947
  14. Sahab S. A., Khan M. S., Sessa S., A result in best approximation theory, J. Approx. Theory 55 (1988), 349–351. (1988) Zbl0676.41031MR0968941
  15. Singh S. P., An application of a fixed point theorem to approximation theory, J. Approx. Theory 25 (1979), 89–90. (1979) Zbl0399.41032MR0526280
  16. Singh S. P., Application of fixed point theorems to approximation theory, in: V. Lakshmikantam (Ed.), Applied Nonlinear Analysis, Academic Press, New York 1979. (1979) MR0537550
  17. Singh S. P., Some results on best approximation in locally convex spaces, J. Approx. Theory 28 (1980), 329–332. (1980) Zbl0444.41018MR0589988
  18. Singh S. P., Watson B., Srivastava P.,, Fixed point theory and best approximation: The KKM-map principle, Vol. 424, Kluwer Academic Publishers 1997. (1997) Zbl0901.47039MR1483076
  19. Smoluk A., Invariant approximations, Mat. Stos. 17 (1981), 17–22 [in Polish]. (1981) MR0658256
  20. Subrahmanyam P. V., An application of a fixed point theorem to best approximations, J. Approx. Theory 20 (1977), 165–172. (1977) MR0445195
  21. Opial Z., Weak convergence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc. 73 (1967), 531–537. (1967) MR0211301

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