On a dual characterisation in best approximation problem

Authors

  • Teodor Precupanu “Al. I. Cuza” University, Iaşi, Romania

DOI:

https://doi.org/10.33993/jnaat372-891

Keywords:

Ky Fan best approximation problem, fixed point, KKM-mapping, strongly continuous function
Abstract views: 206

Abstract

We establish a dual characterization of solutions of Ky Fan best approximation problem and as consequence we obtain an existence criterium under conditions formulated for the weak topology.

Downloads

Download data is not yet available.

References

Barbu, V. and Precupanu, T., Convexity and Optimizaton in Banach Spaces, D. Reidel Publish. Comp., Dordrecht, Boston, Lancaster, 1986.

Ding, X.P. and Tan, K. K., A set-valued generalization of Fan's best approximation theorem, Canad. J. Math., 44, pp. 784-796, 1992, https://doi.org/10.4153/cjm-1992-046-9 DOI: https://doi.org/10.4153/CJM-1992-046-9

Fan, Ky, A generalization of Tychonoff's fixed point theorem, Math. Ann., 142, pp. 305-310, 1961, https://doi.org/10.1007/bf01353421 DOI: https://doi.org/10.1007/BF01353421

Fan, Ky, Extensions of two fixed point theorems of F.E. Browder, Math. Z., 112, pp. 234-240, 1969, https://doi.org/10.1007/bf01110225 DOI: https://doi.org/10.1007/BF01110225

Fan, Ky, Some properties of convex sets related to fixed point theorems, Math. Ann., 266, pp. 519-539, 1984, https://doi.org/10.1007/bf01458545 DOI: https://doi.org/10.1007/BF01458545

Garkavi, A. I., Duality theorems for approximaton by elements of convex sets (in Russian), Uspehi Mat. Nauk, 16, pp. 141-145, 1961.

Kapoor, O. P., On a intersection lemma, J. Math. Anal. Appl., 45, pp. 354-356, 1974, https://doi.org/10.1016/0022-247x(74)90077-8 DOI: https://doi.org/10.1016/0022-247X(74)90077-8

Knaster, B., Kuratowski, C. and Mazurkiewicz, S., Eine Beweis des fixpunktsatzes für n-dimensionale Simplexe, Fund. Math., 14, pp. 132-137, 1929, https://doi.org/10.4064/fm-14-1-132-137 DOI: https://doi.org/10.4064/fm-14-1-132-137

Laurent, P.J., Approximation and Optimization, Herman, Paris, 1972.

Lin, T.C., A note on a theorem by Ky Fan, Canad. Math. Bull., 22, pp. 513-515, 1979, https://doi.org/10.4153/cmb-1979-067-x DOI: https://doi.org/10.4153/CMB-1979-067-x

Lin, T.C., Approximation theorems and fixed point theorems in cones, Proc. Amer. Math. Soc., 102, pp. 502-506, 1988. DOI: https://doi.org/10.1090/S0002-9939-1988-0928968-X

Lin, T.C. and Yen, C.L., Applications of the proximity map to fixed point theorems in Hilbert space, J. Approx. Theory, 52, pp. 141-148, 1988, https://doi.org/10.1016/0021-9045(88)90053-6 DOI: https://doi.org/10.1016/0021-9045(88)90053-6

Precupanu, T., Duality in best approximation problem, An. Şt. Univ. "Al.I. Cuza" Iaşi, s. Matematică, 26, pp. 23-30, 1980.

Reich, S., Approximate selections, best approximations, fixed points and invariant sets, J. Math. Anal. Appl., 62, pp. 104-113, 1978, https://doi.org/10.1016/0022-247x(78)90222-6 DOI: https://doi.org/10.1016/0022-247X(78)90222-6

Roux, D. and Singh, S.P., On a best approximation theorem, Jnanabha, 19, pp. 1-19, 1989.

Sehgal, V.M. and Singh, S.P. A generalization to multifunctions of Fan's best approximation theorem, Proc. Amer. Math. Soc., 102, pp. 534-537, 1988. DOI: https://doi.org/10.1090/S0002-9939-1988-0928974-5

Singh, S. and Watson, B., Best approximation and fixed point theorems, Proc. NATO-ASI on Approximation Theory, Wavelets and Applications, Kluwer Academic Publish., Dordrecht, Boston, London, pp. 285-294, 1995, https://doi.org/10.1007/978-94-015-8577-4_15 DOI: https://doi.org/10.1007/978-94-015-8577-4_15

Singh, S., Watson, B. and Srivastava, P., Fixed Point Theory and Best Approximation: The KKM-map Principle, Kluwer Academic Publish., Dordrecht, Boston, London, 1997, https://doi.org/10.1007/978-94-015-8822-5 DOI: https://doi.org/10.1007/978-94-015-8822-5

Downloads

Published

2008-08-01

How to Cite

Precupanu, T. (2008). On a dual characterisation in best approximation problem. Rev. Anal. Numér. Théor. Approx., 37(2), 191–196. https://doi.org/10.33993/jnaat372-891

Issue

Section

Articles