On a dual characterisation in best approximation problem


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


Ky Fan best approximation problem, fixed point, KKM-mapping, strongly continuous function
Abstract views: 174


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.


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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

Fan, Ky, A generalization of Tychonoff's fixed point theorem, Math. Ann., 142, pp. 305-310, 1961, 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

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

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

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

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

Lin, T.C., Approximation theorems and fixed point theorems in cones, Proc. Amer. Math. Soc., 102, pp. 502-506, 1988.

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

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

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.

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

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




How to Cite

Precupanu, T. (2008). On a dual characterisation in best approximation problem. Rev. Anal. Numér. Théor. Approx., 37(2), 191–196. Retrieved from https://ictp.acad.ro/jnaat/journal/article/view/2008-vol37-no2-art11