On lower semicontinuity of the restricted center multifunction

Devidas Pai

doi:10.33993/jnaat381-904

Authors

Devidas Pai Indian Institute of Technology Bombay, India

DOI:

https://doi.org/10.33993/jnaat381-904

Keywords:

restricted center, restricted center multifunction, lower semicontinuity of multifunction

Abstract views: 196

Abstract

Given a finite dimensional subspace \(V\) and a certain family \({\mathcal F}\) of nonempty closed and bounded subsets of \( {\mathcal C}_0(T,U)\), where \(T\) is a locally compact Hausdorff space and \(U\) is a strictly convex Banach space, we investigate here lower semicontinuity of the restricted center multifunction \(C_{V}:{\mathcal F} \rightrightarrows V.\) In particular, we establish a Haar-like intrinsic characterization of finite dimensional subspaces \(V\) of \({\mathcal C}_0(T,U)\) which yields lower semicontinuity of \(C_{V}.\)

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References

Beer, G. and Pai, D., On convergence of convex sets and relative Chebyshev centers, J. Approx. Theory, 62, pp. 147-169, 1990, https://doi.org/10.1016/0021-9045(90)90029-p DOI: https://doi.org/10.1016/0021-9045(90)90029-P

Blatter, J., Morris P.D. and Wulbert D.E., Continuity of the set-valued metric projection, Math. Ann, 178 , pp. 12-24, 1968, https://doi.org/10.1007/bf01350621 DOI: https://doi.org/10.1007/BF01350621

Blatter, J. and Schumaker, L., The set of continuous selections of a metric projection in C(X),, J. Approx. Theory, 36, pp. 141-155, 1982, https://doi.org/10.1016/0021-9045(82)90061-2 DOI: https://doi.org/10.1016/0021-9045(82)90061-2

Brosowski, B. and Wegmann, R., On the lower semicontinuity of the set-valued metric projection, J. Approx. Theory, 8, pp. 84-100, 1973, https://doi.org/10.1016/0021-9045(73)90033-6 DOI: https://doi.org/10.1016/0021-9045(73)90033-6

Brown, A.L., On continuous selections for metric projections in spaces of continuous functions, J. Funct. Anal., 8, pp. 431-449, 1971, https://doi.org/10.1016/0022-1236(71)90005-x DOI: https://doi.org/10.1016/0022-1236(71)90005-X

Deutsch, F., Continuous selections for metric projections: Some recent progress, in "Approximation Theory V", Chui, C.K., Schumaker, L.L. and Ward, J.D. (eds.), pp. 319-322, Academic Press, New York, 1986.

Deutsch, F., An exposition of recent results on continuous metric selections, in "Numerical Methods of Approximation Theory", Collatz, L., Meinardus, G. and Nurnberger, G. (eds.), ISNM 81, pp. 67-80, Birkhauser Verlag, Basel, 1987. DOI: https://doi.org/10.1007/978-3-0348-6656-9_6

Indira K. and Pai, D., Hausdorff strong uniqueness in simultaneous approximation. Part I, in "Approximation Theory XI: Gatlinburg 2004", Chui, C.K, Neamtu, M. and Schumaker, L.L. (eds.), pp. 101-118, Nashboro Press, Brentwood, TN, USA, 2005.

Lazar, A.J., Morris, P.D. and Wulbert, D.E., Continuous selections for metric projections, J. Funct. Anal., 3, pp. 193-216, 1969, https://doi.org/10.1016/0022-1236(69)90040-8 DOI: https://doi.org/10.1016/0022-1236(69)90040-8

Li, W., Strong uniqueness and Lipschitz continuity of metric projections: A generalization of the Classical Haar theory, J. Approx. Theory, 56 , pp. 164-184, 1989, https://doi.org/10.1016/0021-9045(89)90108-1 DOI: https://doi.org/10.1016/0021-9045(89)90108-1

Li, W., Various continuities of metric projection in C₀(T,X), J. Approx. Theory, 57, pp. 150-168, 1989, https://doi.org/10.1016/0021-9045(89)90053-1 DOI: https://doi.org/10.1016/0021-9045(89)90053-1

Li, W., An intrinsic characterization of lower semicontinuity of the metric projection in C₀(T,X), J. Approx. Theory, 57, pp. 136-149, 1989, https://doi.org/10.1016/0021-9045(89)90052-x DOI: https://doi.org/10.1016/0021-9045(89)90052-X

Li, W., Continuous Selections for Metric Projection and Interpolating Subspaces, in "Approximation & Optimization", Vol. 1, Lang, P., Fankfurt am Main, 1991.

Mach, J., Continuity properties of Chebyshev centers, J. Approx. Theory, 29, pp. 223-230, 1980, https://doi.org/10.1016/0021-9045(80)90127-6 DOI: https://doi.org/10.1016/0021-9045(80)90127-6

Mhaskar, H.N. and Pai, D.V., Fundamentals of Approximation Theory, CRC Press, Boca Raton, Florida, 2000.

Nurnberger, G. and Sommer, M., Weak Chebyshev subspaces and continuous selections for the metric projection, Trans. Amer. Math. Soc., 238, pp. 129-138, 1978, https://doi.org/10.1090/s0002-9947-1978-0482912-9 DOI: https://doi.org/10.1090/S0002-9947-1978-0482912-9

Pai, D.V., Strong uniqueness of best simultaneous approximation, J. Indian Math. Soc. (N.S.), 67, pp. 201-215, 2000.

Pai, D.V. and Indira, K., On well-posedness of some problems in approximation theory, in "Advances in Constructive Approximation", Neamtu, M. and Saff, E.B. (eds.), pp. 371-392, Nashboro Press, Brentwood, TN, 2004.

Pai, D. and Indira, K., Hausdorff strong uniqueness in simultaneous approximation. Part II, in "Frontiers in Interpolation and Approximation", Govil, N.K. Mhaskar, H.N., Mohapatra, R.N., Nashed, Z.and Szabados, J. (eds.), pp. 365-380, Chapman & Hall/CRC, Taylor & Francis Group, Boca Raton, USA, 2007.

Pai, D.V. and Nowroji, P.T., On restricted centers of sets, J. Approx. Theory, 66, pp. 170-189, 1991, https://doi.org/10.1016/0021-9045(91)90119-u DOI: https://doi.org/10.1016/0021-9045(91)90119-U

Panda, B.B. and Kapoor, O.P., On farthest points of sets, J. Math. Anal. Appl., 62, pp. 345-353, 1978, https://doi.org/10.1016/0022-247x(78)90131-2 DOI: https://doi.org/10.1016/0022-247X(78)90131-2

Zuhovickii, S.I. and Steckin, S.B., On the approximation of abstract functions, Amer. Math. Soc. Transl., 16, pp. 401-406, 1960, https://doi.org/10.1090/trans2/016/20 DOI: https://doi.org/10.1090/trans2/016/20