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Existence and uniqueness of a solution for a minimization problem with a generic increasing function

Part of: Existence theories

Published online by Cambridge University Press:  09 April 2009

A. M. Rubinov  and
A. J. Zaslavski
A. M. Rubinov
Affiliation:
School of Information Technology and Mathematical Sciences University of Ballarat Ballarat VIC 3353 Australia e-mail: amr@ballarat.edu.au
A. J. Zaslavski
Affiliation:
Department of Mathematics The Technion-Israel Institute of Technology 32000 Haifa Israel e-mail: ajzasl@techunix.technion.ac.il
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Abstract

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In this paper we study the existence and uniqueness of a solution for minimization problems with generic increasing functions in an ordered Banach space X. The standard approaches are not suitable in such a setting. We propose a new type of perturbation adjusted for the problem under consideration, prove the existence and point out sufficient conditions providing the uniqueness of a solution. These results are proved by assuming that the space X enjoys the following property: each decreasing norm-bounded sequence has a limit. We supply a counterexample, which shows that this property is essential and give a modification of obtained results for the space C(T), which does not possess this property.

Keywords

Increasing functionoptimizationgeneric propertyordered Banach space.

MSC classification

Secondary: 49J27: Problems in abstract spaces 90C30: Nonlinear programming
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 67 , Issue 1 , August 1999 , pp. 85 - 103
Copyright
Copyright © Australian Mathematical Society 1999

References

[1]Beer, G. and Lucchetti, R., ‘Convex optimization and the epi-distance topology’, Trans. Amer. Math. Soc. 327 (1991), 795–813.CrossRefGoogle Scholar
[2]De Blasi, F. S. and Myjak, J., ‘Sur la convergence des approximations successives pour les contractions non linéaires dans un espace de Banach’, C. R. Acad. Sc. Paris 283 (1976), 185–187.Google Scholar
[3]De Blasi, F. S. and Myjak, J., ‘Generic flows generated by continuous vector fields in Banach spaces’, Adv. Math. 50 (1983), 266–280.CrossRefGoogle Scholar
[4]Deville, R., Godefroy, G. and Zizler, V., ‘A smooth variational principle with applications to Hamiltion-Jacobi equations in infinite dimensions’, J. Funct. Anal. 111 (1993), 197–212.Google Scholar
[5]Ioffe, A. D. and Zaslavski, A. J., ‘Variational principles and well-posedness in optimization and calculus of variations’, SIAM J. Control Optimiz., to appear.Google Scholar
[6]Krasnosel'skii, M. A., Positive solutions of operator equations (Noordhof, Leyden, 1964).Google Scholar
[7]Reich, S. and Zaslavski, A. J., ‘Convergence of generic infinite products of nonexpansive and uniformly continuous operators’, Nonlinear Anal., to appear.Google Scholar
[8]Schaefer, H. H., Banach lattices and positive operators (Springer, Berlin, 1974).Google Scholar
[9]Zaslavski, A. J., ‘Existence of solutions of optimal control problems for a generic integrand without convexity assumptions’, Nonlinear Anal., to appear.Google Scholar