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Geometric Variational Principle

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 Abstract: We improve and extend the Ekeland variational principle weakening simultaneously its assumptions. In particular, we prove a geometrical result which combines the Ekeland result with the classical theorem of Weierstrass.
Twórcy
  • Institute of Mathematics, Łódź Technical University, Żwirki 36, 90-924 Łódź, Poland
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Bibliografia
[1] J.-P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhäuser, Boston, 1991.
[2] F. H. Clarke, Optimization and Nonsmooth Analysis, Wiley-Interscience, New York, 1983.
[3] J. Caristi and W. A. Kirk, Mapping theorems in metric and Banach spaces, Bull. Acad. Polon. Sci. 23 (1975), 891-894.
[4] L. de Haan, On Regular Variation and Its Application to the Weak Convergence of Sample Extremes, Mathematical Centre Tracts 32, Amsterdam, 1970.
[5] I. Ekeland, Sur les problèmes variationnels, C. R. Acad. Sci. Paris 275 (1972), 1057-1059.
[6] I. Ekeland, Nonconvex minimization problems, Bull. Amer. Math. Soc. 1 (1979), 443-474.
[7] D. G. de Figueiredo, The Ekeland Variational Principle with Applications and Detours, Springer, Berlin, 1989.
[8] L. Gajek and D. Zagrodny, Existence of maximal points with respect to ordered bipreference relations, J. Optim. Theory Appl. 70 (1992), 355-364.
[9] L. Gajek and D. Zagrodny, Countably orderable sets and their applications in optimization, Optimization 26 (1992), 287-301.
[10] L. Gajek and D. Zagrodny, Weierstrass theorem for monotonically semicontinuous functions, Optimization, to appear.
[11] L. Gajek and D. Zagrodny, Geometric mean value theorems for the Dini derivative, J. Math. Anal. Appl., to appear.
[12] J. Jahn, Existence theorems in vector optimization, J. Optim. Theory Appl. 50 (1986), 397-405.
[13] P. Loridan, ε-Solutions in vector minimization problems, ibid. 43 (1984), 265-276.
[14] J.-P. Penot, The drop theorem, the petal theorem and Ekeland's variational principle, Nonlinear Anal. 10 (1968), 813-822.
[15] R. R. Phelps, Convex Functions, Monotone Operators and Differentiability, Lecture Notes in Math. 1364, Springer, Berlin, 1989.
[16] S. Rolewicz, On a norm scalarization in infinite dimensional Banach space, Control Cybernet. 4 (1975), 85-89.
[17] S. Rolewicz, Metric Linear Spaces, 2nd enlarged ed., PWN-Polish Scientific Publishers and D. Reidel, Warszawa-Dordrecht, 1984.
[18] D. Zagrodny, Approximate mean value theorem for upper subderivatives, Nonlinear Anal. 12 (1988), 1413-1428.
Kolekcja
DML-PL
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