A random variational principle with application to weak Hadamard differentiability of convex integral functionals

De Blasi, Francesco; Georgiev, Pando

doi:10.1090/S0002-9939-01-05990-1

A random variational principle with application to weak Hadamard differentiability of convex integral functionals
HTML articles powered by AMS MathViewer

by Francesco S. De Blasi and Pando Gr. Georgiev PDF

Proc. Amer. Math. Soc. 129 (2001), 2253-2260 Request permission

Abstract:

We present a random version of the Borwein-Preiss smooth variational principle, stating that under suitable conditions, the set of minimizers of a perturbed function depending on a random variable, admits a measurable selection. Two applications are given. The first one shows that if $E$ is a superreflexive Banach space, then any convex continuous integral functional on $L^1(T, \mu ; E)$ from a certain class (in particular the usual $L^1$ norm), is weak Hadamard differentiable on a subset whose complement is $\sigma$-very porous. The second application is a random version of the Caristi fixed point theorem for multifunctions.

References

Jean-Pierre Aubin and Ivar Ekeland, Applied nonlinear analysis, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1984. A Wiley-Interscience Publication. MR 749753
Jonathan M. Borwein and Simon Fitzpatrick, A weak Hadamard smooth renorming of $L_1(\Omega ,\mu )$, Canad. Math. Bull. 36 (1993), no. 4, 407–413. MR 1245313, DOI 10.4153/CMB-1993-055-5
J. M. Borwein and D. Preiss, A smooth variational principle with applications to subdifferentiability and to differentiability of convex functions, Trans. Amer. Math. Soc. 303 (1987), no. 2, 517–527. MR 902782, DOI 10.1090/S0002-9947-1987-0902782-7
Robert Deville, Gilles Godefroy, and Václav Zizler, A smooth variational principle with applications to Hamilton-Jacobi equations in infinite dimensions, J. Funct. Anal. 111 (1993), no. 1, 197–212. MR 1200641, DOI 10.1006/jfan.1993.1009
Robert Deville, Gilles Godefroy, and Václav Zizler, Smoothness and renormings in Banach spaces, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 64, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1993. MR 1211634
J. Diestel and J. J. Uhl Jr., Vector measures, Mathematical Surveys, No. 15, American Mathematical Society, Providence, R.I., 1977. With a foreword by B. J. Pettis. MR 0453964, DOI 10.1090/surv/015
I. Ekeland, On the variational principle, J. Math. Anal. Appl. 47 (1974), 324–353. MR 346619, DOI 10.1016/0022-247X(74)90025-0
Ivar Ekeland, Nonconvex minimization problems, Bull. Amer. Math. Soc. (N.S.) 1 (1979), no. 3, 443–474. MR 526967, DOI 10.1090/S0273-0979-1979-14595-6
C. J. Himmelberg, Measurable relations, Fund. Math. 87 (1975), 53–72. MR 367142, DOI 10.4064/fm-87-1-53-72
A. Kolmogorov, C. Fomin, Elements of function theory, Moscow, Nauka (1987).
K. Kuratowski and C. Ryll-Nardzewski, A general theorem on selectors, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 13 (1965), 397–403 (English, with Russian summary). MR 188994
Robert R. Phelps, Convex functions, monotone operators and differentiability, Lecture Notes in Mathematics, vol. 1364, Springer-Verlag, Berlin, 1989. MR 984602, DOI 10.1007/BFb0089089
David Preiss, R. R. Phelps, and I. Namioka, Smooth Banach spaces, weak Asplund spaces and monotone or usco mappings, Israel J. Math. 72 (1990), no. 3, 257–279 (1991). MR 1120220, DOI 10.1007/BF02773783