Kosmol, P. and Muller-Wichards, D. (2009) *Strong Solvability in Orlicz Space.* Հայաստանի ԳԱԱ Տեղեկագիր. Մաթեմատիկա, 44 (5). pp. 27-70. ISSN 00002-3043

## Abstract

The main objective of the authors is to characterize strong solvability of optimization problems where convergence of the values to the optimum already implies norm-convergence of the approximations to the minimal solution. It turns out that strong solvability can be geometrically characterized by the local uniform convexity of the corresponding convex functional (local uniform convexity being appropriately defined). For bounded functionals we establish that in reflexive Banach spaces strong solvability is characterized by the Frechetdifferentiability of the convex conjugate. These results are based in part on a paper of Asplund and Rockafellar on the duality of A-differentiability and Bconvexity of conjugate pairs of convex functions, where B is the polar of A. Before we apply these results to Orlicz spaces, we turn to E-spaces introduced by Fan and Glicksberg. Using the properties of E-spaces we can show that for finite not purely atomic measures Frechet differentiability of an Orlicz space already implies its reflexivity. The main theorem gives - in 17 equivalent statements - a characterization of strong solvability, local uniform convexity, and Frechet differentiability of the dual space, in case Լփ is reflexive. It is remarkable that all these properties can also be equivalently expressed by the differentiability of Փ or the strict convexity of Փ. In particular, LՓ is an E-space, if LՓ is reflexive and Փ is strictly convex.

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