Continuity of a~multivalued mapping connected with the problem of minimizing a~functional

V. I. Berdyshev

Izvestiya. Mathematics , Tome 16 (1981) no. 3, pp. 431-456

Résumé

Let $X$ and $U$ be locally convex spaces, $\varphi(x,u)$ a proper convex lower semicontinuous functional on $X\times U$ and $t=t(u)\geqslant\inf\{\varphi(x,u)\colon x\in X\}$. This paper gives conditions for the multivalued mapping $$ \Phi_t\colon u\in U\to \Phi_t(u)=\{x\in X\colon\varphi(x,u)\leqslant t\} $$ to be uniformly continuous and satisfy a Lipschitz condition, and determines the relation of $\Phi_t$ with other multivalued mappings, in particular, with a metric projection. On the basis of the functional conjugate to $\varphi$ a mapping conjugate to $\Phi_t$ is introduced and a condition for its upper semicontinuity is presented. The problem of minimizing a homogeneous convex functional on a convex set is considered. Bibliography: 21 titles.