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Evolution equations with lack of convexity. (English) Zbl 0545.46029
Let $\Omega$ be an open subset of a real Hilbert space H, whose norm and scalar product are denoted by $\vert \cdot \vert$ and ($\cdot \vert \cdot)$. If $f:\quad \Omega \to {\bbfR}\cup \{+\infty \}$ is a function, set $$\partial\sp-f(u)=\{\alpha \in H:\lim \quad \inf\sb{v\to u}\frac{f(v)-f(u)-(\alpha \vert v-u)}{\vert v-u\vert}\ge 0\},\quad if\quad f(u)<+\infty;$$ $$\partial\sp-f(u)=\emptyset,\quad if\quad f(u)=+\infty.$$ If f is lower semicontinuous (with respect to the norm topology), f is said to have a $\phi$-monotone subdifferential, if there exists a continuous function $\phi:\quad \Omega \times {\bbfR}\sp 2\to {\bbfR}\sp+$ such that $$(\alpha -\beta \vert u-v)\ge - [\phi(u,f(u),\vert \alpha \vert)+\phi(v,f(v),\vert \beta \vert)]\quad \vert u-v\vert\sp 2$$ whenever $\partial\sp-f(u)\ne \emptyset$, $\partial\sp-f(v)\ne \emptyset$, $\alpha \in \partial\sp-f(u)$, $\beta \in \partial\sp-f(v)$. In this paper some general properties of this class of functions are studied and some theorems of existence, uniqueness, regularity and convergence, concerning the associated evolution equation $U'(t)\in -\partial\sp-f(U(t))$ are proved.

##### MSC:
 46G05 Derivatives, etc. (functional analysis) 46A50 Compactness in topological linear spaces; angelic spaces, etc. 58D25 Differential equations and evolution equations on spaces of mappings 49J27 Optimal control problems in abstract spaces (existence)
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