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Evolution equations with lack of convexity. (English) Zbl 0545.46029

Let Ω be an open subset of a real Hilbert space H, whose norm and scalar product are denoted by |·| and (·|·). If f:Ω{+} is a function, set

- f(u)={αH:liminf vu f(v)-f(u)-(α|v-u) |v-u|0},iff(u)<+;
- f(u)=,iff(u)=+·

If f is lower semicontinuous (with respect to the norm topology), f is said to have a ϕ-monotone subdifferential, if there exists a continuous function ϕ:Ω× 2 + such that

(α-β|u-v)-[ϕ(u,f(u),|α|)+ϕ(v,f(v),|β|)]|u-v| 2

whenever - f(u), - f(v), α - f(u), β - 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)- - f(U(t)) are proved.

MSC:
46G05Derivatives, etc. (functional analysis)
46A50Compactness in topological linear spaces; angelic spaces, etc.
58D25Differential equations and evolution equations on spaces of mappings
49J27Optimal control problems in abstract spaces (existence)