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Monotone trajectories of differential inclusions in Banach spaces. (English) Zbl 0877.34018
Let $$E$$ be a separable Banach space and $$X$$ a nonempty, compact subset of $$E$$. Consider the multivalued operator $$F:[0,T]\times X\to P_{cv,wcp}(E)$$. The paper contains two existence results for the following problem: Given $$x_0$$ in $$X$$, we look for Lipschitz solution of the differential inclusion $w'(t)\in F(t,w(t)),\quad w(0)=x_0,$ which are viable (i.e. $$w(t)\in X$$, for all $$t\in [0,T]$$) and monotone (i.e. for any $$s,t\in [0,t]$$, $$s<t$$ implies $$w(t)\in P(w(s))$$, where $$P(x)= \{y\in X:y\geq x\}$$ defines a preorder on $$X$$).

MSC:
 34A60 Ordinary differential inclusions 49K24 Optimal control problems with differential inclusions (nec./ suff.) (MSC2000)
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