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Multivalued perturbations of $$m$$-accretive differential inclusions. (English) Zbl 0922.47048
Summary: Given an $$m$$-accretive operator $$A$$ in a Banach space $$X$$ and an upper semicontinuous multivalued map $$F:[0,a]\times X\to 2^X$$, we consider the initial value problem $u'\in -Au+ F(t,u)\quad\text{on }[0,a],\quad u(0)= x_0.$ We concentrate on the case when the semigroup generated by $$-A$$ is only equicontinuous and obtain existence of integral solutions if, in particular, $$X^*$$ is uniformly convex and $$F$$ satisfies $\beta(F(t, B))\leq k(t)\beta(B)\quad\text{for all bounded }B\subset X,$ where $$k\in L^1([0,a])$$ and $$\beta$$ denotes the Hausdorff-measure of noncompactness. Moreover, we show that the set of all solutions is a compact $$R_\delta$$-set in this situation. In general, the extra condition on $$X^*$$ is essential as we show by an example in which $$X$$ is not uniformly smooth and the set of all solutions is not compact, but it can be omitted if $$A$$ is single-valued and continuous or – $$A$$ generates a $$C_0$$-semigroup of bounded linear operators.
In the simpler case when – $$A$$ generates a compact semigroup, we give a short proof of existence of solutions, again if $$X^*$$ is uniformly (or strictly) convex. In this situation, we also provide a counterexample in $$\mathbb{R}^4$$ in which no integral solution exists.

##### MSC:
 47H06 Nonlinear accretive operators, dissipative operators, etc. 47E05 General theory of ordinary differential operators 47D06 One-parameter semigroups and linear evolution equations
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