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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 (should also be assigned at least one other classification number in Section 47-XX)
47D06 One-parameter semigroups and linear evolution equations
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