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Multivalued perturbations ofm-accretive differential inclusions

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Given anm-accretive operatorA in a Banach spaceX and an upper semicontinuous multivalued mapF: [0,aX→2X, we consider the initial value problemu′∈−Au+F(t,u) on [0,a],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 andF satisfies β(F(t,B))k(t)β(B) for all boundedBX wherekL 1([0,a]) and β denotes the Hausdorff-measure of noncompactness. Moreover, we show that the set of all solutions is a compactR δ-set in this situation. In general, the extra condition onX* is essential as we show by an example in whichX is not uniformly smooth and the set of all solutions is not compact, but it can be omited ifA is single-valued and continuous or—A generates aC o-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 ifX* is uniformly (or strictly) convex. In this situation we also provide a counter-example in ℝ4 in which no integral solution exists.

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Correspondence to Dieter Bothe.

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The author gratefully acknowledges financial support by DAAD within the scope of the French-German project PROCOPE.

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Bothe, D. Multivalued perturbations ofm-accretive differential inclusions. Israel J. Math. 108, 109–138 (1998).

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