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Existence theorems for nonlinear evolution equations. (English) Zbl 0642.47055

The author establishes a local existence result for integral solutions to abstract evolution equations in a Banach space X, of the form \[ u'(t)+Au(t)\ni (Bu)(t),\quad t\in [0,T];\quad u(\theta)=u_ 0\in \overline{D(A)}, \] where u:[0,T]\(\to X\), A is m-accretive and B:C([0,T];X)\(\to L\) 1(0,T;X) is a nonlinear operator. The key hypothesis is a compactness assumption for the operator B. Several examples of applications are provided, covering a number of known and some new cases. Furthermore, conditions are given in order the found integral solutions be strong.
Reviewer: P.de Mottoni

MSC:

47H20 Semigroups of nonlinear operators
47E05 General theory of ordinary differential operators
47H06 Nonlinear accretive operators, dissipative operators, etc.
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