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Periodic solutions for nonlinear evolution equations in a Banach space. (English) Zbl 0701.34074
Let A: D(A)\(\subseteq X\to 2^ X\) be an operator in a real Banach space X with the property that -A generates a compact semigroup, and A-aI is m- accretive for some \(a>0\). The author proves existence of T-periodic solutions to the nonlinear evolution relation \(u'(t)+Au(t)\ni F(t,u(t))\) \((0\leq t<\infty)\), where \(F: [0,\infty)\times \overline{D(A)}\to X\) is Carathéodory and satisfies \[ \lim_{r\to \infty}(1/r)\sup \{\| F(t,u)\|:\;0\leq t<\infty,\quad \| v\| \leq r\}<a. \]
Reviewer: J.Appell

MSC:
34G20 Nonlinear differential equations in abstract spaces
34C25 Periodic solutions to ordinary differential equations
35K55 Nonlinear parabolic equations
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