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Existence of solutions for a class of nonlinear evolution equations with nonmonotone perturbations. (English) Zbl 0806.34051
The authors study the initial value problem (1) $$u'(t) + Au(t) + G(u(t)) = f(t)$$ $$(0 \leq t \leq T)$$, $$u(0) = u_ 0$$, in a Hilbert space $$H$$; $$A$$ is monotone and hemicontinuous in $$H$$ and $$G:V \to V^*$$, where $$V$$ is reflexive Banach space with $$V \hookrightarrow H \hookrightarrow V^*$$; the function $$f(\cdot)$$ belongs to $$L^ q (0,T;V^*)$$ for some $$q>1$$. The main result (under several additional hypotheses) is an existence theorem where the solution $$u(\cdot)$$ belongs to $$C(0,T;H) \cap L^ p (0,T;V)$$ with $$1/p + 1/q = 1$$. This result generalizes previous work of N. Hirano [Nonlinear Anal., Theory Methods Appl. 13, No. 6, 599-609 (1989; Zbl 0682.34010)], where the range of $$G$$ belongs to $$H$$.

MSC:
 34G20 Nonlinear differential equations in abstract spaces
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References:
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