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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) $u'(t) + Au(t) + G(u(t)) = f(t)$ $(0 \le t \le T)$, $u(0) = u\sb 0$, in a Hilbert space $H$; $A$ is monotone and hemicontinuous in $H$ and $G:V \to V\sp*$, where $V$ is reflexive Banach space with $V \hookrightarrow H \hookrightarrow V\sp*$; the function $f(\cdot)$ belongs to $L\sp q (0,T;V\sp*)$ for some $q>1$. The main result (under several additional hypotheses) is an existence theorem where the solution $u(\cdot)$ belongs to $C(0,T;H) \cap L\sp p (0,T;V)$ with $1/p + 1/q = 1$. This result generalizes previous work of {\it N. Hirano} [Nonlinear Anal., Theory Methods Appl. 13, No. 6, 599-609 (1989; Zbl 0682.34010)], where the range of $G$ belongs to $H$.

34G20Nonlinear ODE in abstract spaces
Full Text: DOI
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