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Zbl 0806.34051
Ahmed, N.U.; Xiang, X.
Existence of solutions for a class of nonlinear evolution equations with nonmonotone perturbations.
(English)
[J] Nonlinear Anal., Theory Methods Appl. 22, No.1, 81-89 (1994). ISSN 0362-546X

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 $H$.
[H.O.Fattorini (Los Angeles)]
MSC 2000:
*34G20 Nonlinear ODE in abstract spaces

Keywords: nonlinear differential equations in abstract spaces; Hilbert space; existence

Citations: Zbl 0682.34010

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