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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
Citations:
Zbl 0682.34010
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References:
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