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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) (0tT), u(0)=u 0 , in a Hilbert space H; A is monotone and hemicontinuous in H and G:VV * , where V is reflexive Banach space with VHV * ; the function f(·) 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(·) belongs to C(0,T;H)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.

34G20Nonlinear ODE in abstract spaces