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Existence and decay of solutions of an abstract second order nonlinear problem. (English) Zbl 1179.34064
This paper is concerned with the global existence and behavior of solutions to initial value problems of the form
$u''(t)+Au(t)+B^\alpha u'(t) =0; \quad u(0)=u_0, u'(0)=u_1,$ where $$A$$ is a monotone nonlinear operator, $$B$$ is a linear operator and $$\alpha$$ is a real number with $$0<\alpha \leq 1$$. The conditions are stated in terms of the monotonicity, differentiability and other properties of $$A$$.

##### MSC:
 34G20 Nonlinear differential equations in abstract spaces 34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
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##### References:
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