Abstract
In this paper we study the following abstract second order differential equation with dissipation in a Hilbert space H: u″+cu′+dA u+kG(u)=P(t), u∈H, t∈R, where c, d and k are positive constants, G:H→H is a Lipschitzian function and P:R→H is a continuous and bounded function. A:D(A)⊂H→H is an unbounded linear operator which is self-adjoint, positive definite and has compact resolvent. Under these conditions we prove that for some values of d, c and k this system has a bounded solution which is exponentially asymptotically stable. Moreover; if P(t) is almost periodic, then this bounded solution is also almost periodic. These results are applied to a very well known second order system partial differential equations; such as the sine-Gordon equation, The suspension bridge equation proposed by Lazer and McKenna, etc.
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