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The main result can be stated roughly as follows: Let u be a Hilbert-space-valued solution of $u'(t) = Au(t) $, $ - \infty < t < \infty $, where A differs from a symmetric linear operator by a suitable nonlinear perturbation. Then either u is constant or else $\sup _t \| {u(t)} \| = \infty $. More generally, A can depend on t, and the equation $u''(t) = A(t)u(t)$ is also considered.

Let A be a linear, closed, densely defined m-accretive operator from a Banach space X to itself, and let T(t), t ⩾ 0, be the semigroup of operators which has −A as its infinitesimal generator. Let B be a nonlinear, continuous, everywhere defined accretive operator from X to itself, and let S(t), t ⩾ 0, be the semigroup of nonlinear operators which has −B as its infinitesimal generator. It is shown that for all x ϵ X, t ⩾ 0, limn → ∞(T(tn) S(tn))nx = U(t)x exists, U(t)x = T(t)x − ∝0t T(t − s) BU(s) x ds, and U(t), t ⩾ 0, is a strongly continuous semigroup of nonlinear contractions on X. It is shown also that −(A + B) is the infinitesimal generator of U(t), t ⩾ 0, and A + B is m-accretive on X.

Weak nonlinear ion-acoustic waves in a plasma consisting of cold ions and warm electrons are investigated by using a nonlinear perturbation method. In the lowest order of perturbation, it is found that the system of equations for the ion-acoustic waves can be reduced to the following single nonlinear dispersive-dissipative equation: \begin{aligned} \frac{\partial u}{\partial\tau}+u\frac{\partial u}{\partial\xi}+\mu\frac{\partial^{3}u}{\partial\xi^{3}}-\nu\frac{\partial}{\partial\xi}\left(\frac{\partial u}{\partial\tau}+u\frac{\partial u}{\partial\xi}\right){=}0, \end{aligned} where u , ξ and τ are normalized velocity of ion-fluid, stretched space and time coordinates, respectively. The parameter µ is a measure of dispersion due to charge separation, whereas ν is a measure of dissipation due to electron-ion collisions. It is shown that this equation has either oscillatory (dispersion-dominant case) or quasi-monotone (dissipation-dominant case) shock wave solutions.

We study the properties of the solutions of one two-frequency system in the whole space. We establish the correspondence between its solutions and the phase trajectories of an autonomous second-order system obtained from the original one by using its group properties. The original system can be treated as a model for the study of the phenomena taking place in multifrequency systems under resonance. This paper is closely related to the investigations in [1].

It is shown that a strongly continuous semi-group of nonlinear nonexpansive operators can be constructed as lim n→∞ ((I+t/nB)−1 (I+t/nB)−1) n whereA is a linearm-accretive operator,B is a nonlinearm-accretive operator, andB satisfies a boundedness condition relative toA.