The introduction of the concept of weak non-linearity is motivated by the effort to determine a certain class of non-linear systems whose properties important in technical appliations coincide with those of asymptotically stable linear systems. A model of the phzsical system described by Eq. (1) is called weakly non-linear (quasilinear) if any two solutions $x_1(t), x_2(t)$ of Eq. (1) sarisfy Eq. (2). A model described by Eq. (1) is weakly non-linear if there exists a way of expressing the right-hand side $f(x,t)$ of this equation in the form (3) so that the inequality (8) is satisfied.
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