Kantorovich theorem and justification of asymptotics of solutions of differential equations with large high-frequency summands
Consider the vector differential equation \[ {dx\over d\tau}= f(x,\omega\tau)+ \sqrt{\omega}\varphi(x,\omega\tau),\tag{\(*\)} \] where \(\omega\) is a large positive parameter and \(f\) and \(\varphi\) are \(T\)-periodic in the second argument. Let \[ \langle g(y,t)\rangle= {1\over T} \int^T_0 g(y,t)\,dt. \] The author considers, along with \((*)\), the ``averaged'' equation \[ {dy\over d\tau}= \langle f(y,t)\rangle+ \Biggl\langle{\partial\varphi(y, t)\over\partial y} \int^t_0 \varphi(y,s)\,ds\Biggr\rangle\equiv \psi(y). \] Let \(y_0\) be a simple root of \(\Psi(y)= 0\). Under some assumptions on \(f\) and \(\varphi\), there are positive numbers \(\omega_0\) (\(\gg 1\)), \(r_0\), \(c_0\) such that for \(\omega> \omega_0\) the ball \(\| x-y_0\|\leq r_0\) contains a unique \(T\omega^{-1}\)-periodic solution \(x_\omega\) of \((*)\) satisfying \[ \| x_\omega- y_0\|\leq c_0 \omega^{-1/2}. \] The author gives a new proof of this result by using a theorem due to L. V. Kantorovich based on Newton's modified method.
scientific article; zbMATH DE number 5659068