Averaging certain quasidifferential equations (Q1569762)

The differential equation \[ \lim_{\Delta \to 0}\frac{1}{\Delta}d(x(\tau + \Delta),g(\Delta, \tau, x(\tau))) = 0 , \] where \(d(. , .)\) is a distance in the locally compact metric space \(W\) and \(g(. , . , .)\) is a mapping in \(W\), is called the quasidifferential equation of a dynamical system. The notion of the quasidifferential equations allows to consider ordinary differential equations, generalized differential equations and differential equations with Hukuhara derivatives from one point of view. The quasidifferential equation of the form \[ d(x(t+\Delta, \varepsilon), q(\Delta,t,\varepsilon t,x(t,\varepsilon),\varepsilon))=o(\varepsilon,\Delta), \] where \(\varepsilon \geq 0\) is a small parameter and the mapping \(g(\varepsilon \Delta,t,x(t, \varepsilon), \varepsilon)= q(\Delta, t, \varepsilon t,x(t, \varepsilon),\varepsilon) \) specifies a quasimotion in the metric space \(W\), is considered. A new procedure of the averaging method of quasidifferential equations with a small parameter is introduced. The new averaging procedure allows the author to prove theorems generalizing known results, i.e. Bogolyubov's theorem and others.