Averaging of a multipoint problem with parameters for an impulsive oscillation system (Q5890224)

The author studies a multipoint boundary value problem for the system NEWLINE\[NEWLINE\dot x=a(x,\varphi,t);\quad \dot \varphi ={\omega(t)\over \varepsilon}+b(x,\varphi,t),\quad t\neq t_j^{(s)},\tag{1}NEWLINE\]NEWLINE NEWLINE\[NEWLINE\Delta x\bigl|_{t=t_j^{(s)}}=\varepsilon f_s(x,\varphi),\quad \Delta \varphi \bigl|_{t= t_j^{(s)}}=\varepsilon g_s(x,\varphi).NEWLINE\]NEWLINE Here, \(s=1,\ldots,l, 0<t_1^{(1)}< \cdots <t_{1}^{(l)}<2\pi \varepsilon, t_{j+1}^{(s)}=t_j^{(s)}+2\pi \varepsilon \) for all \(j\geq 1\) and \(s=1,\ldots,l\), \(t\in [0,L]\), \(a(x,\varphi ,t)\) and \(b(x,\varphi,t)\) are, respectively, \(\mathbb R^n\)- and \(\mathbb R^m\)-valued functions defined on the set \( D\times \mathbb R^m\times [0,L],\) where \(D\) is a bounded domain in \(\mathbb R^n\). These functions are assumed to be uniformly almost-periodic in \(\varphi \). The boundary conditions have the form of \(n+m\) (nonlinear) equations involving the parameter \(\varepsilon \), additional ``spectral'' parameters \(\mu_1,\ldots,\mu _{r_1}\), and values of \(x(t)\) and \(\varphi(t)\) at points \(t_k\), \(k =1,\ldots,r,\) where \(0=t_0<t_1< \cdots <t_r=L\).NEWLINENEWLINENEWLINEThe corresponding averaged system is constructed as follows NEWLINE\[NEWLINE\dot x=\overline a(x,t)+\overline f(x),\quad \dot \varphi ={\omega(t)\over \varepsilon}+\overline b(x,t)+\overline g(x),NEWLINE\]NEWLINE with NEWLINE\[NEWLINE\overline a(x,t):= \lim_{T\to \infty }T^{-m}\int_{0}^T\cdots \int_{0}^{T}a(x,\varphi,t) d\varphi_1\cdots d\varphi_m. NEWLINE\]NEWLINE Boundary conditions for this system have the same form as for (1). A determining equation for the parameters \(\mu_1,\ldots,\mu _{r_1}\) is derived. If this equation has a solution the averaged boundary value problem is soluble. Under certain additional conditions, in a small neighborhood of a solution to the averaged problem there exists a unique solution to the boundary value problem for system (1). An estimate for the difference between these solutions is established. See also the paper of \textit{A. M. Samojlenko} and \textit{Ya. R. Petryshyn} [Ukr. Mat. Zh. 49, 581-589 (1997; Zbl 0889.34020)].