Averaging of a multifrequency boundary-value problem with linearly transformed argument (Q5890223)

The author studies a multipoint boundary value problem for the system NEWLINE\[NEWLINE\dot x=X(t,x,x_\lambda,\varphi,\varphi_\theta,\varepsilon),\quad \dot \varphi ={\omega(t)\over \varepsilon}+Y(t,x,x_\lambda,\varphi,\varphi_\theta,\varepsilon).\tag{1}NEWLINE\]NEWLINE Here, \(x_\lambda :=x(\lambda t)\), \(\varphi_\theta :=\varphi(\theta t)\), \(\lambda,\theta\in [0,1]\); \(X\) and \(Y\) are, respectively, \(\mathbb R^n\)- and \(\mathbb R^m\)-valued functions defined on the set \([0,L]\times D\times D\times T^m\times T^m\times[0,\varepsilon_0],\) where \(D\) is a bounded domain in \(\mathbb R^n\) and \(T^m:=\mathbb R^m/2\pi \mathbb Z^m\) is the standard \(m\)-dimensional torus. The boundary conditions have the form of \(n+m\) (nonlinear) equations involving \(x(t_\alpha), \varphi(t_\alpha), \alpha =1,\ldots,N\), and \(\varepsilon,\) where \(0=t_0<t_1< \cdots <t_N=L\).NEWLINENEWLINENEWLINEThe corresponding averaged system has the form NEWLINE\[NEWLINE\dot x=X_0(t,x,x_\lambda,\varepsilon);\quad \dot \varphi ={\omega(t)\over \varepsilon}+Y_0(t,x,x_\lambda,\varepsilon),NEWLINE\]NEWLINE with NEWLINE\[NEWLINEf_0(t,x,y,\varepsilon):= (2\pi)^{-2m}\int_{T^m\times T^m}f(t,x,y,\varphi,\psi,\varepsilon) d\varphi d\psi.NEWLINE\]NEWLINE The author shows that if there exists a solution to the boundary value problem for the averaged system then, under certain additional conditions, in a small neighborhood of such a solution there exists a unique solution to the boundary value problem for system (1). An estimate for the difference between these solutions is established.