Topological structure of solution sets to asymptotic boundary value problems
The paper studies the topological structure of the solutions set to the semi-linear problem \[ \ddot x(t)+A(t)\dot x(t)+B(t)x(t)\in C(t,x(t),x\dot (t)),\quad l(x,\dot x)=0 \tag{1} \] on the compact interval \([0,m]\), where \(A\), \(B\) are Lebesgue integrable matrix functions, \(C\) is an upper-Carathéodory mapping and \(l\) is a linear bounded operator. The authors prove that the set of solutions of (1) is a nonempty, compact \(AR\)-space. Then, by means of the inverse limit method, problem (1) is investigated on the non-compact interval \([t_0,\infty]\) and it is proved that the set of its solutions is an \(R_\delta\)-set. The information about the structure is employed for obtaining an existence result for nonlinear asymptotic problems. Some illustrating examples are supplied.
scientific article; zbMATH DE number 5651185
scientific article