Bounded solutions in a given set of differential systems
The authors deal with systems of ordinary differential equations of the form \[ \dot y= Ay+ g(t,y,z),\quad \dot z= h(t,y,z),\tag{1} \] where \(A\) is a hyperbolic \(m\times m\)-matrix (i.e. a real constant matrix with all eigenvalues having nonzero real parts). \(g\) and \(h\) are supposed to be continuous vector functions. Using the continuation method, which was developed by \textit{M. Furi} and \textit{P. Pera} [Ann. Pol. Math. 47, 331-346 (1987; Zbl 0656.47052)], the authors prove the existence of at least one bounded (on \(\mathbb{R}\)) solution to (1) lying in a given set. This set is defined by means of strict bounded (on \(\mathbb{R}\)) lower and upper functions to (1), whose existence is assumed. Under the existence of more families of such strict lower and upper functions to (1) a multiplicity result is formulated.
scientific article; zbMATH DE number 1398397