Existence, uniqueness, and stability theorems for bounded solutions of nonlinear differential systems
The author investigates the nature and the existence of bounded solutions of the coupled system \[ x'=f_ 1(t,x)+g_ 1(t,x,y)+h_ 1(t),\quad y'=f_ 2(t,x)+g_ 2(t,x,y)+h_ 2(t) \] regarded as perturbation of the uncoupled system \(x'=f_ 1(t,x)\), \(y'=f_ 2(t,y)\) where \(x,f_ 1,g_ 1,h_ 1\) and \(y,f_ 2,g_ 2,h_ 2\) are \(n_ 1\)- and \(n_ 2\)- vectors, respectively, and \(t\geq 0.\) One of the problems considered by the author is to determine the behavior of the solutions as t tends to infinity. The behavior turns out to depend upon the initial conditions. Some of the results of the author himself and of Perron are generalized in this paper. The basic tools used for investigation are Lyapunov functions and the comparison principle. The paper is well organized and is remarkable both in content and style.
scientific article; zbMATH DE number 3929461
scientific article