Perturbations of nonlinear systems of ordinary differential equations (Q1073239)

Along with the n dimensional system (1) \(y'=f(t,y)\) the perturbed system (2) \(x'=f(t,x)+g(t,x)\) is considered. Certain refined assumptions are made about estimates of the fundamental matrix solution of the variational system with respect to a solution of system (1), and about g. (The function g need not be ''small'', it may be sublinear or linear.) Under these assumptions theorems are proved concerning the boundedness, the convergence to zero, the ''slowly growingness'', the asymptotic stability, and the uniform asymptotic stability of solutions of (2). These theorems have important implications if applied to the case when (1) is linear.