Global existence and asymptotic behavior of solutions of nonlinear differential equations
The global existence and asymptotic behavior of solutions of the systems (1) \(x'(t) = f(t,x)\) and (2) \(y'(t) = A(t)y(t) + f(t,y)\) in \(I_a = [a, \infty)\) is studied. Let \(f \in C^0 (I_a \times \mathbb{C}^n, \mathbb{C}^n)\) satisfy an inequality \(|f(t,x) |\leq \sum^p_1 \lambda_i (t)w_i (|x |)\) with \(0 \leq \lambda_i \in L^1 (I_a)\), \(w_i\) positive and increasing and \(w_{i + 1}/w_i\) increasing in \((0, \infty)\). If \(\int^\infty_a \lambda_p (s)ds\) has an upper bound in terms of \(\lambda_j w_j\) (which is not easily stated), then (i) any solution of (1) with small \(|x (t_0) |\) exists in \(I_a\) and satisfies, for some \(\xi\), \(x(t) \to \xi + 0(\int^\infty_t \lambda (s) ds)\) as \(t \to \infty\), where \(\lambda \in L^1 (I_a)\), and (ii), for all \(\xi\) with \(|\xi |\) small, there exists a solution in \(I_a\) with this asymptotic behavior. If \(|\Phi^{-1-} (t)f(t, \Phi (t)x |\) satisfies the inequality required for \(f\) and if the asymptotic behavior is described by \(y(t) = \Phi (t) [\xi + 0(\int^\infty_t \lambda (s) ds)]\), then (i) and (ii) hold for solutions of (2).
scientific article; zbMATH DE number 706474