Dichotomies and asymptotic formulas for the solutions of differential equations
The paper deals with a linear system (1) \(\dot x= A(t) x\), where \(A(t)\) is an \(n\times n\) continuous matrix defined on \({\mathcal I}= [a, \infty)\). Let \(\Phi (t)\) be a fundamental matrix of (1), \(h(t)\) and \(k(t)\) be continuous functions satisfying the compensation law: \(h(t) h^{-1} (s) k(t) k^{-1} (s)\leq c_1\) \((t\geq s\geq a)\), where \(c_1\) is a positive constant. Suppose that system (1) possesses an \((h, k)\)-dichotomy, that is, there exists a projection \(P\) such that \(|\Phi (t) P\Phi^{-1} (s) |\leq c h(t) h^{-1} (s)\) \((t\geq s\geq a)\), \(|\Phi(t) (E-P) \Phi^{-1} (s)|c k^{-1} (t) k(s)\) \((s\geq 1\geq a)\) [the author, Contributions US-ACH, 1-14 (1992)], where \(c= \text{const}>0\). Let \(b(t)\) be a continuous vector, \(t\in {\mathcal I}\); \(f: {\mathcal I}\times C^n\to C^n\) be a continuous function such that \(|f(t, y_1)- f(t, y_2) |\leq \lambda (t) |y_1- y_2|\) for \((t, y_i)\in {\mathcal I}\times C^n\) \((i= 1,2)\) and for a nonnegative function \(\lambda\in L_1 ({\mathcal I})\) and \(h^{-1} (t) f(t, 0)\in L_1 ({\mathcal I})\). The author solves asymptotic equivalence problems for systems (2) \(\dot x= A(t) x+ b(t)\) and (3) \(\dot y= A(t) y+ b(t) f(t, y)\). Let by \(h_+ =h\) and \(h_-= k^{-1}\) and introduce the Banach spaces \(C_\pm= C_\pm (t_0)\) defined by \(C_\pm (t_0)= \{z\in C [t_0, \infty): |y|_\pm< \infty\}\), where \(t_0\geq a\), \(|y|_\pm= \sup_{t\geq t_0} \{h^{-1}_\pm (t)|y(t) |\}\). The main result of this paper is Theorem 1: Assume that the above-mentioned hypotheses hold. Then there exists a one-to-one continuous correspondence between the solutions \(x_+\in C_+\) (respectively, \(x_-\in C_-\)) of system (2) and the solutions \(y_+\in C_+\) (respectively, \(y_-\in C_-\)) of system (3). Moreover, we have the two formulas \(y_\pm= x_\pm+ |\Phi P|\cdot \widetilde {o} (1)+ h_\pm \cdot o(1)\), \(t\to \infty\), where \(\widetilde {o} (1)\) is a convergent function (as \(t\to \infty\)). If \(h^{-1} (t) \Phi (t) P\to 0\) as \(t\to \infty\) we obtain \(y_\pm= x_\pm+ h_\pm \cdot O(1)\), as \(t\to \infty\).
scientific article; zbMATH DE number 861747