Asymptotically linear solutions of differential equations via Lyapunov functions (Q1049321)

The authors are concerned with the asymptotic behavior of solutions to the second-order nonlinear differential equation \[ x''(t)+f(t,\frac{x}{t})=0,\quad t\geq t_0\geq1, \tag{1} \] where \(f: [t_0,+\infty)\times\mathbb{R}\longrightarrow\mathbb{R}\) is continuous and also to the model case of the Emden-Fowler like variable-separated model equation \[ x''(t)+A(t)x^{2n-1}=0,\quad t\geq t_0\;(n\geq1\,\text{ is an integer}), \tag{2} \] where \(A: [t_0,+\infty)\longrightarrow\mathbb{R}\) is nonnegative and continuous. Under some growth conditions on the nonlinearity \(f\) and then on \(A\), existence of asymptotically linear solutions is proved for equations (1) and (2). By asymptotically linear solution, it is meant a solution \(x\in C^2([t_0,+\infty),\mathbb{R})\) which enjoys the following asymptotic behavior as \(t\to+\infty\) \[ x(t)=x_1t+x_2+o(1)\,\text{ and }\,x'(t)=x_1+o(t^{-1}), \] for some real constants \(x_1, x_2.\) The proofs use integral inequalities and a Liapunov functional.