On a generalization of the Cauchy problem (Q1377968)

The author studies a generalization of the Cauchy problem \[ x^{(n)} =f(t,x, \dots, x^{(n-1)}),\;t\geq T \tag{1} \] \[ x^{(j)} (t_0)= x_0^{(j)}, \quad j=0, \dots, n-1 \tag{2} \] in the case of an infinitely distant point. In this case the conditions (2) are replaced with the conditions of asymptotic approximation \[ \lim_{t\to\infty} \bigl(x^{(j)} (t)-v^{(j)} (t)\bigr) =0, \quad j=0,1, \dots, n-1 \tag{3} \] there \(V:[T,\infty) \to\mathbb{R}\) is a given \(n\)-times continuously differentiable function. Further we consider the weight analog of problem (1), (3). The author proves some existence and uniqueness results. These conditions may be regarded as a (weight integral) analog of the Lipschitz conditions. In the last section examples of the application of the theorems for the Duffing equation and the Riccati equation are given.