On a generalization of the Cauchy problem
type: Item
instance of: scholarly article
title: On a generalization of the Cauchy problem
zbMATH Open document ID: 0893.34002
author: P. A. Zharov
published in: Doklady Mathematics
publication date: 1998-08-31T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 34A12
zbMATH DE Number: 1113108
zbMATH Keywords: Cauchy problems
zbMATH Keywords: asymptotic approximation
reviewed by: A. Yu. Timofeev
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1113108