Global existence and asymptotic behavior of solutions of second-order nonlinear differential equations
type: Item
instance of: scholarly article
title: Global existence and asymptotic behavior of solutions of second-order nonlinear differential equations
zbMATH Open document ID: 0644.34011
DOI: 10.1016/0022-247X(87)90351-9
author: Hiroyuki Usami
published in: Journal of Mathematical Analysis and Applications
publication date: 1987-01-01T00:00:00Z
review text: The author considers the equation (1) \((p(t)y')'=f(t,y,y')\), \(t\geq 0\), under the assumptions: \((A_ 1)p: [0,\infty)\to (0,\infty)\) is continuous and \(P(t)=\int^{t}_{0}1/p(s)ds\to \infty\) as \(t\to \infty\); \((A_ 2)\) \(f: [0,\infty)\times R\times R\to (0,\infty)\) is continuous and f(t,u,v) is nondecreasing in u and v, \((A_ 3)\) Every Cauchy problem for (1) has a unique solution. Using the Schauder-Tychonov fixed point theorem and by the continuous dependence on initial data, the author proves the existence of proper solutions of (1) belonging to separate classes of solutions introduced in the paper. Also the existence of a singular solution of (1) is presented. Next, using the supersolution-subsolution method, the author shows that the above results for (1) can be applied to solve the elliptic exterior Dirichlet problem.
Mathematics Subject Classification ID: 34A34
Mathematics Subject Classification ID: 34E05
Mathematics Subject Classification ID: 35J15
zbMATH DE Number: 4050024
zbMATH Keywords: second order differential equations
zbMATH Keywords: Cauchy problem
zbMATH Keywords: Schauder-Tychonov fixed point theorem
zbMATH Keywords: supersolution-subsolution method
zbMATH Keywords: elliptic exterior Dirichlet problem
reviewed by: Jan Ohriska
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4050024