Existence and uniqueness of initial value problems for a class of second- order differential equations
type: Item
instance of: scholarly article
title: Existence and uniqueness of initial value problems for a class of second- order differential equations
zbMATH Open document ID: 0691.34005
DOI: 10.1016/0022-0396(89)90135-6
author: J. Webb
author: Stewart C. Welsh
published in: Journal of Differential Equations
publication date: 1989-01-01T00:00:00Z
review text: The authors investigate the initial value problem for the second order differential equations of the form (1) \(x''(t)=g(x(t),x'(t),x''(t)),\) which cannot be solved for the highest order derivative. Their work is related to that of Petryshyn who used the theory of A-mappings. The authors' investigations arise from a global bifurcation result for an A- proper operator, where some of the possible behaviour of a global bifurcation can be eliminated if the corresponding initial value problem has a unique solution. The authors claim that if g is Lipschitz in all variables with Lipschitz constant I in the third variable, then solutions of the initial value problem are not unique as it might be expected. The authors prove in section 2 that the solution can be modified on a set of measure zero so that \(x''\) is continuous while in section 3 they give an example to illustrate the theorem proved in this section namely. If \(g(0,0,0)=0\) and the initial date is taken to be zero, then 0 is always a solution of the initial value problem \(x''=g(x,x',x''),\) \(x(0)=A=0\), \(x'(0)=B=0\) which is written as the equation \(Lx=Nx\), \(x\in X\) where \(L,N: X\to Y\) are given by \(Lx=x''\), \(Nx(t)=g(x(t),x'(t),x''(t)).\)
Mathematics Subject Classification ID: 34A12
Mathematics Subject Classification ID: 65L05
zbMATH DE Number: 4130775
zbMATH Keywords: A-mappings
zbMATH Keywords: global bifurcation
reviewed by: F. M. Ragab
MaRDI profile type: Publication
full work available at URL: 0022-0396(89)90135-6
OpenAlex ID: W2028566075
cites work: On the approximation-solvability of equations involving 𝐴-proper and pseudo-𝐴-proper mappings
cites work: Using degree theory for densely defined A-proper maps in the solvability of semilinear equations with unbounded and noninvertible linear part
cites work: Periodic solutions of nonlinear second-order differential equations which are not solvable for the highest derivative
cites work: Topological degree and A-proper operators
cites work: Global results concerning bifurcation for Fredholm maps of index zero with a transversality condition
description: scientific article; zbMATH DE number 4130775