×

Remarks on the uniqueness of second order ODEs. (English) Zbl 1224.34008

Summary: We are concerned with the uniqueness problem for solutions to the second order ODE of the form \(x''+f(x,t)=0\), subject to appropriate initial conditions under the unique assumption that \(f\) is non-decreasing with respect to \(x\) for each \(t\) fixed. We show that there is non-uniqueness in general. On the other hand, several types of reasonable additional assumptions make the problem uniquely solvable. The interest in this problem comes, among others, from the study of oscillations of lumped parameter systems with implicit constitutive relations.

MSC:

34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI EuDML Link

References:

[1] P. Hartman: Ordinary Differential Equations. 2nd ed. with some corrections and additions. S.M. Hartman, Baltimore, 1973. · Zbl 0281.34001
[2] L. Meirovitch: Elements of Vibration Analysis. Second edition. McGraw-Hill, New York, 1986. · Zbl 0359.70039
[3] D. Pražák, K.R. Rajagopal: Mechanical oscillators described by a system of differential-algebraic equations. Submitted.
[4] K.R. Rajagopal: A generalized framework for studying the vibration of lumped parameter systems. Submitted.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.