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Keywords:
second order ODEs; uniqueness of solutions; oscillations
second order ODEs; uniqueness of solutions; oscillations
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 sole 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 other, from the study of oscillations of lumped parameter systems with implicit constitutive relations.
References:
[1] Hartman, P.: Ordinary Differential Equations. 2nd ed. with some corrections and additions. S. M. Hartman Baltimore (1973). MR 0344555 | Zbl 0281.34001
[2] Meirovitch, L.: Elements of Vibration Analysis. Second edition. McGraw-Hill New York (1986).
[3] Pražák, D., Rajagopal, K. R.: Mechanical oscillators described by a system of differential-algebraic equations. Submitted.
[4] Rajagopal, K. R.: A generalized framework for studying the vibration of lumped parameter systems. Submitted.
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