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Asymptotic stability of second order ordinary, functional, and partial differential equations. (English) Zbl 0779.34042

The authors propose a unified treatment of the asymptotic stability for the equations \(u''+a(t)u'+u=0\), \(u_{tt}=u_{xx}-a(t)u_ t\) with \(u(t,0)=u(t,\pi)=0\) and \(u''+a(t)u'+u(t-r)=0\). Their approach is based on a certain transformation of the considered equations into equivalent systems of two differential equations of first order in time. For these systems, suitable Lyapunov functions are defined and a sufficient condition for asymptotic stability expressed in terms of these functions is given. Several cases in which the given sufficient condition is satisfied are indicated. In this way the authors obtain results which are comparable with some of the best classical ones for ordinary differential equations. Finally, corresponding considerations for nonlinear problems are given.
Reviewer: C.Popa (Iaşi)

MSC:

34D20 Stability of solutions to ordinary differential equations
34K20 Stability theory of functional-differential equations
35B35 Stability in context of PDEs
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