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Asymptotic properties of solutions of the \(n\)th order differential equation with delayed argument. (English) Zbl 0686.34072
The nth order nonlinear differential equation with delayed argument of the form \[ (1)\quad a_ n(t)(a_{n-1}(t)(...(a_ 1(t)(a_ 0(t)y(t))')'...)')'+H(t,y(g(t)))=b(t), \] where \(a_ i(t)>0\), b(t), g(t) are continuous on \([t_ 0,\infty)\) and H(t,y) is continuous on \([t_ 0,\infty)\times R\) is investigated. There are found sufficient conditions under which all solutions of the equation (1) are nonoscillatory. Also there are found sufficient conditions so that for every oscillatory solution y(t) of the equation (1) there is \(\lim_{t\to \infty}y(t)=0.\)
Reviewer: M.Rusnák
MSC:
34K99 Functional-differential equations (including equations with delayed, advanced or state-dependent argument)
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
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References:
[1] HRUBINOVÁ A.: Asymptotic properties of solutions of nth order nonlinear differential equations with advanced argument. (in Slovak). Dissertation, Department of mathematics VŠT Košice, 1984.
[2] SINGH B., KUSANO T.: Asymptotic behavior of oscillatory solutions of a difìerential equation with deviating arguments. J. Math. Anal. Appl., 83, 1981, 395-407. · Zbl 0472.34044
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