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Comparison theorems for the asymptotic behavior of solutions of nonlinear difference equations. (English) Zbl 0929.39002
We consider the asymptotic behavior of solutions to the nonlinear difference equation \[ \Delta^my_n+f(n,y_n,y_{n+1},\dots,y_{n+m-1})=b_n,\tag{E} \] where \(m\geq 2\), \(n\in N_{n_0}\), \(b:N_{n_0}\to\mathbb{R}\), and \(f:N_{n_0}\times\mathbb{R}^m\to\mathbb{R}\). Here \(y_n=y(n)\), \(b_n=b(n)\).
It is known that in the case when the function \(f\) is “small” in some sense, then the solutions of Eq. (E) are asymptotically equivalent with the solutions of the following difference equation: \(\Delta^mz_n=b_n\) as \(n\to\infty\).
The purpose of this paper is to consider the above problem using a comparison method differing in a number of ways from those previously applied. Namely, we reduce the problem of asymptotically equivalent solutions to a difference equation of higher order to the boundedness of solutions to some difference equation of first order. In consequence, our conditions imposed on the function \(f\) are less restrictive than those previously required by other authors.

39A11 Stability of difference equations (MSC2000)
Full Text: DOI
[1] Hallam, T.G., Asymptotic expansions in a nonhomogeneous differential equation, Proc. amer. math. soc., 18, 432-438, (1967) · Zbl 0173.10403
[2] Hallam, T.G., Asymptotic behavior of the solutions of an nth order nonhomogeneous ordinary differential equation, Trans. amer. math. soc., 122, 177-194, (1966) · Zbl 0138.33001
[3] Hooker, J.W.; Patula, W.T., A second-order nonlinear difference equation: oscillation and asymptotic behavior, J. math. anal. appl., 91, 9-29, (1983) · Zbl 0508.39005
[4] Ladas, G., On principal solutions of nonlinear differential equations, J. math. anal. appl., 36, 103-109, (1971) · Zbl 0224.34026
[5] Marušiak, P., Note on ladas’ paper on oscillation and asymptotic behavior of solutions of differential equations with retarded argument, J. differential equations, 13, 150-156, (1973) · Zbl 0249.34063
[6] Patula, W.T., Growth, oscillation and comparison theorems for second order linear difference equations, SIAM J. math. anal., 10, 1272-1279, (1979) · Zbl 0433.39005
[7] Popenda, J., On the asymptotic behaviour of the solutions of ann, Ann. polon. math., 44, 95-111, (1984) · Zbl 0553.39002
[8] Popenda, J.; Werbowski, J., On the asymptotic behaviour of the solutions of difference equations of second order, Comment. math., 22, 135-142, (1980) · Zbl 0463.39002
[9] Trench, W.F., Asymptotic behavior of solutions of a linear second-order difference equation, J. comput. appl. math., 41, 95-103, (1992) · Zbl 0758.39004
[10] Werbowski, J., On the asymptotic behaviour of solutions of differential equations with delay, Differentsial’nye uravnenia, 17, 917-920, (1981) · Zbl 0478.34050
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