Asymptotic behavior of solutions to difference equations of neutral type. (English) Zbl 1531.39001
Summary: We present sufficient conditions for the existence of a solution \(x\) to an equation
\[
\Delta^m(x_n - u_nx_{n - k}) = a_nf(x_{n - \tau}) + b_n,
\]
which is “close” to a given solution \(y\) to the linear homogeneous equation of neutral type \(\Delta^m(y_n - \lambda y_{n-k}) = 0\), where \(\lambda\) is the limit of the sequence \(u\). Closeness of solutions to above equations is understood as \(x_n - y_n = \mathrm{o}(\omega_n)\), where \(\omega\) is a given nonincreasing sequence with positive values. Moreover, we establish under which conditions for a given solution \(x\) to \(\Delta^m(x_n - u_nx_{n - k}) = a_nf(x_{n - \tau}) + b_n\) and a given nonincreasing sequence with positive values \(\omega\) there exists a polynomial sequence \(\varphi\) of degree less than \(m\) such that \(x_n = \varphi(n) + \mathrm{o}(\omega_n)\). Presented conditions strongly depend on \(\lambda\).
MSC:
| 39A22 | Growth, boundedness, comparison of solutions to difference equations |
| 39A12 | Discrete version of topics in analysis |
| 34K40 | Neutral functional-differential equations |
Keywords:
difference equation; neutral type equation; prescribed asymptotic behavior; approximative solutionReferences:
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