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Asymptotic behavior of solutions of second-order difference equations of Volterra type. (English) Zbl 1430.39001

Summary: In this paper we investigate the Volterra difference equation of the form \[\Delta(r_n\Delta x_n)=b_n+\sum_{k=1}^nK(n,k)f(x_k).\] We establish sufficient conditions for the existence of a solution \(x\) of the above equation with the property \(x_n=y_n+o(n^s)\), where \(y\) is a given solution of the equation \(\Delta(r_n\Delta y_n)=b_n\) and \(s\) is nonpositive real number. We also obtain sufficient conditions for the existence of asymptotically periodic solutions.

MSC:

39A10 Additive difference equations
39A22 Growth, boundedness, comparison of solutions to difference equations
39A12 Discrete version of topics in analysis
45D05 Volterra integral equations
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