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Asymptotic behavior of solutions of second order nonlinear difference equations. (English) Zbl 0773.39003

Sufficient conditions for the solutions of the perturbed difference equation \(y(n+2)+a(n)y(n+1)+b(n)y(n)=g(n,y(n),y(n+1))\) satisfying the initial relation \(| y(0)|+| y(1)|<\rho\) (where \(\rho\) is a constant) to be presented in the form \(y(n)=(\delta_ 1+o(1))z_ 1(n)+(\delta_ 2+o(1))z_ 2(n)\), where \(z_ 1,z_ 2\) represent the fundamental system of solutions of the unperturbed \((g\equiv 0)\) equation, are given. The case of generalized polynomial perturbations is examined in detail. An interesting Gronwall type inequality is proved.
To the large list of references, the reviewer wants to add two papers on a similar subject: A. Drozdowicz [Glas. Mat., III. Ser. 22(42), 327-333 (1987; Zbl 0654.39001)] and M. Migda [Rad. Mat. 5, 297-309 (1989; Zbl 0702.39002)].

MSC:

39A10 Additive difference equations
26D15 Inequalities for sums, series and integrals
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