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On the existence of solutions of linear Volterra difference equations asymptotically equivalent to a given sequence. (English) Zbl 1250.39002

The authors study asymptotic properties of solutions of a linear Volterra difference equation \[ x_{n+1}=a(n)+b(n)x(n)+\sum_{i=0}^nK(n,i)x(i), \] where \(n\in\mathbb{N}_0=\{0,1,2,\dots\}\), \(x:\mathbb{N}_0\to\mathbb{R}\), \(a:\mathbb{N}_0\to\mathbb{R}\), \(K:\mathbb{N}_0\times\mathbb{N}_0\to\mathbb{R}\), \(b:\mathbb{N}_0\to\mathbb{R}\setminus\{0\}\) is \(\omega\)-periodic, \(\omega\in\mathbb{N}=\{1,2,\dots\}\). Schauder’s fixed point technique is applied to obtain sufficient conditions for the validity of a property of solutions that, for every admissible constant \(c\in\mathbb{R}\), there exists a solution \(x=x(n)\) such that \[ x(n)\sim\left(c+\sum_{i=0}^{n-1}\frac{a(i)}{\beta(i+1)}\right)\beta(n), \] where \(\beta(n)=\prod_{j=0}^{n-1}b(j)\), for \(n\to\infty\), and inequalities for solutions are derived. Relevant comparisons and illustrative examples are given as well.

MSC:

39A10 Additive difference equations
39A06 Linear difference equations
39A22 Growth, boundedness, comparison of solutions to difference equations
39A23 Periodic solutions of difference equations
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