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.


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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