## 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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### References:

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