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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)+ i=0 n K(n,i)x(i),

where n 0 ={0,1,2,}, x: 0 , a: 0 , K: 0 × 0 , b: 0 {0} is ω-periodic, ω={1,2,}. 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, there exists a solution x=x(n) such that

x(n)c+ i=0 n-1 a(i) β(i+1)β(n),

where β(n)= j=0 n-1 b(j), for n, and inequalities for solutions are derived. Relevant comparisons and illustrative examples are given as well.

MSC:
39A10Additive difference equations
39A06Linear equations (difference equations)
39A22Growth, boundedness, comparison of solutions (difference equations)
39A23Periodic solutions (difference equations)
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