Subexponential Solutions of Linear Volterra Difference Equations

Abstract

We study the asymptotic behavior of the solutions of a scalar convolution sum-difference equation. The rate of convergence of the solution is found by determining the asymptotic behavior of the solution of the transient renewal equation.

Bulletin of the Polish Academy of Sciences: Technical Sciences

“…We denote x by ϕ C (•, x 0 ) or ϕ R-L (•, x 0 ), respectively. In Theorem 1 below, we characterize solutions of (10) as solutions of an associated Volterra convolution equation. For the proof of this characterization we use the following Lemma which provides an alternative representation of fractional differences [1,2] and fractional sums [23].…”

“…For the proof of this characterization we use the following Lemma which provides an alternative representation of fractional differences [1,2] and fractional sums [23]. (10) with Caputo forward difference operator if and only if…”

“…and this equation is equivalent to (10) with Riemann-Liouville forward difference operator if and only if…”

“…This approach is considered e.g. in [4,10,38] for general Volterra equations to obtain the exact decay rates of solutions. In [15] and [13], results from [4] are used to determine the exact decay rates of solutions.…”

Boundedness and Asymptotic Stability for the Solution of Homogeneous Volterra Discrete Equations

“…Asymptotic analysis of difference equations of the form (5) or its explicit version often appeared in literature in the last decades. Some of them deal with the convolution case ( , = − ); see, for instance, [6] and the references therein and [7][8][9][10][11][12]. Most of the known results for the nonconvolution case are based on the hypothesis of double summability of the coefficients (∑ +∞ =0 ∑ =0 | , | < +∞); see [1,4,[13][14][15][16][17][18][19].…”