Some stability conditions for scalar Volterra difference equations

Abstract

Abstract. New explicit stability results are obtained for the following scalar linear difference equationand for some nonlinear Volterra difference equations.

Conditional Stability and Asymptotic Behavior of Solutions of Weakly Delayed Linear Discrete Systems in R2

“…For these cases, conditional stability is investigated and formulas are deduced expressing the asymptotic behavior of solutions as → ∞. For further results related to weakly delayed systems, representations of solutions of discrete systems, their stability, and asymptotic behavior, we refer to [1][2][3][4][5][6][7][8][9][10][11] and to the references therein.…”

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

“…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]. Another interesting approach, resembling the study of continuous VIDE (see, e.g., [20,21]), basically requires that the coefficient +1 of (5), assumed to be negative, in some sense "prevails" on the summation of the remaining coefficients .…”

“…Even if each of the equations above can be easily transformed into the other, we read in the literature (see, e.g., [4]) that (1) is the discrete analogue of a Volterra Integrodifferential Equation (VIDE), whereas (2) is seen as the discrete version of a second kind Volterra Integral Equation (VIE). This is due to the fact that the simple position …”

Bounded Solutions to Nonhomogeneous Linear Second-Order Difference Equations

“…Here, we will frequently use various things connected to Equation (1). Some recent applications of this and related solvable equations can be found in [5,6]. Let us also mention that beside showing the solvability of difference equations and systems by finding closed-form formulas for their solutions, in the cases when it is not possible to find them, one can try to find some of their invariants which can be also useful in studying of the long-term behavior of their solutions ( [21,22]).…”