An asymptotic result for some delay difference equations with continuous variable

Abstract

We consider a nonhomogeneous linear delay difference equation with continuous variable and establish an asymptotic result for the solutions. Our result is obtained by the use of a positive root with an appropriate property of the so-called characteristic equation of the corresponding homogeneous linear (autonomous) delay difference equation. More precisely, we show that, for any solution, the limit of a specific integral transformation of it, which depends on a suitable positive root of the characteristic equation, exists as a real number and it is given explicitly in terms of the positive root of the characteristic equation and the initial function.

Some New Aspect of Lyapunov Type Theorems for Stochastic Difference Equations With Continuous Time

“…Rewrite condition (20) in the form E∆V(t + jh 0 ) ≤ bE|x(t + ( j − k)h 0 )| 2 , t ≥ t 0 , j = 0, 1, …. Summing this inequality from j = 0 to j = k, we obtain 2 0 0 0…”

“…One general method of Lyapunov functionals construction was proposed and developed in [4][5][6][7][8][9][10][11] for both stochastic differential equations with aftereffect and stochastic difference equations with discrete time. After some modification of the basic Lyapunov-type stability theorem, this method was also used for stochastic difference equations with continuous time [12][13][14], which are popular enough in researches [15][16][17][18][19][20]. Here some new aspect of Lyapunov type theorems is shown, which allows to simplify an application of the general method of Lyapunov functionals construction for stochastic difference equations with continuous time.…”

Asymptotic behavior of impulsive non-autonomous delay difference equations with continuous variables

“…In particular, delay effects on the asymptotic behavior and other behaviors of this kind of equations have widely been studied in the literature [1][2][3][4][5][6][7].…”