Difference inequality for stability of impulsive difference equations with distributed delays

Abstract

In this paper, we consider a class of impulsive difference equations with distributed delays. By establishing an impulsive delay difference inequality and using the properties of "r-cone" and eigenspace of the spectral radius of non-negative matrices, some new sufficient conditions for global exponential stability of the impulsive difference equations with distributed delays are obtained. An example is given to demonstrate the effectiveness of the theory.

Exponential Stability of Discrete Neural Networks With Non-Instantaneous Impulses, Delays and Variable Connection Weights With Computer Simulation

“…Definition 1. ( [14]) The trivial solution of the system (1) is called globally exponentially stable if there exist constants N > 0 and α ∈ (0, 1) such that for any initial value x 0 the inequality…”

“…One of the most important problems in the theory and application of differential and difference equations is stability. For the basis of the stability theory of difference equations with a delay or without any delay we refer to [1], [4], [6], [9], [12], [13], [14], [15], [16]. A good overview of the basic results and methods for stability investigations of linear autonomous difference equations is given in [7].…”

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“…Such applications include solutions to optimal control problems [15], stability analysis [16,17], robotics [18], fuzzy logic [19,20], difference inequalities [21] or differential equations [22], as far as positive integers are concerned.…”