Abstract
Exponential periodic attractor of impulsive Hopfield-type neural network system with piecewise constant argument
Highlights
CNNs models of the mixed type impulsive differential equation (IDE)-DEPCA can be found in the mathematical literature of the last decades [5, 8, 13, 56, 57]
In [25], the same author investigated some sufficient conditions for the existence, uniqueness and globally exponentially stability of solutions of the following IDEPCAG system with alternately retarded and advanced piecewise constant argument:
In this work we have obtained some sufficient conditions for the existence, uniqueness, periodicity and stability of solutions for the impulsive Hopfield-type neural network system with piecewise constant arguments (1.2)
Summary
U. Akhmet considers the equation x (t) = f (t, x(t), x(γ(t))), where γ(t) is a piecewise constant argument of generalized type, that is, given (tk)k∈Z and (ζk)k∈Z such that tk < tk+1 , ∀k ∈ Z with limk→±∞ tk = ±∞ and tk ≤ ζk ≤ tk+1, if t ∈ Ik = [tk, tk+1) , γ(t) = ζk. These equations are called Differential Equations with Piecewise Constant Argument of Generalized Type (in short DEPCAG) They have continuous solutions, even when γ(t) is not, producing a recursive law on tk i.e., a discrete equation. For instance we distinguish between discrete and continuous models, when the time is considered as discrete or a continuous variable, respectively Another general classification is given by the dynamics of the cells by considering the deterministic or probabilistic behavior. CNNs models of the mixed type IDE-DEPCA can be found in the mathematical literature of the last decades [5, 8, 13, 56, 57]
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