Asymptotic behaviour of solutions of systems of discrete equations via Liapunov type technique
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TLDR
In this article, conditions for the existence of a continuum of solutions, the graphs of which remain within a given prescribed set, are formulated for the general systems of discrete equations Δu ( k ) = F ( k, u (k )).Abstract:
The second Liapunov method serves as a powerful tool for the investigation of the stability of the trivial solution of ordinary differential equations systems and discrete equations systems. In the presented paper, a Liapunov-type qualitative approach is used for the investigation of asymptotic behaviour of the solutions of systems of discrete equations. Conditions for the existence of continuum of solutions, the graphs of which remain within a given prescribed set, are formulated for the general systems of discrete equations Δu ( k ) = F ( k , u ( k )). An additional advantage of the presented approach consists of the fact that no assumption concerning the existence of the trivial solution (or the existence of an equilibrium point) of systems considered is assumed. Moreover, the asymptotic behaviour of solutions of some classes of linear difference systems is given by means of concrete asymptotic formulae. Several illustrative examples are considered, too.read more
Citations
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Journal ArticleDOI
Representation of solutions of linear discrete systems with constant coefficients and pure delay
Josef Diblík,D. Khusainov +1 more
TL;DR: In this paper, the authors developed a method for construction of solutions of linear discrete systems with constant coefficients and with pure delay, which is expressed with the aid of a special function called the discrete matrix delayed exponential having between every two adjoining knots the form of a polynomial.
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Representation of the Solutions of Linear Discrete Systems with Constant Coefficients and Two Delays
Josef Diblík,Blanka Morávková +1 more
TL;DR: In this paper, the authors developed a method for the construction of solutions to initial problems of linear discrete systems with constant coefficients and with two delays where, are fixed,,,, are constant matrices, is a given vector, and is an unknown vector.
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On local stability for a nonlinear difference equation with a non-hyperbolic equilibrium and fading stochastic perturbations
TL;DR: In this paper, the authors considered the nonlinear stochastic difference equation and established a condition on the noise intensity σ and the rate of decay of the tails of the distribution functions F n, under which the convergence of solutions to zero occurs with probability zero.
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Anti-Lyapunov method for systems of discrete equations
TL;DR: In this paper, a general principle which gives a guarantee that the graph of at least one solution stays in a prescribed domain is formulated and an auxiliary tool, called a method of regular polyfacial sets, is developed.
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Combination of liapunov and retract methods in the investigation of the asymptotic behavior of solutions of systems of discrete equations
Josef Diblík,Irena Hlavičková +1 more
TL;DR: In this article, a discussion of the asymptotic behavior of solutions of first order nonlinear difference equations is presented, and it is shown that under appropriate conditions there exists at least one solution of the system considered the graph of which stays in a prescribed domain.
References
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Periodic solutions of first order linear difference equations
Ravi P. Agarwal,J. Popenda +1 more
TL;DR: In this paper, the authors set together various basic statements on the periodicity of the solutions of first order linear difference equations and defined various sequences which are in a sense connected with the concept of periodicity.