We study a frequently investigated class of linear difference equationsΔv(n)=−p(n)v(n−k)with a positive coefficientp(n)and a single delayk. Recently, it was proved that if the functionp(n)is bounded above by a certain function, then there exists a positive vanishing solution of the considered equation, and the upper bound was found. Here we improve this result by finding even the lower bound for the positive solution, supposing the functionp(n)is bounded above and below by certain functions.
This contribution is devoted to the investigation of the asymptotic behavior of delayed difference equations with an integer delay. We prove that under appropriate conditions there exists at least one solution with its graph staying in a prescribed domain. This is achieved by the application of a more general theorem which deals with systems of first-order difference equations. In the proof of this theorem we show that a good way is to connect two techniques—the so-called retract-type technique and Liapunov-type approach. In the end, we study a special class of delayed discrete equations and we show that there exists a positive and vanishing solution of such equations.
ABSTRACT. This contribution concerns the asymptotic behavior of solutions of a first-order difference equation. We are looking for a solution whose graph stays in a given domain. It is supposed that all the boundary points of this domain are the so called points of strict egress. Under this supposition, it has been already proved that the existence of at least one solution the graph of which stays in the given domain is guaranteed. The main aim of this article is to find the concrete value of the initial condition which generates such a solution. The method we introduce resembles the well-known bisection method for finding roots of equations.
This contribution shows an application of Markov chains in digital communication. A random sequence of the symbols 0 and 1 is analyzed by a state machine. The state machine switches to state "0" after detecting an unbroken sequence of w zero symbols (w being a fixed integer), and to state "1" after detecting an unbroken sequence of w ones. The task to find the probabilities of each of these two states after n time steps leads to a Markov chain. We show the construction of the transition matrix and determine the steady-state probabilities for the time-homogeneous case.
The paper deals with a statistical simulation model for a newly proposed feed-forward blind oversampling Clock and Data Recovery circuit with low hardware complexity. Unlike previous published solutions, where the selected sampling phase is constant on a fixed-length window, the new circuit selects the phase upon the occurrence of several consecutive edges in one sampling domain, i.e. the window length changes randomly. The proposed simulation model is based on periodic Markov chain representation of the domain-selection process. The averaged BitError Rate can be simply computed from the steady-state of the chain. Computational complexity is determined by the jitter period length. The model includes random jitter, sinusoidal jitter, and frequency offset of transmit and receive clocks.
This contribution concerns the asymptotic behavior of solutions of systems of difference equations. It reassumes paper [J. Dibl´ık: Anti-Lyapunovmethod for systems of discrete equations, Nonlinear Anal. 57 (2004), 1043-1057], where conditions are described which ensure that the graph of at least one solution of the studied system stays in a prescribed domain. Those conditions include the existence of a so called connecting function for the sets forming the given domain. The connecting function must have certain properties which enable the use of the retract method for the proof of the main result of the cited work. We show that the conditions can be made simpler, without the need to introduce the connecting function.
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