Journal articles: 'Delay difference equation'

1

Tang, X. H., and J. S. Yu. "Oscillation of delay difference equation." Computers & Mathematics with Applications 37, no. 7 (April 1999): 11–20. http://dx.doi.org/10.1016/s0898-1221(99)00083-8.

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2

Davies, Roy O., and A. J. Ostaszewski. "On a Difference-Delay Equation." Journal of Mathematical Analysis and Applications 247, no. 2 (July 2000): 608–26. http://dx.doi.org/10.1006/jmaa.2000.6893.

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3

Wang, Chao, Ravi P. Agarwal, and Donal O’Regan. "δ-Almost Periodic Functions and Applications to Dynamic Equations." Mathematics 7, no. 6 (June 9, 2019): 525. http://dx.doi.org/10.3390/math7060525.

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In this paper, by employing matched spaces for time scales, we introduce a δ -almost periodic function and obtain some related properties. Also the hull equation for homogeneous dynamic equation is introduced and results of the existence are presented. In the sense of admitting exponential dichotomy for the homogeneous equation, the expression of a δ -almost periodic solution for a type of nonhomogeneous dynamic equation is obtained and the existence of δ -almost periodic solutions for new delay dynamic equations is considered. The results in this paper are valid for delay q-difference equations and delay dynamic equations whose delays may be completely separated from the time scale T .

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4

Tang, X. H. "Oscillation for nonlinear delay difference equations." Tamkang Journal of Mathematics 32, no. 4 (December 31, 2001): 275–80. http://dx.doi.org/10.5556/j.tkjm.32.2001.342.

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The oscillatory behavior of the first order nonlinear delay difference equation of the form $$ x_{n+1} - x_n + p_n x_{n-k}^{\alpha} = 0, ~~~ n = 0, 1, 2, \ldots ~~~~~~~ \eqno{(*)} $$ is investigated. A necessary and sufficient condition of oscillation for sublinear equation (*) ($ 0 < \alpha < 1 $) and an almost sharp sufficient condition of oscillation for superlinear equation (*) ($ \alpha > 1 $) are obtained.

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5

Ding, Xiaohua. "Exponential stability of a kind of stochastic delay difference equations." Discrete Dynamics in Nature and Society 2006 (2006): 1–9. http://dx.doi.org/10.1155/ddns/2006/94656.

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We present a Razumilchin-type theorem for stochastic delay difference equation, and use it to investigate the mean square exponential stability of a kind of nonautonomous stochastic difference equation which may also be viewed as an approximation of a nonautonomous stochastic delay integrodifferential equations (SDIDEs), and of a difference equation arises from some of the earliest mathematical models of the macroeconomic trade cycle with the environmental noise.

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Dai, Binxiang, and Na Zhang. "Stability and global attractivity for a class of nonlinear delay difference equations." Discrete Dynamics in Nature and Society 2005, no. 3 (2005): 227–34. http://dx.doi.org/10.1155/ddns.2005.227.

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A class of nonlinear delay difference equations are considered and some sufficient conditions for global attractivity of solutions of the equation are obtained. It is shown that the stability properties, both local and global, of the equilibrium of the delay equation can be derived from those of an associated nondelay equation.

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7

Wei, Zhijian. "Periodicity in a Class of Systems of Delay Difference Equations." Journal of Applied Mathematics 2011 (2011): 1–11. http://dx.doi.org/10.1155/2011/735825.

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We study a system of delay difference equations modeling four-dimensional discrete-time delayed neural networks with no internal decay. Such a discrete-time system can be regarded as the discrete analog of a differential equation with piecewise constant argument. By using semicycle analysis method, it is shown that every bounded solution of this discrete-time system is eventually periodic. The obtained results are new, and they complement previously known results.

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8

Győri, I., G. Ladas, and P. N. Vlahos. "Global attractivity in a delay difference equation." Nonlinear Analysis: Theory, Methods & Applications 17, no. 5 (January 1991): 473–79. http://dx.doi.org/10.1016/0362-546x(91)90142-n.

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9

Sun, Taixiang, Hongjian Xi, and Mingde Xie. "Global stability for a delay difference equation." Journal of Applied Mathematics and Computing 29, no. 1-2 (September 3, 2008): 367–72. http://dx.doi.org/10.1007/s12190-008-0137-1.

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10

Ashyralyev, A., K. Turk, and D. Agirseven. "On the stable difference scheme for the time delay telegraph equation." BULLETIN OF THE KARAGANDA UNIVERSITY-MATHEMATICS 99, no. 3 (September 30, 2020): 105–19. http://dx.doi.org/10.31489/2020m3/105-119.

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The stable difference scheme for the approximate solution of the initial boundary value problem for the telegraph equation with time delay in a Hilbert space is presented. The main theorem on stability of the difference scheme is established. In applications, stability estimates for the solution of difference schemes for the two type of the time delay telegraph equations are obtained. As a test problem, one-dimensional delay telegraph equation with nonlocal boundary conditions is considered. Numerical results are provided.

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11

Kikina, L. K., and I. P. Stavroulakis. "Oscillation Criteria for Second-Order Delay, Difference, and Functional Equations." International Journal of Differential Equations 2010 (2010): 1–14. http://dx.doi.org/10.1155/2010/598068.

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Consider the second-order linear delay differential equationx′′(t)+p(t)x(τ(t))=0,t≥t0, wherep∈C([t0,∞),ℝ+),τ∈C([t0,∞),ℝ),τ(t)is nondecreasing,τ(t)≤tfort≥t0andlimt→∞τ(t)=∞, the (discrete analogue) second-order difference equationΔ2x(n)+p(n)x(τ(n))=0, whereΔx(n)=x(n+1)−x(n),Δ2=Δ∘Δ,p:ℕ→ℝ+,τ:ℕ→ℕ,τ(n)≤n−1, andlimn→∞τ(n)=+∞, and the second-order functional equationx(g(t))=P(t)x(t)+Q(t)x(g2(t)),t≥t0, where the functionsP,Q∈C([t0,∞),ℝ+),g∈C([t0,∞),ℝ),g(t)≢tfort≥t0,limt→∞g(t)=∞, andg2denotes the 2th iterate of the functiong, that is,g0(t)=t,g2(t)=g(g(t)),t≥t0. The most interesting oscillation criteria for the second-order linear delay differential equation, the second-order difference equation and the second-order functional equation, especially in the case whereliminft→∞∫τ(t)tτ(s)p(s)ds≤1/eandlimsupt→∞∫τ(t)tτ(s)p(s)ds<1for the second-order linear delay differential equation, and0<liminft→∞{Q(t)P(g(t))}≤1/4andlimsupt→∞{Q(t)P(g(t))}<1, for the second-order functional equation, are presented.

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12

Ashyralyev, Allaberen, and Deniz Agirseven. "Bounded Solutions of Semilinear Time Delay Hyperbolic Differential and Difference Equations." Mathematics 7, no. 12 (December 2, 2019): 1163. http://dx.doi.org/10.3390/math7121163.

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In this paper, we study the initial value problem for a semilinear delay hyperbolic equation in Hilbert spaces with a self-adjoint positive definite operator. The mean theorem on the existence and uniqueness of a bounded solution of this differential problem for a semilinear hyperbolic equation with unbounded time delay term is established. In applications, the existence and uniqueness of bounded solutions of four problems for semilinear hyperbolic equations with time delay in unbounded term are obtained. For the approximate solution of this abstract differential problem, the two-step difference scheme of a first order of accuracy is presented. The mean theorem on the existence and uniqueness of a uniformly bounded solution of this difference scheme with respect to time stepsize is established. In applications, the existence and uniqueness of a uniformly bounded solutions with respect to time and space stepsizes of difference schemes for four semilinear partial differential equations with time delay in unbounded term are obtained. In general, it is not possible to get the exact solution of semilinear hyperbolic equations with unbounded time delay term. Therefore, numerical results for the solution of difference schemes for one and two dimensional semilinear hyperbolic equation with time delay are presented. Finally, some numerical examples are given to confirm the theoretical analysis.

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13

Bohner, Martin, Srinivasan Geetha, Srinivasan Selvarangam, and Ethiraju Thandapani. "Oscillation of third-order delay difference equations with negative damping term." Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica 72, no. 1 (June 25, 2018): 19. http://dx.doi.org/10.17951/a.2018.72.1.19-28.

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The aim of this paper is to investigate the oscillatory and asymptotic behavior of solutions of a third-order delay difference equation. By using comparison theorems, we deduce oscillation of the difference equation from its relation to certain associated first-order delay difference equations or inequalities. Examples are given to illustrate the main results.

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14

Jiang, Jianchu. "Oscillation of nonlinear delay difference equations." International Journal of Mathematics and Mathematical Sciences 28, no. 5 (2001): 301–6. http://dx.doi.org/10.1155/s0161171201007323.

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15

Sekiguchi, Masaki, Emiko Ishiwata, and Yukihiko Nakata. "Convergence of a Logistic Type Ultradiscrete Model." Discrete Dynamics in Nature and Society 2017 (2017): 1–6. http://dx.doi.org/10.1155/2017/7893049.

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We derive a piecewise linear difference equation from logistic equations with time delay by ultradiscretization. The logistic equation that we consider in this paper has been shown to be globally stable in the continuous and discrete time formulations. Here, we study if ultradiscretization preserves the global stability property, analyzing the asymptotic behaviour of the obtained piecewise linear difference equation. It is shown that our piecewise linear difference equation has a threshold property concerning global attractivity of equilibria, similar to the stable logistic equations with time delay.

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16

Kiventidis, Thomas. "Positive solutions of integrodifferential and difference equations with unbounded delay." Glasgow Mathematical Journal 35, no. 1 (January 1993): 105–13. http://dx.doi.org/10.1017/s0017089500009629.

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AbstractWe establish a necessary and sufficient condition for the existence of a positive solution of the integrodifferential equationwhere nis an increasing real-valued function on the interval [0, α); that is, if and only if the characteristic equationadmits a positive root.Consider the difference equation , where is a sequence of non-negative numbers. We prove this has positive solution if and only if the characteristic equation admits a root in (0, 1). For general results on integrodifferential equations we refer to the book by Burton [1] and the survey article by Corduneanu and Lakshmikantham [2]. Existence of a positive solution and oscillations in integrodifferential equations or in systems of integrodifferential equations recently have been investigated by Ladas, Philos and Sficas [5], Györi and Ladas [4], Philos and Sficas [12], Philos [9], [10], [11].Recently, there has been some interest in the existence or the non-existence of positive solutions or the oscillation behavior of some difference equations. See Ladas, Philos and Sficas [6], [7].The purpose of this paper is to investigate the positive solutions of integrodifferential equations (Section 1) and difference equations (Section 2) with unbounded delay. We obtain also some results for integrodifferential and difference inequalities.

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17

Ekimov, Alexander V., Aleksei P. Zhabko, and Pavel V. Yakovlev. "The stability of differential-difference equations with proportional time delay. I. Linear controlled system." Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control Processes 16, no. 3 (2020): 316–25. http://dx.doi.org/10.21638/11701/spbu10.2020.308.

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The article considers a controlled system of linear differential-difference equations with a linearly increasing delay. Sufficient conditions for the asymptotic stability of such systems are known; however, general conditions for the stabilizability of controlled systems and constructive algorithms for constructing stabilizing controls have not yet been obtained. For a linear differential-difference equation of delayed type with linearly increasing delay, the canonical Zubov transformation is applied and conditions for the stabilization of such systems by static control are derived. An algorithm for checking the conditions for the existence of a stabilizing control and for its constructing is formulated. New theorems on stability analysis of systems of linear differential-difference equations with linearly increasing delay are proven. The results obtained can be applied to the case of systems with several proportional delays.

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18

TIAN, Chuan-Jun, Sui Sun CHENG, and Sheng-Li XIE. "Frequent Oscillation Criteria for a Delay Difference Equation." Funkcialaj Ekvacioj 46, no. 3 (2003): 421–39. http://dx.doi.org/10.1619/fesi.46.421.

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19

Dhar, Probir Kumar, Abhik Mukherjee, and Durjoy Majumder. "Difference Delay Equation-Based Analytical Model of Hematopoiesis." Automatic Control of Physiological State and Function 1 (2012): 1–11. http://dx.doi.org/10.4303/acpsf/235488.

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20

Philos, Ch G. "Oscillations in a nonautonomous delay logistic difference equation." Proceedings of the Edinburgh Mathematical Society 35, no. 1 (February 1992): 121–31. http://dx.doi.org/10.1017/s0013091500005381.

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Consider the nonautonomous delay logistic difference equationwhere (pn)n≧0 is a sequence of nonnegative numbers, (ln)n≧0 is a sequence of positive integers with limn→∞(n−ln) = ∞ and K is a positive constant. Only solutions which are positive for n≧0 are considered. We established a sharp condition under which all solutions of (E0) are oscillatory about the equilibrium point K. Also we obtained sufficient conditions for the existence of a solution of (E0) which is nonoscillatory about K.

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21

Driver, R. D., G. Ladas, and P. N. Vlahos. "Asymptotic behavior of a linear delay difference equation." Proceedings of the American Mathematical Society 115, no. 1 (January 1, 1992): 105. http://dx.doi.org/10.1090/s0002-9939-1992-1111217-0.

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22

Yan, J., and B. Liu. "Asymptotic behavior of a nonlinear delay difference equation." Applied Mathematics Letters 8, no. 6 (November 1995): 1–5. http://dx.doi.org/10.1016/0893-9659(95)00075-2.

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23

Zhou, Yinggao. "Global attractivity in a delay logistic difference equation." Applied Mathematics-A Journal of Chinese Universities 18, no. 1 (March 2003): 53–58. http://dx.doi.org/10.1007/s11766-003-0083-5.

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24

Khatibzadeh, Hadi. "An oscillation criterion for a delay difference equation." Computers & Mathematics with Applications 57, no. 1 (January 2009): 37–41. http://dx.doi.org/10.1016/j.camwa.2008.07.041.

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25

Zhu, Wei, Daoyi Xu, and Zhichun Yang. "Global exponential stability of impulsive delay difference equation." Applied Mathematics and Computation 181, no. 1 (October 2006): 65–72. http://dx.doi.org/10.1016/j.amc.2006.01.015.

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26

Liu, Zeqing, Liangshi Zhao, Jeong Sheok Ume, and Shin Min Kang. "Solvability of a Second Order Nonlinear Neutral Delay Difference Equation." Abstract and Applied Analysis 2011 (2011): 1–24. http://dx.doi.org/10.1155/2011/328914.

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This paper studies the second-order nonlinear neutral delay difference equationΔ[anΔ(xn+bnxn−τ)+f(n,xf1n,…,xfkn)]+g(n,xg1n,…,xgkn)=cn,n≥n0. By means of the Krasnoselskii and Schauder fixed point theorem and some new techniques, we get the existence results of uncountably many bounded nonoscillatory, positive, and negative solutions for the equation, respectively. Ten examples are given to illustrate the results presented in this paper.

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27

Falkena, Swinda K. J., Courtney Quinn, Jan Sieber, and Henk A. Dijkstra. "A delay equation model for the Atlantic Multidecadal Oscillation." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 477, no. 2246 (February 2021): 20200659. http://dx.doi.org/10.1098/rspa.2020.0659.

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A new technique to derive delay models from systems of partial differential equations, based on the Mori–Zwanzig (MZ) formalism, is used to derive a delay-difference equation model for the Atlantic Multidecadal Oscillation (AMO). The MZ formalism gives a rewriting of the original system of equations, which contains a memory term. This memory term can be related to a delay term in a resulting delay equation. Here, the technique is applied to an idealized, but spatially extended, model of the AMO. The resulting delay-difference model is of a different type than the delay differential model which has been used to describe the El Niño Southern Oscillation. In addition to this model, which can also be obtained by integration along characteristics, error terms for a smoothing approximation of the model have been derived from the MZ formalism. Our new method of deriving delay models from spatially extended models has a large potential to use delay models to study a range of climate variability phenomena.

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28

Moremedi, G. M., and I. P. Stavroulakis. "Oscillation Conditions for Difference Equations with a Monotone or Nonmonotone Argument." Discrete Dynamics in Nature and Society 2018 (June 21, 2018): 1–13. http://dx.doi.org/10.1155/2018/9416319.

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Consider the first-order delay difference equation with a constant argument Δxn+pnxn-k=0, n=0,1,2,…, and the delay difference equation with a variable argument Δxn+pnxτn=0, n=0,1,2,…, where p(n) is a sequence of nonnegative real numbers, k is a positive integer, Δx(n)=x(n+1)-x(n), and τ(n) is a sequence of integers such that τ(n)≤n-1 for all n≥0 and limn→∞τ(n)=∞. A survey on the oscillation of all solutions to these equations is presented. Examples illustrating the results are given.

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29

Zhang, B. G., and Pengxiang Yan. "Classification of solutions of delay difference equations." International Journal of Mathematics and Mathematical Sciences 17, no. 3 (1994): 619–23. http://dx.doi.org/10.1155/s016117129400089x.

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30

Wiener, Joseph, and Lokenath Debnath. "A parabolic differential equation with unbounded piecewise constant delay." International Journal of Mathematics and Mathematical Sciences 15, no. 2 (1992): 339–46. http://dx.doi.org/10.1155/s0161171292000425.

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31

Ouncharoen, Rujira, Saowaluck Chasreechai, and Thanin Sitthiwirattham. "On Nonlinear Fractional Difference Equation with Delay and Impulses." Symmetry 12, no. 6 (June 8, 2020): 980. http://dx.doi.org/10.3390/sym12060980.

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In this paper, we establish the existence results for a nonlinear fractional difference equation with delay and impulses. The Banach and Schauder’s fixed point theorems are employed as tools to study the existence of its solutions. We obtain the theorems showing the conditions for existence results. Finally, we provide an example to show the applicability of our results.

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32

Stević, Stevo, Bratislav Iričanin, Witold Kosmala, and Zdeněk Šmarda. "Note on the bilinear difference equation with a delay." Mathematical Methods in the Applied Sciences 41, no. 18 (October 22, 2018): 9349–60. http://dx.doi.org/10.1002/mma.5293.

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33

Kipnis, M. M., and V. V. Malygina. "The Stability Cone for a Matrix Delay Difference Equation." International Journal of Mathematics and Mathematical Sciences 2011 (2011): 1–15. http://dx.doi.org/10.1155/2011/860326.

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We construct a stability cone, which allows us to analyze the stability of the matrix delay difference equation . We assume that and are simultaneously triangularizable matrices. We construct points in which are functions of eigenvalues of matrices , such that the equation is asymptotically stable if and only if all the points lie inside the stability cone.

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34

Zhang, Guang, and Sui Sun Cheng. "Oscillation criteria for a neutral difference equation with delay." Applied Mathematics Letters 8, no. 3 (May 1995): 13–17. http://dx.doi.org/10.1016/0893-9659(95)00022-i.

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35

Li, Xianyi, and Deming Zhu. "Global asymptotic stability for a nonlinear delay difference equation." Applied Mathematics-A Journal of Chinese Universities 17, no. 2 (June 2002): 183–88. http://dx.doi.org/10.1007/s11766-002-0043-5.

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36

Wu, Kaining, Xiaohua Ding, and Liming Wang. "Stability and Stabilization of Impulsive Stochastic Delay Difference Equations." Discrete Dynamics in Nature and Society 2010 (2010): 1–15. http://dx.doi.org/10.1155/2010/592036.

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When an impulsive control is adopted for a stochastic delay difference system (SDDS), there are at least two situations that should be contemplated. If the SDDS is stable, then what kind of impulse can the original system tolerate to keep stable? If the SDDS is unstable, then what kind of impulsive strategy should be taken to make the system stable? Using the Lyapunov-Razumikhin technique, we establish criteria for the stability of impulsive stochastic delay difference equations and these criteria answer those questions. As for applications, we consider a kind of impulsive stochastic delay difference equation and present some corollaries to our main results.

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Liu, Zeqing, Wei Sun, Jeong Sheok Ume, and Shin Min Kang. "Positive Solutions of a Second-Order Nonlinear Neutral Delay Difference Equation." Abstract and Applied Analysis 2012 (2012): 1–30. http://dx.doi.org/10.1155/2012/172939.

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The purpose of this paper is to study solvability of the second-order nonlinear neutral delay difference equationΔ(a(n,ya1n,…,yarn)Δ(yn+bnyn-τ))+f(n,yf1n,…,yfkn)=cn, ∀n≥n0. By making use of the Rothe fixed point theorem, Leray-Schauder nonlinear alternative theorem, Krasnoselskill fixed point theorem, and some new techniques, we obtain some sufficient conditions which ensure the existence of uncountably many bounded positive solutions for the above equation. Five nontrivial examples are given to illustrate that the results presented in this paper are more effective than the existing ones in the literature.

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Ibrahim, Tarek F., Abdul Qadeer Khan, and Abdelhameed Ibrahim. "Qualitative Behavior of a Nonlinear Generalized Recursive Sequence with Delay." Mathematical Problems in Engineering 2021 (August 5, 2021): 1–8. http://dx.doi.org/10.1155/2021/6162320.

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Difference equations are of growing importance in engineering in view of their applications in discrete time-systems used in association with microprocessors. We will check out the global stability and boundedness for a nonlinear generalized high-order difference equation with delay.

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39

Cermák, Jan, Jiří Jánský, and Petr Tomásek. "Two types of stability conditions for linear delay difference equations." Applicable Analysis and Discrete Mathematics 9, no. 1 (2015): 120–38. http://dx.doi.org/10.2298/aadm141009016c.

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The paper discusses asymptotic stability conditions for a four-parameter linear difference equation appearing in the process of discretization of a delay differential equation. We present two types of conditions, which are necessary and sufficient for asymptotic stability of the studied equation. A relationship between both the types of conditions is established and some of their consequences are discussed.

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40

Glagolev, Mikhail V., Aleksandr F. Sabrekov, and Vladimir M. Goncharov. "Delay differential equations as a tool for mathematical modelling of population dynamic." Environmental Dynamics and Global Climate Change 9, no. 2 (November 27, 2018): 40–63. http://dx.doi.org/10.17816/edgcc10483.

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The manuscript constitutes a lecture from a course “Mathematical modelling of biological processes”, adapted to the format of the journal paper. This course of lectures is held by one of authors in Ugra State University. Delay differential equations are widely used in different ecological and biological problems. It has to do with the fact that delay differential equations are able to take into account that different biological processes depend not only on the state of the system at the moment but on the state of the system in previous moments too. The most popular case of using delay differential equations in biology is modelling in population ecology (including the modelling of several interacting populations dynamic, for example, in predator-prey system). Also delay differential equations are considered in demography, immunology, epidemiology, molecular biology (to provide mathematical description of regulatory mechanisms in a cell functioning and division), physiology as well as for modelling certain important production processes (for example, in agriculture). In the beginning of the paper as introduction some basic concepts of differential difference equation theory (delay differential equations are specific type of differential difference equations) is considered and their classification is presented. Then it is discussed in more detail how such an important equations of population dynamic as Maltus equation and logistic (Verhulst-Pearl) equation are transformed into corresponsive delay differential equations – Goudriaan-Roermund and Hutchinson. Then several discussion questions on using of a delay differential equations in biological models are considered. It is noted that in a certain cases using of a delay differential equations lead to an incorrect behavior from the point of view of common sense. Namely solution of Goudriaan-Roermund equation with harvesting, stopped when all species were harvested, shows that spontaneous generation takes place in the system. This incorrect interpretation has to do with the fact that delay differential equations are used to simplify considered models (that are usually are systems of ordinary differential equations). Using of integro-differential equations could be more appropriate because in these equations background could be taken into account in a more natural way. It is shown that Hutchinson equation can be obtained by simplification of Volterra integral equation with a help of Diraq delta function. Finally, a few questions of analytical and numerical solution of delay differential equations are discussed.

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Tyler, Albert V., Linda L. Sebring, Margaret C. Murphy, and Lea F. Murphy. "A Sensitivity Analysis of Deriso's Delay-Difference Equation Using Simulation." Canadian Journal of Fisheries and Aquatic Sciences 42, no. 4 (April 1, 1985): 836–41. http://dx.doi.org/10.1139/f85-107.

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The degree to which the delay-difference equation can track biomass changes of fish stocks was examined using a sensitivity analysis technique in conjunction with a simulation model having explicit age-class structure. The simplicity of the delay-difference equation results from the manner in which it subsumes age-class structure by using two parameters, one for mortality and one for growth. The changes in biomass in a simulated stock were followed closely by the equation when error-free determinations of stock parameters were transferred from the simulated stock to the equation, even when a high degree of density dependence was allowed in the growth rate, and in some cases when there were large differences in age-specific mortality in the simulated stock. When error was induced in the stock parameters, the delay-difference equation was fairly robust in estimating biomass with mortality rate and growth rate simultaneously either above the true value or below the true value. Good biomass estimates also resulted when growth rate was at the true value and mortality rate was above the true value. Poor biomass estimates resulted when mortality was overestimated while growth was underestimated, or vice versa.

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42

Niri, Khadija, and Ioannis P. Stavroulakis. "On the oscillation of the solutions to delay and difference equations." Tatra Mountains Mathematical Publications 43, no. 1 (December 1, 2009): 173–87. http://dx.doi.org/10.2478/v10127-009-0036-3.

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Abstract Consider the first-order linear delay differential equation xʹ(t) + p(t)x(τ(t)) = 0, t≥ t0, (1) where p, τ ∈ C ([t0,∞, ℝ+, τ(t) is nondecreasing, τ(t) < t for t ≥ t0 and limt→∞ τ(t) = ∞, and the (discrete analogue) difference equation Δx(n) + p(n)x(τ(n)) = 0, n= 0, 1, 2,…, (1)ʹ where Δx(n) = x(n + 1) − x(n), p(n) is a sequence of nonnegative real numbers and τ(n) is a nondecreasing sequence of integers such that τ(n) ≤ n − 1 for all n ≥ 0 and limn→∞ τ(n) = ∞. Optimal conditions for the oscillation of all solutions to the above equations are presented.

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43

Sumathy, M., P. Venkata Mohan Reddy, and M. Maria Susai Manuel. "Qualitative Property of Third-Order Nonlinear Neutral Distributed-Delay Generalized Difference Equations." Mathematical Problems in Engineering 2021 (August 23, 2021): 1–12. http://dx.doi.org/10.1155/2021/2875613.

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This paper investigates the qualitative property of third-order nonlinear neutral distributed-delay generalized difference equations. By utilizing Philos-type technique and Riccati transformation, some oscillation criteria are presented to ensure that every solution of this equation oscillates or converges to zero. To illustrate the significance of our main result, we provide a suitable example.

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Yi, Taishan, and Xingfu Zou. "Map dynamics versus dynamics of associated delay reaction–diffusion equations with a Neumann condition." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 466, no. 2122 (April 29, 2010): 2955–73. http://dx.doi.org/10.1098/rspa.2009.0650.

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In this paper, we consider a class of delay reaction–diffusion equations (DRDEs) with a parameter ε >0. A homogeneous Neumann boundary condition and non-negative initial functions are posed to the equation. By letting , such an equation is formally reduced to a scalar difference equation (or map dynamical system). The main concern is the relation of the absolute (or delay-independent) global stability of a steady state of the equation and the dynamics of the nonlinear map in the equation. By employing the idea of attracting intervals for solution semiflows of the DRDEs, we prove that the globally stable dynamics of the map indeed ensures the delay-independent global stability of a constant steady state of the DRDEs. We also give a counterexample to show that the delay-independent global stability of DRDEs cannot guarantee the globally stable dynamics of the map. Finally, we apply the abstract results to the diffusive delay Nicholson blowfly equation and the diffusive Mackey–Glass haematopoiesis equation. The resulting criteria for both model equations are amazingly simple and are optimal in some sense (although there is no existing result to compare with for the latter).

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Zhou, Ying-gao. "Existence of positive solutions in a delay logistic difference equation." Journal of Central South University of Technology 9, no. 2 (June 2002): 142–44. http://dx.doi.org/10.1007/s11771-002-0060-9.

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Liu, Min, and Zhenyu Guo. "Solvability of a Higher-Order Nonlinear Neutral Delay Difference Equation." Advances in Difference Equations 2010 (2010): 1–14. http://dx.doi.org/10.1155/2010/767620.

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Jekl, Jan. "Linear even order homogenous difference equation with delay in coefficient." Electronic Journal of Qualitative Theory of Differential Equations, no. 45 (2020): 1–19. http://dx.doi.org/10.14232/ejqtde.2020.1.45.

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Yu, J. S. "Asymptotic stability for a linear difference equation with variable delay." Computers & Mathematics with Applications 36, no. 10-12 (November 1998): 203–10. http://dx.doi.org/10.1016/s0898-1221(98)80021-7.

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Zhou, Zhan, and Qinqin Zhang. "Global attractivity of a nonautonomous logistic difference equation with delay." Computers & Mathematics with Applications 38, no. 7-8 (October 1999): 57–64. http://dx.doi.org/10.1016/s0898-1221(99)00238-2.

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Matsunaga, H., T. Hara, and S. Sakata. "Global attractivity for a nonlinear difference equation with variable delay." Computers & Mathematics with Applications 41, no. 5-6 (March 2001): 543–51. http://dx.doi.org/10.1016/s0898-1221(00)00297-2.

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Journal articles on the topic 'Delay difference equation'

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