## Asymptotic convergence of the solutions of a discrete equation with two delays in the critical case.(English)Zbl 1220.39004

Summary: A discrete equation $$\Delta y(n) = \beta(n)[y(n - j) - y(n - k)]$$ with two integer delays $$k$$ and $$j$$, $$k > j \geq 0$$ is considered for $$n \rightarrow \infty$$. We assume $$\beta : \mathbb Z^{\infty}_{n_{0}-k} \rightarrow (0, \infty)$$, where $$\mathbb Z^{\infty}_{n_0} = \{ n_0, n_0 + 1, \dots \}$$, $$n_0 \in \mathbb N$$ and $$n \in \mathbb Z^{\infty}_{n_0}$$. Criteria for the existence of strictly monotone and asymptotically convergent solutions for $$n \rightarrow \infty$$ are presented in terms of inequalities for the function $$\beta$$. Results are sharp in the sense that the criteria are valid even for some functions $$\beta$$ with a behavior near the so-called critical value, defined by the constant $$(k - j)^{-1}$$. Among others, it is proved that, for the asymptotic convergence of all solutions, the existence of a strictly monotone and asymptotically convergent solution is sufficient.

### MSC:

 39A12 Discrete version of topics in analysis 39A06 Linear difference equations 34K06 Linear functional-differential equations
Full Text:

### References:

 [1] O. Arino and M. Pituk, “Convergence in asymptotically autonomous functional-differential equations,” Journal of Mathematical Analysis and Applications, vol. 237, no. 1, pp. 376-392, 1999. · Zbl 0936.34064 · doi:10.1006/jmaa.1999.6489 [2] H. Bereketo\uglu and F. Karako\cc, “Asymptotic constancy for impulsive delay differential equations,” Dynamic Systems and Applications, vol. 17, no. 1, pp. 71-83, 2008. · Zbl 1159.34052 [3] I. Györi, F. Karako\cc, and H. Bereketo\uglu, “Convergence of solutions of a linear impulsive differential equations system with many delays,” Dynamics of Continuous, Discrete and Impulsive Systems Series A, Mathematical Analysis, vol. 18, no. 2, pp. 191-202, 2011. · Zbl 1227.34077 [4] H. Bereketo\uglu and M. Pituk, “Asymptotic constancy for nonhomogeneous linear differential equations with unbounded delays,” Discrete and Continuous Dynamical Systems. Series A, supplement, pp. 100-107, 2003. · Zbl 1071.34080 [5] J. Diblík, “Asymptotic convergence criteria of solutions of delayed functional differential equations,” Journal of Mathematical Analysis and Applications, vol. 274, no. 1, pp. 349-373, 2002. · Zbl 1025.34062 · doi:10.1016/S0022-247X(02)00311-6 [6] J. Diblík and M. Rů\vzi, “Convergence of the solutions of the equation y\?(t)=\beta (t)[y(t - \delta ) - y(t - \tau )] in the critical case,” The Journal of Mathematical Analysis and Applications, vol. 331, pp. 1361-1370, 2007. · Zbl 1125.34059 · doi:10.1016/j.jmaa.2006.10.001 [7] E. Messina, Y. Muroya, E. Russo, and A. Vecchio, “Convergence of solutions for two delays Volterra integral equations in the critical case,” Applied Mathematics Letters, vol. 23, no. 10, pp. 1162-1165, 2010. · Zbl 1197.45006 · doi:10.1016/j.aml.2010.05.002 [8] L. Berezansky and E. Braverman, “On oscillation of a food-limited population model with time delay,” Abstract and Applied Analysis, no. 1, pp. 55-66, 2003. · Zbl 1026.34075 · doi:10.1155/S1085337503209040 [9] H. Bereketoglu and A. Huseynov, “Convergence of solutions of nonhomogeneous linear difference systems with delays,” Acta Applicandae Mathematicae, vol. 110, no. 1, pp. 259-269, 2010. · Zbl 1204.39003 · doi:10.1007/s10440-008-9404-2 [10] M. Dehghan and M. J. Douraki, “Global attractivity and convergence of a difference equation,” Dynamics of Continuous, Discrete and Impulsive Systems. Series A. Mathematical Analysis, vol. 16, no. 3, pp. 347-361, 2009. · Zbl 1185.37025 [11] I. Györi and L. Horváth, “Asymptotic constancy in linear difference equations: limit formulae and sharp conditions,” Advances in Difference Equations, vol. 2010, Article ID 789302, 20 pages, 2010. · Zbl 1191.39001 · doi:10.1155/2010/789302 [12] S. Stević, “Global stability and asymptotics of some classes of rational difference equations,” Journal of Mathematical Analysis and Applications, vol. 316, no. 1, pp. 60-68, 2006. · Zbl 1090.39009 · doi:10.1016/j.jmaa.2005.04.077 [13] C. M. Kent, W. Kosmala, and S. Stević, “On the difference equation xn+1=xnxn - 2 - 1,” Abstract and Applied Analysis, vol. 2011, Article ID 815285, 25 pages, 2011. · Zbl 1216.39016 [14] J. Ba\vstinec and J. Diblík, “Subdominant positive solutions of the discrete equation \Delta u(k+n)= - p(k)u(k),” Abstract and Applied Analysis, no. 6, pp. 461-470, 2004. · Zbl 1101.39003 · doi:10.1155/S1085337504306056 [15] R. Medina and M. Pituk, “Nonoscillatory solutions of a second-order difference equation of Poincaré type,” Applied Mathematics Letters, vol. 22, no. 5, pp. 679-683, 2009. · Zbl 1169.39004 · doi:10.1016/j.aml.2008.04.015 [16] B. Z. Vulikh, A Brief Course in the Theory of Functions of a Real Variable (An Introduction to the Theory of the Integral), Mir Publishers, Moscow, Russia, 1976.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.