# Article

 Title: One case of appearance of positive solutions of delayed discrete equations (English) Author: Baštinec, Jaromír Author: Diblík, Josef Language: English Journal: Applications of Mathematics ISSN: 0862-7940 (print) ISSN: 1572-9109 (online) Volume: 48 Issue: 6 Year: 2003 Pages: 429-436 Summary lang: English . Category: math . Summary: When mathematical models describing various processes are analysed, the fact of existence of a positive solution is often among the basic features. In this paper, a general delayed discrete equation $\Delta u(k+n)=f(k,u(k),u(k+1),\dots ,u(k+ n))$ is considered. Sufficient conditions concerning $f$ are formulated in order to guarantee the existence of a positive solution for $k\rightarrow \infty$. An upper estimate for it is given as well. The appearance of the positive solution takes its origin in the nature of the equation considered since the results hold only for delayed equations (i.e. for $n>0$) and not for the case of an ordinary equation (with $n=0$). (English) Keyword: positive solution Keyword: nonlinear discrete delayed equation MSC: 39A10 MSC: 39A11 MSC: 39A12 idZBL: Zbl 1099.39001 idMR: MR2025296 DOI: 10.1023/B:APOM.0000024484.15065.1c . Date available: 2009-09-22T18:14:59Z Last updated: 2020-07-02 Stable URL: http://hdl.handle.net/10338.dmlcz/134541 . Reference: [1] R. P. Agarwal: Difference Equations and Inequalities, Theory, Methods, and Applications, 2nd ed.Marcel Dekker, New York, 2000. MR 1740241 Reference: [2] J. Baštinec, J.  Diblík, B. Zhang: Existence of bounded solutions of discrete delayed equations.In: Proceedings of the Sixth International Conference on Difference Equations and Applications, Augsburg 2001, Taylor $\&$ Francis (eds.), In print. Reference: [3] J.  Diblík: A criterion for existence of positive solutions of systems of retarded functional differential equations.Nonlinear Anal. 38 (1999), 327–339. MR 1705781 Reference: [4] J.  Diblík, M.  Růžičková: Existence of positive solutions of a singular initial problem for nonlinear system of differential equations.Rocky Mountain J.  Math, In print. Reference: [5] Y.  Domshlak, I. P. Stavroulakis: Oscillation of first-order delay differential equations in a critical case.Appl. Anal. 61 (1996), 359–371. MR 1618248, 10.1080/00036819608840464 Reference: [6] S. N. Elaydi: An Introduction to Difference Equations, 2nd ed.Springer, New York, 1999. Zbl 0930.39001, MR 1711587 Reference: [7] Á. Elbert, I. P. Stavroulakis: Oscillation and non-oscillation criteria for delay differential equations.Proc. Amer. Math. Soc. 123 (1995), 1503–1510. MR 1242082, 10.1090/S0002-9939-1995-1242082-1 Reference: [8] L. H. Erbe, Q. Kong, B. G. Zhang: Oscillation Theory for Functional Differential Equations.Marcel Dekker, New York, 1995. MR 1309905 Reference: [9] K. Gopalsamy: Stability and Oscillations in Delay Differential Equations of Population Dynamics.Kluwer Academic Publishers, Dordrecht, 1992. Zbl 0752.34039, MR 1163190 Reference: [10] I. Györi, G.  Ladas: Oscillation Theory of Delay Differential Equations.Clarendon Press, Oxford, 1991. MR 1168471 Reference: [11] A. Tineao: Asympotic classification of the positive solutions of the nonautonomous two-competing species problem.J.  Math. Anal. Appl. 214 (1997), 324–348. MR 1475573 Reference: [12] Xue-Zhong He, K. Gopalsamy: Persistence, attractivity, and delay in facultative mutualism.J.  Math. Anal. Appl. 215 (1997), 154–173. MR 1478857, 10.1006/jmaa.1997.5632 .

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