# zbMATH — the first resource for mathematics

One case of appearance of positive solutions of delayed discrete equations. (English) Zbl 1099.39001
The authors study the existence of a positive solution of the nonlinear delay discrete equation $\Delta u(k+n)=f\big (k,u(k),u(k+1),\dots ,u(k+n)\big ), \quad k\in {\mathbb Z},\;k\geq a.$ They derive a sharp sufficient condition for the existence of such solution. The result is a special case with $$b(k)\equiv 0$$ and $$c(k):=\sqrt {k}\, \big ( \frac {n}{n+1} \big )^k$$ of the existence and boundedness result.

##### MSC:
 39A10 Additive difference equations 39A11 Stability of difference equations (MSC2000)
##### Keywords:
positive solution; nonlinear delay difference equation
Full Text:
##### References:
  R. P. Agarwal: Difference Equations and Inequalities, Theory, Methods, and Applications, 2nd ed. Marcel Dekker, New York, 2000.  J. Baštinec, J. Diblík, B. Zhang: Existence of bounded solutions of discrete delayed equations. Proceedings of the Sixth International Conference on Difference Equations and Applications, Augsburg 2001, Taylor $$\&$$ Francis (eds.), In print.  J. Diblík: A criterion for existence of positive solutions of systems of retarded functional differential equations. Nonlinear Anal. 38 (1999), 327-339. · Zbl 0935.34061  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.  Y. Domshlak, I. P. Stavroulakis: Oscillation of first-order delay differential equations in a critical case. Appl. Anal. 61 (1996), 359-371. · Zbl 0882.34069  S. N. Elaydi: An Introduction to Difference Equations, 2nd ed. Springer, New York, 1999. · Zbl 0930.39001  Á. Elbert, I. P. Stavroulakis: Oscillation and non-oscillation criteria for delay differential equations. Proc. Amer. Math. Soc. 123 (1995), 1503-1510. · Zbl 0828.34057  L. H. Erbe, Q. Kong, B. G. Zhang: Oscillation Theory for Functional Differential Equations. Marcel Dekker, New York, 1995.  K. Gopalsamy: Stability and Oscillations in Delay Differential Equations of Population Dynamics. Kluwer Academic Publishers, Dordrecht, 1992. · Zbl 0752.34039  I. Györi, G. Ladas: Oscillation Theory of Delay Differential Equations. Clarendon Press, Oxford, 1991. · Zbl 0780.34048  A. Tineao: Asympotic classification of the positive solutions of the nonautonomous two-competing species problem. J. Math. Anal. Appl. 214 (1997), 324-348.  Xue-Zhong He, K. Gopalsamy: Persistence, attractivity, and delay in facultative mutualism. J. Math. Anal. Appl. 215 (1997), 154-173. · Zbl 0893.34036
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.