Siafarikas, Panayiotis D. (ed.), International conference on differential, difference equations and their applications, July 1–5, 2002, Patras, Greece. Cairo: Hindawi Publishing Corporation (ISBN 977-5945-14-3/hbk). 207-216 (2004).

Summary: A delayed discrete equation

${\Delta}u(k+n)=-p\left(k\right)u\left(k\right)$ with positive coefficient

$p$ is considered. Sufficient conditions with respect to

$p$ are formulated in order to guarantee the existence of positive solutions if

$k\to \infty $. As a tool of the proof of corresponding result, the method described in the authors’ previous papers is used [cf.

*J. Diblík*, Comput. Math. Appl. 42, No. 3–5, 515–528 (2001;

Zbl 0999.39005);

*J. Baštinec, J. Diblík* and

*B. Zhong*, CRC Press. 359–366 (2004;

Zbl 1065.39006)]. Except for the fact of the existence of positive solutions, their upper estimation is given. The analysis shows that every positive solution of the indicated family of positive solutions tends to zero (if

$k\to \infty $) with the speed not smaller than the speed characterized by the function

$\sqrt{k}\xb7{(n/(n+1))}^{k}$. A comparison with the known results is given and some open questions are discussed.