Acta Univ. Agric. Silvic. Mendelianae Brun. 2011, 59(7), 245-248 | DOI: 10.11118/actaun201159070245

Summation comparison theorems for half-linear second order difference equations on finite interval

Robert Mařík
Ústav matematiky, Mendelova univerzita v Brně, Zemědělská 1, 613 00 Brno, Česká republika

In the paper, new comparison theorems for the half-linear difference equation
Δ(RkΦ(Δzk)) + CkΦ(zk+1) = 0, Φ(u) = |u|p-2 u, p > 1,
are derived. We show that if a solution of this equation has a generalized zero on the discrete interval [a, b], then the same holds for a solution of its majorant. The main tool used in the paper is the variational technique which relates nonexistence of a solution with a generalized zero with nonegativity of the p-degree functional defined on the suitable class of admissible functions.

Keywords: difference equation, second order, focal point, half-linear equation, p-degree functional, free end point
Grants and funding:

Author is supported by the Grant P201/10/1032 of the Czech Science Foundation.

Received: May 16, 2011; Published: January 26, 2014  Show citation

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Mařík, R. (2011). Summation comparison theorems for half-linear second order difference equations on finite interval. Acta Universitatis Agriculturae et Silviculturae Mendelianae Brunensis, 59(7), 245-248. doi: 10.11118/actaun201159070245
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