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One case of appearance of positive solutions of delayed discrete equations

Jaromír Baštinec; Josef Diblík

Applications of Mathematics (2003)

  • Volume: 48, Issue: 6, page 429-436
  • ISSN: 0862-7940

Abstract

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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 Δ u ( k + n ) = f ( k , u ( k ) , u ( k + 1 ) , ⋯ , u ( k + n ) ) is considered. Sufficient conditions concerning f are formulated in order to guarantee the existence of a positive solution for k → ∞ . 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 ).

How to cite

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Baštinec, Jaromír, and Diblík, Josef. "One case of appearance of positive solutions of delayed discrete equations." Applications of Mathematics 48.6 (2003): 429-436. <http://eudml.org/doc/33158>.

@article{Baštinec2003,
abstract = {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 $n=0$).},
author = {Baštinec, Jaromír, Diblík, Josef},
journal = {Applications of Mathematics},
keywords = {positive solution; nonlinear discrete delayed equation; positive solution; nonlinear delay difference equation},
language = {eng},
number = {6},
pages = {429-436},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {One case of appearance of positive solutions of delayed discrete equations},
url = {http://eudml.org/doc/33158},
volume = {48},
year = {2003},
}

TY - JOUR
AU - Baštinec, Jaromír
AU - Diblík, Josef
TI - One case of appearance of positive solutions of delayed discrete equations
JO - Applications of Mathematics
PY - 2003
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 48
IS - 6
SP - 429
EP - 436
AB - 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 $n=0$).
LA - eng
KW - positive solution; nonlinear discrete delayed equation; positive solution; nonlinear delay difference equation
UR - http://eudml.org/doc/33158
ER -

References

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  9. Stability and Oscillations in Delay Differential Equations of Population Dynamics, Kluwer Academic Publishers, Dordrecht, 1992. (1992) Zbl0752.34039MR1163190
  10. Oscillation Theory of Delay Differential Equations, Clarendon Press, Oxford, 1991. (1991) MR1168471
  11. Asympotic classification of the positive solutions of the nonautonomous two-competing species problem, J.  Math. Anal. Appl. 214 (1997), 324–348. (1997) MR1475573
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