Geodesic
Oscillation of a nonlinear difference equation with several delays
Mathematica Bohemica, Tome 128 (2003) no. 3, pp. 309-317

Voir la notice de l'article provenant de la source Czech Digital Mathematics Library

MR Zbl
In this paper we consider the nonlinear difference equation with several delays \[ (ax_{n+1}+bx_{n})^k-(cx_{n})^k+\sum \limits _{i=1}^{m} p_{i}(n)x^k_{n-\sigma _{i}}=0 \] where $a,b,c\in (0,\infty )$, $k=q/r, q, r$ are positive odd integers, $m$, $\sigma _{i}$ are positive integers, $\lbrace p_{i}(n)\rbrace $, $i=1,2,\dots ,m, $ is a real sequence with $p_{i}(n)\ge 0$ for all large $n$, and $\liminf _{n\rightarrow \infty }p_{i}(n)=p_{i}\infty $, $i=1,2,\dots ,m$. Some sufficient conditions for the oscillation of all solutions of the above equation are obtained.
In this paper we consider the nonlinear difference equation with several delays \[ (ax_{n+1}+bx_{n})^k-(cx_{n})^k+\sum \limits _{i=1}^{m} p_{i}(n)x^k_{n-\sigma _{i}}=0 \] where $a,b,c\in (0,\infty )$, $k=q/r, q, r$ are positive odd integers, $m$, $\sigma _{i}$ are positive integers, $\lbrace p_{i}(n)\rbrace $, $i=1,2,\dots ,m, $ is a real sequence with $p_{i}(n)\ge 0$ for all large $n$, and $\liminf _{n\rightarrow \infty }p_{i}(n)=p_{i}\infty $, $i=1,2,\dots ,m$. Some sufficient conditions for the oscillation of all solutions of the above equation are obtained.
DOI : 10.21136/MB.2003.134176
Classification : 39A10, 39A11
Keywords: nonlinear difference equtions; oscillation; eventually positive solutions; characteristic equation 
Luo, X. N.; Zhou, Yong; Li, C. F. Oscillation of a nonlinear difference equation with several delays. Mathematica Bohemica, Tome 128 (2003) no. 3, pp. 309-317. doi: 10.21136/MB.2003.134176
@article{10_21136_MB_2003_134176,
     author = {Luo, X. N. and Zhou, Yong and Li, C. F.},
     title = {Oscillation of a nonlinear difference equation with~several delays},
     journal = {Mathematica Bohemica},
     pages = {309--317},
     year = {2003},
     volume = {128},
     number = {3},
     doi = {10.21136/MB.2003.134176},
     mrnumber = {2012607},
     zbl = {1055.39015},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.21136/MB.2003.134176/}
}
TY  - JOUR
AU  - Luo, X. N.
AU  - Zhou, Yong
AU  - Li, C. F.
TI  - Oscillation of a nonlinear difference equation with several delays
JO  - Mathematica Bohemica
PY  - 2003
SP  - 309
EP  - 317
VL  - 128
IS  - 3
UR  - http://geodesic.mathdoc.fr/articles/10.21136/MB.2003.134176/
DO  - 10.21136/MB.2003.134176
LA  - en
ID  - 10_21136_MB_2003_134176
ER  - 
%0 Journal Article
%A Luo, X. N.
%A Zhou, Yong
%A Li, C. F.
%T Oscillation of a nonlinear difference equation with several delays
%J Mathematica Bohemica
%D 2003
%P 309-317
%V 128
%N 3
%U http://geodesic.mathdoc.fr/articles/10.21136/MB.2003.134176/
%R 10.21136/MB.2003.134176
%G en
%F 10_21136_MB_2003_134176

[1] R. P. Agarwal, P. J. Y. Wong: Advanced Topics in Difference Equations. Kluwer Academic Publishers, 1997. | MR

[2] I. Györi, G. Ladas: Oscillation Theory of Delay Differential Equations with Applications. Oxford Science Publications, 1991. | MR

[3] G. H. Hardy, J. E. Littlewood, G. Polya: Inequalities. Second edition, Cambridge University Press, 1952. | MR

[4] B. Li: Discrete oscillations. J. Differ. Equations Appl. 2 (1996), 389–399. | DOI | MR | Zbl

[5] S. T. Liu, S. S. Cheng: Oscillation criteria for a nonlinear difference equation. Far East J. Math. Sci. 5 (1997), 387–392. | MR

[6] J. Yang, X. P. Guan et al.: Nonexistence of positive solution of certain nonlinear difference equations. Acta Math. Appl. Sinica 23 (2000), 562–567. (Chinese) | MR

[7] J. S. Yu, B. G. Zhang, X. Z. Qian: Oscillations of delay differential equations with oscillatory coefficients. J. Math. Anal. Appl. 177 (1993), 432–444. | DOI | MR

[8] Y. Zhou, B. G. Zhang: Oscillations of delay differential equations in a critical state. Comput. Math. Appl. 39 (2000), 71–89. | DOI | MR

[9] B. G. Zhang, Yong Zhou: Oscillation of delay differential equations with several delays. (to appear).

[10] B. G. Zhang, Y. J. Sun: Existence of nonoscillatory solutions of a class of nonlinear difference equations with a forced term. Math. Bohem. 126 (2001), 639–647. | MR

Cité par Sources :