| Title:
|
Oscillation conditions for difference equations with several variable arguments (English) |
| Author:
|
Chatzarakis, George E. |
| Author:
|
Kusano, Takaŝi |
| Author:
|
Stavroulakis, Ioannis P. |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
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2464-7136 (online) |
| Volume:
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140 |
| Issue:
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3 |
| Year:
|
2015 |
| Pages:
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291-311 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Consider the difference equation $$ \Delta x(n)+\sum _{i=1}^{m}p_{i}(n)x(\tau _{i}(n))=0,\quad n\geq 0\quad \bigg [\nabla x(n)-\sum _{i=1}^{m}p_{i}(n)x(\sigma _{i}(n))=0,\quad n\geq 1\bigg ], $$ where $(p_{i}(n))$, $1\leq i\leq m$ are sequences of nonnegative real numbers, $\tau _{i}(n)$ [$\sigma _{i}(n)$], $1\leq \break i\leq m$ are general retarded (advanced) arguments and $\Delta $ [$\nabla $] denotes the forward (backward) difference operator $\Delta x(n)=x(n+1)-x(n)$ [$\nabla x(n)=x(n)-x(n-1)$]. New oscillation criteria are established when the well-known oscillation conditions $$ \limsup _{n\rightarrow \infty }\sum _{i=1}^{m}\sum _{j=\tau (n)}^{n}p_{i}(j)>1 \quad \bigg [\limsup _{n\rightarrow \infty }\sum _{i=1}^{m}\sum _{j=n}^{\sigma (n)}p_{i}(j)>1\bigg ] $$ and $$ \liminf _{n\rightarrow \infty }\sum _{i=1}^{m}\sum _{j=\tau _{i}(n)}^{n-1}p_{i}(j)>\frac {1}{\rm e} \quad \bigg [\liminf _{n\rightarrow \infty }\sum _{i=1}^{m}\sum _{j=n+1}^{\sigma _{i}(n)}p_{i}(j)>\frac {1}{\rm e}\bigg ] $$ are not satisfied. Here $\tau (n)=\max _{1\leq i\leq m}\tau _{i}(n)$ $[ \sigma (n)=\min _{1\leq i\leq m}\sigma _{i}(n) ]$. The results obtained essentially improve known results in the literature. Examples illustrating the results are also given. (English) |
| Keyword:
|
difference equation |
| Keyword:
|
retarded argument |
| Keyword:
|
advanced argument |
| Keyword:
|
oscillatory solution |
| Keyword:
|
nonoscillatory solution |
| MSC:
|
39A10 |
| MSC:
|
39A21 |
| idZBL:
|
Zbl 06486940 |
| idMR:
|
MR3397258 |
| DOI:
|
10.21136/MB.2015.144396 |
| . |
| Date available:
|
2015-09-03T10:51:25Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/144396 |
| . |
| Reference:
|
[1] Agarwal, R. P., Bohner, M., Grace, S. R., O'Regan, D.: Discrete Oscillation Theory.Hindawi Publishing Corporation New York (2005). Zbl 1084.39001, MR 2179948 |
| Reference:
|
[2] Baštinec, J., Berezansky, L., Diblík, J., Šmarda, Z.: A final result on the oscillation of solutions of the linear discrete delayed equation $\Delta x(n)=-p(n)x(n-k)$ with a positive coefficient.Abstr. Appl. Anal. 2011 (2011), Article No. 586328, 28 pages. Zbl 1223.39008, MR 2824906 |
| Reference:
|
[3] Baštinec, J., Diblík, J.: Remark on positive solutions of discrete equation $\Delta u(k+n)=$ $-p(k)u(k)$.Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods (electronic only) 63 (2005), e2145--e2151. Zbl 1224.39002, 10.1016/j.na.2005.01.007 |
| Reference:
|
[4] Baštinec, J., Diblík, J., Šmarda, Z.: Existence of positive solutions of discrete linear equations with a single delay.J. Difference Equ. Appl. 16 (2010), 1047-1056. Zbl 1207.39014, MR 2722821, 10.1080/10236190902718026 |
| Reference:
|
[5] Berezansky, L., Braverman, E.: On existence of positive solutions for linear difference equations with several delays.Adv. Dyn. Syst. Appl. 1 (2006), 29-47. Zbl 1124.39002, MR 2287633 |
| Reference:
|
[6] Chatzarakis, G. E., Koplatadze, R., Stavroulakis, I. P.: Optimal oscillation criteria for first order difference equations with delay argument.Pac. J. Math. 235 (2008), 15-33. Zbl 1153.39010, MR 2379767, 10.2140/pjm.2008.235.15 |
| Reference:
|
[7] Chatzarakis, G. E., Manojlovic, J., Pinelas, S., Stavroulakis, I. P.: Oscillation criteria of difference equations with several deviating arguments.Yokohama Math. J. 60 (2014), 13-31. Zbl 1318.39011, MR 3328615 |
| Reference:
|
[8] Chatzarakis, G. E., Philos, C. G., Stavroulakis, I. P.: An oscillation criterion for linear difference equations with general delay argument.Port. Math. (N.S.) 66 (2009), 513-533. Zbl 1186.39010, MR 2567680, 10.4171/PM/1853 |
| Reference:
|
[9] Chatzarakis, G. E., Pinelas, S., Stavroulakis, I. P.: Oscillations of difference equations with several deviated arguments.Aequationes Math. 88 (2014), 105-123. Zbl 1306.39007, MR 3250787, 10.1007/s00010-013-0238-2 |
| Reference:
|
[10] Erbe, L. H., Zhang, B. G.: Oscillation of discrete analogues of delay equations.Differ. Integral Equ. 2 (1989), 300-309. Zbl 0723.39004, MR 0983682 |
| Reference:
|
[11] Fukagai, N., Kusano, T.: Oscillation theory of first order functional-differential equations with deviating arguments.Ann. Mat. Pura Appl. (4) 136 (1984), 95-117. Zbl 0552.34062, MR 0765918 |
| Reference:
|
[12] Grammatikopoulos, M. K., Koplatadze, R., Stavroulakis, I. P.: On the oscillation of solutions of first-order differential equations with retarded arguments.Georgian Math. J. 10 (2003), 63-76. Zbl 1051.34051, MR 1990688, 10.1515/GMJ.2003.63 |
| Reference:
|
[13] Győri, I., Ladas, G.: Oscillation Theory of Delay Differential Equations: With Applications.Oxford Mathematical Monographs Clarendon Press, Oxford (1991). Zbl 0780.34048, MR 1168471 |
| Reference:
|
[14] Lakshmikantham, V., Trigiante, D.: Theory of Difference Equations: Numerical Methods and Applications.Mathematics in Science and Engineering 181 Academic Press, Boston (1988). Zbl 0683.39001, MR 0939611 |
| Reference:
|
[15] Li, X., Zhu, D.: Oscillation of advanced difference equations with variable coefficients.Ann. Differ. Equations 18 (2002), 254-263. Zbl 1010.39001, MR 1940383 |
| Reference:
|
[16] Luo, X. N., Zhou, Y., Li, C. F.: Oscillation of a nonlinear difference equation with several delays.Math. Bohem. 128 (2003), 309-317. Zbl 1055.39015, MR 2012607 |
| Reference:
|
[17] Stavroulakis, I. P.: Oscillation criteria for delay and difference equations with non-monotone arguments.Appl. Math. Comput. 226 (2014), 661-672. Zbl 1354.34120, MR 3144341, 10.1016/j.amc.2013.10.041 |
| Reference:
|
[18] Tang, X. H., Yu, J. S.: Oscillations of delay difference equations.Hokkaido Math. J. 29 (2000), 213-228. Zbl 0958.39015, MR 1745511, 10.14492/hokmj/1350912965 |
| Reference:
|
[19] Tang, X. H., Yu, J. S.: Oscillation of delay difference equation.Comput. Math. Appl. 37 (1999), 11-20. Zbl 0937.39012, MR 1688201, 10.1016/S0898-1221(99)00083-8 |
| Reference:
|
[20] Tang, X. H., Zhang, R. Y.: New oscillation criteria for delay difference equations.Comput. Math. Appl. 42 (2001), 1319-1330. Zbl 1002.39022, MR 1861531, 10.1016/S0898-1221(01)00243-7 |
| Reference:
|
[21] Yan, W., Meng, Q., Yan, J.: Oscillation criteria for difference equation of variable delays.Dyn. Contin. Discrete Impuls. Syst. Ser. A, Math. Anal. 13A (2006), 641-647. MR 2219618 |
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