Oscillation conditions for difference equations with several variable arguments

Chatzarakis, George E. ; Kusano, Takaŝi ; Stavroulakis, Ioannis P.

Mathematica Bohemica, Tome 140 (2015) no. 3, pp. 291-311

Abstract (VO)

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.

Détail

DOI : 10.21136/MB.2015.144396

Classification : 39A10, 39A21
Keywords: difference equation; retarded argument; advanced argument; oscillatory solution; nonoscillatory solution

Bibliographie

[1] Agarwal, R. P., Bohner, M., Grace, S. R., O'Regan, D.: Discrete Oscillation Theory. Hindawi Publishing Corporation New York (2005). | MR | Zbl

[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. | MR | Zbl

[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. | DOI | Zbl

[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. | DOI | MR | Zbl

[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. | MR | Zbl

[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. | DOI | MR | Zbl

[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. | MR | Zbl

[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. | DOI | MR | Zbl

[9] Chatzarakis, G. E., Pinelas, S., Stavroulakis, I. P.: Oscillations of difference equations with several deviated arguments. Aequationes Math. 88 (2014), 105-123. | DOI | MR | Zbl

[10] Erbe, L. H., Zhang, B. G.: Oscillation of discrete analogues of delay equations. Differ. Integral Equ. 2 (1989), 300-309. | MR | Zbl

[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. | MR | Zbl

[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. | DOI | MR | Zbl

[13] Győri, I., Ladas, G.: Oscillation Theory of Delay Differential Equations: With Applications. Oxford Mathematical Monographs Clarendon Press, Oxford (1991). | MR | Zbl

[14] Lakshmikantham, V., Trigiante, D.: Theory of Difference Equations: Numerical Methods and Applications. Mathematics in Science and Engineering 181 Academic Press, Boston (1988). | MR | Zbl

[15] Li, X., Zhu, D.: Oscillation of advanced difference equations with variable coefficients. Ann. Differ. Equations 18 (2002), 254-263. | MR | Zbl

[16] Luo, X. N., Zhou, Y., Li, C. F.: Oscillation of a nonlinear difference equation with several delays. Math. Bohem. 128 (2003), 309-317. | MR | Zbl

[17] Stavroulakis, I. P.: Oscillation criteria for delay and difference equations with non-monotone arguments. Appl. Math. Comput. 226 (2014), 661-672. | DOI | MR | Zbl

[18] Tang, X. H., Yu, J. S.: Oscillations of delay difference equations. Hokkaido Math. J. 29 (2000), 213-228. | DOI | MR | Zbl

[19] Tang, X. H., Yu, J. S.: Oscillation of delay difference equation. Comput. Math. Appl. 37 (1999), 11-20. | DOI | MR | Zbl

[20] Tang, X. H., Zhang, R. Y.: New oscillation criteria for delay difference equations. Comput. Math. Appl. 42 (2001), 1319-1330. | DOI | MR | Zbl

[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