George E. Chatzarakis, Department of Electrical and Electronic Engineering Educators, School of Pedagogical and Technological Education, 14121 N. Heraklion, Athens, Greece, e-mail: geaxatz@otenet.gr, gea.xatz@aspete.gr; Takaŝi Kusano, Department of Mathematics, Faculty of Science, Hiroshima University, 1-3-1 Kagamiyama, Higashi-Hiroshima, 739-8526, Japan, e-mail: kusanot@zj8.so-net.ne.jp; Ioannis P. Stavroulakis, Department of Mathematics, University of Ioannina, P.O. Box 1186, 45110 Ioannina, Greece, e-mail: ipstav@cc.uoi.gr
Abstract: Consider the difference equation
\Delta x(n)+\sum_{i=1}^mp_i(n)x(\tau_i(n))=0,\quad n\geq0\quad\bigg[\nabla x(n)-\sum_{i=1}^mp_i(n)x(\sigma_i(n))=0,\quad n\geq1\bigg],
where $(p_i(n))$, $1\leq i\leq m$ are sequences of nonnegative real numbers, $\tau_i(n)$ [$\sigma_i(n)$], $1\leq
i\leq m$ 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)}^np_i(j)>1 \quad\biggl[\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)>\frac1 e \quad\biggl[\liminf_{n\rightarrow\infty}\sum_{i=1}^m\sum_{j=n+1}^{\sigma_i(n)}p_i(j)>\frac1 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.
Keywords: difference equation; retarded argument; advanced argument; oscillatory solution; nonoscillatory solution
Classification (MSC 2010): 39A10, 39A21
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