文章基本信息

标题：Oscillation criteria for difference equations with non-monotone arguments
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作者：George E Chatzarakis ; Leonid Shaikhet
期刊名称：Advances in Difference Equations
印刷版ISSN：1687-1839
电子版ISSN：1687-1847
出版年度：2017
卷号：2017
期号：1
页码：62
DOI：10.1186/s13662-017-1119-0
语种：English
出版社：Hindawi Publishing Corporation
摘要：This paper is concerned with the oscillatory behavior of first-order retarded [advanced] difference equation of the form Δ x ( n ) + p ( n ) x ( τ ( n ) ) = 0 , n ∈ N 0 [ ∇ x ( n ) − q ( n ) x ( σ ( n ) ) = 0 , n ∈ N ] , $$ \Delta x(n)+p(n)x\bigl(\tau (n)\bigr)=0, \quad n\in \mathbb{N} _( \qquad \bigl[\nabla x(n)-q(n)x\bigl(\sigma (n)\bigr)=0, \ n\in \mathbb{N} \bigr], $$ where ( p ( n ) ) n ≥ 0 $(p(n))_{n\geq 0}$ [ ( q ( n ) ) n ≥ 1 ] $[(q(n))_{n\geq 1}]$ is a sequence of nonnegative real numbers and τ ( n ) $\tau (n)$ [ σ ( n ) ] $[\sigma (n)]$ is a non-monotone sequence of integers such that τ ( n ) ≤ n − 1 $\tau (n)\leq n-1$ , for n ∈ N 0 $n\in \mathbb{N}_($ and lim n → ∞ τ ( n ) = ∞ $\lim_{n\rightarrow \infty }\tau (n)=\infty $ [ σ ( n ) ≥ n + 1 , for n ∈ N ] $[\sigma (n)\geq n+1,\mbox{ for }n\in \mathbb{N}]$ . Sufficient conditions, involving limsup, which guarantee the oscillation of all solutions are established. These conditions improve all previous well-known results in the literature. Also, using algorithms on MATLAB software, examples illustrating the significance of the results are given.
关键词：difference equations ; non-monotone arguments ; retarded arguments ; advanced arguments ; oscillation ; Grönwall inequality