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Zbl 1063.34057
Stavroulakis, I.P.
Oscillation criteria for delay, difference and functional equations. (English)
[J] Funct. Differ. Equ. 11, No. 1-2, 163-183 (2004). ISSN 0793-1786

The paper is a survey on the most interesting oscillation criteria for the following type of equations:\par 1) the first-order linear delay differential equation $x'(t)+p(t)x(\tau(t))=0$, $t>t_0$, with emphasis to the case $$ 0<\liminf_{t\rightarrow \infty}\int_{\tau(t)}^t p(s)\,ds\le {1\over e}\quad \text{ and}\quad \limsup_{t\rightarrow \infty}\int_{t-\tau}^t p(s)\,ds<1\,; $$ 2) the difference equation $x_{n+1}-x_n+p_n x_{n-k}=0$, $n=0,1,2,\dots,$ with emphasis to the case $$ \liminf_{n\rightarrow\infty}\sum_{i=n-k}^{n-1}\le\left(k\over k+1\right)^{k+1} \quad\text{ and}\quad \limsup_{n\rightarrow\infty}\sum_{i=n-k^n} p_i<1\,; $$ 3) the functional equation $x(g(t))=P(t)x(t)+Q(t) x(g^2(t))$, $t\ge t_0$, with emphasis to the case $$ 0<\liminf_{t\rightarrow\infty}\{Q(t)P(g(t))\}\le{1\over 4}\quad\text{ and}\quad \limsup_{t\rightarrow\infty}\{Q(t)P(g(t))\}<1.$$
[Ivan Ginchev (Varna)]

MSC 2000:: *34K11 Oscillation theory of functional-differential equations
39A12 Discrete version of topics in analysis
39B22 Functional equations for real functions

Keywords: oscillations; delay differential equations; difference equations; functional equations

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