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Oscillation theorems for second-order nonlinear delay difference equations

Author： Saker S. H.

Publisher： Springer Publishing Company

ISSN： 0031-5303

Source： Periodica Mathematica Hungarica, Vol.47, Iss.1-2, 2003-01, pp. : 201-213

Disclaimer: Any content in publications that violate the sovereignty, the constitution or regulations of the PRC is not accepted or approved by CNPIEC.

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Abstract

By means of Riccati transformation technique, we establish some new oscillation criteria for second-order nonlinear delay difference equation[Delta left(p_{n}left( Delta x_{n}right) ^{gamma }right)+q_{n}f(x_{n-sigma })=0,%quad n=0,1,2,dots,] when $\sum\limits_{n=0}^{\infty }\left( \frac{1}{p_{n}}\right) ^{\frac{1}{\gamma }}=\infty$ . When $\sum\limits_{n=0}^{\infty }\left( \frac{1}{p_{n}}\right) ^{\frac{1}{\gamma }}<\infty$ we present some sufficient conditions which guarantee that, every solution oscillates or converges to zero. When $\sum\limits_{n=0}^{\infty }\left( \frac{1}{p_{n}}\right) ^{\frac{1}{\gamma }}=\infty$ holds, our results do not require the nonlinearity to be nondecreasing and are thus applicable to new classes of equations to which most previously known results are not.

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