MATHEMATICA BOHEMICA, Vol. 125, No. 4, pp. 421-430 (2000)

On the oscillation of certain difference equations

S. R. Grace, H. A. El-Morshedy

S. R. Grace, Department of Engineering Mathematics, Faculty of Engineering, Cairo University, Orman, Giza 12000, Egypt
H. A. El-Morshedy, Department of Mathematics, Damietta Faculty of Science, New Damietta 34517, Egypt

Abstract: In this paper we study the oscillation of the difference equations of the form $$ \Delta ^{2}x_{n}+p_{n}\Delta x_{n}+f(n, x_{n-g}, \Delta x_{n-h})=0, $$ in comparison with certain difference equations of order one whose oscillatory character is known. The results can be applied to the difference equation $$ \Delta ^{2}x_{n}+p_{n}\Delta x_{n}+q_{n}|x_{n-g}|^{\lambda }|\Delta x_{n-h}|^{\mu }\sgn x_{n-g}=0, $$ where $\lambda $ and $\mu $ are real constants, $\lambda >0$ and $\mu \geq 0$.

Keywords: oscillation, delay difference equations, forced equations

Classification (MSC2000): 39A10, 39A12

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