Comparison results and linearized oscillations for higher-order difference equations
Consider the difference equationsΔmxn+(−1)m+1pnf(xn−k)=0, n=0,1,… (1)andΔmyn+(−1)m+1qng(yn−ℓ)=0, n=0,1,…. (2)We establish a comparison result according to which, when m is odd, every solution of Eq.(1) oscillates provided that every solution of Eq.(2) oscillates and, when m is even,...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
1992-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Subjects: | |
| Online Access: | http://dx.doi.org/10.1155/S0161171292000152 |
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| Summary: | Consider the difference equationsΔmxn+(−1)m+1pnf(xn−k)=0, n=0,1,… (1)andΔmyn+(−1)m+1qng(yn−ℓ)=0, n=0,1,…. (2)We establish a comparison result according to which, when m is odd, every solution of Eq.(1) oscillates provided that every solution of Eq.(2) oscillates and, when m is even, every bounded solution of Eq.(1) oscillates provided that every bounded solution of Eq.(2) oscillates. We also establish a linearized oscillation theorem according to which, when m is odd, every solution of Eq.(1) oscillates if and only if every solution of an associated linear equationΔmzn+(−1)m+1pzn−k=0, n=0,1,… (*)oscillates and, when m is even, every bounded solution of Eq.(1) oscillates if and only if every bounded solution of (*) oscillates. |
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| ISSN: | 0161-1712 1687-0425 |