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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| Format: | Article |
| Language: | English |
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Wiley
1992-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
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| Online Access: | http://dx.doi.org/10.1155/S0161171292000152 |
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| author | G. Ladas C. Qian |
| author_facet | G. Ladas C. Qian |
| author_sort | G. Ladas |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-109db6287ffc4976a2e59074b61663a2 |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 1992-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-109db6287ffc4976a2e59074b61663a22025-02-03T01:33:09ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04251992-01-0115112914210.1155/S0161171292000152Comparison results and linearized oscillations for higher-order difference equationsG. Ladas0C. Qian1Department of Mathematics, The University of Rhode Island, Kingston 02881-0816, R.I., USADepartment of Mathematics, The University of Rhode Island, Kingston 02881-0816, R.I., USAConsider 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.http://dx.doi.org/10.1155/S0161171292000152linearized oscillationshigher order difference equationscomparison results. |
| spellingShingle | G. Ladas C. Qian Comparison results and linearized oscillations for higher-order difference equations International Journal of Mathematics and Mathematical Sciences linearized oscillations higher order difference equations comparison results. |
| title | Comparison results and linearized oscillations for higher-order difference equations |
| title_full | Comparison results and linearized oscillations for higher-order difference equations |
| title_fullStr | Comparison results and linearized oscillations for higher-order difference equations |
| title_full_unstemmed | Comparison results and linearized oscillations for higher-order difference equations |
| title_short | Comparison results and linearized oscillations for higher-order difference equations |
| title_sort | comparison results and linearized oscillations for higher order difference equations |
| topic | linearized oscillations higher order difference equations comparison results. |
| url | http://dx.doi.org/10.1155/S0161171292000152 |
| work_keys_str_mv | AT gladas comparisonresultsandlinearizedoscillationsforhigherorderdifferenceequations AT cqian comparisonresultsandlinearizedoscillationsforhigherorderdifferenceequations |