Periodic solutions of nonlinear second-order difference equations
<p/> <p>We establish conditions for the existence of periodic solutions of nonlinear, second-order difference equations of the form <it>y</it>(<it>t</it> + 2) + <it>by</it> (<it>t</it> + 1) + <it>cy</it>(<it>t</it>)...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
SpringerOpen
2005-01-01
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| Series: | Advances in Difference Equations |
| Online Access: | http://www.advancesindifferenceequations.com/content/2005/718682 |
| Summary: | <p/> <p>We establish conditions for the existence of periodic solutions of nonlinear, second-order difference equations of the form <it>y</it>(<it>t</it> + 2) + <it>by</it> (<it>t</it> + 1) + <it>cy</it>(<it>t</it>) = <it>f</it> (<it>y</it>(<it>t</it>)), where <it>f</it>: ℝ → ℝ and <it>β</it> > 0 is continuous. In our main result we assume that <it>f</it> exhibits sublinear growth and that there is a constant <it>uf</it> (<it>u</it>) > 0 such that |<it>u</it>| ≥ <it>β</it> whenever <it>c</it> = 1. For such an equation we prove that if <it>N</it> is an odd integer larger than one, then there exists at least one <it>N</it>-periodic solution unless all of the following conditions are simultaneously satisfied: |<it>b</it>| < 2, <it>N across</it><sup>-1</sup>(-<it>b</it>/2), and <it>π</it> is an even multiple of <it>c</it> ≠ 0.</p> |
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| ISSN: | 1687-1839 1687-1847 |