Multiplicity results for a generalized Sturm-Liouville dynamical system on time scales
<p>Abstract</p> <p>By applying the fixed point theorem in cones, some new and general results on the existence of positive solution to second order generalized Sturm-Liouville dynamical system on time scale <inline-formula><m:math name="1687-1847-2011-24-i1" xmln...
| Main Author: | |
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| Format: | Article |
| Language: | English |
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SpringerOpen
2011-01-01
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| Series: | Advances in Difference Equations |
| Subjects: | |
| Online Access: | http://www.advancesindifferenceequations.com/content/2011/1/24 |
| _version_ | 1828419312817274880 |
|---|---|
| author | Zhang Youwei |
| author_facet | Zhang Youwei |
| author_sort | Zhang Youwei |
| collection | DOAJ |
| description | <p>Abstract</p> <p>By applying the fixed point theorem in cones, some new and general results on the existence of positive solution to second order generalized Sturm-Liouville dynamical system on time scale <inline-formula><m:math name="1687-1847-2011-24-i1" xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mi>T</m:mi> </m:math> </inline-formula></p> <p><display-formula><m:math name="1687-1847-2011-24-i2" xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow> <m:mfenced separators="" open="{" close=""> <m:mrow> <m:mtable class="gathered"> <m:mtr> <m:mtd> <m:mtable equalrows="false" columnlines="none none none none none none none none none none none none none none none none none none none" equalcolumns="false" class="array"> <m:mtr> <m:mtd class="array" columnalign="center"> <m:mtable class="gathered"> <m:mtr> <m:mtd> <m:msubsup> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mi>Δ</m:mi> <m:mi>Δ</m:mi> </m:mrow> </m:msubsup> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>t</m:mi> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:mo class="MathClass-bin">+</m:mo> <m:msub> <m:mrow> <m:mi>h</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>t</m:mi> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:msub> <m:mrow> <m:mi>f</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>t</m:mi> <m:mo class="MathClass-punc">,</m:mo> <m:msub> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>t</m:mi> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:mo class="MathClass-punc">,</m:mo> <m:msub> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>t</m:mi> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:mo class="MathClass-punc">,</m:mo> <m:msubsup> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mi>Δ</m:mi> </m:mrow> </m:msubsup> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>t</m:mi> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:mo class="MathClass-punc">,</m:mo> <m:msubsup> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> <m:mrow> <m:mi>Δ</m:mi> </m:mrow> </m:msubsup> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>t</m:mi> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:mo class="MathClass-rel">=</m:mo> <m:mn>0</m:mn> <m:mo class="MathClass-punc">,</m:mo> </m:mtd> </m:mtr> <m:mtr> <m:mtd> <m:msubsup> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> <m:mrow> <m:mi>Δ</m:mi> <m:mi>Δ</m:mi> </m:mrow> </m:msubsup> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>t</m:mi> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:mo class="MathClass-bin">+</m:mo> <m:msub> <m:mrow> <m:mi>h</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>t</m:mi> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:msub> <m:mrow> <m:mi>f</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>t</m:mi> <m:mo class="MathClass-punc">,</m:mo> <m:msub> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>t</m:mi> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:mo class="MathClass-punc">,</m:mo> <m:msub> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>t</m:mi> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:mo class="MathClass-punc">,</m:mo> <m:msubsup> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mi>Δ</m:mi> </m:mrow> </m:msubsup> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>t</m:mi> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:mo class="MathClass-punc">,</m:mo> <m:msubsup> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> <m:mrow> <m:mi>Δ</m:mi> </m:mrow> </m:msubsup> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>t</m:mi> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:mo class="MathClass-rel">=</m:mo> <m:mn>0</m:mn> <m:mo class="MathClass-punc">,</m:mo> </m:mtd> </m:mtr> <m:mtr> <m:mtd/> </m:mtr> </m:mtable> </m:mtd> <m:mtd class="array" columnalign="center"> <m:mi>t</m:mi> <m:mo class="MathClass-rel">∈</m:mo> <m:msub> <m:mrow> <m:mrow> <m:mo class="MathClass-open">[</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>t</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo class="MathClass-punc">,</m:mo> <m:msub> <m:mrow> <m:mi>t</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> </m:mrow> <m:mo class="MathClass-close">]</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:mi>T</m:mi> </m:mrow> </m:msub> <m:mo class="MathClass-punc">,</m:mo> </m:mtd> </m:mtr> <m:mtr> <m:mtd class="array" columnalign="center"/> </m:mtr> </m:mtable> </m:mtd> </m:mtr> <m:mtr> <m:mtd> <m:mtable equalrows="false" columnlines="none none none none none none none none none none none none none none none none none none none" equalcolumns="false" class="array"> <m:mtr> <m:mtd class="array" columnalign="center"> <m:mtable class="gathered"> <m:mtr> <m:mtd> <m:mi>a</m:mi> <m:msub> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mi>i</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>t</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:mo class="MathClass-bin">-</m:mo> <m:mi>b</m:mi> <m:msubsup> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mi>i</m:mi> </m:mrow> <m:mrow> <m:mi>Δ</m:mi> </m:mrow> </m:msubsup> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>t</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:mo class="MathClass-rel">=</m:mo> <m:msubsup> <m:mrow> <m:mo mathsize="big"> ∑</m:mo> </m:mrow> <m:mrow> <m:mi>k</m:mi> <m:mo class="MathClass-rel">=</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mi>m</m:mi> <m:mo class="MathClass-bin">-</m:mo> <m:mn>2</m:mn> </m:mrow> </m:msubsup> <m:msub> <m:mrow> <m:mi>a</m:mi> </m:mrow> <m:mrow> <m:mi>k</m:mi> </m:mrow> </m:msub> <m:msub> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mi>i</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>ξ</m:mi> </m:mrow> <m:mrow> <m:mi>k</m:mi> </m:mrow> </m:msub> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:mo class="MathClass-punc">,</m:mo> </m:mtd> </m:mtr> <m:mtr> <m:mtd> <m:mi>c</m:mi> <m:msub> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mi>i</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>σ</m:mi> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>t</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:mo class="MathClass-bin">+</m:mo> <m:mi>d</m:mi> <m:msubsup> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mi>i</m:mi> </m:mrow> <m:mrow> <m:mi>Δ</m:mi> </m:mrow> </m:msubsup> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>σ</m:mi> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>t</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:mo class="MathClass-rel">=</m:mo> <m:msubsup> <m:mrow> <m:mo mathsize="big"> ∑</m:mo> </m:mrow> <m:mrow> <m:mi>k</m:mi> <m:mo class="MathClass-rel">=</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mi>m</m:mi> <m:mo class="MathClass-bin">-</m:mo> <m:mn>2</m:mn> </m:mrow> </m:msubsup> <m:msub> <m:mrow> <m:mi>b</m:mi> </m:mrow> <m:mrow> <m:mi>k</m:mi> </m:mrow> </m:msub> <m:msub> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mi>i</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>ξ</m:mi> </m:mrow> <m:mrow> <m:mi>k</m:mi> </m:mrow> </m:msub> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:mo class="MathClass-punc">,</m:mo> </m:mtd> </m:mtr> <m:mtr> <m:mtd/> </m:mtr> </m:mtable> </m:mtd> <m:mtd class="array" columnalign="center"> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>i</m:mi> <m:mo class="MathClass-rel">=</m:mo> <m:mn>1</m:mn> <m:mo class="MathClass-punc">,</m:mo> <m:mn>2</m:mn> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:mo class="MathClass-punc">,</m:mo> </m:mtd> </m:mtr> <m:mtr> <m:mtd class="array" columnalign="center"/> </m:mtr> </m:mtable> </m:mtd> </m:mtr> <m:mtr> <m:mtd/> </m:mtr> </m:mtable> </m:mrow> </m:mfenced> </m:mrow> </m:math> </display-formula></p> <p>are obtained. The first-order Δ-derivatives are involved in the nonlinear terms explicitly.</p> <p><b>Mathematics Subject Classification (2000) </b>39A10</p> |
| first_indexed | 2024-12-10T14:50:14Z |
| format | Article |
| id | doaj.art-3c0c8d81d0664228bfa69eab0170f87c |
| institution | Directory Open Access Journal |
| issn | 1687-1839 1687-1847 |
| language | English |
| last_indexed | 2024-12-10T14:50:14Z |
| publishDate | 2011-01-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Advances in Difference Equations |
| spelling | doaj.art-3c0c8d81d0664228bfa69eab0170f87c2022-12-22T01:44:27ZengSpringerOpenAdvances in Difference Equations1687-18391687-18472011-01-012011124Multiplicity results for a generalized Sturm-Liouville dynamical system on time scalesZhang Youwei<p>Abstract</p> <p>By applying the fixed point theorem in cones, some new and general results on the existence of positive solution to second order generalized Sturm-Liouville dynamical system on time scale <inline-formula><m:math name="1687-1847-2011-24-i1" xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mi>T</m:mi> </m:math> </inline-formula></p> <p><display-formula><m:math name="1687-1847-2011-24-i2" xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow> <m:mfenced separators="" open="{" close=""> <m:mrow> <m:mtable class="gathered"> <m:mtr> <m:mtd> <m:mtable equalrows="false" columnlines="none none none none none none none none none none none none none none none none none none none" equalcolumns="false" class="array"> <m:mtr> <m:mtd class="array" columnalign="center"> <m:mtable class="gathered"> <m:mtr> <m:mtd> <m:msubsup> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mi>Δ</m:mi> <m:mi>Δ</m:mi> </m:mrow> </m:msubsup> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>t</m:mi> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:mo class="MathClass-bin">+</m:mo> <m:msub> <m:mrow> <m:mi>h</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>t</m:mi> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:msub> <m:mrow> <m:mi>f</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>t</m:mi> <m:mo class="MathClass-punc">,</m:mo> <m:msub> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>t</m:mi> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:mo class="MathClass-punc">,</m:mo> <m:msub> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>t</m:mi> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:mo class="MathClass-punc">,</m:mo> <m:msubsup> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mi>Δ</m:mi> </m:mrow> </m:msubsup> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>t</m:mi> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:mo class="MathClass-punc">,</m:mo> <m:msubsup> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> <m:mrow> <m:mi>Δ</m:mi> </m:mrow> </m:msubsup> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>t</m:mi> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:mo class="MathClass-rel">=</m:mo> <m:mn>0</m:mn> <m:mo class="MathClass-punc">,</m:mo> </m:mtd> </m:mtr> <m:mtr> <m:mtd> <m:msubsup> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> <m:mrow> <m:mi>Δ</m:mi> <m:mi>Δ</m:mi> </m:mrow> </m:msubsup> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>t</m:mi> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:mo class="MathClass-bin">+</m:mo> <m:msub> <m:mrow> <m:mi>h</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>t</m:mi> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:msub> <m:mrow> <m:mi>f</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>t</m:mi> <m:mo class="MathClass-punc">,</m:mo> <m:msub> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>t</m:mi> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:mo class="MathClass-punc">,</m:mo> <m:msub> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>t</m:mi> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:mo class="MathClass-punc">,</m:mo> <m:msubsup> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mi>Δ</m:mi> </m:mrow> </m:msubsup> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>t</m:mi> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:mo class="MathClass-punc">,</m:mo> <m:msubsup> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> <m:mrow> <m:mi>Δ</m:mi> </m:mrow> </m:msubsup> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>t</m:mi> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:mo class="MathClass-rel">=</m:mo> <m:mn>0</m:mn> <m:mo class="MathClass-punc">,</m:mo> </m:mtd> </m:mtr> <m:mtr> <m:mtd/> </m:mtr> </m:mtable> </m:mtd> <m:mtd class="array" columnalign="center"> <m:mi>t</m:mi> <m:mo class="MathClass-rel">∈</m:mo> <m:msub> <m:mrow> <m:mrow> <m:mo class="MathClass-open">[</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>t</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo class="MathClass-punc">,</m:mo> <m:msub> <m:mrow> <m:mi>t</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> </m:mrow> <m:mo class="MathClass-close">]</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:mi>T</m:mi> </m:mrow> </m:msub> <m:mo class="MathClass-punc">,</m:mo> </m:mtd> </m:mtr> <m:mtr> <m:mtd class="array" columnalign="center"/> </m:mtr> </m:mtable> </m:mtd> </m:mtr> <m:mtr> <m:mtd> <m:mtable equalrows="false" columnlines="none none none none none none none none none none none none none none none none none none none" equalcolumns="false" class="array"> <m:mtr> <m:mtd class="array" columnalign="center"> <m:mtable class="gathered"> <m:mtr> <m:mtd> <m:mi>a</m:mi> <m:msub> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mi>i</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>t</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:mo class="MathClass-bin">-</m:mo> <m:mi>b</m:mi> <m:msubsup> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mi>i</m:mi> </m:mrow> <m:mrow> <m:mi>Δ</m:mi> </m:mrow> </m:msubsup> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>t</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:mo class="MathClass-rel">=</m:mo> <m:msubsup> <m:mrow> <m:mo mathsize="big"> ∑</m:mo> </m:mrow> <m:mrow> <m:mi>k</m:mi> <m:mo class="MathClass-rel">=</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mi>m</m:mi> <m:mo class="MathClass-bin">-</m:mo> <m:mn>2</m:mn> </m:mrow> </m:msubsup> <m:msub> <m:mrow> <m:mi>a</m:mi> </m:mrow> <m:mrow> <m:mi>k</m:mi> </m:mrow> </m:msub> <m:msub> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mi>i</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>ξ</m:mi> </m:mrow> <m:mrow> <m:mi>k</m:mi> </m:mrow> </m:msub> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:mo class="MathClass-punc">,</m:mo> </m:mtd> </m:mtr> <m:mtr> <m:mtd> <m:mi>c</m:mi> <m:msub> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mi>i</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>σ</m:mi> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>t</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:mo class="MathClass-bin">+</m:mo> <m:mi>d</m:mi> <m:msubsup> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mi>i</m:mi> </m:mrow> <m:mrow> <m:mi>Δ</m:mi> </m:mrow> </m:msubsup> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>σ</m:mi> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>t</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:mo class="MathClass-rel">=</m:mo> <m:msubsup> <m:mrow> <m:mo mathsize="big"> ∑</m:mo> </m:mrow> <m:mrow> <m:mi>k</m:mi> <m:mo class="MathClass-rel">=</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mi>m</m:mi> <m:mo class="MathClass-bin">-</m:mo> <m:mn>2</m:mn> </m:mrow> </m:msubsup> <m:msub> <m:mrow> <m:mi>b</m:mi> </m:mrow> <m:mrow> <m:mi>k</m:mi> </m:mrow> </m:msub> <m:msub> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mi>i</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>ξ</m:mi> </m:mrow> <m:mrow> <m:mi>k</m:mi> </m:mrow> </m:msub> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:mo class="MathClass-punc">,</m:mo> </m:mtd> </m:mtr> <m:mtr> <m:mtd/> </m:mtr> </m:mtable> </m:mtd> <m:mtd class="array" columnalign="center"> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>i</m:mi> <m:mo class="MathClass-rel">=</m:mo> <m:mn>1</m:mn> <m:mo class="MathClass-punc">,</m:mo> <m:mn>2</m:mn> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:mo class="MathClass-punc">,</m:mo> </m:mtd> </m:mtr> <m:mtr> <m:mtd class="array" columnalign="center"/> </m:mtr> </m:mtable> </m:mtd> </m:mtr> <m:mtr> <m:mtd/> </m:mtr> </m:mtable> </m:mrow> </m:mfenced> </m:mrow> </m:math> </display-formula></p> <p>are obtained. The first-order Δ-derivatives are involved in the nonlinear terms explicitly.</p> <p><b>Mathematics Subject Classification (2000) </b>39A10</p>http://www.advancesindifferenceequations.com/content/2011/1/24Sturm-Liouville dynamical systemConeFixed pointPositive solutionTime scales |
| spellingShingle | Zhang Youwei Multiplicity results for a generalized Sturm-Liouville dynamical system on time scales Advances in Difference Equations Sturm-Liouville dynamical system Cone Fixed point Positive solution Time scales |
| title | Multiplicity results for a generalized Sturm-Liouville dynamical system on time scales |
| title_full | Multiplicity results for a generalized Sturm-Liouville dynamical system on time scales |
| title_fullStr | Multiplicity results for a generalized Sturm-Liouville dynamical system on time scales |
| title_full_unstemmed | Multiplicity results for a generalized Sturm-Liouville dynamical system on time scales |
| title_short | Multiplicity results for a generalized Sturm-Liouville dynamical system on time scales |
| title_sort | multiplicity results for a generalized sturm liouville dynamical system on time scales |
| topic | Sturm-Liouville dynamical system Cone Fixed point Positive solution Time scales |
| url | http://www.advancesindifferenceequations.com/content/2011/1/24 |
| work_keys_str_mv | AT zhangyouwei multiplicityresultsforageneralizedsturmliouvilledynamicalsystemontimescales |