Solvability for fractional order boundary value problems at resonance
<p>Abstract</p> <p>In this paper, by using the coincidence degree theory, we consider the following boundary value problem for fractional differential equation</p> <p> <display-formula> <m:math name="1687-2770-2011-20-i1" xmlns:m="http://www.w3.o...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
SpringerOpen
2011-01-01
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| Series: | Boundary Value Problems |
| Subjects: | |
| Online Access: | http://www.boundaryvalueproblems.com/content/2011/1/20 |
| _version_ | 1818326267438563328 |
|---|---|
| author | Hu Zhigang Liu Wenbin |
| author_facet | Hu Zhigang Liu Wenbin |
| author_sort | Hu Zhigang |
| collection | DOAJ |
| description | <p>Abstract</p> <p>In this paper, by using the coincidence degree theory, we consider the following boundary value problem for fractional differential equation</p> <p> <display-formula> <m:math name="1687-2770-2011-20-i1" xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow> <m:mfenced separators="" open="{" close=""> <m:mrow> <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:msubsup> <m:mrow> <m:mi>D</m:mi> </m:mrow> <m:mrow> <m:msup> <m:mrow> <m:mn>0</m:mn> </m:mrow> <m:mrow> <m:mo class="MathClass-bin">+</m:mo> </m:mrow> </m:msup> </m:mrow> <m:mrow> <m:mi>α</m:mi> </m:mrow> </m:msubsup> <m:mi>x</m:mi> <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-rel">=</m:mo> <m:mi>f</m:mi> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>t</m:mi> <m:mo class="MathClass-punc">,</m:mo> <m:mi>x</m:mi> <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:msup> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mrow> <m:mi>′</m:mi> </m:mrow> </m:msup> <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:msup> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mrow> <m:mo class="MathClass-op">″</m:mo> </m:mrow> </m:msup> <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-punc">,</m:mo> </m:mtd> <m:mtd class="array" columnalign="center"> <m:mi>t</m:mi> <m:mo class="MathClass-rel">∈</m:mo> <m:mrow> <m:mo class="MathClass-open">[</m:mo> <m:mrow> <m:mn>0</m:mn> <m:mo class="MathClass-punc">,</m:mo> <m:mn>1</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:mi>x</m:mi> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mn>0</m:mn> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:mo class="MathClass-rel">=</m:mo> <m:mi>x</m:mi> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mn>1</m:mn> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:mo class="MathClass-punc">,</m:mo> </m:mtd> <m:mtd class="array" columnalign="center"> <m:msup> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mrow> <m:mi>′</m:mi> </m:mrow> </m:msup> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mn>0</m:mn> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:mo class="MathClass-rel">=</m:mo> <m:msup> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mrow> <m:mo class="MathClass-op">″</m:mo> </m:mrow> </m:msup> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mn>0</m:mn> </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 class="array" columnalign="center"/> </m:mtr> </m:mtable> </m:mrow> </m:mfenced> </m:mrow> </m:math> </display-formula> </p> <p>where <inline-formula> <m:math name="1687-2770-2011-20-i2" xmlns:m="http://www.w3.org/1998/Math/MathML"><m:msubsup> <m:mrow> <m:mi>D</m:mi> </m:mrow> <m:mrow> <m:msup> <m:mrow> <m:mn>0</m:mn> </m:mrow> <m:mrow> <m:mo class="MathClass-bin">+</m:mo> </m:mrow> </m:msup> </m:mrow> <m:mrow> <m:mi>α</m:mi> </m:mrow> </m:msubsup> </m:math> </inline-formula> denotes the Caputo fractional differential operator of order <it>α</it>, 2 < <it>α </it>≤ 3. A new result on the existence of solutions for above fractional boundary value problem is obtained.</p> <p> <b>Mathematics Subject Classification (2000): </b>34A08, 34B15.</p> |
| first_indexed | 2024-12-13T11:57:40Z |
| format | Article |
| id | doaj.art-3b5784fd583f414ea679c1c122abad9e |
| institution | Directory Open Access Journal |
| issn | 1687-2762 1687-2770 |
| language | English |
| last_indexed | 2024-12-13T11:57:40Z |
| publishDate | 2011-01-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Boundary Value Problems |
| spelling | doaj.art-3b5784fd583f414ea679c1c122abad9e2022-12-21T23:47:10ZengSpringerOpenBoundary Value Problems1687-27621687-27702011-01-012011120Solvability for fractional order boundary value problems at resonanceHu ZhigangLiu Wenbin<p>Abstract</p> <p>In this paper, by using the coincidence degree theory, we consider the following boundary value problem for fractional differential equation</p> <p> <display-formula> <m:math name="1687-2770-2011-20-i1" xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow> <m:mfenced separators="" open="{" close=""> <m:mrow> <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:msubsup> <m:mrow> <m:mi>D</m:mi> </m:mrow> <m:mrow> <m:msup> <m:mrow> <m:mn>0</m:mn> </m:mrow> <m:mrow> <m:mo class="MathClass-bin">+</m:mo> </m:mrow> </m:msup> </m:mrow> <m:mrow> <m:mi>α</m:mi> </m:mrow> </m:msubsup> <m:mi>x</m:mi> <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-rel">=</m:mo> <m:mi>f</m:mi> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>t</m:mi> <m:mo class="MathClass-punc">,</m:mo> <m:mi>x</m:mi> <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:msup> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mrow> <m:mi>′</m:mi> </m:mrow> </m:msup> <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:msup> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mrow> <m:mo class="MathClass-op">″</m:mo> </m:mrow> </m:msup> <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-punc">,</m:mo> </m:mtd> <m:mtd class="array" columnalign="center"> <m:mi>t</m:mi> <m:mo class="MathClass-rel">∈</m:mo> <m:mrow> <m:mo class="MathClass-open">[</m:mo> <m:mrow> <m:mn>0</m:mn> <m:mo class="MathClass-punc">,</m:mo> <m:mn>1</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:mi>x</m:mi> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mn>0</m:mn> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:mo class="MathClass-rel">=</m:mo> <m:mi>x</m:mi> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mn>1</m:mn> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:mo class="MathClass-punc">,</m:mo> </m:mtd> <m:mtd class="array" columnalign="center"> <m:msup> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mrow> <m:mi>′</m:mi> </m:mrow> </m:msup> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mn>0</m:mn> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:mo class="MathClass-rel">=</m:mo> <m:msup> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mrow> <m:mo class="MathClass-op">″</m:mo> </m:mrow> </m:msup> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mn>0</m:mn> </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 class="array" columnalign="center"/> </m:mtr> </m:mtable> </m:mrow> </m:mfenced> </m:mrow> </m:math> </display-formula> </p> <p>where <inline-formula> <m:math name="1687-2770-2011-20-i2" xmlns:m="http://www.w3.org/1998/Math/MathML"><m:msubsup> <m:mrow> <m:mi>D</m:mi> </m:mrow> <m:mrow> <m:msup> <m:mrow> <m:mn>0</m:mn> </m:mrow> <m:mrow> <m:mo class="MathClass-bin">+</m:mo> </m:mrow> </m:msup> </m:mrow> <m:mrow> <m:mi>α</m:mi> </m:mrow> </m:msubsup> </m:math> </inline-formula> denotes the Caputo fractional differential operator of order <it>α</it>, 2 < <it>α </it>≤ 3. A new result on the existence of solutions for above fractional boundary value problem is obtained.</p> <p> <b>Mathematics Subject Classification (2000): </b>34A08, 34B15.</p>http://www.boundaryvalueproblems.com/content/2011/1/20Fractional differential equationsboundary value problemsresonancecoincidence degree theory |
| spellingShingle | Hu Zhigang Liu Wenbin Solvability for fractional order boundary value problems at resonance Boundary Value Problems Fractional differential equations boundary value problems resonance coincidence degree theory |
| title | Solvability for fractional order boundary value problems at resonance |
| title_full | Solvability for fractional order boundary value problems at resonance |
| title_fullStr | Solvability for fractional order boundary value problems at resonance |
| title_full_unstemmed | Solvability for fractional order boundary value problems at resonance |
| title_short | Solvability for fractional order boundary value problems at resonance |
| title_sort | solvability for fractional order boundary value problems at resonance |
| topic | Fractional differential equations boundary value problems resonance coincidence degree theory |
| url | http://www.boundaryvalueproblems.com/content/2011/1/20 |
| work_keys_str_mv | AT huzhigang solvabilityforfractionalorderboundaryvalueproblemsatresonance AT liuwenbin solvabilityforfractionalorderboundaryvalueproblemsatresonance |