Solvability for fractional order boundary value problems at resonance

• Main Authors: Hu Zhigang, Liu Wenbin
• Format: Article
• Language: English
• Published: SpringerOpen 2011-01-01
• Series: Boundary Value Problems
• Subjects: Fractional differential equations; boundary value problems; resonance; coincidence degree theory
• Online Access: http://www.boundaryvalueproblems.com/content/2011/1/20

<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>&#945;</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>&#8242;</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">&#8243;</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">&#8712;</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>&#8242;</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">&#8243;</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>&#945;</m:mi> </m:mrow> </m:msubsup> </m:math> </inline-formula> denotes the Caputo fractional differential operator of order <it>&#945;</it>, 2 &lt; <it>&#945; </it>&#8804; 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>