On existence and uniqueness of positive solutions to a class of fractional boundary value problems
<p>Abstract</p> <p>The purpose of this paper is to investigate the existence and uniqueness of positive solutions for the following fractional boundary value problem</p> <p><display-formula><m:math name="1687-2770-2011-25-i1" xmlns:m="http://www....
| 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/2011/1/25 |
| Summary: | <p>Abstract</p> <p>The purpose of this paper is to investigate the existence and uniqueness of positive solutions for the following fractional boundary value problem</p> <p><display-formula><m:math name="1687-2770-2011-25-i1" xmlns:m="http://www.w3.org/1998/Math/MathML"><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>u</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-bin">+</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>u</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: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:mspace width="1em" class="quad"/> <m:mn>0</m:mn> <m:mo class="MathClass-rel"><</m:mo> <m:mi>t</m:mi> <m:mo class="MathClass-rel"><</m:mo> <m:mn>1</m:mn> <m:mo class="MathClass-punc">,</m:mo> </m:mtd> </m:mtr> <m:mtr> <m:mtd class="array" columnalign="center"> <m:mi>u</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>u</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-rel">=</m:mo> <m:msup> <m:mrow> <m:mi>u</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: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:math> </display-formula></p> <p>where 2 < <it>α </it>≤ 3 and <inline-formula><m:math name="1687-2770-2011-25-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> is the Riemann-Liouville fractional derivative.</p> <p>Our analysis relies on a fixed-point theorem in partially ordered metric spaces. The autonomous case of this problem was studied in the paper [Zhao et al., Abs. Appl. Anal., to appear], but in Zhao et al. (to appear), the question of uniqueness of the solution is not treated.</p> <p>We also present some examples where we compare our results with the ones obtained in Zhao et al. (to appear).</p> <p><b>2010 Mathematics Subject Classification</b>: 34B15</p> |
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| ISSN: | 1687-2762 1687-2770 |