Solvability of boundary value problems with Riemann-Stieltjes Δ-integral conditions for second-order dynamic equations on time scales at resonance
oleh: Li Yongkun, Shu Jiangye
| Format: | Article |
|---|---|
| Diterbitkan: | SpringerOpen 2011-01-01 |
Deskripsi
<p>Abstract</p> <p>In this paper, by making use of the coincidence degree theory of Mawhin, the existence of the nontrivial solution for the boundary value problem with Riemann-Stieltjes Δ-integral conditions on time scales at resonance</p> <p><display-formula><m:math name="1687-1847-2011-42-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:msup> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mrow> <m:mi>Δ</m:mi> <m:mi>Δ</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-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>Δ</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:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:mo class="MathClass-bin">+</m:mo> <m:mi>e</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:mtd> <m:mtd class="array" columnalign="center"> <m:mstyle class="text"> <m:mtext class="textsf" mathvariant="sans-serif">a</m:mtext> </m:mstyle> <m:mo class="MathClass-punc">.</m:mo> <m:mstyle class="text"> <m:mtext class="textsf" mathvariant="sans-serif">e</m:mtext> </m:mstyle> <m:mo class="MathClass-punc">.</m:mo> <m:mspace width="2.77695pt" class="tmspace"/> <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:mn>0</m:mn> <m:mo class="MathClass-punc">,</m:mo> <m:mi>T</m:mi> </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:msup> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mrow> <m:mi>Δ</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:mtd class="array" columnalign="center"> <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:munderover accentunder="false" accent="false"> <m:mrow> <m:mo class="MathClass-op"> ∫ </m:mo> </m:mrow> <m:mrow> <m:mn>0</m:mn> </m:mrow> <m:mrow> <m:mi>T</m:mi> </m:mrow> </m:munderover> <m:msup> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mrow> <m:mi>σ</m:mi> </m:mrow> </m:msup> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>s</m:mi> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:mi>Δ</m:mi> <m:mi>g</m:mi> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>s</m:mi> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> </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>is established, where <inline-formula><m:math name="1687-1847-2011-42-i2" xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mi>f</m:mi> <m:mo class="MathClass-punc">:</m:mo> <m:msub> <m:mrow> <m:mrow> <m:mo class="MathClass-open">[</m:mo> <m:mrow> <m:mn>0</m:mn> <m:mo class="MathClass-punc">,</m:mo> <m:mi>T</m:mi> </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-bin">×</m:mo> <m:mi>ℝ</m:mi> <m:mo class="MathClass-bin">×</m:mo> <m:mi>ℝ</m:mi> <m:mo class="MathClass-rel">→</m:mo> <m:mi>ℝ</m:mi> </m:math> </inline-formula> satisfies the Carathéodory conditions and <inline-formula><m:math name="1687-1847-2011-42-i3" xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mi>e</m:mi> <m:mo class="MathClass-punc">:</m:mo> <m:msub> <m:mrow> <m:mrow> <m:mo class="MathClass-open">[</m:mo> <m:mrow> <m:mn>0</m:mn> <m:mo class="MathClass-punc">,</m:mo> <m:mi>T</m:mi> </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-rel">→</m:mo> <m:mi>ℝ</m:mi> </m:math> </inline-formula> is a continuous function and <inline-formula><m:math name="1687-1847-2011-42-i4" xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mi>g</m:mi> <m:mo class="MathClass-punc">:</m:mo> <m:msub> <m:mrow> <m:mrow> <m:mo class="MathClass-open">[</m:mo> <m:mrow> <m:mn>0</m:mn> <m:mo class="MathClass-punc">,</m:mo> <m:mi>T</m:mi> </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-rel">→</m:mo> <m:mi>ℝ</m:mi> </m:math> </inline-formula> is an increasing function with <inline-formula><m:math name="1687-1847-2011-42-i5" xmlns:m="http://www.w3.org/1998/Math/MathML"><m:msubsup> <m:mrow> <m:mo class="MathClass-op">∫ </m:mo> </m:mrow> <m:mrow> <m:mn>0</m:mn> </m:mrow> <m:mrow> <m:mi>T</m:mi> </m:mrow> </m:msubsup> <m:mi>Δ</m:mi> <m:mi>g</m:mi> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>s</m:mi> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:mo class="MathClass-rel">=</m:mo> <m:mn>1</m:mn> </m:math> </inline-formula>. An example is given to illustrate the main results.</p>