Lösbarkeit des überlinearen Neumann-Problems bei Resonanz an der ersten Eigenwertstelle
摘要
Untersuchung der Existenz von Lösungen für eine Klasse nichtlinearer überlinearer diskreter Neumann-Randwertprobleme zweiter Ordnung <inline-formula><alternatives><math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M5"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mo>-</mo><msup><mrow><mi>Δ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>u</mi><mfenced separators="|"><mrow><mi>t</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></mfenced><mo>=</mo><msup><mrow><mfenced separators="|"><mrow><msup><mrow><mi>u</mi></mrow><mrow><mo>+</mo></mrow></msup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></mrow></mfenced></mrow><mrow><mi>p</mi></mrow></msup><mo>+</mo><mi>f</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>,</mo><mo> </mo><mo> </mo><mo> </mo><mi>t</mi><mo>∈</mo><msub><mrow><mfenced open="[" close="]" separators="|"><mrow><mn mathvariant="normal">1</mn><mo>,</mo><mi>T</mi></mrow></mfenced></mrow><mrow><mi mathvariant="double-struck">Z</mi></mrow></msub><mo>,</mo></mtd></mtr><mtr><mtd><mi>Δ</mi><mi>u</mi><mfenced separators="|"><mrow><mn mathvariant="normal">0</mn></mrow></mfenced><mo>=</mo><mi>Δ</mi><mi>u</mi><mfenced separators="|"><mrow><mi>T</mi></mrow></mfenced><mo>=</mo><mn mathvariant="normal">0</mn><mo>,</mo></mtd></mtr></mtable></mrow></mfenced></math><graphic specific-use="big" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="alternativeImage/8D25821C-1890-499b-A479-F24AE129D65C-M005.jpg"><?fx-imagestate width="69.68067169" height="12.10733318"?></graphic><graphic specific-use="small" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="alternativeImage/8D25821C-1890-499b-A479-F24AE129D65C-M005c.jpg"><?fx-imagestate width="69.68067169" height="12.10733318"?></graphic></alternatives></inline-formula>, wobei <inline-formula><alternatives><math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M6"><mi>T</mi><mo>≥</mo><mn mathvariant="normal">2</mn></math><graphic specific-use="big" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="alternativeImage/8D25821C-1890-499b-A479-F24AE129D65C-M006.jpg"><?fx-imagestate width="8.46666718" height="2.62466669"?></graphic><graphic specific-use="small" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="alternativeImage/8D25821C-1890-499b-A479-F24AE129D65C-M006c.jpg"><?fx-imagestate width="8.46666718" height="2.62466669"?></graphic></alternatives></inline-formula> eine ganze Zahl ist, <inline-formula><alternatives><math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M7"><mi>p</mi><mo>></mo><mn mathvariant="normal">1</mn></math><graphic specific-use="big" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="alternativeImage/8D25821C-1890-499b-A479-F24AE129D65C-M007.jpg"><?fx-imagestate width="8.38199997" height="3.47133350"?></graphic><graphic specific-use="small" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="alternativeImage/8D25821C-1890-499b-A479-F24AE129D65C-M007c.jpg"><?fx-imagestate width="8.38199997" height="3.47133350"?></graphic></alternatives></inline-formula>, erfüllt die Funktion <inline-formula><alternatives><math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M8"><mi>f</mi><mtext> </mtext></math><graphic specific-use="big" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="alternativeImage/8D25821C-1890-499b-A479-F24AE129D65C-M008.jpg"><?fx-imagestate width="2.53999996" height="3.47133350"?></graphic><graphic specific-use="small" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="alternativeImage/8D25821C-1890-499b-A479-F24AE129D65C-M008c.jpg"><?fx-imagestate width="2.53999996" height="3.47133350"?></graphic></alternatives></inline-formula> die Bedingung <inline-formula><alternatives><math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M9"><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>t</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>T</mi></mrow></munderover></mstyle><mrow><mtext> </mtext><mi>f</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mo><</mo><mn mathvariant="normal">0</mn></math><graphic specific-use="big" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="alternativeImage/8D25821C-1890-499b-A479-F24AE129D65C-M009.jpg"><?fx-imagestate width="17.44133186" height="8.29733276"?></graphic><graphic specific-use="small" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="alternativeImage/8D25821C-1890-499b-A479-F24AE129D65C-M009c.jpg"><?fx-imagestate width="17.44133186" height="8.29733276"?></graphic></alternatives></inline-formula>. Unter Verwendung der topologischen Gradtheorie wurde gezeigt, dass das Problem mindestens eine Lösung besitzt. Darüber hinaus wurde die zugehörige Methode auf diskrete Gleichungssysteme zweiter Ordnung mit ähnlicher nichtlinearer Struktur erweitert.
关键词
diskrete Gleichungen; Resonanz; überlinear; topologischer Grad
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