Разрешимость сверхлинейной задачи Неймана при резонансе на первом собственном значении
摘要
Исследование существования решения класса сверхлинейных дискретных краевых задач типа Неймана второго порядка <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> , где <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> — целое число, <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> , функция <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> удовлетворяет условию <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>. Используя теорию топологической степени, доказано существование как минимум одного решения задачи. Кроме того, соответствующий метод обобщен на системы дискретных уравнений второго порядка с подобной нелинейной структурой.
关键词
дискретные уравнения; резонанс; сверхлинейность; топологическая степень
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