Multiple Homoclinic Solutions for Second‐Order Perturbed Hamiltonian Systems. Issue 2 (25th June 2013)

• Record Type: Journal Article
• Authors: Ye, Yiwei; Tang, Chun‐Lei
• Is Part Of: Studies in applied mathematics. Volume 132:Issue 2(2014)
• Page Start: 112
• Page End: 137
• Publication Date: 2013-06-25
• Subjects: Mathematics -- Periodicals; Mathématiques -- Périodiques; Mathématique; Mathématique appliquée; Ressource Internet (Descripteur de forme); Périodique électronique (Descripteur de forme); 519
• DOI: 10.1111/sapm.12023
• Languages: English
• ISSNs: 0022-2526

Abstract

<abstract abstract-type="main"> <title> <x xml:space="preserve">Abstract</x> </title> <p>In this paper, we study the second‐order perturbed Hamiltonian systems <disp-formula content-type="mathematics" id="sapm12023-disp-0001"><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg42b8xn9p" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="block" altimg="urn:x-wiley:00222526:sapm12023:equation:sapm12023-math-0001" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mover accent="true"><mml:mi>u</mml:mi><mml:mo>̈</mml:mo></mml:mover><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>−</mml:mo><mml:mi>λ</mml:mi><mml:mi>L</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mi>u</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mi>∇</mml:mi><mml:mi>W</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>, </mml:mo><mml:mi>u</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>, </mml:mo></mml:mrow></mml:math></alternatives></disp-formula>where <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg42b8xn5g" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:00222526:sapm12023:equation:sapm12023-math-0002" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>λ</mml:mi><mml:mo>≥</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math></alternatives> is a parameter, <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg42b8xn61" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:00222526:sapm12023:equation:sapm12023-math-0003" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>L</mml:mi><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives> is positive definite for all <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg42b8xn7k" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:00222526:sapm12023:equation:sapm12023-math-0004" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>t</mml:mi><mml:mo>∈</mml:mo><mml:mi mathvariant="double-struck">R</mml:mi></mml:mrow></mml:math></alternatives> but unnecessarily uniformly positive definite for <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg42b8xn84" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:00222526:sapm12023:equation:sapm12023-math-0005" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>t</mml:mi><mml:mo>∈</mml:mo><mml:mi mathvariant="double-struck">R</mml:mi></mml:mrow></mml:math></alternatives>, and <italic>W</italic> is either asymptotically quadratic or superquadratic in <italic>x</italic> as <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg42b8xnb7" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:00222526:sapm12023:equation:sapm12023-math-0006" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>|</mml:mo><mml:mi>x</mml:mi><mml:mo>|</mml:mo><mml:mo>→</mml:mo><mml:mi>∞</mml:mi></mml:mrow></mml:math></alternatives>. Based on variational methods, we prove the existence of at least two nontrivial homoclinic solutions for the above system when <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg42b8xncs" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:00222526:sapm12023:equation:sapm12023-math-0007" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>f</mml:mi><mml:mo>∈</mml:mo><mml:msup><mml:mi>L</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mi mathvariant="double-struck">R</mml:mi><mml:mo>, </mml:mo><mml:msup><mml:mi mathvariant="double-struck">R</mml:mi><mml:mi>N</mml:mi></mml:msup><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:mo>∖</mml:mo></mml:mrow><mml:mfenced open="{" close="}"><mml:mn>0</mml:mn></mml:mfenced></mml:mrow></mml:math></alternatives> small enough.</p> </abstract>