Existence and uniqueness of limit cycles for generalized φ-Laplacian Liénard equations

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• 作者：S. Pé ; rez-Gonzá ; lez^a ; ^{setperez@mat.uab.cat" class="auth_mail" title="E-mail the corresponding author} ; J. Torregrosa^b ; ^{torre@mat.uab.cat" class="auth_mail" title="E-mail the corresponding author} ; P.J. Torres^c ; ^{ptorres@ugr.es" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Existence and uniqueness ; Periodic orbits ; Limit cycles ; φ-Laplacian Lié ; nard equations ; Generalized Lié ; nard equations
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2016
• 出版时间：15 July 2016
• 年：2016
• 卷：439
• 期：2
• 页码：745-765
• 全文大小：517 K

文摘

The Liénard equation class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16002225&_mathId=si1.gif&_user=111111111&_pii=S0022247X16002225&_rdoc=1&_issn=0022247X&md5=b2f6ddf9cc6d3473ea35e2d16e5b1bd0" title="Click to view the MathML source">x^″+f(x)x^′+g(x)=0class="mathContainer hidden">class="mathCode">$x^{″}+f(x)x^{\prime }+g(x)=0$ appears as a model in many problems of science and engineering. Since the first half of the 20th century, many papers have appeared providing existence and uniqueness conditions for limit cycles of Liénard equations. In this paper we extend some of these results for the case of the generalized φ  -Laplacian Liénard equation, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16002225&_mathId=si2.gif&_user=111111111&_pii=S0022247X16002225&_rdoc=1&_issn=0022247X&md5=5ee8ed3d76bedd6bcbcb57fe79dce0dd" title="Click to view the MathML source">(φ(x^′))^′+f(x)ψ(x^′)+g(x)=0class="mathContainer hidden">class="mathCode">${(\phi (x^{\prime }))}^{\prime }+f(x)\psi (x^{\prime })+g(x)=0$. This generalization appears when derivations of the equation different from the classical one are considered. In particular, the relativistic van der Pol equation, class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16002225&_mathId=si3.gif&_user=111111111&_pii=S0022247X16002225&_rdoc=1&_issn=0022247X&md5=b6890e393b3d3d57e825a021e9efbd78">class="imgLazyJSB inlineImage" height="32" width="298" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16002225-si3.gif">class="mathContainer hidden">class="mathCode">${(x^{\prime }/\sqrt{1-{(x^{\prime }/c)}^2})}^{\prime }+\mu (x^2-1)x^{\prime }+x=0$, has a unique periodic orbit when class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16002225&_mathId=si4.gif&_user=111111111&_pii=S0022247X16002225&_rdoc=1&_issn=0022247X&md5=e542643ff8f2967a49f07224fa3ae0ff" title="Click to view the MathML source">μ≠0class="mathContainer hidden">class="mathCode">$\mu \ne 0$.