Equivalences between nonuniform exponential dichotomy and admissibility

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• 作者：Linfeng Zhou^a ; ^{zlf63math@163.com" class="auth_mail" title="E-mail the corresponding author} ; Kening Lu^b ; ^{klu@math.byu.edu" class="auth_mail" title="E-mail the corresponding author} ; Weinian Zhang^a ; ^{matzwn@126.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：34D09 ; 34C11
• 刊名：Journal of Differential Equations
• 出版年：2017
• 出版时间：5 January 2017
• 年：2017
• 卷：262
• 期：1
• 页码：682-747
• 全文大小：838 K

文摘

Relationship between exponential dichotomies and admissibility of function classes is a significant problem for hyperbolic dynamical systems. It was proved that a nonuniform exponential dichotomy implies several admissible pairs of function classes and conversely some admissible pairs were found to imply a nonuniform exponential dichotomy. In this paper we find an appropriate admissible pair of classes of Lyapunov bounded functions which is equivalent to the existence of nonuniform exponential dichotomy on half-lines class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616303084&_mathId=si1.gif&_user=111111111&_pii=S0022039616303084&_rdoc=1&_issn=00220396&md5=062e290a94f09fb64fb6b8a5bdfcfc80" title="Click to view the MathML source">R_±class="mathContainer hidden">class="mathCode">${\mathbb{R}}_{\pm }$ separately, on both half-lines class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616303084&_mathId=si1.gif&_user=111111111&_pii=S0022039616303084&_rdoc=1&_issn=00220396&md5=062e290a94f09fb64fb6b8a5bdfcfc80" title="Click to view the MathML source">R_±class="mathContainer hidden">class="mathCode">${\mathbb{R}}_{\pm }$ simultaneously, and on the whole line class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616303084&_mathId=si173.gif&_user=111111111&_pii=S0022039616303084&_rdoc=1&_issn=00220396&md5=b08bee803470d4019c8237fd40d36ae9" title="Click to view the MathML source">Rclass="mathContainer hidden">class="mathCode">$\mathbb{R}$. Additionally, the maximal admissibility is proved in the case on both half-lines class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616303084&_mathId=si1.gif&_user=111111111&_pii=S0022039616303084&_rdoc=1&_issn=00220396&md5=062e290a94f09fb64fb6b8a5bdfcfc80" title="Click to view the MathML source">R_±class="mathContainer hidden">class="mathCode">${\mathbb{R}}_{\pm }$ simultaneously.