Exponential dichotomy for evolution families on the real line
<p>We give necessary and sufficient conditions for uniform exponential dichotomy of evolution families in terms of the admissibility of the pair <mml:math> <mml:mrow><mml:mo>(</mml:mo> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> &...
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Hindawi Limited
2006-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://www.hindawi.com/GetArticle.aspx?doi=10.1155/AAA/2006/31641 |
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doaj-a12caa60d5114c89a50db6b8a94abf6f2020-11-24T22:19:43ZengHindawi LimitedAbstract and Applied Analysis1085-33752006-01-012006Exponential dichotomy for evolution families on the real line<p>We give necessary and sufficient conditions for uniform exponential dichotomy of evolution families in terms of the admissibility of the pair <mml:math> <mml:mrow><mml:mo>(</mml:mo> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> <mml:mrow><mml:mo>(</mml:mo> <mml:mrow> <mml:mi>ℝ</mml:mi><mml:mo>,</mml:mo><mml:mi>X</mml:mi> </mml:mrow> <mml:mo>)</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:msup> <mml:mi>L</mml:mi> <mml:mi>q</mml:mi> </mml:msup> <mml:mrow><mml:mo>(</mml:mo> <mml:mrow> <mml:mi>ℝ</mml:mi><mml:mo>,</mml:mo><mml:mi>X</mml:mi> </mml:mrow> <mml:mo>)</mml:mo></mml:mrow> </mml:mrow> <mml:mo>)</mml:mo></mml:mrow> </mml:math>. We show that the admissibility of the pair <mml:math> <mml:mrow><mml:mo>(</mml:mo> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> <mml:mrow><mml:mo>(</mml:mo> <mml:mrow> <mml:mi>ℝ</mml:mi><mml:mo>,</mml:mo><mml:mi>X</mml:mi> </mml:mrow> <mml:mo>)</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:msup> <mml:mi>L</mml:mi> <mml:mi>q</mml:mi> </mml:msup> <mml:mrow><mml:mo>(</mml:mo> <mml:mrow> <mml:mi>ℝ</mml:mi><mml:mo>,</mml:mo><mml:mi>X</mml:mi> </mml:mrow> <mml:mo>)</mml:mo></mml:mrow> </mml:mrow> <mml:mo>)</mml:mo></mml:mrow> </mml:math> is equivalent to the uniform exponential dichotomy of an evolution family if and only if <mml:math> <mml:mi>p</mml:mi><mml:mo>≥</mml:mo><mml:mi>q</mml:mi> </mml:math>. As applications we obtain characterizations for uniform exponential dichotomy of semigroups.</p>http://www.hindawi.com/GetArticle.aspx?doi=10.1155/AAA/2006/31641 |
| collection |
DOAJ |
| language |
English |
| format |
Article |
| sources |
DOAJ |
| title |
Exponential dichotomy for evolution families on the real line |
| spellingShingle |
Exponential dichotomy for evolution families on the real line Abstract and Applied Analysis |
| title_short |
Exponential dichotomy for evolution families on the real line |
| title_full |
Exponential dichotomy for evolution families on the real line |
| title_fullStr |
Exponential dichotomy for evolution families on the real line |
| title_full_unstemmed |
Exponential dichotomy for evolution families on the real line |
| title_sort |
exponential dichotomy for evolution families on the real line |
| publisher |
Hindawi Limited |
| series |
Abstract and Applied Analysis |
| issn |
1085-3375 |
| publishDate |
2006-01-01 |
| description |
<p>We give necessary and sufficient conditions for uniform exponential dichotomy of evolution families in terms of the admissibility of the pair <mml:math> <mml:mrow><mml:mo>(</mml:mo> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> <mml:mrow><mml:mo>(</mml:mo> <mml:mrow> <mml:mi>ℝ</mml:mi><mml:mo>,</mml:mo><mml:mi>X</mml:mi> </mml:mrow> <mml:mo>)</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:msup> <mml:mi>L</mml:mi> <mml:mi>q</mml:mi> </mml:msup> <mml:mrow><mml:mo>(</mml:mo> <mml:mrow> <mml:mi>ℝ</mml:mi><mml:mo>,</mml:mo><mml:mi>X</mml:mi> </mml:mrow> <mml:mo>)</mml:mo></mml:mrow> </mml:mrow> <mml:mo>)</mml:mo></mml:mrow> </mml:math>. We show that the admissibility of the pair <mml:math> <mml:mrow><mml:mo>(</mml:mo> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> <mml:mrow><mml:mo>(</mml:mo> <mml:mrow> <mml:mi>ℝ</mml:mi><mml:mo>,</mml:mo><mml:mi>X</mml:mi> </mml:mrow> <mml:mo>)</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:msup> <mml:mi>L</mml:mi> <mml:mi>q</mml:mi> </mml:msup> <mml:mrow><mml:mo>(</mml:mo> <mml:mrow> <mml:mi>ℝ</mml:mi><mml:mo>,</mml:mo><mml:mi>X</mml:mi> </mml:mrow> <mml:mo>)</mml:mo></mml:mrow> </mml:mrow> <mml:mo>)</mml:mo></mml:mrow> </mml:math> is equivalent to the uniform exponential dichotomy of an evolution family if and only if <mml:math> <mml:mi>p</mml:mi><mml:mo>≥</mml:mo><mml:mi>q</mml:mi> </mml:math>. As applications we obtain characterizations for uniform exponential dichotomy of semigroups.</p> |
| url |
http://www.hindawi.com/GetArticle.aspx?doi=10.1155/AAA/2006/31641 |
| _version_ |
1725777834286252032 |