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| QUASI-LOCAL CONJUGACY THEOREMS IN BANACH SPACES |
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Citation： |
ZHANG Weirong,MA Jipu.QUASI-LOCAL CONJUGACY THEOREMS IN BANACH SPACES[J].Chinese Annals of Mathematics B,2005,26(4):551~558 |
| Page view： 1087
Net amount： 744 |
Authors： |
ZHANG Weirong; MA Jipu |
Foundation： |
Project supported by the National Natural Science Foundation of China (No.10271053). |
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| Abstract： |
Let $f:U(x_0)\subset E\longrightarrow F$ be a $C^{1}$ map and $f^{\prime}(x_0)$ be the Frechet derivative of $f$ at $x_0$. In local analysis of nonlinear functional analysis, implicit function theorem, inverse function theorem, local surjectivity theorem, local injectivity theorem, and the local conjugacy theorem are well known. Those theorems are established by using the properties: $f^{\prime}(x_0)$ is double splitting and $R(f^{\prime}(x))\cap N(T_{0}^{+})=\{0\}$ near $x_0.$ However, in infinite dimensional Banach spaces, $f^{\prime}(x_0)$ is not always double splitting (i.e., the generalized inverse of $f^{\prime}(x_0)$ does not always exist), but its bounded outer inverse of $f^{\prime}(x_0)$ always exists.
Only using the $C^{1}$ map $f$ and the outer inverse $\t$ of $f^{\prime}(x_0)$, the authors obtain two quasi-local conjugacy theorems, which imply the local conjugacy theorem if $x_0$ is a locally fine point of $f$. Hence the quasi-local conjugacy theorems generalize the local conjugacy theorem in Banach spaces. |
Keywords： |
Frechet derivative, Quasi-local conjugacy theorems, Outer inverse, Local conjugacy theorem |
Classification： |
47H |
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