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| Quasi-Normal Structure and Fixed Point of Nonlinear Mappings |
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Citation： |
Zhao Hanbin.Quasi-Normal Structure and Fixed Point of Nonlinear Mappings[J].Chinese Annals of Mathematics B,1982,3(6):779~788 |
| Page view： 838
Net amount： 883 |
Authors： |
Zhao Hanbin; |
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| Abstract： |
Let (X,|| ||) be a Banach space. For $\Omega \subset X^*$ and $x\in X$ we introduce the following notations (p\geq 1 and n\in N)
$|X|_{\Omega _p(n)}=sup{(\sum\limits_{f\in F} |f(x)|^p)^{1/p}:F \subset \Omega,|F|\leq n$
$|X|_{\Omega _\infty}=sup{|f(x)|:f\in \Omega}$
A convex subset E of X is said to have guasi-normal structure whenever there exists a norm 1 | on A which satisfies the following conditions;
(i) E has norinal structure relative to the norm ||| |||.
(ii) There exist $\Omega \subset X^*$, p\geq 1 and \theta \in (0,1], such that
$|x|_{\Omega _p(2) \leq |||x||| \leq ||x||}$ for x\in E and |||x|||<||x||
implies $2^1/p |x|_\Omega_\infty \geq \theta ||x||+(1-\theta)|||x|||$
or (ii)' There exist \Omega \subset X^*,p\geq 1 and \alpha \in [1,4^1/p) such that for all x\in E, |x|_\Omega_\rho(4)\leq |||x|||,||x||=max{|||x|||,\alpha|x|_\Omega_\infty} and for any countable subset w of \Omega
$sup{\sum\limits _{\delt\in w |f(x)|^p:x\in E}<+\infty$
We notice that a set with normal steucture must have quasi-normal structure and there exist sets without normal structure which quasi-normal structure.
The main result of the present paper is as follows.
Theorem. Let (X, || ||) be a Banach space, E a weak compact convex nonempty subset of X with quasi-normal structure. Let T be a mapping of E in to itself. If there exists a sequence {x_n} in any T-invariant convex subset of E such that
$lim_{n\rightarrow \infty} ||x_n-x_n+1||=lim_{n\rightarrow \infty}||x_n-Tx_n||=0$
and
$lim_{n\rightarrow \infty} ||y-x_n||=\delta(\bar co{x_n}),for y\in \bar co({c_n})$
limll2/-*?ll=3(coK}), for y€co({xa}),
then the mapping T has a fixed point in E,
In particular, if the mapping T satisfies
$||Tx-Ty||\leq max{||x-y||,1/2(||x-Ty||+||y-Tx||)},for x,y\in E$
then the mapping T has a fixed point in E. |
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