| Ni LUQUN,YAO JINGQI,ZHAO HANZHANG.[J].数学年刊A辑,1980,1(1):63~74 |
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| SOME FIXED POINT THEOREMS OF CONTRACTIVETYPE MAPFINGS |
| Received:July 24, 1978 |
| DOI： |
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| 1. Let X be the conjugate of a separable Banach space satifying the *-Opial
condition, i. e., if \[\{ {x_n}\} \subset x,{x_n}\mathop \to \limits^{{w^*}} {x_\infty },{x_\infty } \ne y\], then\[\mathop {\overline {\lim } }\limits_{n \to \infty } ||{x_n} - {x_\infty }|| < \mathop {\overline {\lim } }\limits_{n \to \infty } ||{x_n} - y||\]
for rxample \[X = {l_1}\]
Let K be a nonempty weak* closed convex subset of X.
The main results are:
Theorem 1. Suppose T is a ooniinuons mappings of K into itself such that for
every \[x,y \in K\]，\[||Tx - Ty|| \le a||x - y|| + b\{ ||x - Tx|| + ||y - Ty||\} + c\{ ||x - Ty|| + ||y - Tx||\} \]
where real numbers \[a,b,c \ge 0\] and \[a + 2b + 2c = 1\]. Suppose also K is bounded.Then T has at least one fixed point in K.
Theorem 2. Let T be a mapping of K into itself, and \[a(x,y),b(x,y),c(x,y)\]be real functions such that for all\[x,y \in K\]
\[||Tx - Ty|| \le a(x,y)||x - y|| + b(x,y)\{ ||x - Tx|| + ||y - Ty||\} + c(x,y)\{ ||x - Ty|| + ||y - Tx||\} \]
and \[a(x{\rm{y}},y){\rm{ + }}2b(x,y){\rm{ + }}2c(x,y) \le 1\]
Suppose there exists \[x \in K\] such that \[O(x) = \{ {T^n}x\} _{n = 1}^\infty \] is bounded and
\[\mathop {\inf }\limits_{y,z \in o(x)} c(y,z) > 0\]
Then T has at least one fixed point z in K and \[{T^n}x\mathop \to \limits^{{w^*}} z\].
2. We denote \[CL(x) = \{ A;nonempty{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} closed{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} subset{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} of{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} X\} \]
\[K(x) = A;nonempty{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} closed{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} subset{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} of{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} x\} \]
here X is a complete metric space with metric d.
On \[CL(x)\] and \[K(x)\] we introduce the generalized Hausdorff distance \[H(,)\],
The main results are:
Theorem 3. Suppose \[\{ T,S\} \] is a pair of set-valued mappings of X into \[CL(x)\],which satisfies the following condition:
\[H(Tx,Sy) \le hMax\{ d(x,y),D(x,Tx),D(y,Sy),\frac{1}{2}[D(x,Sy) + D(y,Tx)]\} \]
for each \[x,y \in K\], where 0 |
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