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| ON THE EQUIVALENCE PRINCIPLE FOR CONTRACTIVE MAPPINGS |
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Citation： |
Ding Xieping.ON THE EQUIVALENCE PRINCIPLE FOR CONTRACTIVE MAPPINGS[J].Chinese Annals of Mathematics B,1984,5(2):145~152 |
| Page view： 673
Net amount： 802 |
Authors： |
Ding Xieping; |
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| Abstract： |
The main result is:
Theorem 1. Let T be a continuous selfmapping of a complete metric space $\[(X,d)\]$ and have the unique fixed point property. If there exists $\[n:X \to N\]$ (the set of all positive integers) which is locsally bounded such that for each $\[x \in X\]$ and for all $\[r \in N,r \ge n(x)\]$,
$$\[\begin{array}{*{20}{c}}
{D({O_T}({T^{n(x)}}x,0,r) \le \varphi (D({O_T}(x,0,r))),or}\{D({O_T}({T^{n(x)}}x,0,r) \le \psi (D({O_T}(x,0,r))),}
\end{array}\]$$
where $\[\varphi ,\psi \]$ are contractive gauge functions,then
(a) T has a unique fixed point $\[{x^*}\]$;
(b) For each $\[x \in X,{T^n}x \to {x^*}\]$ as $\[n \to \infty \]$.
(c) There exists a neighborhood $\[U({x^*})\]$ of x* sueh that $\[\mathop {\lim }\limits_{n \to \infty } {T^n}(U({x^*})) = \{ {x^*}\} \]$;
(d) x* is stable;
(e) For any given $\[C \in (0,1)\]$ there exists a metric d* topologically equivalent to d such that T is a Banach contraction under d* with Lipsohitz constant C.
By Theorem 1 it is shown that many contractive type mappings defined in [1—26] are topologically equivalent to each other. |
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