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Fixed points of rotative Lipschitzian mappings. (English) Zbl 0535.47031
Let X be a closed convex subset of a Banach space. A mapping $$T:X\to X$$ is said to be in class $$\Phi(n,a,k,X)$$ if T has Lipschitz constant k and satisfies for fixed $$N\in {\mathbb{N}}$$ and $$a\in {\mathbb{R}}$$, 0$$\leq a\leq n$$, the condition $$\| T^ nx-x\| \leq a\| Tx-x\|,$$ $$x\in X$$. Such mappings are called (lipschitzian) rotative. For $$k=1$$, the condition roughly means that the relative distances between consecutive iterates of T must shorten or that these iterations are not situated along (metrical) lines. In an earlier paper [Can. Math. Bull. 24, 113-115 (1981 Zbl 0461.47027)] the authors proved that if $$T\in \Phi(n,a,1,X)$$ for some $$n\in {\mathbb{N}}$$ and $$a<n$$, then T always has a fixed point. In the present paper the authors investigate the case $$k>1$$. Let $$\gamma_ n(a) = \inf \{k:$$ there exist $$X$$ and $$T$$ such that $$T\in\Phi(n,a,k,X)$$ and $$\text{Fix}T=\emptyset\}$$, the authors show that for any $$n\in{\mathbb{N}}$$ and $$a<n$$, $$\gamma_ n(a)>1$$. (Thus if $$k<\gamma_ n(a)$$, $$T$$ has at least one fixed point.) It is also shown that in an arbitrary Banach space, $\gamma_2(a) \geq \frac{(2-a+\sqrt{(2-a)^ 2+a^ 2)}}2.$
Reviewer: W.A.Kirk

##### MSC:
 47H10 Fixed-point theorems
##### Keywords:
lipschitzian mapping; rotative mapping; fixed point theorem
Zbl 0461.47027
Full Text:
##### References:
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