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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

47H10 Fixed-point theorems
Zbl 0461.47027
Full Text: DOI
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