zbMATH — the first resource for mathematics

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
Zbl 0461.47027
Full Text:
References:
  Alspach D. E.,A fixed point free nonexpansive map. Preprint. · Zbl 0468.47036  Baillon J. B.,Quelques aspects de la theorie des points fixes dans les espaces de Banach.–I. Preprint. · Zbl 0414.47040  Browder F. E.,Nonexpansive nonlinear operators in Banach spaces. Proc. Nat. Acad. Sci. USA 54 (1965), 1041–1044. · Zbl 0128.35801  Goebel K.,Convexity of balls and fixed points theorems for mappings with nonexpansive square. Compositio Math. 22 (1970), 269–274. · Zbl 0202.12802  Goebel K.,On the minimal displacement of points under Lipschitzian mappings. Pacific J. Math. 45 (1973), 151–163. · Zbl 0265.47046  Goebel K., Kirk W. A.,A fixed point theorem for transformations whose iterates have uniform Lipschitz constant. Studia Math. 47 (1973), 135–140. · Zbl 0265.47044  Koebel K., Koter M.,A remark on nonexpansive mappings. Canad. Math. Bull. 24 (1981), 113–115. · Zbl 0461.47027  Goehde D.,Zum Prinzip der kontraktiven Abbildung. Math. Nachr. 30 (1965), 251–258. · Zbl 0127.08005  Kirk W. A.,A fixed point theorem for mappings which do not increase distances. Amer. Math. Monthly 72 (1965), 1004–1006. · Zbl 0141.32402  Kirk W. A.,A fixed point theorem for mappings with a nonexpansive iterate. Proc. Amer. Math. Soc. 29 (1971), 294–298. · Zbl 0213.41303  Lifschitz E. A.,Fixed points theorems for operators in strongly convex spaces. Voronež. Gos. Univ. Trudy Mat. Fak. 16 (1975), 23–28.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.