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Fixed points of Lipschitzian 2-rotative mappings. (English) Zbl 0634.47053
Let X be a closed and convex subset of a Banach space B. Let \(\Phi\) (n,a,k,X) denote the class of all k-Lipschitzian maps \(T: X\to X\) with the property that there exist \(n\in N\), \(a<n\), such that for every \(x\in X\) \[ (*)\quad \| T^ nx-x\| \leq a\| Tx-x\|. \] The transformation T satisfying (*) will be called n-rotative with constant a. In this paper, the author studies this class of maps with \(k>1\) and \(n=2\). A typical result can be stated as follows:
Theorem. Let \(T\in \Phi (2,a,k,X)\), \(X\subset B\), and \([1-\delta_ B(2/k)]k<2-a\), where \(\delta_ B\) denotes the modulus of convexity of the space B. Then T has at least one fixed point.
The above result generalizes a theorem of K. Goebel [Compos. math. 22, 269-274 (1970; Zbl 0202.128)]. Many other related results for a family of Lipschitzian rotative mappings and also for weakly rotative mappings have also been obtained. Illustrative examples are provided.
Reviewer: M.S.Khan

47H10 Fixed-point theorems
Zbl 0202.128