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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

##### MSC:
 47H10 Fixed-point theorems
Zbl 0202.128