On some fixed point theorems on uniformly convex Banach spaces. (English) Zbl 0780.47047

Veeramani, P.

J. Math. Anal. Appl. 167, No. 1, 160-166 (1992).

The author introduces the concept of \(T\)-regularity and generalizes a well known fixed point theorem of F. E. Browder [Proc. Nat. Acad. Sci. USA 54, 1041-1043 (1965; Zbl 0128.358)]. Let \(X\) be a vector space and \(A\) be a subset of \(X\). \(A\) is said to be a \(T\)-regular set if and only if
(i) \(T: A\to A\); (ii) \({1\over 2}(x+Tx)\in A\), for each \(x\) in \(A\).
Main Theorem. Let \(K\) be a nonempty weakly compact \(T\)-regular subset of a uniformly convex Banach space \(X\). Further for each weakly closed \(T\)- regular subset \(F\) of \(K\), with \(\delta(F)>0\), there exists some \(\beta(F)\), \(0<\beta(F)<1\), such that \[ \| Tx-Ty\|\leq max\{\| x-y\|, \beta\delta(F)\} \] for all \(x,y\in F\). Then \(T\) has a fixed point in \(K\).
The proof is based on a Zorn’s lemma. From the concept of \(T\)-regularity and other results of the author, theorems of C. L. Outlaw [Pac. J. Math. 30, 747-750 (1969; Zbl 0174.193)] and A. A. Gillespie and B. B. Williams [J. Math. Anal. Appl. 74, 382-387 (1980; Zbl 0448.47039)] can be deduced as special cases.

Reviewer: A.A.Melentsov (Sverdlovsk)

MSC:

47H10

Fixed-point theorems

Keywords:

weakly compact \(T\)-regular subset of a uniformly convex Banach space; Zorn’s lemma; \(T\)-regularity

Citations:

Zbl 0128.358; Zbl 0174.193; Zbl 0448.47039

References:

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[2]	Gillespie, A. A.; Williams, B. B., Some theorems on fixed points in Lipschitz and Kannan type mappings, J. Math. Anal. Appl., 74, 382-387 (1980) · Zbl 0448.47039
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