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

MSC:
47H10 Fixed-point theorems
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References:
[1] Browder, F.E, Nonexpansive nonlinear operations in a Banach space, (), 1041-1043 · Zbl 0128.35801
[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
[3] Ishikawa, S, Fixed points and iteration of a nonexpansive mapping in a Banach space, (), 65-71, No. 1 · Zbl 0352.47024
[4] Kirk, W.A, A fixed point theorem for mapping which do not increase distances, Amer. math monthly, 72, 1004-1006, (1965) · Zbl 0141.32402
[5] Outlaw, C.L, Mean value iteration of nonexpansive mappings in a Banach space, Pacific J. math., 30, 747-750, (1969) · Zbl 0179.19801
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[7] Singh, S.P, Application of a fixed point theorem to approximation theory, J. approx. theory, 25, 88-89, (1979) · Zbl 0399.41032
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