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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
Full Text:
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 [6] Pai, D.V; Veeramani, P, On some fixed point theorems in Banach spaces, Internat. J. math. math. sci., 5, No. 1, 113-122, (1982) · Zbl 0492.47031 [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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