Invariant approximations, generalized -contractions, and -subweakly commuting maps
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We present common fixed point theory for generalized contractive R-subweakly commuting maps and obtain some results on invariant approximation. 1. Introduction and preliminaries Let S be a subset of a normed space X = (X, · ) and T and I self-mappings of X. Then T is called (1) nonexpansive on S if Tx − T y ≤ x − y for all x, y ∈ S; (2) Inonexpansive on S if Tx − T y ≤ Ix − I y for all x, y ∈ S; (3) I-contraction on S if there exists k ∈ [0,1) such that Tx − T y ≤ kIx − I y for all x, y ∈ S. The set of fixed points of T (resp., I) is denoted by F(T) (resp., F(I)). The set S is called (4) pstarshaped with p ∈ S if for all x ∈ S, the segment [x, p] joining x to p is contained in S (i.e., kx + (1 − k)p ∈ S for all x ∈ S and all real k with 0 ≤ k ≤ 1); (5) convex if S is pstarshaped for all p ∈ S. The convex hull co(S) of S is the smallest convex set in X that contains S, and the closed convex hull clco(S) of S is the closure of its convex hull. The mapping T is called (6) compact if clT(D) is compact for every bounded subset D of S. The mappings T and I are said to be (7) commuting on S if ITx = TIx for all x ∈ S; (8) R-weakly commuting on S [7] if there exists R ∈ (0, ∞) such that TIx − ITx ≤ RTx − Ix for all x ∈ S. Suppose S ⊂ X is p-starshaped with p ∈ F(I) and is both T- and I-invariant. Then T and I are called (8) R-subweakly commuting on S [11] if there exists R ∈ (0, ∞) such that TIx − ITx ≤ Rdist(Ix,[Tx, p]) for all x ∈ S, where dist(Ix,[Tx, p]) = inf {Ix − z : z ∈ [Tx, p]}. Clearly commutativity implies R-subweak commutativity, but the converse may not be true (see [11]). The set PS (x) = { y ∈ S : y − x = dist(x,S)} is called the set of best approximants to x ∈ X out of S, where dist(x,S) = inf { y − x : y ∈ S}. We define CSI (x) = {x ∈ S : Ix ∈ PS (x)} and denote by 0 the class of closed convex subsets of X containing 0. For S ∈ 0 , we define Sx = {x ∈ S : x ≤ 2 x}. It is clear that PS (x) ⊂ Sx ∈ 0 . In 1963, Meinardus [6] employed the Schauder fixed point theorem to establish the existence of invariant approximations. Afterwards, Brosowski [2] obtained the following extension of the Meinardus result. Copyright © 2005 Hindawi Publishing Corporation Fixed Point Theory and Applications 2005:1 (2005) 79–86 DOI: 10.1155/FPTA.2005.79
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Invariant approximations
Theorem 1.1. Let T be a linear and nonexpansive self-mapping of a normed space X, S ⊂ X such that T(S) ⊂ S, and x ∈ F(T). If PS (x) is nonempty, compact, and convex, then PS (x)∩F(T) = ∅. Singh [15] observed that Theorem 1.1 is still true if the linearity of T is dropped and PS (x) is only starshaped. He further remarked, in [16], that Brosowski’s theorem remains valid if T is nonexpansive only on PS (x)∪{ x}. Then Hicks and Humphries [5] improved Singh’s result by weakening the assumption T(S) ⊂ S to T(∂S) ⊂ S; here ∂S denotes the boundary of S. On the other hand, Subrahmanyam [18] generalized the Meinardus result as follows. Theorem 1.2. Let T be a nonexpansive self-mapping of X, S a finite
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