Invariant Approximations, Generalized I-contractions, and R-subweakly Commuting Maps

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− Ty‖ ≤ ‖x− y‖ for all x, y ∈ S; (2) Inonexpansive on S if ‖Tx − Ty‖ ≤ ‖Ix − I y‖ for all x, y ∈ S; (3) I-contraction on S if there exists k ∈ [0,1) such that ‖Tx − Ty‖ ≤ k‖Ix − 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‖ ≤ R‖Tx − Ix‖ for all x ∈ S. Suppose S ⊂ X is p-starshaped with p ∈ F(I) and is both Tand 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(x̂) = {y ∈ S : ‖y− x̂‖ = dist(x̂,S)} is called the set of best approximants to x̂ ∈ X out of S, where dist(x̂,S) = inf{‖y− x̂‖ : y ∈ S}. We define C S(x̂) = {x ∈ S : Ix ∈ PS(x̂)} and denote by 0 the class of closed convex subsets of X containing 0. For S∈ 0, we define Sx̂ = {x ∈ S : ‖x‖ ≤ 2‖x̂‖}. It is clear that PS(x̂) ⊂ Sx̂ ∈ 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.