Let C be a nonempty closed convex subset of a real Banach space X which has a uniformly Gâteaux differentiable norm. Let T ∈ ΓC and f ∈ΠC. Assume that {xt} converges strongly to a fixed point z of T as t→ 0, where xt is the unique element of C which satisfies xt = t f (xt) + (1− t)Txt. Let {αn} and {βn} be two real sequences in (0,1) which satisfy the following conditions: (C1) limn→∞αn = 0; (C...

Let K be a nonempty closed convex subset of a real Banach space E and T be a Lipschitz pseudocontractive self-map of K with F (T ) := {x ∈ K : Tx = x} 6= ∅. An iterative sequence {xn} is constructed for which ||xn − Txn|| → 0 as n → ∞. If, in addition, K is assumed to be bounded, this conclusion still holds without the requirement that F (T ) 6= ∅. Moreover, if, in addition, E has a uniformly G...

LetK be a closed convex subset of a real Banach space E, T : K → K is continuous pseudocontractive mapping, and f : K → K is a fixed L-Lipschitzian strongly pseudocontractive mapping. For any t ∈ (0,1), let xt be the unique fixed point of t f + (1− t)T . We prove that if T has a fixed point and E has uniformly Gâteaux differentiable norm, such that every nonempty closed bounded convex subset of...

Let E be a real Banach space with uniformly Gâteaux differentiable norm possessing uniform normal structure. K is a nonempty bounded closed convex subset of E, and { } ( ) ... , 2 , 1 = n Tn is a sequence of − n k Lipschitzian nonexpansive mappings from K into itself such that 1 lim = ∞ → n n k and ( ) 0 1 / ≠ ∞ = n n T F ∩ and f be a contraction on K. Under sutiable conditions on sequence { },...

This note describes the pleasant features that accrue in weighted settings when the partial sums of the operator-valued Fourier series corresponding to a multiplier function ψ : T → C are uniformly bounded in operator norm. This circle of ideas also includes a Tauberiantype condition on the multiplier function ψ sufficient to insure such uniform boundedness of partial sums. These considerations...

Let K be a nonempty closed convex subset of a reflexive and strictly convex Banach space E with a uniformly Gâteaux differentiable norm, F {T h : h ≥ 0} a generalized asymptotically nonexpansive self-mapping semigroup of K, and f : K → K a fixed contractive mapping with contractive coefficient β ∈ 0, 1 . We prove that the following implicit and modified implicit viscosity iterative schemes {xn}...

Let E be a reflexive Banach space with a uniformly Gâteaux differentiable norm. Suppose that every weakly compact convex subset of E has the fixed point property for nonexpansive mappings. Let C be a nonempty closed convex subset of E, f : C → C a contractive mapping or a weakly contractive mapping , and T : C → C nonexpansive mapping with the fixed point set F T / ∅. Let {xn} be generated by a...

Let K be a nonempty closed convex subset of a real Banach space E which has a uniformly Gâteaux differentiable norm. Assume that K is a sunny nonexpansive retract of E with Q as the sunny nonexpansive retraction. Let Ti : K → E, i = 1, 2, · · · , N be a family of nonexpansive mappings which are weakly inward with F = ⋂N i=1 F (Ti) 6= ∅. Let f : K → K be a fixed contractive mapping. For given x0...

Let E be a real reflexive Banach space which has uniformly Gâteaux differentiable norm. Let K be aclosed convex subset of E which is also a sunny nonexpansive retract of E, and T : K → E be nonexpansive mapping satisfying the weakly inward condition and F (T ) = {x ∈ K, Tx = x} 6= ∅, and f : K → K be a contractive mapping. Suppose that x0 ∈ K, {xn} is defined by { xn+1 = αnf(xn) + (1− αn)((1− δ...