C.E. Chidume

Strong and weak convergence theorems for asymptotically nonexpansive mappings

Abstract Suppose K is a nonempty closed convex nonexpansive retract of a real uniformly convex Banach space E with P as a nonexpansive retraction. Let T :K→E be an asymptotically nonexpansive nonself-map with sequence {kn}n⩾1⊂[1,∞), limkn=1, F(T):={x∈K: Tx=x}≠∅ . Suppose {xn}n⩾1 is generated iteratively by x 1 ∈K, x n+1 =P (1−α n )x n +α n T(PT) n−1 x n , n⩾1, where {αn}n⩾1⊂(0,1) is such that ϵ 0. It is proved that (I−T) is demiclosed at 0. Moreover, if ∑n⩾1(kn2−1) x ∗ ∈F(T) is proved. If T is not assumed to be completely continuous but E also has a Frechet differentiable norm, then weak convergence of {xn} to some x ∗ ∈F(T) is obtained.

Approximation of fixed points of strongly pseudocontractive mappings

Let E be a real Banach space with a uniformly convex dual, and let K be a nonempty closed convex and bounded subset of E. Let T: K → K be a continuous strongly pseudocontractive mapping of K into itself. Let {c n } n=1 ∞ be a real sequence satisfying: (i) 0 < C n < 1 FOR ALL N ≥ 1; (II) ∑ n=1 ∞ c n = ∞; and (iii) ∑ n=1 ∞ c n b(c n ) < ∞, where b: [0, ∞) → [0, ∞) is some continuous nondecreasing function satisfying b(0) = 0, b(ct) ≤ cb(t) for all c ≥ 1. Then the sequence {x n } n=1 ∞ generated by x 1 ∈ K, x n+1 = (1 − c n )x n + c n Tx n , n ≥ 1, converges strongly to the unique fixed point of T. A related result deals with the Ishikawa iteration scheme when T is Lipschitzian and strongly pseudoconactive

Iterative approximation of fixed points of Lipschitzian strictly pseudocontractive mappings

Suppose X = Lp (or lp), p > 2, and K is a nonempty closed convex bounded subset of X. Suppose T: K -* K is a Lipschitzian strictly pseudo-contractive mapping of K into itself. Let {C, }?00 be a real sequence satisfying: (i) 0 1, (ii) Eoo1 Cn = oo, and (iii) 0n= 2n 1, converges strongly to afixed point of T in K.

Fixed point iteration for pseudocontractive maps

Let K be a compact convex subset of a real Hilbert space, H; T : K → K a continuous pseudocontractive map. Let {an}, {bn}, {cn}, {an}, {b ′ n} and {cn} be real sequences in [0,1] satisfying appropriate conditions. For arbitrary x1 ∈ K, define the sequence {xn}∞n=1 iteratively by xn+1 = anxn + bnTyn + cnun; yn = a ′ nxn + b ′ nTxn + c ′ nvn, n ≥ 1, where {un}, {vn} are arbitrary sequences in K. Then, {xn}∞n=1 converges strongly to a fixed point of T . A related result deals with the convergence of {xn}∞n=1 to a fixed point of T when T is Lipschitz and pseudocontractive. Our theorems also hold for the slightly more general class of continuous hemicontractive nonlinear maps.

An iterative process for nonlinear Lipschitzian strongly accretive mappings in Lp spaces

Abstract Suppose X = L p (or l p ), p ⩾ 2. Let T : X → X be a Lipschitzian and strongly accretive map with constant k ϵ (0, 1) and Lipschitz constant L . Define S : X → X by Sx = f − Tx − x . Let {C n } n = 1 ∞ be a real sequence satisfying: 1. (i) 0 C n ⩽ k [( p − 1) L 2 + 2 k − 1] −1 for each n , 2. (ii) ∑ n C n = ∞. Then, for arbitrary x 0 ϵ X , the sequence x n + 1 = (1 − C n )x n + C n SX n , n ⩾ 0 converges strongly to the unique solution of Tx = f . Moreover, if C n = k [( p − 1) L 2 + 2 k − 1] −1 for each n , then, ‖x n + 1 − q‖ ⩽ θ n 2 ‖x 1 − q‖ , where q denotes the solution of Tx = f and θ = (1 − k [( p − 1) L 2 + 2 k − 1] −1 ) ϵ (0, 1). A related result deals with the iterative approximation of Lipschitz strongly pseudocontractive maps in X .

Fixed Point Iterations For Strictly Hemi-Contractive Maps In Uniformly Smooth Banach Spaces

It is proved that the Mann iteration process converges strongly to the fixed point of a strictly hemi-contractive map in real uniformly smooth Banach spaces. The class of strictly hemi-contractive maps includes all strictly pseudo-contractive maps with nonempty fixed point sets. A related result deals with the Ishikawa iteration scheme when the mapping is Lipschitzian and strictly hemi-contractive. Our theorems generalize important known results.

Approximating fixed points of total asymptotically nonexpansive mappings

We introduce a new class of asymptotically nonexpansive mappings and study approximating methods for finding their fixed points. We deal with the Krasnoselskii-Mann-type iterative process. The strong and weak convergence results for self-mappings in normed spaces are presented. We also consider the asymptotically weakly contractive mappings.

Global iteration schemes for strongly pseudo-contractive maps

Suppose E is a real uniformly smooth Banach space, K is a nonempty closed convex and bounded subset of E, and T: K -K is a strong pseudo-contraction. It is proved that if T has a fixed point in K then both the Mann and the Ishikawa iteration processes, for an arbitrary initial vector in K, converge strongly to the unique fixed T. No continuity assumption is necessary for this convergence. Moreover, our iteration parameters are independent of the geometry of the underlying Banach space and of any property of the operator.

Convergence Theorems for Mappings Which Are Asymptotically Nonexpansive in the Intermediate Sense

Abstract Suppose K is a nonempty closed convex nonexpansive retract of a real uniformly convex Banach space E with P as a nonexpansive retraction. Let T : K → E be a non-self mapping which is asymptotically nonexpansive in the intermediate sense with F(T) ≔ {x ∈ K : Tx = x} ≠ ∅. A demiclosed principle for T is proved. Moreover, if T is completely continuous, an iterative sequence {x n } is constructed which converges strongly to some x* ∈ F(T). If T is not assumed to be completely continuous but the dual E* of E is assumed to have the Kadec–Klee property, then {x n } converges weakly to some x* ∈ F(T). The operator P which plays a central role in our proofs is, in this case, the Banach space analogue of the proximity map in Hilbert spaces.

Approximation methods for nonlinear operator equations

Let E be a real normed linear space and A : E → E be a uniformly quasi-accretive map. For arbitrary x 1 E E define the sequence x n E E by x n+1 := x n - α n Ax n , n > 1, where {an} is a positve real sequence satisfying the following conditions: (i) Σα n = ∞; (ii) lim α n = 0. For x* E N(A):= {x E E: Ax = 0}, assume that σ:= inf n ∈ N0 φ(∥x n+1 -x*∥)/∥x n+1 - x*∥ > 0 and that ∥Ax n+1 - Ax n ∥ → 0, where No:= {n ∈ N (the set of all positive integers): x n+1 ¬= x*} and ψ: [0,∞) → [0,∞) is a strictly increasing function with ψ(0) = 0. It is proved that a Mann-type iteration process converges strongly to x*. Furthermore if, in addition, A is a uniformly continuous map, it is proved, without the condition on σ, that the Mann-type iteration process converges strongly to x*. As a consequence, corresponding convergence theorems for fixed points of hemi-contractive maps are proved.