Strong and weak convergence theorems for asymptotically nonexpansive mappings

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 {k_n}_n1[1,∞), limk_n=1, F(T):={xK: Tx=x}≠. Suppose {x_n}_n1 is generated iteratively by

where {α_n}_n1(0,1) is such that ε<1−α_n<1−ε for some ε>0. It is proved that (I−T) is demiclosed at 0. Moreover, if ∑_n1(k_n²−1)<∞ and T is completely continuous, strong convergence of {x_n} to some x^*F(T) is proved. If T is not assumed to be completely continuous but E also has a Fréchet differentiable norm, then weak convergence of {x_n} to some x^*F(T) is obtained.     

Strong convergence of an iterative method for non‐expansive mappings

Let E be a real reflexive Banach space having a weakly continuous duality mapping J_φ with a gauge function φ, and let K be a nonempty closed convex subset of E. Suppose that T is a non‐expansive mapping from K into itself such that F (T) ≠ ??. For an arbitrary initial value x₀ ∈ K and fixed anchor u ∈ K, define iteratively a sequence {x_n } as follows: x_{n +1} = α_n u + β_n x_n + γ_n Tx_n , n ≥ 0, where {α_n }, {β_n }, {γ_n } ? (0, 1) satisfies α_n +β_n + γ_n = 1, (C 1) lim_{n →∞} α_n = 0, (C 2) ∑^∞_{n =1} α_n = ∞ and (B) 0 < lim inf_{n →∞} β_n ≤ lim sup_{n →∞} β_n < 1. We prove that {x_n } converges strongly to Pu as n → ∞, where P is the unique sunny non‐expansive retraction of K onto F (T). We also prove that the same conclusions still hold in a uniformly convex Banach space with a uniformly Gâteaux differentiable norm or in a uniformly smooth Banach space. Our results extend and improve the corresponding ones by C. E. Chidume and C. O. Chidume [Iterative approximation of fixed points of non‐expansive mappings, J. Math. Anal. Appl. 318 , 288–295 (2006)], and develop and complement Theorem 1 of T. H. Kim and H. K. Xu [Strong convergence of modified Mann iterations, Nonlinear Anal. 61 , 51–60 (2005)]. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Convergence of a hybrid algorithm for a reversible semigroup of nonlinear operators in Banach spaces

The purpose of this paper is to study hybrid iterative schemes of Halpern types for a semigroup ℑ={T(s):s∈S} of relatively nonexpansive mappings on a closed and convex subset C of a Banach space with respect to a sequence {μ_n} of asymptotically left invariant means defined on an appropriate invariant subspace of l^∞(S). We prove that given a certain sequence {α_n} in [0,1], x∈C, we can generate an iterative sequence {x_n} which converges strongly to Π_F(ℑ)x where Π_F(ℑ)x is the generalized projection from C onto the fixed point set F(ℑ). Our main result is even new for the case of a Hilbert space.     

Strong convergence theorem for uniformly L-Lipschitzian asymptotically pseudocontractive mapping in real Banach space

Let E be a real Banach space. Let K be a nonempty closed and convex subset of E, a uniformly L-Lipschitzian asymptotically pseudocontractive mapping with sequence {kn}_n?0⊂[1,+∞), limn→∞kn=1 such that F(T)≠∅. Let {αn}_n?0⊂[0,1] be such that _∑n?0αn=∞, and _∑n?0αn(kn−1)<∞. Suppose {xn}_n?0 is iteratively defined by xn+1=(1−αn)xn+αnTnxn, n?0, and suppose there exists a strictly increasing continuous function , ?(0)=0 such that 〈Tnx−x^∗,j(x−x^∗)〉?kn‖x−x^∗‖2−?(‖x−x^∗‖), ∀x∈K. It is proved that {xn}_n?0 converges strongly to x^∗∈F(T). It is also proved that the sequence of iteration {xn} defined by xn+1=anxn+bnTnxn+cnun, n?0 (where {un}_n?0 is a bounded sequence in K and {an}_n?0, {bn}_n?0, {cn}_n?0 are sequences in [0,1] satisfying appropriate conditions) converges strongly to a fixed point of T.     

Approximating common fixed points of asymptotically nonexpansive mappings by composite algorithm in Banach spaces

Let E be a uniformly convex Banach space and K a nonempty convex closed subset which is also a nonexpansive retract of E. Let T ₁, T ₂ and T ₃: K → E be asymptotically nonexpansive mappings with {k _n }, {l _n } and {j _n }. [1, ∞) such that Σ _n=1 ^∞ (k _n − 1) < ∞, Σ _n=1 ^∞ (l _n − 1) < ∞ and Σ _n=1 ^∞ (j _n − 1) < ∞, respectively and F nonempty, where F = {x ∈ K: T _1x = T _2x = T ₃ x} = x} denotes the common fixed points set of T ₁, _{T 2} and T ₃. Let {α _n }, {α′ _n } and {α″ _n } be real sequences in (0, 1) and ∈ ≤ {α _n }, {α′ _n }, {α″ _n } ≤ 1 − ∈ for all n ∈ N and some ∈ > 0. Starting from arbitrary x ₁ ∈ K define the sequence {x _n } by

New iterations with errors for approximating common fixed points for two generalized asymptotically quasi-nonexpansive nonself-mappings

Let X be a real uniformly convex Banach space and C a nonempty closed convex nonexpansive retract of X with P as a nonexpansive retraction. Let T ₁, T ₂: C → X be two uniformly L-Lipschitzian, generalized asymptotically quasi-nonexpansive non-self-mappings of C satisfying condition A′ with sequences {k _n ⁽ⁱ⁾ } and {δ _n ⁽ⁱ⁾ } ? [1, ∞),, i = 1, 2, respectively such that Σ _n=1 ^∞ (k _n ⁽ⁱ⁾ ? 1) < ∞, Σ _n=1 ⁽ⁱ⁾ δ _n ⁽ⁱ⁾ < ∞, and F = F(T ₁) ∩ F(T ₂) ≠ ?. For an arbitrary x ₁ ∈ C, let {x _n } be the sequence in C defined by $$ \begin{gathered} y_n = P\left( {\left( {1 - \beta _n - \gamma _n } \right)x_n + \beta _n T_2 \left( {PT_2 } \right)^{n - 1} x_n + \gamma _n v_n } \right), \hfill \\ x_{n + 1} = P\left( {\left( {1 - \alpha _n - \lambda _n } \right)y_n + \alpha _n T_1 \left( {PT_1 } \right)^{n - 1} x_n + \lambda _n u_n } \right), n \geqslant 1, \hfill \\ \end{gathered} $$ where {α _n }, {β _n }, {γ _n } and {λ _n } are appropriate real sequences in [0, 1) such that Σ _n=1 ^∞ ] γ _n < ∞, Σ _n=1 ^∞ λ _n < ∞, and {u _n }, }v _n } are bounded sequences in C. Then {x _n } and {y _n } converge strongly to a common fixed point of T ₁ and T ₂ under suitable conditions.     

Approximation Methods for a Common Fixed Point of a Finite Family of Nonexpansive Mappings

Let K be a nonempty closed and convex subset of a real Banach space E. Let T: K → E be a continuous pseudocontractive mapping and f:K → E a contraction, both satisfying weakly inward condition. Then for t ? (0, 1), there exists a sequence {y _t } ? K satisfying the following condition: y _t  = (1 ? t)f(y _t ) + tT(y _t ). Suppose further that {y _t } is bounded or F(T) ≠  and E is a reflexive Banach space having weakly continuous duality mapping J _? for some gauge ?. Then it is proved that {y _t } converges strongly to a fixed point of T, which is also a solution of certain variational inequality. Moreover, an explicit iteration process that converges strongly to a common fixed point of a finite family of nonexpansive mappings and hence to a solution of a certain variational inequality is constructed.     

Mann iterative process for pseudocontractive mappings

Let K be a nonempty, closed and convex subset of a real Banach space E. Let T:K→K be a strictly pseudocontractive map. For a fixed x ₀∈K, define a sequence {x _n } by x _n+1=(1?α _n )x _n +α _n Tx _n , where {α _n } is a real sequence defined in [0,1] satisfying the following conditions (i) $\sum_{n=0}^{\infty }\alpha _{n}=\infty $ , (ii) lim? _n→∞ α _n =0. Then lim?inf? _n→∞‖x _n ?Tx _n ‖=0. If, in addition, T is demicompact, then {x _n } converges strongly to some fixed point of T. Remark 8 is important.     

Multipliers of classes of cauchy transforms

The analytic map g on the unit disk D is said to induce a multiplication operator L from the Banach space X to the Banach space Y if L(f)=f·g∈Y for all f∈X. For z ∈ D and α>0 the families of weighted Cauchy transforms F_α are defined by ?(z) = ∫_T K_x ^α (z)dμ(x) where μ(x) is complex Borel measures, x belongs to the unit circle T and the kernel K_x (z) = (1- xz)^?1. In this article we will explore the relationship between the compactness of the multiplication operator L acting on F ₁ and the complex Borel measures μ(x). We also give an estimate for the essential norm of L     

Concave iteration semigroups of linear continuous set-valued functions

Let {F ^t : t ≥ 0} be a concave iteration semigroup of linear continuous set-valued functions defined on a convex cone K with nonempty interior in a Banach space X with values in cc(K). If we assume that the Hukuhara differences F ⁰(x) ? F ^t (x) exist for x ∈ K and t > 0, then D _t F ^t (x) = (?1)F ^t ((?1)G(x)) for x ∈ K and t ≥ 0, where D _t F ^t (x) denotes the derivative of F ^t (x) with respect to t and $G(x) = \mathop {\lim }\limits_{s \to 0} {{\left( {F^0 \left( x \right) - F^s \left( x \right)} \right)} \mathord{\left/ {\vphantom {{\left( {F^0 \left( x \right) - F^s \left( x \right)} \right)} {\left( { - s} \right)}}} \right. \kern-0em} {\left( { - s} \right)}}$ for x ∈ K.     

On the noise sensitivity of monotone functions

It is known that for all monotone functions f : {0, 1}ⁿ → {0, 1}, if x ∈ {0, 1}ⁿ is chosen uniformly at random and y is obtained from x by flipping each of the bits of x independently with probability ? = n^?α, then P[f(x) ≠ f(y)] < cn^?α+1/2, for some c > 0. Previously, the best construction of monotone functions satisfying P[f_n(x) ≠ f_n(y)] ≥ δ, where 0 < δ < 1/2, required ? ≥ c(δ)n^?α, where α = 1 ? ln 2/ln 3 = 0.36907 …, and c(δ) > 0. We improve this result by achieving for every 0 < δ < 1/2, P[f_n(x) ≠ f_n(y)] ≥ δ, with:

• ? = c(δ)n^?α for any α < 1/2, using the recursive majority function with arity k = k(α);
• ? = c(δ)n^?1/2log^tn for t = log₂ = .3257 …, using an explicit recursive majority function with increasing arities; and
• ? = c(δ)n^?1/2, nonconstructively, following a probabilistic CNF construction due to Talagrand.

We also study the problem of achieving the best dependence on δ in the case that the noise rate ? is at least a small constant; the results we obtain are tight to within logarithmic factors. © 2003 Wiley Periodicals, Inc. Random Struct. Alg., 23: 333–350, 2003     

Approximate fixed point sequences and convergence theorems for asymptotically pseudocontractive mappings

Let K be a nonempty closed convex and bounded subset of a real Banach space E and T:K→K be uniformly L-Lipschitzian, uniformly asymptotically regular with sequence {ε_n}, and asymptotically pseudocontractive with constant {k_n}, where {k_n} and {ε_n} satisfy certain mild conditions. Let a sequence {x_n} be generated from x₁∈K by x_n+1:=(1−λ_n)x_n+λ_nTⁿx_n−λ_nθ_n(x_n−x₁), for all integers n?1, where {λ_n} and {θ_n} are real sequences satisfying appropriate conditions, then ‖x_n−Tx_n‖→0 as n→∞. Moreover, if E is reflexive, and has uniform normal structure with coefficient N(E) and L<N(E)^1/2 and has a uniformly Gâteaux differentiable norm, and T satisfies an additional mild condition, then {x_n} also converges strongly to a fixed point of T.     

非扩张映射和广义变分不等式的粘滞逼近法

应用已提出的非扩张映射的粘滞逼近方法,给定初值x_0∈C,考虑一般迭代过程{x_n},g(x_(n+1))=α_nf(x_n)+(1-α_n)SP_C(g(x_n)-λ_nAx_n),n≥0,其中{α_n}■(0,1),S:C→C是非扩张映射,C是实Hilbert空间H的非空闭凸子集.在{α_n}满足合适的条件下可证明,{x_n}强收敛到非扩张映射的不动点集和广义变分不等式解的公共元,且满足某变分不等式.     

Viscosity Approximation Methods for Multivalued Mappings in Banach Spaces

Let K be a nonempty closed and convex subset of a real reflexive Banach space X that has weakly sequentially continuous duality mapping J. Let T: K → K be a multivalued non-expansive non-self-mapping satisfying the weakly inwardness condition as well as the condition T(y) = {y} for any y ∈ F(T) and such that for a contraction f: K → K and any t ∈ (0, 1), there exists x _t  ∈ K satisfying x _t  ∈ tf(x _t ) + (1 ? t)Tx _t . Then it is proved that {x _t } ? K converges strongly to a fixed point of T, which is also a solution of certain variational inequality. Moreover, the convergence of two explicit methods are also investigated.     

无限簇非扩张非自映象公共不动点的黏性逼近法

设E是具有一致Gateaux可微范数的严格凸的自反的Banach空间,K是E的非空闭凸子集而且是E的sunny非扩张收缩核.设f:K→K是一压缩映象,P:E→K是一sunny非扩张保核收缩,{T_n}_n~∞1:K→E是一可数无限簇非扩张非自映象且■是[0,1]中的非负数列.考虑下列迭代序列■其中W_n是由P,T_n,T_(n-1),…,T_1和λ_n,λ_(n-1),…,λ_1,■n≥1生成的W-映象.该文在较弱条件下用黏性逼近方法证明了迭代序列{x_n}强收敛于p∈F且p是下列变分不等式〈(I-f)p,j(p-x~*)〉≤0,■x~*∈F的唯一解.     

非线性算子具误差的隐迭代程序的强收敛性

设K是一致凸Banach空间中的非空闭凸子集,T_i:K→K(i=1,2,…,N)是有限族完全渐近非扩张映象.对任意的x_0∈K,具误差的隐迭代序列{x_n}为:x_n=α_nx_n-1+β_nT_n~kx_n+γ_nu_n,n≥1,其中{α_n},{β_n},{γ_n}■[0,1]满足α_n+β_n+γ_n=1,{u_n}是K中的有界序列.在一定的条件下,该文建立了隐迭代序列{x_n}的强收敛性.得到隐迭代序列{x_n}强收敛于有限族完全渐近非扩张映象公共不动点的充要条件.所得结果改进和推广了Shahzad与Zegeye,Zhou与Chang,Chang,Tan,Lee与Chan等人的相应结果.     

A strong convergence theorem for asymptotically nonexpansive mappings in Banach spaces

Let C be a closed, convex subset of a uniformly convex Banach space whose norm is uniformly Gâteaux differentiable and let T be an asymptotically nonexpansive mapping from C into itself such that the set F (T) of fixed points of T is nonempty. Let {a_n} be a sequence of real numbers with 0 ￡ a_n ￡ 10 \leq a_n \leq 1, and let x and x₀ be elements of C. In this paper, we study the convergence of the sequence {x_n} defined by¶¶x_n+1=a_n x + (1-a_n) [1/(n+1)] ?_j=0ⁿ T^j x_n   x_{n+1}=a_n x + (1-a_n) {1\over n+1} \sum\limits_{j=0}^n T^j x_n\quad for n=0,1,2,...  . n=0,1,2,\dots \,.     

Approximation Numbers of Operators on Normed Linear Spaces

In [1], B?ttcher et. al. showed that if T is a bounded linear operator on a separable Hilbert space H, {e_j}_j=1^￥H, \{e_{j}\}_{j=1}^{\infty} is an orthonormal basis of H and P_n is the orthogonal projection onto the span of {e_j}_j=1ⁿ\{e_{j}\}_{j=1}^{n}, then for each k ? \mathbbNk \in {\mathbb{N}}, the sequence {s_k(P_nTP_n)}\{s_{k}(P_{n}TP_{n})\} converges to s_k(T), where for a bounded operator A on H, s_k(A) denotes the kth approximation number of A, that is, s_k(A) is the distance from A to the set of all bounded linear operators of rank at most k − 1. In this paper we extend the above result to more general cases. In particular, we prove that if T is a bounded linear operator from a separable normed linear space X to a reflexive Banach space Y and if {P_n} and {Q_n} are sequences of bounded linear operators on X and Y, respectively, such that ||P_n|| ||Q_n|| ￡ 1\|P_n\| \|Q_n\| \leq 1 for all n ? \mathbbNn \in {\mathbb{N}} and {Q_nTP_n} converges to T under the weak operator topology, then {s_k(Q_nTP_n)}\{s_{k}(Q_{n}TP_{n})\} converges to s_k(T). We also obtain a similar result for the case of any normed linear space Y which is the dual of some separable normed linear space. For compact operators, we give this convergence of s_k(Q_nTP_n)s_{k}(Q_{n}TP_{n}) to s_k(T) with separability assumptions on X and the dual of Y. Counter examples are given to show that the results do not hold if additional assumptions on the space Y are removed. Under separability assumptions on X and Y, we also show that if there exist sequences of bounded linear operators {P_n} and {Q_n} on X and Y respectively such that (i) Q_nTP_nQ_{n}TP_{n} is compact, (ii) ||P_n|| ||Q_n|| ￡ 1\|P_{n}\| \|Q_{n}\| \leq 1 and (iii) {Q_nTP_n}\{Q_{n}TP_{n}\} converges to T in the weak operator topology, then {s_k(Q_nTP_n)}\{s_k(Q_{n}TP_{n})\} converges to s_k(T) if and only if s_k(T) = s_k(T￠)s_{k}(T) = s_{k}(T^\prime). This leads to a generalization of a result of Hutton [3], proved for compact operators between normed linear spaces.