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Convergence theorem for an iterative algorithm of λstrict pseudocontraction
Fixed Point Theory and Applications volume 2011, Article number: 95 (2011)
Abstract
In this article, we prove strong convergence of sequence generated by the following iteration sequence for a class of Lipschitzian pseudocontractive mapping T:
whenever {α_{ n }} and {β_{ n }} satisfy the appropriate conditions.
2000 AMS Subject Classification: 47H06; 47J05; 47J25; 47H10; 47H17.
1. Introduction
Let T be a pseudocontractive mapping defined on a real smooth Banach space E. We consider the problem of finding a solution z ∈ E of the fixed point equation x = Tx. One classical way to study pseudocontractive mappings is to use a strong pseudocontraction to approximate a pseudocontractive mapping T. More precisely take t ∈ (0, 1) and u ∈ E define a strong pseudocontraction T_{ t }by T_{ t }x = tu + (1  t)Tx. In [1, Corollary 2],Deimling proves that T_{ t }has a unique fixed point x_{ t }, i.e.,
This implicit iteration was introduced by Browder [2] for a nonexpansive mapping T in Hilbert space. Halpern [3] was the first who introduced the following explicit iteration scheme for a nonexpansive mapping T which was referred to as Halpern iteration: for u, x_{0} ∈ K, α_{ n }∈ [0, 1],
Convergence of this two schemes have been studied by many researchers with various types of additional conditions. For the studies of a nonexpansive mapping T, see Bruck [4, 5], Reich [6, 7], SongXu [8], TakahashiUeda [9], Suzuki [10], and many others. For the studies of a continuous pseudocontractive mapping T, see MoralesJung [11], Schu [12], ChidumeZegeye [13], ChidumeUdomene [14], Udomene [15], ChidumeOfoedu [16], ChenSongZhou [17, 18], Song [19–21], SongChen [22, 23] and others. The following results play a key role in proving strong convergence of Halpern iteration.
Theorem 1.1 [11, 22, 23] Let E be a reflexive Banach space which has both the fixed point property for nonexpansive selfmappings and a uniformly Gâteaux differentiable norm or be a reflexive and strictly convex Banach space with a uniformly Gâteaux differentiable norm. Assume that K is a nonempty, closed and convex subset of E. Suppose that T is a continuous pseudocontractive mapping from K into E with F\left(T\right)\ne \varnothing. Then, as t → 0, x_{ t }, defined by (1.1) converges strongly to a fixed point of T.
Theorem 1.2. [22] Let K be nonempty, closed and convex subset of a Banach space E with a uniformly Gâteaux differentiable norm and let T : K → K be a continuous pseudocontractive mapping with a fixed point. Assume that there exists a bounded sequence {x_{ n }} such that lim_{n→∞}∥x_{ n } Tx_{ n }∥ = 0 and p = lim_{t→0}z_{ t }exists, where {z_{ t }} is defined by (1.1). Then,
Mann [24] introduced the following iteration for T in a Hilbert space:
where {α_{ n }} is a sequence in [0, 1]. Latterly, Reich [25] studied this iteration in a uniformly convex Banach space with a Fré chet differentiable norm, and obtained that if T has a fixed point and {\sum}_{n=0}^{\infty}{\alpha}_{n}\left(1{\alpha}_{n}\right)=\infty, then the sequence {x_{ n }} converges weakly to a fixed point of T. This Mann's iteration process has extensively been studied over the last 20 years for constructions of fixed points of nonlinear mappings and for solving nonlinear operator equations involving monotone, accretive and pseudocontractive operators (see, e.g., [16, 26–34] and others). In an infinitedimensional Hilbert space, the classical Mann's iteration algorithm (1.3) has, in general, only weak convergence, even for nonexpansive mappings. In order to get strong convergence result, one has to modify the Mann's iteration algorithm. Several attempts have been made and many important results have been reported (see, e.g., [12–16, 35–37] and others). Recently, Zhou [37] obtained strong convergence theorem for the following iterative sequence in a 2uniformly smooth Banach space: for u, x_{0} ∈ E and λstrict pseudocontraction T,
where {α_{ n }}, {β_{ n }} and {γ_{ n }} in (0, 1) satisfy:

(i)
a\le {\alpha}_{n}\le \frac{\lambda}{{K}^{2}} for some a > 0 and for all n ≥ 0;

(ii)
\underset{n\to \infty}{lim}{\beta}_{n}=0 and \sum _{n=1}^{\infty}{\beta}_{n}=\infty;

(iii)
\underset{n\to \infty}{lim}{\alpha}_{n+1}{\alpha}_{n}=0;

(iv)
0<\underset{n\to \infty}{liminf}{\gamma}_{n}\le \underset{n\to \infty}{limsup}{\gamma}_{n}<1.
Very recently, Zhang and Su [38] extended Zhou's results to quniformly smooth Banach space. However, the above results excluded γ_{ n }≡ 0 and {\gamma}_{n}\equiv \frac{1}{n+1}.
In this article, we deal with iterative schemes generated by the following iterative sequence (in (1.4), γ_{ n }≡ 0) for λstrict pseudocontraction T:
and obtain its strong convergence whenever {α_{ n }} and {β_{ n }} satisfy the following conditions:

(i)
{\alpha}_{n}\in \left[a,\frac{\lambda}{{K}^{2}}\right] such that \sum _{n=1}^{\infty}{\alpha}_{n+1}{\alpha}_{n}<\infty;

(ii)
\underset{n\to \infty}{lim}{\beta}_{n}=0,\sum _{n=1}^{\infty}{\beta}_{n}=\infty and \sum _{n=1}^{\infty}{\beta}_{n+1}{\beta}_{n}<\infty.
Our result not only complements and develops corresponding ones of Zhou [37, Theorem 2.3] (see also Zhang and Su [38, Theorem 4.1], where γ_{ n }≡ 0), but also extend main result of ChidumeChidume [35] and KimXu [36] from nonexpansive mappings to λstrict pseudocontractions.
2. Preliminaries
Throughout this article, a Banach space E will always be over the real scalar field. We denote its norm by ∥ · ∥ and its dual space by E*. The value of x* ∈ E* at y ∈ E is denoted by 〈y, x〉 and the normalized duality mapping from E into 2^{E}* is denoted by J, that is, J(x) = {f ∈ E* : 〈x, f〉 = ∥x∥∥f∥, ∥x∥ = ∥f∥}. Let F(T) = {x ∈ E : Tx = x} be the set of all fixed point of a mapping T.
Recall that a mapping T with domain D(T) and range R(T) in Banach space E is called strongly pseudocontractive if, for all x, y ∈ D(T), there exist k ∈ (0, 1) and j(x  y) ∈ J(x  y) such that
or, equivalently,
while T is said to be pseudocontractive if (2.1) or (2.2) holds for k = 1. A mapping T is said to be Lipschitzian if, for all x, y ∈ K, there exists L > 0 such that
A mapping T is called nonexpansive if L = 1 and, further, T is said to be contractive if L < 1. An important class of mappings closely related to the class of pseudocontractive mappings is that of accretive mappings. A mapping A is accretive if and only if (I  A) is pseudocontractive. The accretive mappings were independently introduced by Browder [39] and Kato [40] in 1967. The importance of these mappings is well known. A mapping T is called λstrictly pseudocontractive, if for all x, y ∈ D(T), there exists λ ∈ (0, 1) and j(x  y) ∈ J(x  y) such that
It is obvious that λstrictly pseudocontractive mapping is Lipschitzian with L=\frac{\lambda +1}{\lambda}. The class of nonexpansive mappings is a subclass of strictly pseudocontractive mappings in Hilbert space, but the converse implication may be false. We remark that the class of strongly pseudocontractive mappings is independent from the class of λstrict pseudocontractions. This can be seen from the existing examples (see, e.g., [30, 37]).
Let S(E) := {x ∈ E; ∥x∥ = 1} denote the unit sphere of a Banach space E. The space E is said to have (i) a Gâteaux differentiable norm (we also say that E is smooth), if the limit
exists for each x, y ∈ S(E); (ii) a uniformly Gâteaux differentiable norm, if for any y in S(E), the limit (2.4) is uniformly attained for x ∈ S(E); (iii) a Fréchet differentiable norm, if for any x ∈ S(E), the limit (2.4) is attained uniformly for y ∈ S(E); (iv) a uniformly Fréchet differentiable norm (we also say that E is uniformly smooth), if the limit (2.4) is attained uniformly for all (x, y) ∈ S(E) × S(E); (v) fixed point property for nonexpansive selfmappings, if each nonexpansive selfmapping defined on any bounded, closed convex subset K of E has at least one fixed point. Let ρ_{ E }: [0, ∞) → [0, ∞) be the modulus of smoothness of E defined by
Let q > 1. A Banach space E is said to be quniformly smooth, if there exists a fixed constant c > 0 such that ρ_{ E }(t) < ct^{q}. It is well known that E is uniformly smooth if and only if \underset{t\to 0}{lim}\frac{{\rho}_{E}\left(t\right)}{t}=0. If E is quniformly smooth, then E is uniformly smooth, and hence the norm of E is uniformly Fréchet differentiable, in particular, the norm of E is Fréchet differentiable. Typical example of uniformly smooth Banach spaces is L_{ p }(p > 1). More precisely, L_{ p }is min{p, 2}uniformly smooth for every p > 1.
Lemma 2.1.(Zhou [37]) Let E be a real 2uniformly smooth Banach space with the best smooth constant K, C be a nonempty subset of E, and let T : C → C be a λstrict pseudocontraction. For any α ∈ (0, 1), we define T_{ α }= (1  α)x + αTx. Then, as \alpha \in \left(0,\left(\right)close="]">\frac{\lambda}{{K}^{2}}\right.\n is nonexpansive such that F(T_{ α }) = F(T).
Lemma 2.2. (Liu [34] and Xu [41]) Let {a_{ n }} be a sequence of nonnegative real numbers satisfying the property:
where {t_{ n }}, {b_{ n }} and {c_{ n }} satisfy the restrictions:
(i) \sum _{n=0}^{\infty}{t}_{n}=\infty; (ii) \sum _{n=0}^{\infty}{b}_{n}<+\infty; (iii) \underset{n\to \infty}{limsup}{c}_{n}\le 0.
Then, {a_{ n }} converges to zero as n → ∞.
3. Main result
Theorem 3.1 Let E be a real 2uniformly smooth Banach space with the best smooth constant K and let C be a nonempty, closed and convex subset of E. Suppose that T : C → C is a λstrict pseudocontraction with F\left(T\right)\ne \varnothing. Given u, x_{0} ∈ C, a sequence {x_{ n }} is generated by
where {β_{ n }} and {α_{ n }} in (0, 1) satisfy the following control conditions:
(i) {\alpha}_{n}\in \left[a,\frac{\lambda}{{K}^{2}}\right] for some constant a\in \left(0,\frac{\lambda}{{K}^{2}}\right) such that \sum _{n=1}^{\infty}{\alpha}_{n+1}{\alpha}_{n}<\infty;
(ii) \underset{n\to \infty}{lim}{\beta}_{n}=0,\sum _{n=1}^{\infty}{\beta}_{n}=\infty and \sum _{n=1}^{\infty}{\beta}_{n+1}{\beta}_{n}<\infty.
Then, {x_{ n }} converges strongly to a fixed point of T.
Proof. The proof will be divided into four steps.
Step 1. The sequence {x_{ n }} is bounded. Let {T}_{{\alpha}_{n}}={\alpha}_{n}T+\left(1{\alpha}_{n}\right)I. Then, {T}_{{\alpha}_{n}} is nonexpansive for every n by Lemma 2.1 and so, for p ∈ F(T), we have
Consequently, both {x_{ n }} and {y_{ n }} are bounded. This implies the boundedness of {Tx_{ n }} from the inequality \left\rightT{x}_{n}p\left\right\le \frac{1+\lambda}{\lambda}\left\right{x}_{n}p\left\right.
Let M > 0 be a constant such that M ≥ sup_{n∈ℕ}{∥u∥, ∥x_{ n }∥, ∥Tx_{ n }∥}.
Step 2. Since {y}_{n}={T}_{{\alpha}_{n}}{x}_{n}={\alpha}_{n}T{x}_{n}+\left(1{\alpha}_{n}\right){x}_{n}, then
Furthermore, we have
From (3.1), it follows
Substituting (3.2) into (3.3) yields
From the assumptions on {α_{ n }} and {β_{ n }} and using Lemma 2.3, we conclude that
From the definition of x_{ n }and since lim_{n→∞}β_{ n }= 0, it follows
Combining (3.4), we have
Thus, we obtain
Step 3. There exists z ∈ F(T) such that
Since E is 2uniformly smooth, then E is a reflexive Banach space which has both the fixed point property for nonexpansive selfmappings and a uniformly Gâteaux differentiable norm. Then, from Theorem 1.1, as t → 0, x_{ t }, defined by (1.1) converges strongly to a fixed point z of T. The desired conclusion follows from Theorem 1.2.
Step 4. \underset{n\to \infty}{lim}{x}_{n}=z. In fact,
which implies that
and hence lim_{n→∞}∥x_{ n } z∥ = 0 because of Lemma 2.2. This completes the proof.
Remark 1. Theorem 3.1 is applicable to l_{ p }and L_{ p }for all p ≥ 2, however, we do not know whether it works for L_{ p }for 1 < p < 2.
Remark 2. In Theorem 3.1, if the condition {\sum}_{n=1}^{\infty}{\beta}_{n+1}{\beta}_{n}<\infty is replaced by \underset{n\to \infty}{lim}\frac{{\beta}_{n+1}}{{\beta}_{n}}=1, the conclusion still holds.
Remark 3. Theorem 3.1 not only complements and develops corresponding result of Zhou [37, Theorem 3.2] (see also Zhang and Su [38, Theorem 4.1] where γ_{ n }≡ 0), but also extend main result of ChidumeChidume [35] and KimXu [36] from nonexpansive mappings to λstrict pseudocontractions.
Corollary 3.2 Let E be a reflexive Banach space which has both the fixed point property for nonexpansive selfmappings and a uniformly Gâteaux differentiable norm and let C be a nonempty, closed and convex subset of E. Suppose that T : C → C is a nonexpansive mapping with F\left(T\right)\ne \varnothing. Given u, x_{0} ∈ C, a sequence {x_{ n }} is generated by (3.1), where {α_{ n }} and {β_{ n }} in (0,1) satisfy the following control conditions:
(i) α_{ n }∈ (0, 1) such that \sum _{n=1}^{\infty}{\alpha}_{n+1}{\alpha}_{n}<\infty;
(ii) \underset{n\to \infty}{lim}{\beta}_{n}=0,\sum _{n=1}^{\infty}{\beta}_{n}=\infty;
(iii) either \sum _{n=1}^{\infty}{\beta}_{n+1}{\beta}_{n}<\infty or \underset{n\to \infty}{lim}\frac{{\beta}_{n+1}}{{\beta}_{n}}=1.
Then, {x_{ n }} converges strongly to a fixed point of T.
Proof. Let {T}_{{\alpha}_{n}}={\alpha}_{n}T+\left(1{\alpha}_{n}\right)I. Clearly, {T}_{{\alpha}_{n}} is nonexpansive and F\left(T\right)=F\left({T}_{{\alpha}_{n}}\right) for each n. Therefore, following the same proof technique of Theorem 3.1, the desired result is obtained.
Remark 4. Theorem 3.1 of ChidumeChidume [35] and Theorem 1 of KimXu [36] can be regarded as a special case of Corollary 3.2, respectively. In fact, if α_{ n }≡ δ ∈ (0, 1) in Corollary 3.2, then Theorem 3.1 of ChidumeChidume [35] is reached; if in Corollary 3.2, E is a uniformly smooth Banach space and the conditions lim_{n→∞}a_{ n }= 1 and {\sum}_{n=1}^{\infty}\left(1{\alpha}_{n}\right)=\infty are added, then Theorem 1 of KimXu [36] is obtained.
References
Deimling K: Zero of accretive operators. Manuscripta Math 1974, 13: 365–374.
Browder FE: Fixedpoint theorems for noncompact mappings in Hilbert space. Proc Natl Acad Sci USA 1965, 53: 1272–1276.
Halpern B: Fixed points of nonexpansive maps. Bull Amer Math Soc 1967, 73: 957–961.
Bruck RE: A simple proof of the mean ergodic theorem for nonlinear contractions in Banach spaces. Israel J Math 1979, 32: 107–116.
Bruck RE: On the convex approximation property and the asymptotic behavior of nonlinear contractions in Banach spaces. Israel J Math 1981, 38: 304–314.
Reich S: Approximating zeros of accretive operators. Proc Amer Math Soc 1975, 51: 381–384.
Reich S: Strong convergence theorems for resolvents of accretive operators in Banach spaces. J Math Anal Appl 1980, 75: 287–292.
Song Y, Xu S: Strong convergence theorems for nonexpansive semigroup in Banach spaces. J Math Anal Appl 2008, 338: 152–161.
Takahashi W, Ueda Y: On Reich's strong convergence for resolvents of accretive operators. J Math Anal Appl 1984, 104: 546–553.
Suzuki T: Strong convergence theorems for infinite families of nonexpansive mappings in general Banach spaces. Fixed Point Theory Appl 2005,2005(1):103–123.
Morales CH, Jung JS: Convergence of paths for pseudocontractive mappings in Banach spaces. Proc Amer Math Soc 2000, 128: 3411–3419.
Schu J: Approximating fixed points of Lipschitzian pseudocontractive mappings. Houston J Math 1993, 19: 107–115.
Chidume CE, Zegeye H: Approximate fixed point sequences and convergence theorems for Lipschitz pseudocontractive maps. Proc Amer Math Soc 2004, 132: 831–840.
Chidume CE, Udomene A: Strong convergence theorems for uniformly continuous pseudocontractive maps. J Math Anal Appl 2006, 323: 88–99.
Udomene A: Path convergence, approximation of fixed points and variational solutions of Lipschitz pseudocontractions in Banach spaces. Nonlinear Anal 2007, 67: 2403–2414.
Chidume CE, Ofoedu EU: A new iteration process for generalized Lipschitz pseudocontractive and generalized Lipschitz accretive mappings. Nonlinear Anal 2007, 67: 307–315.
Chen R, Song Y, Zhou H: Viscosity approximation methods for continuous pseudocontractive mappings. Acta Math Sin Chin Ser 2006, 49: 1275–1278.
Chen R, Song Y, Zhou H: Convergence theorems for implicit iteration process for a finite family of continuous pseudocontractive mappings. J Math Anal Appl 2006, 314: 701–709.
Song Y: A note on the paper "A new iteration process for generalized Lipschitz pseudocontractive and generalized Lipschitz accretive mappings". Nonlinear Anal 2008, 68: 3047–3049.
Song Y: On a Mann type implicit iteration process for continuous pseudocontractive mappings. Nonlinear Anal 2007, 67: 3058–3063.
Song Y: Strong convergence of viscosity approximation methods with strong pseudocontraction for Lipschitz pseudocontractive mappings. Positivity 2009, 13: 643–655.
Song Y, Chen R: Convergence theorems of iterative algorithms for continuous pseudocontractive mappings. Nonlinear Anal 2007, 67: 486–497.
Song Y, Chen R: An approximation method for continuous pseudocontractive mappings. J Inequal Appl 2006: 1–9. Article ID 28950,
Mann WR: Mean value methods in iteration. Proc Amer Math Soc 1953, 4: 506–510.
Reich S: Weak convergence theorems for nonexpansive mappings in Banach spaces. J Math Anal Appl 1979, 67: 274–276.
Chidume CE, Moore C: Fixed point iteration for pseudocontractive maps. Proc Amer Math Soc 1999,127(4):1163–1170.
Chidume CE: Iterative approximation of Lipschitz strictly pseudocontractive mappings. Proc Amer Math Soc 1987,99(2):283–288.
Chidume CE: Approximation of fixed points of strongly pseudocontractive mappings. Proc Amer Math Soc 1994,120(2):545–551.
Chidume CE: Global iteration schemes for strongly pseudocontractive maps. Proc Amer Math Soc 1998,126(9):2641–2649.
Chidume CE, Mutangadura SA: An example on the Mann iteration method for Lipschitz pseudocontractions. Proc Amer Math Soc 2001,129(8):2359–2363.
Deng L: On Chidume's open problems. J Math Anal Appl 1993,174(2):441–449.
Deng L, Ding XP: Iterative approximation of Lipschitz strictly pseudocontractive mappings in uniformly smooth Banach spaces. Nonlinear Anal 1995,24(7):981–987.
Hicks TL, Kubicek JR: On the Mann iteration process in Hilbert space. J Math Anal Appl 1977, 59: 498–504.
Liu LS: Ishikawa and Mann iteration process with errors for nonlinear strongly accretive mappings in Banach spaces. J Math Anal Appl 1995, 194: 114–125.
Chidume CE, Chidume CO: Iterative approximation of fixed points of nonexpansive mappings. J Math Anal Appl 2006, 318: 288–295.
Kim TH, Xu HK: Strong convergence of modified Mann iterations. Nonlinear Anal 2005, 61: 51–60.
Zhou H: Convergence theorems for λ strict pseudocontractions in 2uniformly smooth Banach spaces. Nonlinear Anal 2008,69(9):3160–3173.
Zhang H, Su Y: Convergence theorems for strict pseudocontractions in quniformly smooth Banach spaces. Nonlinear Anal 2009, 71: 4572–4580.
Browder FE: Nonlinear equations of evolution and nonlinear accretive operators in Banach spaces. Bull Amer Math Soc 1967, 73: 470–475. Part 2 (1976)
Kato T: Nonlinear semigroups and evolution equations. J Math Soc Japan 1967, 19: 508–520.
Xu HK: Iterative algorithms for nonlinear operators. J London Math Soc 2002, 66: 240–256.
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The authors were thank the editor and the anonymous referee for useful comments and valuable suggestions on the language and structure of our manuscript.
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YS carried out the iteration studies, participated in the sequence alignment, and drafted the manuscript. XC participated in the design of the study and performed the algorithmic analysis and revised the manuscript. All authors read and approved the final manuscript.
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Chai, X., Song, Y. Convergence theorem for an iterative algorithm of λstrict pseudocontraction. Fixed Point Theory Appl 2011, 95 (2011). https://doi.org/10.1186/16871812201195
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DOI: https://doi.org/10.1186/16871812201195
Keywords
 λstrict pseudocontraction
 2uniformly smooth Banach space
 modified Mann iteration
 strong convergence