Journal of Computational and Applied Mathematics 206 (2007) 814–825www.elsevier.com/locate/cam

Iterative construction of fixed points of nonself-mappingsin Banach spaces

Lu-Chuan Zenga, Tamaki Tanakab,1, Jen-Chih Yaoc,∗,2

aDepartment of Mathematics, Shanghai Normal University, Shanghai 200234, ChinabDepartment of Mathematical Science, Graduate School of Science and Technology, Niigata University, Niigata 950-2181, Japan

cDepartment of Applied Mathematics, National Sun Yat-sen University, 804 Kaohsiung, Taiwan

Received 7 March 2005; received in revised form 2 May 2006

Abstract

The purpose of this paper is to study the iterative methods for constructing fixed points of nonself-mappings in Banach spaces.The concept of the class of asymptotically QG-weakly contractive nonself-mappings is introduced and a new iterative algorithm forfinding fixed points of this class of mappings is studied. Several strong convergence results on this algorithm are established underdifferent conditions.© 2006 Elsevier B.V. All rights reserved.

MSC: 47H06; 47H09; 7H10; 47H14; 49M25

Keywords: Iterative method; Fixed-point problem; Retraction; Nonself-mapping

1. Introduction and preliminaries

Let B be a real uniformly convex and uniformly smooth Banach space [18] with norm ‖ · ‖ and let B∗ be its dualspace with the dual norm ‖ · ‖∗. By 〈·, ·〉 we denote the generalized duality pairing between B and B∗ (in other words,〈x, �〉 is the value of � at x). The normalized duality mapping from B into 2B∗

is defined by

J (x) = {� ∈ B∗ : 〈x, �〉 = ‖x‖2 = ‖�‖2∗}, x ∈ B. (1.1)

Recall that the space B is called uniformly convex if for any given � > 0, there exists � > 0 such that for all x, y ∈ B

with ‖x‖�1, ‖y‖�1, and ‖x − y‖ = �, the inequality ‖x + y‖�2(1 − �) holds. The function

�B(�) = inf{1 − 2−1‖x + y‖ : ‖x‖ = ‖y‖ = 1, ‖x − y‖ = �} (1.2)

∗ Corresponding author. Tel.: +886 7 5316171, +886 7 5313661; fax: +886 7 5319479.E-mail addresses: zenglc@hotmail.com (L.-C. Zeng), yaojc@math.nsysu.edu.tw (J.-C. Yao).

1 This research was partially supported by the Teaching and ResearchAward Fund for OutstandingYoung Teachers in Higher Education Institutionsof MOE, China and the Dawn Program Foundation in Shanghai.

2 This research was partially supported by a grant from the National Science Council.

0377-0427/$ - see front matter © 2006 Elsevier B.V. All rights reserved.doi:10.1016/j.cam.2006.08.028

L.-C. Zeng et al. / Journal of Computational and Applied Mathematics 206 (2007) 814–825 815

is called the modulus of convexity of the space B. Also, recall that the space B is called uniformly smooth if for anygiven � > 0, there exists � > 0 such that the inequality

2−1(‖x + y‖ + ‖x − y‖) − 1��‖y‖ (1.3)

holds for all x, y ∈ B with ‖x‖ = 1 and ‖y‖��. The function

�B(�) = sup{2−1(‖x + y‖ + ‖x − y‖) − 1 : ‖x‖ = 1, ‖y‖ = �} (1.4)

is called the modulus of smoothness of the space B.It is known that the space B is uniformly convex if and only if �B(�) > 0, ∀� > 0. Moreover, it is also known that the

space B is uniformly smooth if and only if

lim�→0

hB(�) = lim�→0

�B(�)

�= 0. (1.5)

In addition, B is uniformly convex (respectively, uniformly smooth) if and only if B∗ is uniformly smooth (respectively,uniformly convex).

The following estimates, established in [1,2], will be used in the proofs of the convergence theorems below.If ‖x‖�R and ‖y‖�R, then

〈x − y, J (x) − J (y)〉�2LR2�B

(4‖y − x‖

R

), (1.6)

〈x − y, J (x) − J (y)〉�(2L)−1R2�B

(‖y − x‖2R

), (1.7)

where 1 < L < 1.7 is the Figiel constant [12,23]. Furthermore in a uniformly smooth Banach space, we also have forall x, y ∈ B,

‖J (x) − J (y)‖∗ �8RhB(16LR−1‖x − y‖). (1.8)

Now we recall the definitions of nonexpansive mappings, weakly contractive mappings and asymptotically nonex-pansive mappings (see, e.g., [3,5,8,13–17]).

Definition 1.1. A mapping A : G → B is said to be nonexpansive on a nonempty closed convex subset G of a Banachspace B if for all x, y ∈ G,

‖Ax − Ay‖�‖x − y‖. (1.9)

Definition 1.2 (Alber et al. [8]). A mapping A : G → B is said to be weakly contractive of class C�(t) on a nonemptyclosed convex subset G of a Banach space B if there exists a continuous and increasing function �(t) defined on R+such that � is positive on R+\{0}, �(0) = 0, limt→+∞ �(t) = +∞, and for all x, y ∈ G,

‖Ax − Ay‖�‖x − y‖ − �(‖x − y‖). (1.10)

Let R+ denotes the set of all nonnegative real numbers.

Definition 1.3 (Goebel and Kirk [13]). A self-mapping T : G → G is said to be asymptotically nonexpansive ona nonempty closed convex subset G of a Banach space B if there exists a sequence {kn} of positive numbers withlimn→∞ kn = 1 such that

‖T nx − T ny‖�kn‖x − y‖ (1.11)

for all x, y ∈ C and each integer n�1.

In the sequel, we need to use the concept of a nonexpansive retraction [14,9,19].

816 L.-C. Zeng et al. / Journal of Computational and Applied Mathematics 206 (2007) 814–825

Definition 1.4. Let G be a nonempty closed convex subset of B. A mapping QG : B → G is said to be

(i) a retraction onto G if Q2G = QG;

(ii) a nonexpansive retraction if it also satisfies the inequality

‖QGx − QGy‖�‖x − y‖ ∀x, y ∈ B. (1.12)

Remark 1.1. Now we give some examples on nonexpansive retractions QG.

Example 1. If C is a nonempty closed convex subset of a real Hilbert space H, then the metric projection PC of Honto C is a nonexpansive retraction.

Example 2. Let B be a reflexive and strictly convex Banach space and S : D(S) ⊂ B → 2B is m-accretive, i.e.,〈y1 − y2, j (x1 − x2)〉�0 for each j (x1 − x2) ∈ J (x1 − x2), xi ∈ D(S), yi ∈ Sxi , i = 1, 2 and R(I + �S) = B

for all � > 0 where I is the identity operator on B and R(T ) denotes the range of the operator T. Then there existsa nonexpansive retraction Q of B onto co(D(S)), the closed convex hull of the domain of S. In particular, if B is auniformly smooth Banach space of which the dual B∗ has a Fréchet differentiable norm, then such a nonexpansiveretraction Q is defined by Qx = limr→0+ Jr(x) for each x ∈ B where Jr(x) = (I + rS)−1(x); see [20,21]. In thiscase, D(S) is convex; see [22]. We recall that a Banach space B is said to have a Fréchet differentiable norm if for eachx ∈ U , limt→0 (‖x + ty‖ − ‖x‖)/t exists and is attained uniformly in U where U is the unit sphere of B.

Example 3. As pointed out in [10], a nonempty bounded convex subset C of lp (1 < p < ∞), is a nonexpansive retractof the space if and only if C is the intersection of a countable collection half spaces, each of which is the range of anorm 1 linear projection. A characterization for the Lp-spaces is given in [11].

Now, we introduce the concept of asymptotically QG-weakly contractive mappings.

Definition 1.5. Let G be a nonempty closed convex subset of B such that a nonexpansive retraction QG : B → G

exists. A mapping A : G → B is said to be asymptotically QG-weakly contractive of class C�(t) if there exist asequence {kn} ⊂ [1, ∞) ⊂ R+ with limn→∞ kn = 1, and a continuous and increasing function �(t) defined on R+which is positive on R+\{0} with �(0) = 0 and limt→+∞ �(t) = +∞ such that

‖A(QGA)n−1x − A(QGA)n−1y‖�kn‖x − y‖ − �(‖x − y‖) (1.13)

for all x, y ∈ G and each integer n�1.

Remark 1.2. Let G be a nonempty closed convex subset of B such that a nonexpansive retraction QG : B → G exists.Then each weakly contractive mapping A of class C�(t) on G is asymptotically QG-weakly contractive of class C�(t).Indeed, we can see that

‖A(QGA)n−1x − A(QGA)n−1y‖�‖x − y‖ − �(‖x − y‖) (1.14)

for all x, y ∈ G and each integer n�1. First, whenever n = 1, it follows from the definition of weakly contractivemapping that (1.14) holds. Second, suppose that (1.14) holds for n�1. Then we observe that

‖A(QGA)nx − A(QGA)ny‖�‖(QGA)nx − (QGA)ny‖ − �(‖(QGA)nx − (QGA)ny‖)�‖QGA(QGA)n−1x − QGA(QGA)n−1y‖�‖A(QGA)n−1x − A(QGA)n−1y‖.

By induction, we obtain

‖A(QGA)nx − A(QGA)ny‖�‖A(QGA)n−1x − A(QGA)n−1y‖�‖x − y‖ − �(‖x − y‖).

This shows that (1.14) holds for n + 1 and hence (1.14) is true for all positive integers.

L.-C. Zeng et al. / Journal of Computational and Applied Mathematics 206 (2007) 814–825 817

We observe that there exists asymptotically QG-weakly contractive mapping of class C�(t) which is not weaklycontractive of class C�(t). To see this, consider the following example. Let B = R2, G = {(u, 0) : u ∈ R} andQG : R2 → G be defined by QG(u, v) = (u, 0) for each (u, v) ∈ R2. Then QG is a nonexpansive retraction of R2

onto G. Let A : G → R2 be defined by A(u, v)= (v, u), kn = 1 + 1/n for n�1 and �(t)= t . Then A is asymptoticallyQG-weakly contractive of class c�(t). Note that ‖Ax − Ay‖ = ‖x − y‖ for x, y ∈ G. Hence A cannot be weaklycontractive of class C�(t).

Remark 1.3. It is clear that each asymptotically QG-weakly contractive self-mapping A of class C�(t) on G is anasymptotically nonexpansive self-mapping on G.

Recently, Alber et al. [8] studied descent-like approximation methods and proximal methods of the retraction typefor solving fixed-point problems with nonself-mappings in Hilbert and Banach spaces. They proved strong and weakconvergences for weakly contractive and nonexpansive mappings, respectively.

In this paper, the retraction descent-like approximation method is extended to develop a new iterative algorithmfor finding fixed points of asymptotically weakly contractive nonself-mappings. Several strong convergence resultson this algorithm are established under different conditions. Because asymptotically QG-weakly contractive nonself-mappings are a new class of nonlinear operators in Banach spaces even in Hilbert spaces, and because our recursionxn+1 =QG[xn −n(xn −A(QGA)nxn)] (n�0) is very different from the one xn+1 =QG(xn −n(xn −Axn)) (n�0)

in [8], so it is worth emphasizing that our results are new and novel in Banach spaces even in Hilbert spaces.

2. Recursive inequalities

We will often use the following facts concerning numerical recursive inequalities (see [4,6,7]).

Lemma 2.1. Let {�k} be a sequence of nonnegative numbers and {k} a sequence of positive numbers such that

∞∑n=0

n = ∞. (2.1)

Let the recursive inequality

�n+1 ��n − n�(�n), n = 0, 1, 2, . . . , (2.2)

hold where �(�) is a continuous strictly increasing function for all ��0 with �(0) = 0. Then:

(1) �n → 0 as n → ∞;(2) the estimate of convergence rate

�n ��−1

⎛⎝�(�0) −

n−1∑j=0

j

⎞⎠ (2.3)

is satisfied where � is defined by the formula �(t) = ∫dt/�(t) and �−1 is its inverse function.

Lemma 2.2. Let {�k} and {�k} be sequences of nonnegative numbers and {k} a sequence of positive numbers satisfyingconditions (2.1) and

�n

n

→ 0 as n → ∞. (2.4)

Let the recursive inequality

�n+1 ��n − n�(�n) + �n, n = 0, 1, 2, . . . , �0 = �̄, (2.5)

818 L.-C. Zeng et al. / Journal of Computational and Applied Mathematics 206 (2007) 814–825

be given, where �(�) is a continuous strictly increasing function for all ��0 with �(0) = 0. Then:

(1) �n → 0 as n → ∞;(2) there exists a subsequence {�nl

} ⊂ {�n}, l = 1, 2, . . . , such that

�nl��−1

(1∑nl

m=0 m

+ �nl

nl

),

�nl+1 ��−1

(1∑nl

m=0 m

+ �nl

nl

)+ �nl

,

�n ��nl+1 −n−1∑

m=nl+1

m

ϑm

, nl + 1 < n < nl+1, ϑm =m∑

i=0

i ,

�n+1 � �̄ −n∑

m=0

m

ϑm

� �̄, 1�n�n1 − 1,

1�n1 �smax = max

{s :

s∑m=0

m

ϑm

� �̄

}. (2.6)

3. Modified retraction method and its convergence

Let G be a nonempty closed convex subset of a Banach space B and A a weakly contractive mapping from G into B ofthe class C�(t) with a strictly increasing function �(t). Recently, Alber et al. [8] considered and studied the followingretraction descent-like approximation method:

xn+1 = QG(xn − n(xn − Axn)), n = 0, 1, 2, . . . ,

for finding a fixed point of the weakly contractive mapping A, where QG is a nonexpansive retraction of B onto the setG, and {n} is a sequence of positive numbers in [0, 1].

Now the above retraction descent-like approximation method is extended to develop a new iterative algorithm forcomputing a fixed point of an asymptotically QG-weakly contractive mapping in a Banach space B.

Algorithm 3.1 (Modified retraction descent-like approximation method). Let A : G → B be an asymptotically QG-weakly contractive mapping of the class C�(t). Let {n} be a sequence of positive numbers in [0, 1]. The sequence{xn}, starting at any initial point x0 ∈ G, is generated by the following iterative scheme:

xn+1 = QG[xn − n(xn − A(QGA)nxn)], n = 0, 1, 2, . . . . (3.1)

Remark 3.1. If the mapping A in Algorithm 3.1 is additionally a self-mapping of G, then the iterative scheme (3.1)can be rewritten as

xn+1 = (1 − n)xn + nAn+1xn, n = 0, 1, 2, . . . .

In this case, since A is also an asymptotically nonexpansive mapping, the iterative scheme (3.1) is equivalent to themodified Mann iterative process considered and studied by Tan and Xu [24].

Theorem 3.1. Let {n} be a sequence of positive numbers such that∑∞

n=0 n = ∞. Let G be a nonempty boundedclosed convex subset of a uniformly convex and uniformly smooth Banach space B such that a nonexpansive retractionQG : B → G exists. Let A : G → B be an asymptotically QG-weakly contractive mapping of the class C�(t). Suppose

L.-C. Zeng et al. / Journal of Computational and Applied Mathematics 206 (2007) 814–825 819

that the mapping A has a (unique) fixed point x∗ ∈ G. Then:

(i) the iterative sequence {xn} generated by (3.1), starting at any initial point x0 ∈ G, converges in norm to x∗ asn → ∞;

(ii) there exists a subsequence {xnl} ⊂ {xn}, l = 1, 2, . . . , such that

‖xnl− x∗‖��−1

(1∑nl

m=0 m

+ (knl+1 − 1) diam(G)

), (3.2)

‖xnl+1 − x∗‖��−1

(1∑nl

m=0 m

+ (knl+1 − 1) diam(G)

)+ nl

(knl+1 − 1) diam(G), (3.3)

‖xn − x∗‖�‖xnl+1 − x∗‖ −n−1∑

m=nl+1

m

ϑm

, nl + 1 < n < nl+1, ϑm =m∑

i=0

i , (3.4)

‖xn+1 − x∗‖�‖x0 − x∗‖ −n∑

m=0

m

ϑm

�‖x0 − x∗‖, 1�n�n1 − 1, (3.5)

1�n1 �smax = max

{s :

s∑m=0

m

ϑm

�‖x0 − x∗‖}

, (3.6)

where diam(G) is the diameter of the set G.

Proof. Consider the sequence {xn} generated by (3.1). We have for all n�0,

‖xn+1 − x∗‖ = ‖QG[xn − n(xn − A(QGA)nxn)] − QGx∗‖�‖xn − n(xn − A(QGA)nxn) − x∗‖= ‖(1 − n)xn + nA(QGA)nxn − (1 − n)x

∗ − nx∗‖

�(1 − n)‖xn − x∗‖ + n‖A(QGA)nxn − A(QGA)nx∗‖�(1 − n)‖xn − x∗‖ + nkn+1‖xn − x∗‖ − n�(‖xn − x∗‖)= ‖xn − x∗‖ − n�(‖xn − x∗‖) + n(kn+1 − 1)‖xn − x∗‖�‖xn − x∗‖ − n�(‖xn − x∗‖) + n(kn+1 − 1) diam(G). (3.7)

For each n�0, put �n = ‖xn − x∗‖, n = n, �n = n(kn+1 − 1) diam(G) and

�0 = �̄ = ‖x0 − x∗‖.

Then inequality (3.7) can be rewritten as

�n+1 ��n − n�(�n) + �n, n = 0, 1, 2, . . . , �0 = �̄.

It is easy to see that∑∞

n=0 n = ∞ and

limn→∞

�n

n

= limn→∞ (kn+1 − 1) diam(G) = 0.

Therefore, the estimates (3.2)–(3.6) follow immediately from Lemma 2.2. �

Theorem 3.2. Let {n} be a sequence of positive numbers in [0, 1] such that∑∞

n=0 n = ∞. Let G be a nonemptyclosed convex subset of a uniformly convex and uniformly smooth Banach space B such that a nonexpansive retraction

820 L.-C. Zeng et al. / Journal of Computational and Applied Mathematics 206 (2007) 814–825

QG : B → G exists. Let A be a weakly contractive mapping from G into B of the class C�(t). Suppose that the mappingA has a (unique) fixed point x∗ ∈ G. Then:

(i) the iterative sequence {xn} generated by (3.1), starting at any initial point x0 ∈ G, converges in norm to x∗ asn → ∞;

(ii) ‖xn − Axn‖ → 0 as n → ∞;(iii) the following estimate of the convergence rate holds

‖xn − x∗‖��−1

⎛⎝�(‖x0 − x∗‖) −

n−1∑j=0

j

⎞⎠ ,

where �(t) is defined by the formula �(t) = ∫dt/�(t) and �−1 is its inverse function.

Proof. Consider the sequence {xn} generated by (3.1). We have for all n�0,

‖xn+1 − x∗‖ = ‖QG[xn − n(xn − A(QGA)nxn)] − QGx∗‖�‖xn − n(xn − A(QGA)nxn) − x∗‖= ‖(1 − n)xn + nA(QGA)nxn − (1 − n)x

∗ − nx∗‖

�(1 − n)‖xn − x∗‖ + n‖xn − x∗‖ − n�(‖xn − x∗‖)= ‖xn − x∗‖ − n�(‖xn − x∗‖). (3.8)

Hence, the claims (i) and (iii) follow from (3.8) and Lemma 2.1. Observe from Remark 1.3 that

‖xn − Axn‖�‖xn − x∗‖ + ‖A(QGA)1−1xn − A(QGA)1−1x∗‖�2‖xn − x∗‖ → 0 as n → ∞.

This shows that (ii) holds. �

The following theorem provides other estimates of the convergence rate.

Theorem 3.3. Let {n} be a sequence of positive numbers such that∑∞

n=0 n =∞, n �, and limn→∞ n = 0. LetG be a nonempty bounded closed convex subset of a uniformly convex and uniformly smooth Banach space B such thata nonexpansive retraction QG : B → G exists. Let A : G → B be an asymptotically QG-weakly contractive mappingof the class C�(t). Suppose that the mapping A has a (unique) fixed point x∗ ∈ G. Then the iterative sequence {xn}generated by (3.1), starting at any initial point x0 ∈ G, converges in norm to x∗. Moreover, there exist a subsequence{xnl

} ⊂ {xn}, l = 1, 2, . . ., and constants K > 0 and R > 0 such that

‖xnl− x∗‖2 ��−1

1

(1∑nl

j=0 j

+ �nl

nl

), (3.9)

where

�1(t) = √t�(

√t), �n = 2LR2�B(8KR−1n) + n(kn+1 − 1)K2, (3.10)

‖xnl+1 − x∗‖2 ��−11

(1∑nl

j=0 j

+ �nl

nl

)+ �nl

, (3.11)

‖xn − x∗‖2 �‖xnl+1 − x∗‖2 −n−1∑

m=nl+1

m

ϑm

, nl + 1 < n < nl+1, ϑm =m∑

i=0

i , (3.12)

L.-C. Zeng et al. / Journal of Computational and Applied Mathematics 206 (2007) 814–825 821

‖xn+1 − x∗‖2 �‖x0 − x∗‖2 −n∑

m=0

m

ϑm

, 0�n�n1 − 1, (3.13)

0�n1 �smax = max

{s :

s∑m=0

m

ϑm

�‖x0 − x∗‖2

}. (3.14)

Proof. At first, put

K =(

supn�0

‖xn − x∗‖)

·(

supn�0

kn+1

).

Since G is bounded, we have K < ∞. Hence, we deduce that for all n�0, ‖xn − x∗‖�K ,

‖xn − A(QGA)nxn‖�‖xn − x∗‖ + ‖A(QGA)nxn − x∗‖�(1 + kn+1)‖xn − x∗‖�2K ,

and

‖xn‖�‖xn − x∗‖ + ‖x∗‖�K + ‖x∗‖ = C.

Set n=xn−n(xn−A(QGA)nxn). Since QG is a nonexpansive retraction, A is asymptotically QG-weakly contractiveand ‖x‖2 is a convex functional, we have

‖xn+1 − x∗‖2 �‖xn − n(xn − A(QGA)nxn) − x∗‖2 = ‖ n − x∗‖2

�‖xn − x∗‖2 − 2n〈xn − A(QGA)nxn, J ( n − x∗)〉= ‖xn − x∗‖2 − 2n〈xn − A(QGA)nxn, J (xn − x∗)〉

+ 2〈 n − xn, J ( n − x∗) − J (xn − x∗)〉. (3.15)

Since we have the following estimate:

〈A(QGA)nxn − xn, J (xn − x∗)〉 = − 〈xn − x∗, J (xn − x∗)〉 + 〈A(QGA)nxn − x∗, J (xn − x∗)〉� − ‖xn − x∗‖2 + ‖A(QGA)nxn − x∗‖‖xn − x∗‖� − ‖xn − x∗‖2 + [kn+1‖xn − x∗‖ − �(‖xn − x∗‖)]‖xn − x∗‖= (kn+1 − 1)‖xn − x∗‖2 − �(‖xn − x∗‖)‖xn − x∗‖,

from (3.15) we get

‖xn+1 − x∗‖2 �‖xn − x∗‖2 − 2n�(‖xn − x∗‖)‖xn − x∗‖+ 2〈 n − xn, J ( n − x∗) − J (xn − x∗)〉 + 2n(kn+1 − 1)‖xn − x∗‖2. (3.16)

On the other hand, by (1.6), if ‖x‖�R and ‖y‖�R, then

〈x − y, J (x) − J (y)〉�2LR2�B(4R−1‖x − y‖). (3.17)

Now, set R = C + 2K . Observe that for all n�0, ‖xn − x∗‖�K �R, and

‖ n − x∗‖�‖xn − x∗‖ + n‖xn − A(QGA)nxn‖�K + 2K �R

(hence, it is easy to see that ‖xn‖�R and ‖ n‖�R). Thus, it follows from (3.17) that

〈 n − xn, J ( n − x∗) − J (xn − x∗)〉�2LR2�B(8KR−1n). (3.18)

822 L.-C. Zeng et al. / Journal of Computational and Applied Mathematics 206 (2007) 814–825

Substituting (3.18) into (3.16), we derive

‖xn+1 − x∗‖2 �‖xn − x∗‖2 − 2n�(‖xn − x∗‖)‖xn − x∗‖ + 2�n,

where �n = 2LR2�B(8KR−1n) + n(kn+1 − 1)K2. It is obvious that

�n

n

→ 0 as n → ∞,

because B is uniformly smooth and kn → 1 as n → ∞. Thus, we get for �n = ‖xn − x∗‖2 the following recursiveinequality:

�n+1 ��n − 2n�1(�n) + 2�n. (3.19)

The strong convergence of {xn} to x∗ and the estimates (3.9)–(3.14) now follow from Lemma 2.2. �

Remark 3.2. If the mapping A in Theorem 3.3 is additionally a self-mapping of G, then according to [15,Theorem 1], we know that A has a fixed point in G. Then in Theorem 3.3 the assumption that the mapping A hasa (unique) fixed point x∗ ∈ G can be removed.

Theorem 3.4. Let {n} be a sequence of positive numbers such that∑∞

n=0 n = ∞, n �, and limn→∞ n = 0.Let G be a nonempty closed convex subset of a uniformly convex and uniformly smooth Banach space B such that anonexpansive retraction QG : B → G exists. Let A be a weakly contractive mapping from G into B of the class C�(t).Suppose that the mapping A has a (unique) fixed point x∗ ∈ G. Then the iterative sequence generated by (3.1), startingat any initial point x0 ∈ G, converges in norm to x∗. Moreover, there exist a subsequence {xnl

} ⊂ {xn}, l = 1, 2, . . .,and constants K > 0 and R > 0 such that

‖xnl− x∗‖2 ��−1

1

(1∑nl

j=0 j

+ �nl

nl

), (3.20)

where

�1(t) = √t�(

√t), �n = 2LR2�B(8KR−1n), (3.21)

‖xnl+1 − x∗‖2 ��−11

(1∑nl

j=0 j

+ �nl

nl

)+ �nl

, (3.22)

‖xn − x∗‖2 �‖xnl+1 − x∗‖2 −n−1∑

m=nl+1

m

ϑm

, nl + 1 < n < nl+1, ϑm =m∑

i=0

i , (3.23)

‖xn+1 − x∗‖2 �‖x0 − x∗‖2 −n∑

m=0

m

ϑm

, 0�n�n1 − 1, (3.24)

0�n1 �smax = max

{s :

s∑m=0

m

ϑm

�‖x0 − x∗‖2

}. (3.25)

Proof. The proof follows the idea in Alber et al. [8, Theorem 3.2]. By (3.8),

‖xn+1 − x∗‖�‖xn − x∗‖�‖x0 − x∗‖ = K

for all n�0. Thus ‖xn − A(QGA)nxn‖�2K and

‖xn‖�‖xn − x∗‖ + ‖x∗‖�K + ‖x∗‖ = C.

L.-C. Zeng et al. / Journal of Computational and Applied Mathematics 206 (2007) 814–825 823

Set n =xn −n(xn −A(QGA)nxn). Since QG is a nonexpansive retraction and ‖x‖2 is a convex functional, we have

‖xn+1 − x∗‖2 �‖xn − n(xn − A(QGA)nxn) − x∗‖2 = ‖ n − x∗‖2

�‖xn − x∗‖2 − 2n〈xn − A(QGA)nxn, J ( n − x∗)〉= ‖xn − x∗‖2 − 2n〈xn − A(QGA)nxn, J (xn − x∗)〉

+ 2〈 n − xn, J ( n − x∗) − J (xn − x∗)〉. (3.26)

Since we have the following estimate:

〈A(QGA)nxn − xn, J (xn − x∗)〉 = − 〈xn − x∗, J (xn − x∗)〉 + 〈A(QGA)nxn − x∗, J (xn − x∗)〉� − ‖xn − x∗‖2 + ‖A(QGA)nxn − x∗‖‖xn − x∗‖� − ‖xn − x∗‖2 + [‖xn − x∗‖ − �(‖xn − x∗‖)]‖xn − x∗‖= − �(‖xn − x∗‖)‖xn − x∗‖,

from (3.26) we get

‖xn+1 − x∗‖2 �‖xn − x∗‖2 − 2n�(‖xn − x∗‖)‖xn − x∗‖ + 2〈 n − xn, J ( n − x∗) − J (xn − x∗)〉. (3.27)

On the other hand, by (1.6), if ‖x‖�R and ‖y‖�R, then

〈x − y, J (x) − J (y)〉�2LR2�B(4R−1‖x − y‖).Therefore,

‖xn+1 − x∗‖2 �‖xn − x∗‖2 − 2n�(‖xn − x∗‖)‖xn − x∗‖ + 2�n,

where

�n = 〈 n − xn, J ( n − x∗) − J (xn − x∗)〉�2LR2�B(8KR−1n).

Here we used the estimates ‖ n‖�C + 2K = R and ‖xn‖�C�R. It is obvious that

�n

n

→ 0 as n → ∞,

because B is uniformly smooth. Thus, we get for �n = ‖xn − x∗‖2 the following recursive inequality:

�n+1 ��n − 2n�1(�n) + 2�n.

The strong convergence of {xn} to x∗ and the estimates (3.20)–(3.25) now follow from Lemma 2.2. �

Let {n} be a sequence of positive numbers. Let G be a nonempty closed convex subset of B such that a nonexpansiveretraction QG : B → G exists. Let A : G → B be an asymptotically QG-weakly contractive mapping of the classC�(t). Next, we will study the iterative method (3.1) with perturbed mappings An : G → B:

yn+1 = QG(yn − n(yn − Anyn)), n = 0, 1, 2, . . . , (3.28)

provided that the sequence {An} satisfies the following condition:

‖Anv − A(QGA)nv‖�hng(‖v‖) + �n ∀v ∈ G. (3.29)

Remark 3.3. A simple example of the sequence {An} is stated as follows:

An = A(QGA)n, n = 0, 1, 2, . . . .

Theorem 3.5. Let {n} be a sequence of positive numbers in [0, 1] such that∑∞

n=0 n =∞ and �n > 0. Let G bea nonempty closed convex subset of a uniformly convex and uniformly smooth Banach space B such that a nonexpansive

824 L.-C. Zeng et al. / Journal of Computational and Applied Mathematics 206 (2007) 814–825

retraction QG : B → G exists. Let A : G → B be an asymptotically QG-weakly contractive mapping of the classC�(t). Suppose that the mapping A has a unique fixed point x∗ ∈ G, and that there exist sequences of positive numbers{�n} and {hn} converging to 0 as n → ∞, and a positive function g(t) defined on R+ and bounded on bounded subsetsof R+ such that (3.29) is satisfied for all n�0. If the sequence {yn} generated by (3.28) and starting at any initialpoint y0 ∈ G is bounded, then it converges in norm to the point x∗. If, in addition, limn→∞ n = 0, then there exist asubsequence {ynl

} ⊂ {yn}, l = 1, 2, . . . , converging to x∗ as l → ∞ and a constant K > 0 such that

‖ynl− x∗‖2 ��−1

1

(1∑nl

j=1 j

+ hnlg(K) + �nl

+ (knl+1 − 1)K

). (3.30)

Proof. Using the proof similar to that of Theorem 3.1, we get, for all n�0, the inequality

‖yn+1 − x∗‖ = ‖QG(yn − n(yn − Anyn)) − QGx∗‖�‖yn − n(yn − Anyn) − x∗‖= ‖(1 − n)yn + nAnyn − (1 − n)x

∗ − nx∗‖

�(1 − n)‖yn − x∗‖ + n‖Anyn − A(QGA)nx∗‖�(1 − n)‖yn − x∗‖ + n‖A(QGA)nyn − A(QGA)nx∗‖ + n‖Anyn − A(QGA)nyn‖�(1 − n)‖yn − x∗‖ + n[kn+1‖yn − x∗‖ − �(‖yn − x∗‖)] + n(hng(‖yn‖) + �n)

= ‖yn − x∗‖ − n�(‖yn − x∗‖) + n[hng(‖yn‖) + �n + (kn+1 − 1)‖yn − x∗‖]. (3.31)

By our assumptions, there exists K > 0 such that ‖yn − x∗‖�K and ‖yn‖�K. Hence,

‖yn+1 − x∗‖�‖yn − x∗‖ − n�(‖yn − x∗‖) + n[hng(K) + �n + (kn+1 − 1)K]for some constant C1 > 0. Since

n[hng(K) + �n + (kn+1 − 1)K]n

→ 0 as n → ∞, (3.32)

we conclude again by Lemma 2.2 that yn → x∗. The estimate (3.30) is obtained as in the proof of Theorem 3.3. �

Corollary 3.1. Let {n} be a sequence of positive numbers in [0, 1] such that∑∞

n=0 n =∞ and �n > 0. Let G bea nonempty closed convex subset of a uniformly convex and uniform smooth Banach space B such that a nonexpansiveretraction QG : B → G exists. Let A : G → B be a weakly contractive mapping of the class C�(t) and let x∗ ∈ G, beits unique fixed point. Suppose that there exist sequences of positive numbers {�n} and {hn} converging to 0 as n → ∞,and a positive function g(t) defined on R+ and bounded on bounded subsets of R+ such that (3.29) is satisfied for alln�0. If the sequence {yn} generated by (3.28) and starting at any initial point y0 ∈ G is bounded, then it convergesin norm to the point x∗. If, in addition, limn→∞ n = 0, then there exist a subsequence {ynl

} ⊂ {yn}, l = 1, 2, . . . ,

converging to x∗ as l → ∞ and a constant K > 0 such that

‖ynl− x∗‖2 ��−1

1

(1∑nl

j=1 j

+ hnlg(K) + �nl

). (3.33)

Proof. The conclusion follows directly from Theorem 3.5. �

Acknowledgment

The authors express their sincere thanks to the referee for many valuable and helpful suggestions for the improvementof the paper. In particular, Example 3 in Remark 1.1 was provided by the referee.

L.-C. Zeng et al. / Journal of Computational and Applied Mathematics 206 (2007) 814–825 825

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