An implicit iterative algorithm with errors for two families of generalized asymptotically nonexpansive mappings

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R E S E A R C H

Open Access

An implicit iterative algorithm with errors for two

families of generalized asymptotically

nonexpansive mappings

Ravi P Agarwal

1

, Xiaolong Qin

2

and Shin Min Kang

3*

* Correspondence: smkang@gnu. ac.kr

3_{Department of Mathematics and}

RINS, Gyeongsang National University, Jinju 660-701, Korea Full list of author information is available at the end of the article

Abstract

In this paper, an implicit iterative algorithm with errors is considered for two families of generalized asymptotically nonexpansive mappings. Strong and weak

convergence theorems of common fixed points are established based on the implicit iterative algorithm.

Mathematics Subject Classification (2000)47H09 · 47H10 · 47J25

Keywords:Asymptotically nonexpansive mapping, common fixed point, implicit iterative algorithm, generalized asymptotically nonexpansive mapping

1 Introduction

In nonlinear analysis theory, due to applications to complex real-world problems, a growing number of mathematical models are built up by introducing constraints which can be expressed as subproblems of a more general problem. These constraints can be given by fixed-point problems, see, for example, [1-3]. Study of fixed points of non-linear mappings and its approximation algorithms constitutes a topic of intensive research efforts. Many well-known problems arising in various branches of science can be studied by using algorithms which are iterative in their nature. The well-known convex feasibility problem which captures applications in various disciplines such as image restoration, computer tomography, and radiation therapy treatment planning is to find a point in the intersection of common fixed point sets of a family of nonexpan-sive mappings, see, for example, [3-5].

For iterative algorithms, the most oldest and simple one is Picard iterative algorithm. It is known thatTenjoys a unique fixed point, and the sequence generated in Picard iterative algorithm can converge to the unique fixed point. However, for more general nonexpansive mappings, Picard iterative algorithm fails to convergence to fixed points of nonexpansive even that it enjoys a fixed point.

Recently, Mann-type iterative algorithm and Ishikawa-type iterative algorithm (implicit and explicit) have been considered for the approximation of common fixed points of nonlinear mappings by many authors, see, for example, [6-24]. A classical convergence theorem of nonexpansive mappings has been established by Xu and Ori [23]. In 2006, Chang et al. [6] considered an implicit iterative algorithm with errors for asymptotically nonexpansive mappings in a Banach space. Strong and weak convergence theorems are

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established. Recently, Cianciaruso et al. [9] considered an Ishikawa-type iterative algo-rithm for the class of asymptotically nonexpansive mappings. Strong and weak conver-gence theorems are also established. In this paper, based on the class of generalized asymptotically nonexpansive mappings, an Ishikawa-type implicit iterative algorithm with errors for two families of mappings is considered. Strong and weak convergence theorems of common fixed points are established. The results presented in this paper mainly improve the corresponding results announced in Chang et al. [6], Chidume and Shahzad [7], Cianciaruso et al. [9], Guo and Cho [10], Khan et al. [12], Plubtieng et al. [14], Qin et al. [15], Shzhzad and Zegeye [18], Thakur [21], Thianwan and Suantai [22], Xu and Ori [23], Zhou and Chang [24].

2 Preliminaries

Let Cbe a nonempty closed convex subset of a Banach space E. LetT :C® Cbe a mapping. Throughout this paper, we use F(T) to denote the fixed point set of T.

Recall the following definitions.

T is said to benonexpansiveif

Tx−Ty≤x−y, ∀x,y∈C.

T is said to beasymptotically nonexpansiveif there exists a positive sequence {hn}⊂

[1, ∞) with hn®1 as n® ∞such that

Tnx−Tny≤hnx−y, ∀x,y∈C,n≥1.

It is easy to see that every nonexpansive mapping is asymptotically nonexpansive with the asymptotical sequence {1}. The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [25] in 1972. It is known that if Cis a nonempty bounded closed convex subset of a uniformly convex Banach space E, then every asymptotically nonexpansive mapping on Chas a fixed point. Further, the setF(T) of fixed points of T is closed and convex. Since 1972, a host of authors have studied weak and strong convergence problems of implicit iterative processes for such a class of mappings.

T is said to beasymptotically nonexpansive in the intermediate sense if it is continu-ous and the following inequality holds:

lim sup n→∞ x,ysup∈C

(Tnx−Tny − x−y)≤0. ₍₂_:₁₎

Puttingξn= max{0, supx,yÎC(||Tnx- Tny|| - ||x- y||)}, we see thatξn® 0 asn®∞.

Then, (2.1) is reduced to the following:

Tnx−Tny≤x−y+ξn, ∀x,y∈C,n≥1.

The class of asymptotically nonexpansive mappings in the intermediate sense was introduced by Bruck et al. [26] (see also Kirk [27]) as a generalization of the class of asymptotically nonexpansive mappings. It is known that ifCis a nonempty closed con-vex and bounded subset of a real Hilbert space, then every asymptotically nonexpan-sive self mapping in the intermediate sense has a fixed point; see [28] more details.

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lim sup n→∞

sup x,y∈C

(Tnx−Tny −hnx−y)≤0. ₍₂:₂₎

Putting ξn= max{0, supx,yÎC(||Tnx- Tny|| -hn||x- y||)}, we see thatξn®0 asn®

∞. Then, (2.2) is reduced to the following:

Tnx−Tny≤hnx−y+ξn, ∀x,y∈C,n≥1.

We remark that ifhn≡ 1, then the class of generalized asymptotically nonexpansive

mappings is reduced to the class of asymptotically nonexpansive mappings in the intermediate.

In 2006, Chang et al. [6] considered the following implicit iterative algorithms for a finite family of asymptotically nonexpansive mappings {T1, T2,...,TN} with {an} a real

sequence in (0, 1), {un} a bounded sequence inCand an initial pointx0 ÎC:

x1=α1x0+ (1−α1)T1x1+u1,

x2=α2x1+ (1−α2)T2x2+u2,

· · ·

xN=αNxN−1+ (1−αN)TNxN+uN,

xN+1=αN+1xN+ (1−αN+1)Tn1xN+1+uN+1,

· · ·

x2N=α2Nx2N−1+ (1−α2N)TN2x2N+u2N,

x2N+1=α2N+1x2N+ (1−α2N+1)T13x2N+1+u2N+1,

· · ·.

The above table can be rewritten in the following compact form:

xn=αnxn−1+ (1−αn)T_i(n)j(n)xn+un, ∀n≥1,

where for each n≥ 1 fixed,j(n) - 1 denotes the quotient of the division ofn by N

and i(n) the rest, i.e., n= (j(n) - 1)N+i(n).

Based on the implicit iterative algorithm, they obtained, under the assumption thatC+

C⊂C, weak and strong convergence theorems of common fixed points for a finite family of asymptotically nonexpansive mappings {T1,T2,...,TN}; see [6] for more details.

Recently, Cianciaruso et al. [9] considered a Ishikawa-like iterative algorithm for the class of asymptotically nonexpansive mappings in a Banach space. To be more precise, they introduced and studied the following implicit iterative algorithm with errors.

yn= (1−βn−δn)xn+βnTi(n)j(n)xn+δnvn,

xn= (1−αn−γn)xn−1+αnT j(n)

i(n)yn+γnun, ∀n≥1,

(2:3)

where {an}, {bn}, {gn}, and {δn} are real number sequences in [0,1], {un} and {vn} are

bounded sequence in C. Weak and strong convergence theorems are established in a uniformly convex Banach space; see [29] for more details.

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x0∈C,

x1=α1x0+β1T1(α1x1+β1S1x1+γ 1v1) +γ1u1,

x2=α2x1+β2T2(α2x2+β2S2x2+γ 2v2) +γ2u2,

· · ·

xN=αNxN−1+βNTN(αNxN+βNSNxN+γ NvN) +γNuN,

xN+1=αN+1xN+βN+1TN+1(αN+1xN+1+βN+1SN+1xN+1 +γ _N+1vN+1) +γN+1uN+1,

· · ·

x2N=α2Nx2N−1+β2NT2N(α2Nx2N+β2NS2Nx2N+γ 2Nv2N) +γ2Nu2N,

x2N+1=α2N+1x2N+β2N+1T2N+1(α2N+1x2N+1+β2N+1S2N+1x2N+1 +γ _2N+1v2N+1) +γ2N+1u2N+1,

· · ·,

where {an}, {bn}, {gn}, {αn}, {βn}, and {γn} are sequences in [0,1] such that αn+βn+γn=αn+βn+γn= 1for eachn≥1. We have rewritten the above table in the

following compact form:

xn=αnxn−1+βnTi(n)j(n)(αnxn+βnS j(n)

i(n)xn+γnvn) +γnun, n≥1,

where for each n≥ 1 fixed,j(n) - 1 denotes the quotient of the division ofn by N

and i(n) the rest, i.e., n= (j(n) - 1)N+i(n).

Putting yn=αnxn+βnSnxn+γnvn, we have the following composite iterative

algo-rithm:

yn=αnxn+βnS j(n)

i(n)xn+γ nvn,

xn=αnxn−1+βnT_i(n)j(n)yn+γnun, n≥1.

(2:4)

We remark that the implicit iterative algorithm (2.4) is general which includes (2.3) as a special case.

Now, we show that (2.4) can be employed to approximate fixed points of generalized asymptotically nonexpansive mappings which is assumed to be Lipschitz continuous. Let Ti be a Li_t-Lipschitz generalized asymptotically nonexpansive mapping with a

sequence{hi_n} ⊂[1,∞)such thathi_n→1asn®∞andSibe aLi_s-Lipschitz generalized

asymptotically nonexpansive mapping with sequences{kin} ⊂[1,∞)such thatkin→1

as n® ∞for each 1≤i≤N. Define a mappingWn:C®Cby

Wn(x) =αnxn−1+βnTj(n)i(n)(αnx+βnS j(n)

i(n)x+γnvn) +γnun, ∀n≥1.

It follows that

Wn(x)−Wn(y)

≤βnT_i(n)j(n)(αnx+βnS j(n)

i(n)x+γ nvn)−T j(n)

i(n)(αny+βnS j(n)

i(n)y+γ nvn)

≤βnL(αnx−y+βnS j(n) i(n)x−S

j(n) i(n)y)

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where

L= maxL1_t,. . .,LN_t ,L1_s,. . .LN_s . (2:5)

If βnL(αn+βnL)<1for all n≥1, then Wnis a contraction. By Banach contraction

mapping principal, we see that there exists a unique fixed point xnÎCsuch that

xn=Wn(xn) =αnxn−1+βnT_i(n)j(n)(αnx+βnS j(n)

i(n)x+γ nvn) +γnun, ∀n≥1.

That is, the implicit iterative algorithm (2.4) is well defined.

The purpose of this paper is to establish strong and weak convergence theorem of fixed points of generalized asymptotically nonexpansive mappings based on (2.4).

Next, we recall some well-known concepts.

LetE be a real Banach space and UE= {xÎ E: ||x|| = 1}.E is said to be uniformly

convex if for anyεÎ(0, 2] there existsδ> 0 such that for anyx,y ÎUE,

x−y≥ε implies x+y 2

≤1−δ.

It is known that a uniformly convex Banach space is reflexive and strictly convex. Recall that E is said to satisfyOpial’s condition[30] if for each sequence {xn} in E,

the condition that the sequencexn®xweakly implies that

lim inf

n→∞ xn−x<lim infn→∞ xn−y

for all y Î E andy ≠x. It is well known [30] that each lp (1≤ p< ∞) and Hilbert spaces satisfy Opial’s condition. It is also known [29] that any separable Banach space can be equivalently renormed to that it satisfies Opial’s condition.

Recall that a mapping T : C® Cis said to bedemiclosed at the origin if for each sequence {xn} inC, the condition xn®x0 weakly andTxn® 0 strongly impliesTx0 =

0.Tis said to besemicompactif any bounded sequence {xn} inCsatisfying limn®∞||xn

- Txn|| = 0 has a convergent subsequence.

In order to prove our main results, we also need the following lemmas.

Lemma 2.1. [20]Let {an}, {bn}and{cn}be three nonnegative sequences satisfying the

following condition:

an+1≤(1 +bn)an+cn, ∀n≥n0,

where n0 is some nonnegative integer. If∞_n=0cn<∞and∞n=0bn<∞,thenlimn®∞ anexists.

Lemma 2.2. [17]Let E be a uniformly convex Banach space and0 <l≤tn≤h< 1for

all n≥1.Suppose that{xn}and{yn}are sequences of E such that

lim sup

n→∞ xn≤r, lim supn→∞ yn≤r

and

lim

n→∞tnxn+ (1−tn)yn=r

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The following lemma can be obtained from Qin et al. [31] or Sahu et al. [32] immediately.

Lemma 2.3.Let C be a nonempty closed convex subset of a real Hilbert space H. Let T :C®C be a Lipschitz generalized asymptotically nonexpansive mapping. Then I-T is demiclosed at origin.

3 Main results

Now, we are ready to give our main results in this paper.

Theorem 3.1.Let E be a real uniformly convex Banach space and C be a nonempty closed convex subset of E. Let Ti: C®C be a uniformlyLit-Lipschitz and generalized asymptotically nonexpansive mapping with a sequence{hi

n} ⊂[1,∞),wherehin→1as n

® ∞and Si:C®C be a uniformlyLis-Lipschitz and generalized asymptotically nonex-pansive mapping with a sequence{ki

n} ⊂[1,∞),wherekni →1as n® ∞for each1≤i ≤ N. Assume that_F =N_i=1F(Ti) N_i=1F(Si)=∅.Let{un}, {vn}be bounded sequences

in C and en= max{hn,kn},wherehn= sup{hin: 1≤i≤N}andkn= sup{kin: 1≤i≤N}. Let {an}, {bn}, {gn}, {αn}, {βn}and {γn}be sequences in [0,1] such that αn+βn+γn=αn+βn+γn= 1for each n ≥1.Let {xn}be a sequence generated in (2.4).

Put μin= max{0, supx,y∈C(Tinx−Tiny −hinx−y)}and νi

n= max{0, supx,y∈C(Snix−Sniy −kinx−y)}. Let ξn = max{μn, νn}, where

νn= max{νni : 1≤i≤N}and νn= max{νni : 1≤i≤N}. Assume that the following restrictions are satisfied:

(a)∞_n=1γn<∞and∞n=1γ n<∞;

(b)∞_n=1(en−1)<∞and∞n=1ξn<∞;

(c)βnL(αn+βnL)<1,where L is defined in(2.5);

(d)there exist constantsl,hÎ(0, 1)such thatl≤an,αn≤η.

Then

lim

n→∞xn−Trxn= limn→∞xn−Srxn= 0, ∀r∈ {1, 2,. . .,N}.

Proof. Fixing f ∈F, we see that yn−f=αnxn+βnS

j(n)

i(n)xn+γ nvn−f

≤αnxn−f +βnS j(n)

i(n)xn−f +γ nvn−f

≤αnxn−f +βnej(n)xn−f +βnξj(n)+γ nvn−f

≤ej(n)xn−f +βnξj(n)+γ nvn−f

(3:1)

and

xn−f

=αnxn−1+βnTj(n)_i(n)yn+γnun−f

≤αnxn−1−f +βnTj(n)_i(n)yn−f +γnun−f

≤αnxn−1−f +βnej(n)yn−f +βnξj(n)+γnun−f .

≤αnxn−1−f +(1−αn)ej(n)yn−f +βnξj(n)+γnun−f .

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Substituting (3.1) into (3.2), we see that

xn−f

≤αnxn−1−f +(1−αn)ej(n)(ej(n)xn−f +βnξj(n)+ γ nvn−f ) +βnξj(n)+γnun−f .

≤αnxn−1−f +(1−αn)e2j(n)xn−f +(1 +ej(n))ξj(n) +ej(n)γ nvn−f +γnun−f .

Notice that∞_n=1(en−1)<∞. We see from the restrictions (b) and (d) that there

exists a positive integern0such that

(1−αn)e2j(n)≤R<1, ∀n≥n0,

whereR= (1−λ)(1 +₂₋λ_2λ). It follows that

xn−f≤

1 + (1−αn)(e 2 j(n)−1) 1−(1−αn)e2

j(n)

xn−1−f

+(1 +ej(n))ξj(n)+ej(n)γ nvn−f +γnun−f 1−(1−αn)e2

j(n)

≤

1 +(1 +M1)(ej(n)−1) 1−R

xn−1−f

+(1 +M1)ξj(n)+M1M2γ n+M3γn

1−R ,

(3:3)

whereM1= supn≥1{en},M2 = supn≥1{||vn-f||}, andM3= supn≥1{||un-f||}. In view of

Lemma 2.1, we see that limn®∞ ||xn -f|| exists for each f ∈F. This implies that the

sequence {xn} is bounded. Next, we assume that limn®∞ ||xn-f|| =d> 0. From (3.1),

we see that

T_i(n)j(n)yn−f+γn(un−T_i(n)j(n)yn)

≤T_i(n)j(n)yn−f +γnun−T_i(n)j(n)yn

≤ej(n)yn−f +ξj(n)+γnun−Tj(n)_i(n)yn

≤e2_j(n)xn−f +ej(n)ξj(n)+ej(n)γ nvn−f +ξj(n) +γnun−Tj(n)_i(n)yn.

This implies from the restrictions (a) and (b) that

lim sup n→∞

T_i(n)j(n)yn−f+γn(un−T_i(n)j(n)yn)≤d.

Notice that

xn−1−f+γn(un−Ti(n)j(n)yn)≤xn−1−f +γnun−Tj(n)i(n)yn.

This shows from the restriction (a) that

lim sup n→∞

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On the other hand, we have

d= lim

n→∞xn−f

= lim

n→∞αn(xn−1−f+γn(un−T

j(n) i(n)yn))

+ (1−αn)(T_i(n)j(n)yn−f +γn(un−Tj(n)_i(n)yn)).

It follows from Lemma 2.2 that

lim n→∞T

j(n)

i(n)yn−xn−1= 0. (3:4)

Notice that

xn−xn−1≤βnT j(n)

i(n)yn−xn−1+γnun−xn−1.

It follows from (3.4) and the restriction (a) that

lim

n→∞xn−xn−1= 0. (3:5)

This implies that

lim

n→∞xn−xn+l= 0, ∀l= 1, 2,. . .,N. (3:6)

Notice that

xn−f+γn(vn−Sj(n)_i(n)xn)≤xn−f +γnvn−Sj(n)_i(n)xn

and

Sj(n)_i(n)xn−f+γ n(vn−Sj(n)i(n)xn)

≤Sj(n)_i(n)xn−f +γ nvn−Sj(n)i(n)xn

≤ej(n)xn−f +ξj(n)+γ nvn−S j(n) i(n)xn.

which in turn imply that

lim sup n→∞

xn−f+γn(vn−Sj(n)_i(n)xn)≤d

and

lim sup n→∞

Sj(n)_i(n)xn−f +γn(vn−Sj(n)_i(n)xn)≤d.

xn−f =αnxn−1+βnTj(n)_i(n)yn+γnun−f

≤αnxn−1−f +βnT_i(n)j(n)yn−f +γnun−f

≤αnxn−1−T_i(n)k(n)yn+T_i(n)j(n)yn−f +γnun−f

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from which it follows that lim infn®∞ ||yn- f||≥d. In view of (3.1), we see that lim

supn®∞||yn-f||≤d. This proves that

lim

n→∞yn−f =d.

Notice that

lim

n→∞yn−f = limn→∞αn(xn−f +γ n(vn−S

j(n) i(n)xn))

+ (1−αn)(Sj(n)_i(n)xn−f+γ n(vn−Sj(n)_i(n)xn)).

This implies from Lemma 2.2 that

lim n→∞S

j(n)

i(n)xn−xn= 0. (3:7)

T_i(n)j(n)xn−xn

≤T_i(n)j(n)xn−T_i(n)j(n)yn+T_i(n)j(n)yn−xn−1+xn−1−xn

≤Lxn−yn+Tj(n)_i(n)yn−xn−1+xn−1−xn

≤LβnS j(n)

i(n)xn−xn+Lγ nvn−xn+T_i(n)j(n)yn−xn−1 +xn−1−xn.

This combines with (3.4), (3.5), and (3.7) gives that

lim n→∞T

j(n)

i(n)xn−xn= 0. (3:8)

Since n= (j(n) - 1)N+i(n), wherei(n)Î{1, 2,...,N}, we see that

xn−Si(n)xn ≤xn−Sj(n)_i(n)xn+Sj(n)_i(n)xn−Si(n)xn

≤xn−Sj(n)_i(n)xn+LSj(n)_i(n)−1xn−xn

≤xn−Sj(n)_i(n)xn+L(S_i(n)j(n)−1xn−Sj(n)_i(n₋−_N)1xn−N

+Sj(n)_i(n₋−_N)1xn−N−xn−N+xn−N−xn).

(3:9)

Notice that

j(n−N) =j(n)−1 and i(n−N) =i(n).

This in turn implies that

Sj(n)_i(n)−1xn−S j(n)−1

i(n−N)xn−N=S j(n)−1 i(n) xn−S

j(n)−1 i(n) xn−N

≤Lxn−xn−N

(3:10)

and

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Substituting (3.10) and (3.11) into (3.9) yields that

xn−Si(n)xn ≤xn−Sj(n)_i(n)xn+L(Lxn−xn−N

+Sj(n_i(n−₋N)_N)xn−N−xn−N).

It follows from (3.6) and (3.7) that

lim

n→∞xn−Si(n)xn= 0. (3:12)

In particular, we see that ⎧

⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

limj_→∞xjN+1−S1xjN+1= 0, limj_→∞xjN+2−S2xjN+2= 0,

.. .

limj→∞xjN+N−SNxjN+N= 0.

For anyr,s= 1, 2,...,N, we obtain that xjN+s−SrxjN+s

≤xjN+s−xjN+r+xjN+r−SrxjN+r+SrxjN+r−SrxjN+s

≤(1 +L)xjN+s−xjN+r+xjN+r−SrxjN+r.

Lettingj®∞, we arrive at

lim

j→∞xjN+s−SrxjN+s = 0,

which is equivalent to

lim

n→∞xn−Srxn= 0. (3:13)

Notice that

xn−Ti(n)xn ≤xn−T_i(n)j(n)xn+T_i(n)j(n)xn−Ti(n)xn

≤xn−T_i(n)j(n)xn+LTj(n)_i(n)−1xn−xn

≤xn−T_i(n)j(n)xn+L(T_i(n)j(n)−1xn−Tj(n)_i(n₋−_N)1xn−N

+T_i(nj(n)₋−_N)1xn−N−xn−N +xn−N−xn).

(3:14)

T_i(n)j(n)−1xn−T_i(nj(n)₋−_N)1xn−N=T_i(n)j(n)−1xn−T_i(n)j(n)−1xn−N

≤Lxn−xn−N

(3:15)

and

T_i(nj(n)₋−_N)1xn−N−xn−N =T j(n−N)

i(n−N)xn−N−xn−N. (3:16)

Substituting (3.15) and (3.16) into (3.14) yields that

xn−Ti(n)xn ≤xn−T_i(n)k(n)xn+L(Lxn−xn−N

+T_i(nk(n₋−_N)N)xn−N−xn−N).

It follows from (3.6) and (3.8) that

lim

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In particular, we see that ⎧

⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

limk→∞xjN+1−T1xjN+1= 0, limk→∞xjN+2−T2xjN+2= 0,

.. .

limk_→∞xjN+N−TNxjN+N= 0.

For anyr,s= 1, 2,...,N, we obtain that

xjN+s−TrxjN+s

≤xjN+s−xjN+r+xjN+r−TrxjN+r+TrxjN+r−TrxjN+s

≤(1 +L)xjN+s−xjN+r+xjN+r−TrxjN+r.

Lettingj®∞, we arrive

lim

j→∞xjN+s−TrxjN+s= 0,

which is equivalent to

lim

n→∞xn−Trxn= 0. (3:18)

This completes the proof. □

Now, we are in a position to give weak convergence theorems.

Theorem 3.2.Let E be a real Hilbert space and C be a nonempty closed convex subset of E. Let Ti:C®C be a uniformlyLi_t-Lipschitz and generalized asymptotically

nonex-pansive mapping with a sequence{hi

n} ⊂[1,∞),wherehin→1as n®∞and Si:C®C

be a uniformlyLi

s-Lipschitz and generalized asymptotically nonexpansive mapping with a sequence {ki

n} ⊂[1,∞), where kin→1as n ® ∞ for each 1 ≤ i ≤ N. Assume that

F =N_i=1F(Ti) N_i=1F(Si)=∅.Let{un}, {vn}be bounded sequences in C and en= max

{hn,kn},wherehn= sup{hin: 1≤i≤N}andkn= sup{kin: 1≤i≤N}.Let{an}, {bn}, {gn},

{βn},{βn}and{γn}be sequences in[0,1]such thatαn+βn+γn=αn+βn+γn= 1for each n

≥ 1. Let {xn} be a sequence generated in (2.4). Put

νi

n= max{0, supx,y∈C(Snix−Sniy −kinx−y)}and νi

(a)∞_n=1γn<∞and∞n=1γ n<∞;

(b)∞_n=1(en−1)<∞and

_∞

n=1ξn<∞;

(d)there exist constantsl,hÎ(0, 1)such thatl≤an,αn≤η.

Then the sequence {xn}converges weakly to some point inF.

Proof. SinceEis a Hilbert space and {xn} is bounded, we can obtain that there exists

a subsequence{xnp}of {xn} converges weakly tox* Î C. It follows from (3.13) and

(3.18) that

lim

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SinceI -Srand I- Tr are demiclosed at origin by Lemma 2.3, we see that x∗∈F.

Next, we show that the whole sequence {xn} converges weakly tox*. Suppose the

con-trary. Then there exists some subsequence{xnq}of {xn} such that{xnq}converges weakly tox**Î C. In the same way, we can show thatx∗∗∈F. Notice that we have proved that limn®∞||xn - f|| exists for each f ∈F. By virtue of Opial’s condition of E, we

have

lim inf

p→∞ xnp −x∗<lim inf

p→∞ xnp−x∗∗

= lim inf

q→∞ xnq−x

∗∗

<lim inf

q→∞ xnq−x∗.

This is a contradiction. Hence, x* =x**. This completes the proof.□

Remark 3.3. Theorem 3.2 which includes the corresponding results announced in Chang et al. [6], Chidume and Shahzad [7], Guo and Cho [10], Plubtieng et al. [14], Qin et al. [15], Thakur [21], Thianwan and Suantai [22], Xu and Ori [23], and Zhou and Chang [24] as special cases mainly improves the results of Cianciaruso et al. [9] in the following aspects.

(1) Extend the mappings from one family of mappings to two families of mappings; (2) Extend the mappings from the class of asymptotically nonexpansive mappings to the class of generalized asymptotically nonexpansive mappings.

If Sr = Ifor each r Î {1, 2,..., N} andγn= 0, then Theorem 3.2 is reduced to the

following.

Corollary 3.4.Let E be a real Hilbert space and C be a nonempty closed convex sub-set of E. Let Ti:C®C be a uniformlyLit-Lipschitz and generalized asymptotically non-expansive mapping with a sequence{hi

n} ⊂[1,∞),wherehin→1as n®∞for each1≤ i ≤ N. Assume that F=N_i=1F(Ti)=∅. Let {un} be a bounded sequence in C and

hn= sup{hin: 1≤i≤N}. Let{an}, {bn}and{gn}be sequences in[0,1]such that an+bn

+gn= 1for each n ≥1.Let{xn}be a sequence generated in the following process:

x0∈C, xn=αnxn−1+βnT_i(n)j(n)xn+γnun, n≥1. (3:19)

Put μin= max{0, supx,y∈C(Tnix−Tiny −hinx−y)}. Let μn= max{μin: 1≤i≤N}. Assume that the following restrictions are satisfied:

(a)∞_n=1γn<∞;

(b)∞_n=1(hn−1)<∞and∞n=1μn<∞;

(c)bnL< 1, whereL= max{Li_t : 1≤i≤N};

(d)there exist constantsl,hÎ(0, 1)such thatl≥an,αn≤η.

Then the sequence {xn}converges weakly to some point inF.

Next, we are in a position to state strong convergence theorems in a Banach space.

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asymptotically nonexpansive mapping with a sequence{hi

® ∞and Si:C®C be a uniformlyLi_s-Lipschitz and generalized asymptotically

nonex-pansive mapping with a sequence{ki_n} ⊂[1,∞),whereki_n→1as n® ∞for each1≤i ≤ N. Assume thatF =N_i=1F(Ti) N_i=1F(Si)=∅.Let{un}, {vn}be bounded sequences

(a)∞_n=1γn<∞and

_∞

n=1γ n<∞;

(b)∞_n=1(en−1)<∞and∞n=1ξn<∞;

If one of mappings in{T1,T2,...,TN}or one of mappings in{S1,S2,...,SN}are

semicom-pact, then the sequence{xn}converges strongly to some point inF.

Proof. Without loss of generality, we may assume thatS1 are semicompact. It follows from (3.13) that

lim

n→∞xn−S1xn= 0.

By the semicompactness of S1, we see that there exists a subsequence{xnp}of {xn}

such thatxnp →w∈Cstrongly. From (3.13) and (3.18), we have

w−Srw≤w−xnp +xnp−Srxnp +Srxnp −Srw

and

w−Trw≤w−xnp +xnp −Trxnp +Trxnp−Trw

SinceSrandTrare Lipshcitz continuous, we obtain that w∈F. From Theorem 3.1,

we know that limn®∞||xn- f|| exists for each f ∈F. That is, limn®∞||xn- w|| exists.

From xnp →w, we have

lim

n→∞xn−w= 0.

This completes the proof of Theorem 3.5.□

following.

Corollary 3.6. Let E be a real uniformly convex Banach space and C be a nonempty closed convex subset of E. Let Ti: C®C be a uniformlyLit-Lipschitz and generalized asymptotically nonexpansive mapping with a sequence{hi

® ∞ for each 1 ≤ i ≤ N. Assume that _F=N_i=1F(Ti)=∅. Let {un} be a bounded

sequence in C and hn= sup{hin: 1≤i≤N}. Let {an}, {bn} and {gn} be sequences in

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(3.19). Put μin= max{0, supx,y∈C(Tinx−Tniy −hinx−y)}. Let μn= max{μin: 1≤i≤N}. Assume that the following restrictions are satisfied:

(a)∞_n=1γn<∞;

(b)∞_n=1(hn−1)<∞and∞n=1μn<∞;

If one of mappings in {T1,T2,...,TN}is semicompact, then the sequence{xn}converges

strongly to some point inF.

Theorem 3.7.Let E be a real uniformly convex Banach space and C be a nonempty closed convex subset of E. Let Ti: C®C be a uniformlyLit-Lipschitz and generalized asymptotically nonexpansive mapping with a sequence{hi

® ∞and Si:C®C be a uniformlyLi_s-Lipschitz and generalized asymptotically

nonex-pansive mapping with a sequence{ki_n} ⊂[1,∞),whereki_n→1as n® ∞for each1≤i ≤ N. Assume thatF =N_i=1F(Ti) N_i=1F(Si)=∅.Let{un}, {vn}be bounded sequences

(a)∞_n=1γn<∞and

_∞

n=1γ n<∞;

(b)∞_n=1(en−1)<∞and∞n=1ξn<∞;

(d)there exist constantsl,hÎ(0, 1) such thatl≥an,.αn≤η.

If there exists a nondecreasing function g: [0, ∞)®[0,∞)with g(0) = 0and g(m) > 0

for all mÎ(0, ∞)such that

max

1≤r≤N{x−Srx}+ max1≤r≤N{x−Trx} ≥g(dist(x,F)), ∀x∈C,

then the sequence{xn}converges strongly to some point in F.

Proof. In view of (3.13) and (3.18) that g(dist(xn,F))→0, which implies

dist(xn,F)→0. Next, we show that the sequence {xn} is Cauchy. In view of (3.3), we

obtain by putting

an=

(1 +M1)(ej(n)−1)

1−R and bn=

(1 +M1)ξj(n)+M1M2γ n+M3γn 1−R

that

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It follows, for any positive integers m,n, wherem>n>n0, that

xm−p≤Bxn−p+B

∞

i=n+1

bi+bm,

whereB= exp{∞_n=1an}. It follows that

xn−xm ≤xn−f +xm−f

≤(1 +B)xn−f +B

∞

i=n+1

bi+bm.

Taking the infimum over all f ∈F, we arrive at

xn−xm≤(1 +B)dist(xn,F) +B

∞

i=n+1

bi+bm.

In view of∞_n=1bn<∞anddist(xn,F)→0, we see that {xn} is a Cauchy sequence

in Cand so {xn} converges strongly to somex* Î C. SinceTrandSrare Lipschitz for

each r Î {1, 2,..., N}, we see that_F is closed. This in turn implies thatx∗∈F. This completes the proof. □

following.

Corollary 3.8. Let E be a real uniformly convex Banach space and C be a nonempty closed convex subset of E. Let Ti: C®C be a uniformlyLi_t-Lipschitz and generalized

asymptotically nonexpansive mapping with a sequence{hi_n} ⊂[1,∞),wherehi_n→1as n

® ∞ for each 1 ≤ i ≤ N. Assume that F=N_i=1F(Ti)=∅. Let {un} be a bounded

sequence in C and hn= sup{hin: 1≤i≤N}. Let {an}, {bn} and {gn} be sequences in

[0,1] such thatan +bn +gn = 1 for each n ≥ 1.Let {xn}be a sequence generated in

(3.19). Put μin= max{0, supx,y∈C(Tinx−Tniy −hinx−y)}. Let μn= max{μin: 1≤i≤N}. Assume that the following restrictions are satisfied:

(a)∞_n=1γn<∞;

(b)∞_n=1(hn−1)<∞and∞n=1μn<∞;

(d)there exist constantsl,hÎ(0, 1)such that,l≥an,αn≤η.

If there exists a nondecreasing function g: [0, ∞)®[0,∞)with g(0) = 0and g(m) > 0

for all mÎ(0, ∞)such that

max

1≤r≤N{x−Trx} ≥g(dist(x,F)), ∀x∈C,

then the sequence{xn}converges strongly to some point in F.

Acknowledgements

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Author details

1_{Department of Mathematics, Texas A&M University - Kingsville, Kingsville, TX 78363-8202, USA}2_{School of}

Mathematics and Information Sciences, North China University of Water Resources and Electric Power, Zhengzhou 450011, China3_{Department of Mathematics and RINS, Gyeongsang National University, Jinju 660-701, Korea}

Authors’contributions

All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.

Competing interests

The authors declare that they have no competing interests.

Received: 13 April 2011 Accepted: 27 September 2011 Published: 27 September 2011

References

1. Izmaelov, AF, Solodov, MV: An active set Newton method for mathematical program with complementary constraints. SIAM J Optim.19, 1003_–1027 (2008). doi:10.1137/070690882

2. Kaufman, DE, Smith, RL, Wunderlich, KE: User-equilibrium properties of fixed points in dynamic traffic assignment. Transportation Res Part C.6, 1–16 (1998). doi:10.1016/S0968-090X(98)00005-9

3. Kotzer, T, Cohen, N, Shamir, J: Image restoration by an ovel method of parallel projectio onto constraint sets. Optim Lett.20, 1172–1174 (1995). doi:10.1364/OL.20.001172

4. Bauschke, HH, Borwein, JM: On projection algorithms for solving convex feasibility problems. SIAM Rev.38, 367–426 (1996). doi:10.1137/S0036144593251710

5. Youla, DC: Mathematical theory of image restoration by the method of convex projections. In: Stark H (ed.) Image Recovery: Theory and Applications. pp. 29–77. Academic Press, Florida, USA (1987)

6. Chang, SS, Tan, KK, Lee, HWJ, Chan, CK: On the convergence of implicit iteration process with error for a finite family of asymptotically nonexpansive mappings. J Math Anal Appl.313, 273–283 (2006). doi:10.1016/j.jmaa.2005.05.075 7. Chidume, CE, Shahzad, N: Strong convergence of an implicit iteration process for a finite family of nonexpansive

mappings. Nonlinear Anal.62, 1149–1156 (2005). doi:10.1016/j.na.2005.05.002

8. Cho, SY, Kang, SM: Approximation of fixed points of pseudocontraction semigroups based on a viscosity iterative process. Appl Math Lett.24, 224–228 (2011). doi:10.1016/j.aml.2010.09.008

9. Cianciaruso, F, Marino, G, Wang, X: Weak and strong convergence of the Ishikawa iterative process for a finite family of asymptotically nonexpansive mappings. Appl Math Comput.216, 3558_–3567 (2010). doi:10.1016/j.amc.2010.05.001 10. Guo, W, Cho, YJ: On the strong convergence of the implicit iterative processes with errors for a finite family of

asymptotically nonexpansive mappings. Appl Math Lett.21, 1046_–1052 (2008). doi:10.1016/j.aml.2007.07.034 11. Hao, Y, Cho, SY, Qin, X: Some weak convergence theorems for a family of asymptotically nonexpansive nonself

mappings. Fixed Point Theory Appl2010, 11 (2010). Article ID 218573

12. Khan, SH, Yildirim, I, Ozdemir, M: Convergence of an implicit algorithm for two families of nonexpansive mappings. Comput Math Appl.59, 3084–3091 (2010). doi:10.1016/j.camwa.2010.02.029

13. Kim, JK, Nam, YM, Sim, JY: Convergence theorems of implicit iterative sequences for a finite family of asymptotically quasi-nonexpansive type mappings. Nonlinear Anal.71, e2839–e2848 (2009). doi:10.1016/j.na.2009.06.090

14. Plubtieng, S, Ungchittrakool, K, Wangkeeree, R: Implicit iterations of two finite families for nonexpansive mappings in Banach spaces. Numer Funct Anal Optim.28, 737–749 (2007). doi:10.1080/01630560701348525

15. Qin, X, Cho, YJ, Shang, M: Convergence analysis of implicit iterative algorithms for asymptotically nonexpansive mappings. Appl Math Comput.210, 542–550 (2009). doi:10.1016/j.amc.2009.01.018

16. Qin, X, Kang, SM, Agarwal, RP: On the convergence of an implicit iterative process for generalized asymptotically quasi-nonexpansive mappings. Fixed Point Theory Appl2010, 19 (2010). Article ID 714860

17. Schu, J: Weak and strong convergence of fixed points of asymptotically nonexpansive mappings. Bull Austral Math Soc.

43, 153–159 (1991). doi:10.1017/S0004972700028884

18. Shzhzad, N, Zegeye, H: Strong convergence of an implicit iteration process for a finite family of genrealized asymptotically quasi-nonexpansive maps. Appl Math Comput.189, 1058–1065 (2007). doi:10.1016/j.amc.2006.11.152 19. Su, Y, Qin, X: General iteration algorithm and convergence rate optimal model for common fixed points of

nonexpansive mappings. Appl Math Comput.186, 271_–278 (2007). doi:10.1016/j.amc.2006.07.101

20. Tan, KK, Xu, HK: Approximating fixed points of nonexpansive mappings by the Ishikawa iterative process. J Math Anal Appl.178, 301_–308 (1993). doi:10.1006/jmaa.1993.1309

21. Thakur, BS: Weak and strong convergence of composite implicit iteration process. Appl Math Comput.190, 965–997 (2007). doi:10.1016/j.amc.2007.01.101

22. Thianwan, S, Suantai, S: Weak and strong convergence of an implicity iteration process for a finite family of nonexpansive mappings. Sci Math Japon.66, 221–229 (2007)

23. Xu, HK, Ori, RG: An implicit iteration process for nonexpansive mappings. Numer Funct Anal Optim.22, 767–773 (2001). doi:10.1081/NFA-100105317

24. Zhou, YY, Chang, SS: Convergence of implicit iterative process for a finite of asymptotically nonexpansive mappings in Banach spaces. Numer Funct Anal Optim.23, 911–921 (2002). doi:10.1081/NFA-120016276

25. Goebel, K, Kirk, WA: A fixed point theorem for asymptotically nonexpansive mappings. Proc Amer Math Soc.35, 171–174 (1972). doi:10.1090/S0002-9939-1972-0298500-3

26. Bruck, RE, Kuczumow, T, Reich, S: Convergence of iterates of asymptotically nonexpansive mappings in Banach spaces with the uniform opial property. Colloq Math.65, 169–179 (1993)

27. Kirk, WA: Fixed point theorems for non-Lipschitzian mappings of asymptotically nonexpansive type. Israel J Math.17, 339–346 (1974). doi:10.1007/BF02757136

28. Xu, HK: Existence and convergence for fixed points of mappings of asymptotically nonexpansive type. Nonlinear Anal.

(17)

29. van Dulst, D: Equivalent norms and the fixed point property for nonexpansive mapping. J Lond Math Soc.25, 139–144 (1982). doi:10.1112/jlms/s2-25.1.139

30. Opial, Z: Weak convergence of the sequence of successive appproximations for nonexpansive mappings. Bull Amer Math Soc.73, 591_–597 (1967). doi:10.1090/S0002-9904-1967-11761-0

31. Qin, X, Cho, SY, Kim, JK: Convergence theorems on asymptotically pseudocontractive mappings in the intermediate sense. Fixed Point Theory Appl2010, 14 (2010). Article ID 186874

32. Sahu, DR, Xu, HK, Yao, JC: Asymptotically strict pseudocontractive mappings in the intermediate sens. Nonlinear Anal.

70, 3502_–3511 (2009). doi:10.1016/j.na.2008.07.007

doi:10.1186/1687-1812-2011-58

Cite this article as:Agarwalet al.:An implicit iterative algorithm with errors for two families of generalized

asymptotically nonexpansive mappings.Fixed Point Theory and Applications20112011:58.

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