Relaxed Composite Implicit Iteration Process for Common Fixed Points of a Finite Family of Strictly Pseudocontractive Mappings

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Fixed Point Theory and Applications Volume 2009, Article ID 402602,16pages doi:10.1155/2009/402602

Research Article

Relaxed Composite Implicit Iteration Process for

Common Fixed Points of a Finite Family of Strictly

Pseudocontractive Mappings

L. C. Ceng,

1

_{David S. Shyu,}

2

_{and J. C. Yao}

3

1_{Department of Mathematics, Shanghai Normal University, Shanghai 200234, China}

2_{Department of Finance, National Sun Yat-Sen University, Kaohsiung 80424, Taiwan}

3_{Department of Applied Mathematics, National Sun Yat-Sen University, Kaohsiung 804, Taiwan}

Correspondence should be addressed to J. C. Yao,[email protected]

Received 26 November 2008; Accepted 28 May 2009

Recommended by Lai-Jiu Lin

We propose a relaxed composite implicit iteration process for finding approximate common fixed points of a finite family of strictly pseudocontractive mappings in Banach spaces. Several convergence results for this process are established.

Copyrightq2009 L. C. Ceng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction and Preliminaries

LetEbe a real Banach space, and letE∗be its dual space. Denote byJthe normalized duality mapping fromEinto 2E∗defined by

Jx ϕ∈E∗_:_{x, ϕ}_x2_ϕ2_, _∀_x_∈_E, _1.1

where·,·is the generalized duality pairing betweenEandE∗. IfEis smooth, thenJis single valued and continuous from the norm topology ofEto the weak∗topology ofE∗.

A mappingT with domainDTand rangeRTinEis calledλ-strictly pseudocon-tractive in the terminology of Browder and Petryshyn1, if there exists a constantλ >0 such that

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for allx, y ∈ DTand alljx−y ∈ Jx−y. Without loss of generality, we may assume

λ∈0,1. IfIdenotes the identity operator, then1.2can be written in the form

I−Tx−I−Ty, jx−y ≥λI−Tx−I−Ty2 _1.3

for allx, y∈DTand alljx−y∈Jx−y. In1.2and1.3, the positive numberλ >0 is said to be a strictly pseudocontractive constant.

The class of strictly pseudocontractive mappings has been studied by several authors

see, e.g.,1–10. It is shown in4that a strictly pseudocontractive map isL-Lipschitzian

i.e.,Tx−Ty ≤Lx−y,∀x, y∈DTfor someL >0. Indeed, it follows immediately from

1.3that

x−y ≥λI−Tx−I−Ty ≥λTx−Ty − x−y, 1.4

and henceTx−Ty ≤Lx−y,∀x, y ∈DTwhereL 11/λ. It is clear that in Hilbert spaces the important class of nonexpansive mappingsmappingsT for whichTx−Ty ≤

x−y,∀x, y∈DTis a subclass of the class of strictly pseudocontractive maps.

Let K be a nonempty convex subset of E, and let {Ti}N_i₁ be a finite family of nonexpansive self-maps ofK. In11, Xu and Ori introduced the following implicit iteration process; for any initial x0 ∈ K and {αn}∞_n₁ ⊂ 0,1, the sequence {xn}∞_n₁ is generated as follows:

x1α1x0 1−α1T1x1,

x2α2x1 1−α2T2x2,

.. .

xNαNxN−1 1−αNTNxN, xN1αN1xN 1−αN1T1xN1,

.. .

1.5

The scheme is expressed in a compact form as

xnαnxn−1 1−αnTnxn, n≥1, 1.6

whereT_n T_nmodN. Moreover, they proved the following convergence theorem in a Hilbert

space.

Theorem 1.1see11. LetHbe a Hilbert space, and letKbe a nonempty closed convex subset of H. Let{T_i}N_i₁beNnonexpansive self-maps ofKsuch thatCN_i₁FT_i_/∅whereFT_i {x∈ K:Tixx}. Letx0∈K, and let{αn}∞_n₁be a sequence in0,1, such thatlimn→ ∞αn0. Then the

sequence{xn}defined implicitly by1.6converges weakly to a common fixed point of the mappings

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Subsequently, Osilike 12 extended their results from nonexpansive mappings to strictly pseudocontractive mappings and derived the following convergence theorems in Hilbert and Banach spaces.

Theorem 1.2 see12. Let H be a real Hilbert space, and let K be a nonempty closed convex subset ofH. Let{T_i}N_i₁beNstrictly pseudocontractive self-maps ofKsuch thatCN_i₁FT_i_/∅, whereFTi {x ∈ K : Tix x}. Letx0 ∈ K, and let{αn}∞_n₁ be a sequence in0,1such that

limn→ ∞αn0. Then the sequence{xn}∞n1defined by1.6converges weakly to a common fixed point of the mappings{T_i}N_i₁.

Theorem 1.3see12. LetEbe a real Banach space, and letKbe a nonempty closed convex subset ofE. Let{Ti}N_i₁beNstrictly pseudocontractive self-maps ofKsuch thatCN_i₁FTi/∅, where FTi {x∈K:Tixx}, and let{αn}∞n1be a real sequence satisfying the conditions:

i0< αn<1;

ii ∞_n₁1−αn ∞; iii ∞_n₁1−α_n2<∞.

Letx0∈K, and let{xn}∞_n₁be defined by1.6. Then

ilimn→ ∞xn−pexists for allp∈F; iilim inf_n_{→ ∞}x_n−T_nx_n0.

LetK be a nonempty closed convex subset of a real Banach spaceE. Very recently, Su and Li13introduced a new implicit iteration process forNstrictly pseudocontractive self-maps{T_i}N_i₁ofK:

xnαnxn−1 1−αnTnyn, ynβnxn−1

1−βnTnxn, n1,2, . . . , 1.7

that is,

xnαnxn−1 1−αnTn

βnxn−1

1−βnTnxn, n≥1, 1.8

whereTnTnmodNand{αn},{βn} ⊂0,1. First, they established the following convergence

theorem.

Theorem 1.4 13, Theorem 2.1. Let E be a real Banach space, and let K be a nonempty closed convex subset of E. Let {T_i}N_i₁ be N strictly pseudocontractive self-maps of K such that C N

i1FTi/∅, whereFTi {x ∈ K : Tix x}, and let{αn}∞n1,{βn}∞n1 ⊂ 0,1be two real sequences satisfying the conditions:

i ∞_n₁1−αn ∞; ii ∞_n₁1−α_n2<∞; iii ∞_n₁1−βn<∞;

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Forx0∈K, let{xn}∞_n₁be defined by1.8. Then

ilimn→ ∞xn−pexists for allp∈C;

iilim inf_n_{→ ∞}x_n−T_nx_n0.

Second, they derived the following result by usingTheorem 1.4.

Theorem 1.5 13, Theorem 2.2. Let K be a nonempty closed convex subset of a real Banach spaceE, letT be a semicompact strictly pseudocontractive self-map ofK such thatFT/∅, where FT {x∈K:Txx}, and let{α_n} ⊂0,1be a real sequence satisfying the conditions:

i ∞_n₁1−αn ∞; ii ∞_n₁1−αn2<∞.

Then forx0∈K, the sequence{x_n}defined by Mann iterative process,

xnαnxn−1 1−αnTxn−1, n≥1, 1.9

converges strongly to a fixed point ofT.

On the other hand, Zeng and Yao 14 introduced a new implicit iteration scheme with perturbed mapping for approximation of common fixed points of a finite family of nonexpansive self-maps of a real Hilbert space H and established some convergence theorems for this implicit iteration scheme. To be more specific, let{Ti}Ni1be a finite family of

nonexpansive self-maps ofH, and letF:H → Hbe a mapping such that for some constants

κ, η > 0; F is a κ-Lipschitz and η-strongly monotone mapping. Let {αn}∞_n₁ ⊂ 0,1 and

{λn}∞n1 ⊂0,1and take a fixed numberμ∈0,2η/κ2. The authors proposed the following

implicit iteration process with perturbed mappingF.

For an arbitrary initial pointx0∈H, the sequence{xn}∞_n₁is generated as follows:

x1α1x0 1−α1T1x1−λ1μFT1x1,

x2α2x1 1−α2T2x2−λ2μFT2x2,

.. .

xNαNxN−1 1−αN

TNxN−λNμFTNxN, xN1αN1xN 1−αN1T1xN1−λN1μFT1xN1,

.. .

1.10

The scheme is expressed in a compact form as

xnαnxn−1 1−αn

Tnxn−λnμFTnxn, n≥1. 1.11

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Theorem 1.614, Theorem 2.1. LetHbe a real Hilbert space, and letF:H → Hbe a mapping such that for some constantsκ, η >0;F isκ-Lipschitz andη-strongly monotone. Let{T_i}N_i₁beN nonexpansive self-maps ofHsuch thatCN_i₁FTi_/∅. Letμ∈0,2η/κ2_{, let}_x0_∈_H_,_{_λn_}∞

n1⊂ 0,1, and let{αn}∞_n₁ ⊂ 0,1satisfying the conditions: ∞_n₁λn <∞andα≤ αn ≤ β, n ≥1, for someα, β∈0,1. Then the sequence{x_n}∞_n₁defined by1.11converges weakly to a common fixed point of the mappings{T_i}N_i₁.

The aboveTheorem 1.6extendsTheorem 1.1from the implicit iteration process1.6

to the implicit iteration scheme1.11with perturbed mapping.

LetEbe a real Banach space, and letKbe a nonempty convex subset ofE. Recall that a mappingF : K → Kis said to beδ-strongly accretive if there exists a constantδ ∈ 0,1 such that

Fx−Fy, jx−y ≥δx−y2 _1.12

for allx, y∈Kand alljx−y∈Jx−y.

Proposition 1.7. LetXbe a real Banach space, and letF:K → Kbe a mapping:

iifFisλ-strictly pseudocontractive, thenFis a Lipschitz mapping with constantL 1 1/λ.

iiifFis bothλ-strictly pseudocontractive andδ-strongly accretive withλδ≥1, thenI−F is nonexpansive.

Proof. It is easy to see that statement i immediately follows from the definition of strict pseudocontraction. Now utilizing the definitions of strict pseudocontraction and strong accretivity, we obtain

λI−Fx−I−Fy2_≤_x₋_y2₋_Fx₋_{Fy, j}_x₋_y_≤₁₋_δ_x₋_y2_. _1.13

Sinceλδ≥1,

I−Fx−I−Fy ≤

1−δ

λ x−y ≤ x−y, 1.14

and henceI−Fis nonexpansive.

LetEbe a real Banach space, and letK be a nonempty convex subset ofEsuch that

K−K ⊂ K. Let{Ti}Ni1 beNstrictly pseudocontractive self-maps ofK, and letF :K → K

be a perturbed mapping which is bothδ-strongly accretive andλ-strictly pseudocontractive withδλ≥1. In this paper we introduce a general implicit iteration process as follows:

xnαnxn−1 1−αnTnyn−λnFTnyn, ynβnxn−1

1−βnTnxn, n1,2, . . . , 1.15

whereTn TnmodN, and{αn},{βn},{λn} ⊂0,1. In particular, wheneverλn ≡0, it is easy to

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LetL≥1 denote common Lipschitz constant ofNstrictly pseudocontractive self-maps

{Ti}Ni1ofK. SinceKis a nonempty convex subset ofEsuch thatK−K ⊂K, for eachn≥1,

the operator

Snxαnxn−1 1−αnTnβnxn−1

1−β_nT_nx−λ_nFT_nβ_nx_n₋1

1−β_nT_nx

αnxn−1 1−αnI−λnFTn

βnxn−1

1−βnTnx

αnxn−1 1−αn1−λnIλnI−FTn

βnxn−1

1−βnTnx

1.16

mapsKinto itself.

UtilizingProposition 1.7, we have

Snx−Sny, jx−y

1−α_n1−λ_nIλ_nI−FT_nβ_nx_n₋1

1−β_nT_nx

−1−λnIλnI−FTnβnxn−1

1−βnTny, jx−y

≤1−αn1−λnIλnI−FTnβnxn−1

1−βnTnx

−1−λ_nIλ_nI−FT_nβ_nx_n₋1

1−β_nT_nyx−y

≤1−αn1−λnTnβnxn−1

1−βnTnx−Tnβnxn−1

1−βnTny

λnI−FTnβnxn−1

1−βnTnx−I−FTnβnxn−1

1−βnTny

× x−y

≤1−αnTnβnxn−1

1−βnTnx−Tnβnxn−1

1−βnTnyx−y

≤1−αnLβnxn−1

1−βnTnx−βnxn−1

1−βnTnyx−y

1−αn1−βnLTnx−Tnyx−y

≤1−α_n1−β_nL2x−y2

1.17

for allx, y∈K. Thus,Snis strongly pseudocontractive, if1−αn1−βnL2_<_{1 for each}_n_≥_1.

SinceSnis also Lipschitz mapping, it follows from12,15,16thatSnhas a unique fixed point

xn∈K, that is, for eachn≥1

xn αnxn−1 1−αn1−λnIλnI−FTn

βnxn−1

1−βnTnxn. 1.18

Therefore, if1−α_n1−β_nL2_<₁_,_∀_n_≥_{1, then the composite implicit iteration process}_1.15

with perturbed mapping can be employed for the approximation of common fixed points of

Nstrictly pseudocontractive self-maps ofK.

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Lemma 1.8see8. Let{an}∞_n₁,{bn}∞_n₁, and{ n}∞_n₁be sequences of nonnegative real numbers satisfying the inequality

an1≤1 nanbn, n≥1. 1.19

If _∞

n1

n<∞, ∞ n1

bn<∞, 1.20

thenlimn→ ∞anexists.

The following definition can be found, for example, in13.

Definition 1.9. LetD be a closed subset of a real Banach spaceE, and let T : D → D be a mapping.T is said to be semicompact if, for any bounded sequence{xn}inD such that

xn−Txn → 0n → ∞, there must exist a subsequence{xni} ⊂ {xn}such thatxni → x∗∈

D.

2. Main Results

We are now in a position to prove our main results in this paper.

Theorem 2.1. LetEbe a real Banach space, and letKbe a nonempty closed convex subset ofEsuch thatK−K ⊂ K. LetF : K → Kbe a perturbed mapping which is bothδ-strongly accretive and λ-strictly pseudocontractive withδλ≥1. Let{Ti}N_i₁beNstrictly pseudocontractive self-maps of Ksuch thatC N_i₁FTi/∅, whereFTi {x∈ K :Tix x}, and let{αn}∞_n₁,{βn}∞_n₁, and

{λn}∞n1be three real sequences in0,1satisfying the conditions: i ∞_n₁1−αn ∞;

ii ∞_n₁1−αn2<∞; iii ∞_n₁1−βn<∞; iv ∞_n₁λ_n1−α_n<∞;

v 1−α_n1−β_nL2_<₁_,_∀_n_≥₁_{, where}_L_≥₁_{is the common Lipschitz constant of}_{_T i}Ni1. Forx0∈K, let{x_n}∞_n₁be defined by

xnαnxn−1 1−αn

Tnyn−λnFTnyn, ynβnxn−1

1−β_nT_nx_n, n1,2, . . . , 2.1

whereT_nT_nmodN, then

ilim_n_{→ ∞}x_n−pexists for allp∈C; iilim infn→ ∞xn−Tnxn0.

Proof. First, since each strictly pseudocontractive mapping is a Lipschitz mapping, there exists a constantL≥1 such that

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It is now well knownsee, e.g.,15that

xy2_≤_x2₂_{y, j}_x_y _2.3

for allx, y∈Eand alljxy∈Jxy. Takep∈Carbitrarily. Then it follows from2.1

that

xn−pαnxn−1 1−αnTnyn−λnFTnyn−p

αnxn−1−p

1−αnTnyn−λnFTnyn−p

αnxn−1−p

1−αn1−λnIλnI−FTnyn−p

αnxn−1−p

1−α_n1−λ_nT_ny_n−pλ_nI−FT_ny_n−p

αnxn−1−p

1−αn1−λnTnyn−TnpλnI−FTnyn−I−FTnp

λnI−FTnp−p

αnxn−1−p

1−αn1−λnTnyn−TnpλnI−FTnyn−I−FTnp

−1−αnλnFp.

2.4

Utilizing2.3, we obtain

xn−p2_≤_α2

nxn−1−p221−αn1−λn

Tnyn−Tnp, jxn−p

21−αnλnI−FTnyn−I−FTnp, jxn−p

−21−α_nλ_nFp, jx_n−p

≤α2

nxn−1−p221−αn1−λn

×Tnyn−Tnxn, jxn−pTnxn−Tnp, jxn−p

21−αnλnI−FTnyn−I−FTnxn, jxn−p

I−FTnxn−I−FTnp, jxn−p

−21−αnλnFp, jxn−p.

2.5

Since eachTi, i1,2, . . . , N, is strictly pseudocontractive, there existsλ∈0,1such that

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Thus, utilizingProposition 1.7iiwe know from2.5that

xn−p2≤α2nxn−1−p221−αn1−λn

×Lyn−xnxn−pxn−p2₋_λ_xn₋_Tnxn2

21−αnλnLyn−xnxn−pLxn−p2

−21−αnλnFp, jxn−p

≤α2

nxn−1−p221−αn

×Lyn−xnxn−pxn−p2−λxn−Tnxn2

21−αnλnLxn−p221−αnλnFpxn−p

≤α2

21−αnλnLxn−p2 1−αnλnFp2xn−p2

≤α2

3L1−αnλnxn−p2 1−αnλnFp2.

2.7

From2.1, we also have that

yn−xn

≤βnxn−xn−1

1−β_nx_n−T_nx_n

≤βn1−αnTnyn−λnFTnyn−xn−1

1−βnxn−Tnxn

Tnyn−λnFTnyn−xn−1

≤ Tnyn−λnFTnyn−pxn−1−p

1−λnTnyn−pλnI−FTnyn−pxn−1−p

1−λnTnyn−pλnI−FTnyn−I−Fp−λnFpxn−1−p

≤1−λ_nT_ny_n−pλ_nT_ny_n−pλ_nFpx_n₋1−p

Tnyn−pλnFpxn−1−p

≤Lyn−pλnFpxn−1−p

≤Lβn1xn−1−pL2

1−βnxn−pλnFp.

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SinceTiis a Lipschitz mapping with constantL, we have

xn−Tnxn ≤ xn−pTnxn−p ≤L1xn−p. 2.9

Substituting2.8and2.9into2.7, we deduce that

xn−p2≤α2nxn−1−p221−αn2LβnLβn1xn−1−pxn−p

21−α_n2L3_β

n1−βnxn−p2

21−α_n1−β_nLL1x_n−p2

21−αnxn−p2−21−αnλxn−Tnxn2

2L1−αn2λnβnFpxn−p

3L1−αnλnxn−p2 1−αnλnFp2

≤α2

nxn−1−p221−αn2Lβn

Lβn1xn−1−pxn−p

21−α_n2L3β_n1−β_nx_n−p2

21−α_n1−β_nLL1x_n−p2

21−α_nx_n−p2−21−α_nλx_n−T_nx_n2

4L1−αnλnxn−p22L1−αnλnFp2,

2.10

and hence

1−21−αn2L3βn1−βn−21−αn1−βnLL1−4L1−αnλn−21−αn

xn−p2

≤α2

nxn−1−p221−αn2Lβn

Lβn1xn−1−pxn−p

−21−αnλxn−Tnxn22L1−αnλnFp2.

2.11

Setting

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we conclude from2.11that

xn−p2≤ α

2 n

1−21−α_n−b_nxn−1−p

221−αn

2_Lβn_Lβn₁

1−21−α_n−b_n xn−1−pxn−p

− 21−αnλ

1−21−αn−bnxn−Tnxn

2 1−αnλn

1−21−αn−bn2LF

p2_.

2.13

Thus

xn−p2 _≤

1 1−αn

2_bn

1−21−αn−bn

xn−1−p2

21−α_n2Lβ_nLβ_n1

1−21−αn−bn xn−1−pxn−p

−21−α_nλx_n−T_nx_n2 1−αnλn

1−21−αn−bn2LF

p2_.

2.14

Since

1−21−αn−bn1−1−αn221−αnL3βn1−βn21−βnLL1 4Lλn

2.15

and{αn}∞n1,{βn}∞n1,{λn}∞n1⊂0,1, we get

221−αnL3βn1−βn21−βnLL1 4Lλn≤6L2L32LL1. 2.16

SettingM16L2L3₂_LL₁_{, it follows from condition}_ii_{that lim}

n→ ∞1−αn 0 and

so there must exist a natural numberN1such that for alln≥N1,

1

1−21−α_n−b_n <2. 2.17

Therefore, it follows from2.14that

xn−p2_≤₁₂₁₋_αn2_bn_xn

−1−p2

221−α_n2Lβ_nLβ_n1x_n₋1−pxn−p

−21−αnλxn−Tnxn24LFp21−αnλn.

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In order to consider the second term on the right-hand side of2.18, we will prove that{xn} is bounded. Indeed, utilizing2.8and2.9and simplifying these inequalities, we have

xn−p2

xn−p, jxn−p

αnxn−1−p, j

xn−p 1−αnTnyn−λnFTnyn−p, jxn−p

αnxn−1−p, j

xn−p 1−αn1−λn

×Tnyn−Tnxn, jxn−pTnxn−Tnp, jxn−p

1−αnλnI−FTnyn−I−FTnp, jxn−p −1−αnλnFp, jxn−p

≤αnxn−1−pxn−p 1−αn1−λn

Lyn−xnxn−pLxn−p2

1−α_nλ_nLy_n−x_nx_n−pLx_n−p2 1−α_nλ_nFpx_n−p

≤αnxn−1−pxn−p 1−αn

Lyn−xnxn−pLxn−p2

1−αnλnFpxn−p

≤αnL1−αn2βnLβn1xn−1−pxn−p

1−αnLL31−αn2βn1−βnL1−αn1−βnL1xn−p2

Lβn1−αn2λnFpxn−p 1−αnλnFpxn−p

1−αnLL31−αn2βn1−βnL1−αn1−βnL1xn−p2

L11−αnλnFpxn−p,

2.19

and hence

1−1−αnL−L31−αn2βn1−βn−L1−αn1−βnL1xn−p2

L11−αnλnFpxn−p.

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This implies that

xn−p

≤ αnL1−αn

2_β

nLβn1

1−1−αnL−L3₁₋_αn2_βn₁₋_βn₋_L₁₋_αn₁₋_βn_L₁xn−1−p

1−αnλn

1−1−αnL−L3₁₋_αn2_βn₁₋_βn₋_L₁₋_αn₁₋_βn_L₁L1F

p

≤

1L

3₁₋_αn2_βn₁₋_βn_L₁₋_αn₁₋_βn_L₁ _L₁₋_αn2_βn_Lβn₁

1−1−α_nL−L3₁₋_α

n2βn1−βn−L1−αn1−βnL1

xn−1−p

1−αnλn

1−1−α_nL−L3₁₋_α

n2βn1−βn−L1−αn1−βnL1

L1Fp.

2.21

Now, we consider the second term on the right-hand side of2.21. Since{αn},{βn} ⊂0,1, we have

1−αnLL31−αnβn1−βnL1−βnL1≤1−αnLL3LL1. 2.22

Since limn→ ∞1−αn 0, there exists a natural numberN2≥N1such that for alln≥N2,

1−1−αnL−L31−αn2βn1−βn−L1−αn1−βnL1≥ 1

2. 2.23

Again, it follows from condition{αn},{βn} ⊂0,1that

L3₁₋_αn2_βn₁₋_βn_L₁₋_αn₁₋_βn_L₁ _L₁₋_αn2_βn_Lβn₁

≤L3₁₋_α

n2L1−βnL1 L1−αn2L1.

2.24

Therefore, it follows from2.21that

xn−p ≤

12L3₁₋_α

n2L1−βnL1 L1−αn2L1

xn−1−p

21−αnλnL1Fp.

2.25

According to conditionsii–iv, we can readily see that

∞

n1

2L3₁₋_α

n2L1−βnL1 L1−αn2L1

<∞,

∞

n1

21−αnλnL1Fp<∞.

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Thus, in terms of Lemma 1.8 we deduce that limn→ ∞xn − p exists, and hence {xn} is

bounded.

Now, we consider the second term on the right-hand side of 2.18. Since {xn} is bounded, and{βn}∞_n₁⊂0,1, there exists a constantM2>0 and a natural numberN3≥N2 such that for alln≥N3,

221−αn2LβnLβn1xn−1−pxn−p ≤21−αn2M2. 2.27

Thus, it follows from2.18that

xn−p2≤

121−αn2bn

xn−1−p221−αn2M2

−21−αnλxn−Tnxn24LFp21−αnλn.

2.28

Since{xn}is bounded, there exists a constantM3 > 0 such thatxn−p2 _≤ _M3_{. It follows}

from2.28that

2λ

n

jN1

1−α_jx_j−T_jx_j2≤ x_N−p2M3

n

jN1

21−α_j2b_j

2M2

n

jN1

1−α_j24LFp2

n

jN1

1−α_jλ_j,

2.29

and hence

2λ

∞

nN1

1−αnxn−Tnxn2≤ xN−p2M3

∞

nN1

21−αn2bn

2M2

∞

nN1

1−α_n24LFp2

∞

nN1

1−α_nλ_n.

2.30

Utilizing conditionsii–iv, we know from2.30that

2λ

∞

n1

1−αnxn−Tnxn2<∞. 2.31

Since ∞_n₁1−αn ∞, we have

lim inf

n→ ∞ xn−Tnxn0. 2.32

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The iterative scheme1.15becomes the explicit version as follows, wheneverβn≡1:

xnαnxn−1 1−αnTnxn−1−λnFTnxn−1, n≥1. 2.33

In the case whenN1,2.33is the Mann iteration process as follows:

xnαnxn−1 1−αnTxn−1−λnFTxn−1, n≥1. 2.34

The conclusion of Theorem 2.1 remains valid for the iteration processes2.33 and

2.34. Furthermore, we have the following result.

Theorem 2.2. LetEbe a real Banach space, and letKbe a nonempty closed convex subset ofEsuch thatK−K ⊂K. LetF :K → Kbe a perturbed mapping which is bothδ-strongly accretive andλ -strictly pseudocontractive withδλ≥1. LetTbe a semicompact strictly pseudocontractive self-map ofKsuch thatFT/∅, whereFT {x∈K:Txx}, and let{αn}∞n1and{λn}∞n1be two real sequences in0,1satisfying the conditions:

i ∞_n₁1−αn ∞; ii ∞_n₁1−αn2<∞; iii ∞_n₁λn1−αn<∞.

Then Mann iteration process2.34converges strongly to a fixed point ofT.

Proof. Since

lim inf

n→ ∞ xn−Txn0, 2.35

there exists a subsequence{n_k}of{n}such that

lim

k→ ∞xnk−Txnk0. 2.36

By the semicompactness ofT, there must exist a subsequence{xn_ki}of{xn_k}such that

lim

i→ ∞xnki p0. 2.37

It follows from2.36thatp0 Tp0, and hencep0 ∈FT. Since limn→ ∞xn−p0exists, we

have

lim

n→ ∞xn−p0ilim→ ∞xnki −p00. 2.38

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Acknowledgments

The first author was partially supported by the National Science Foundation of China

10771141, Ph.D. Program Foundation of Ministry of Education of China20070270004, and Science and Technology Commission of Shanghai Municipality grant075105118, Leading Academic Discipline Project of Shanghai Normal University DZL707, Shanghai Leading Academic Discipline Project S30405 and Innovation Program of Shanghai Municipal Education Commission09ZZ133. The third author was partially supported by the Grant NSF 97-2115-M-110-001

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