Strong Convergence Theorems for a Common Fixed Point of a Finite Family of Pseudocontractive Mappings

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Volume 2012, Article ID 405315,17pages doi:10.1155/2012/405315

Research Article

Strong Convergence Theorems for

a Common Fixed Point of a Finite Family of

Pseudocontractive Mappings

O. A. Daman and H. Zegeye

Department of Mathematics, University of Botswana, Private Bag 00704, Gaborone, Botswana

Correspondence should be addressed to H. Zegeye,habtuzh@yahoo.com

Received 16 May 2012; Accepted 12 July 2012

Academic Editor: Billy Rhoades

Copyrightq2012 O. A. Daman and H. Zegeye. 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.

It is our purpose, in this paper, to prove strong convergence of Halpern-Ishikawa iteration method to a common fixed point of finite family of Lipschitz pseudocontractive mappings. There is no compactness assumption imposed either on C or on T. The results obtained in this paper improve most of the results that have been proved for this class of nonlinear mappings.

1. Introduction

LetCbe a nonempty subset of a real Hilbert spaceH. The mappingT : C → H is called

Lipschitz if there existsL≥0 such that

_Tx₋_Ty_≤_L_x₋_y_, _∀_{x, y}_∈_C. 1.1

IfL 1, thenT is called nonexpansive, and ifL < 1, thenT is called a contraction. It

follows from1.1that every contraction mapping is nonexpansive and every nonexpansive

mapping is Lipschitz.

A mappingT :C → His calledα-strictly pseudocontractive1if for allx, y ∈Cthere

existsα∈0,1such that

x−y, Tx−Ty≤x−y2₋_α_I₋_T_x₋_I₋_T_y2_. _1.2

A mappingTis called pseudocontractive if

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We note that1.2and1.3can be equivalently written as

_Tx₋_Ty2_≤_x₋_y2_α_I₋_T_x₋_I₋_T_y2_, _∀_{x, y}_∈_C. _1.4

_Tx₋_Ty2 _≤_x₋_y2_I₋_T_x₋_I₋_T_y2_, _∀_{x, y}_∈_C, _1.5

respectively.

We observe from1.4and 1.5that every nonexpansive mapping is α-strict

pseu-docontractive mapping and everyα-strict pseudocontractive mapping is pseudocontractive

mapping, and hence class of pseudocontractive mappings is a more general class of mappings. Furthermore, pseudocontractive mappings are related with the important class

of nonlinear monotone mappings, where a mappingAwith domainDAand rangeRAin

His called monotone if the inequality

x−y, Ax−Ay≥0, 1.6

holds for everyx, y ∈ DA. We note thatT is pseudocontractive if and only ifA : I−T

is monotone, and hence a fixed point of T, FT : {x ∈ DT : Tx x}is a zero of A,

NA : {x∈ DA :Ax 0}. It is now well knownsee, e.g.,2that ifAis monotone,

then the solutions of the equation Ax 0 correspond to the equilibrium points of some

evolution systems. Consequently, many researchers have made eﬀorts to obtain iterative

methods for approximating fixed points ofT, whenT is pseudocontractivesee, e.g.,3–10

and the references contained therein.

LetCbe a closed subset of a Hilbert spaceH, and letT :C → Cbe a contraction. Then

the Picard iteration method given by

x0∈C, xn1Txn, n≥1, 1.7

converges to the unique fixed point of T. However, this Picard iteration method may not

always converge to a fixed point ofT, whenT is nonexpansive mapping. We can take, for

example,T to be the anticlockwise rotation of the unit disk inR2_{with the Euclidean norm}

about the origin of coordinate of an angle, say,θ.

The scheme that has been used to approximate fixed points of nonexpansive mappings

is the Mann iteration method5given by

x0∈C, xn1 1−αnxnαnTxn, n≥0, 1.8

where{αn}is a real sequence in the interval0,1satisfying certain conditions. But it is worth

mentioning that the Mann iteration process does not always converge strongly to a fixed

point of nonexpansive mappingT. One has to impose compactness assumption onCe.g.,

Cis compactor on T e.g.,T is semicompactto get strong convergence of Mann iteration

method to a fixed point of nonexpansive self-mapT see, e.g.,11,12.

We also note that eﬀorts to approximate a fixed point of a Lipschitz pseudocontractive

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of a compact convex subset of a Hilbert space with a unique fixed point for which no Mann

sequence converges by Chidume and Mutangadura13. This leads now to our next concern.

Can we construct an iterative sequence for approximating fixed point of the Lipschitz pseudocontractive mappings?

In 1974, Ishikawa14introduced an iteration process which converges to a fixed point

of Lipschitz pseudocontractive self-mapT ofC, whenCis compact. In fact, he proved the

following theorem.

Theorem I. If C is a compact convex subset of a Hilbert space H, T : C → C is a Lipschitz

pseudocontractive mapping andx0is any point ofC, then the sequence{x_n}_n_≥₀converges strongly to a fixed point ofT, where{xn}is defined iteratively for each integern≥0 by

xn1 1−αnxnαnTyn, yn 1−βnxnβnTxn, 1.9

here{α_n},{β_n}are sequences of positive numbers satisfying the conditions

i0≤αn ≤βn<1, ii_nlim_{→ ∞}βn0, iii

n≥0

αnβn∞. _1.10

We observe that Theorem I imposes compactness assumption onC, and it is still an

open problem whether or not scheme1.9, known as the Ishikawa iterative method, can be used

to approximate fixed points of Lipschitz pseudocontractive mappings without compactness

assumption onCor onT.

In order to obtain a strong convergence theorem for pseudocontractive mappings

without the compactness assumption, Zhou15established the hybrid Ishikawa algorithm

for Lipschitz pseudocontractive mappings as follows:

yn 1−αnxnαnTxn,

zn1−βnxnβnTyn,

Cn

z∈C: zn−z 2≤ xn−z 2

−αnβn 1−2αn−L2αn2

xn−Txn 2

,

Qn{z∈C:xn−z, x0−xn ≥0},

xn1PCn∩Qnx0, n≥1.

1.11

He proved that the sequence {xn} defined by1.11 converges strongly to PFTx0,

wherePCis the metric projection fromHintoC.

Recently, several authorssee, e.g., 16–18 also used the hybrid Mann and hybrid

Ishikawa algorithm methods to obtain strong convergence to a fixed point of Lipschitz pseudocontractive mappings. But it is worth mentioning that the hybrid schemes are not easy

to compute. They involve computation ofC_nandQ_nfor eachn≥1.

Another iteration scheme was introduced and studied by Chidume and Zegeye19

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LetK be a convex nonempty subset of real Banach spaceE, and letT :K → Kbe a

mapping. From arbitraryx1 ∈K, define{x_n}_n_≥₁by

xn1 1−λnxnλnTxn−λnθnxn−x1, n∈N, 1.12

where {λn}n≥1 and {θn}n≥1 are real sequences in0,1 satisfying the following conditions:

ilim_n_{→ ∞}θ_n 0; iiλ_n oθ_n; iii ∞_n₁λ_nθ_n ∞; iv lim_n_{→ ∞}θ_n₋1/θ_n −

1/λnθn 0, λn1 θn < 1. Examples of real sequences which satisfy these conditions

areλ_n 1/n1a andθ_n 1 /n1b, where 0< b < aandab < 1. They proved the

following theorem.

Theorem CZ. LetCbe a nonempty closed convex subset of a reflexive real Banach spaceEwith a

uniformly Gˆateaux diﬀerentiable norm. LetT : C → Cbe a Lipschitz pseudocontractive mapping with Lipschitz constant L > 0 and FT/∅. Suppose every closed convex and bounded subset of

Khas the fixed point property for nonexpansive self-mappings. Let a sequence{x_n}_n_≥₁be generated iteratively by1.12. Then{x_n}_n_≥₁converges strongly to a fixed point ofT.

Theorem CZ solves the open problem of approximating fixed point of Lipschitz pseudocontractive mappings that has been in the air for many years. However, it is still an open problem whether or not this scheme can be used to approximate a common fixed point of a family of Lipschitz pseudocontractive mappings. Moreover, we observe that the

conditions on the real sequences{θn}and{λn}excluded the natural choice,θn 1/n1

andλ_n1/n1.

Our concern now is the following: can we construct an iterative sequence for a common

fixed point of a family of Lipschitz pseudocontractive mappings?

For a sequence{α_n}of real numbers in0,1and an arbitraryu∈C, let the sequence

{xn}inCbe iteratively defined byx0∈C:

xn1αnu 1−αnTxn, n≥1. 1.13

The recursion formula 1.13 known as Halpern scheme was first introduced in 1967 by

Halpern 20 in the framework of Hilbert spaces. He proved that {xn} convergs strongly

to a fixed point of nonexpansive self-mappingTofC.

Recently, considerable research eﬀorts have been devoted to developing iterative

methods for approximating a common fixed point of a family of several nonlinear mappings

see, e.g.,4,21,22. In 1996, Bauschke3introduced the following Halpern-type iterative

process for approximating a common fixed point for a finite family ofNnonexpansive

self-mappings. In fact, he proved the following theorem.

Theorem B. Let C be a nonempty closed convex subset of a Hilbert spaceH, and letT1, T2, . . . , T_N

be a finite family of nonexpansive mappings of C into itself with F : FT1T_N· · ·T2 · · ·

FTN−2· · ·T1TN/∅. Let{αn} be a real sequence in0,1which satisfies certain mild conditions.

Given points,x0∈C, let{x_n}be generated by

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whereTn TnmodN. Then{xn}converges strongly toPFu, wherePFu : H → F is the metric projection.

But it is worth mentioning that it is still an open problem whether or not this scheme can be used to approximate a common fixed points of Lipschitz pseudocontractive mappings?

In 2008, Zhou22studied weak convergence of an implicit scheme to a common fixed

point of finite family of pseudocontractive mappings. More precisely, he proved the following theorem.

Theorem Z. LetEbe a real uniformly convex Banach space with a Frˆechet diﬀerentiable norm. Let

Cbe a closed convex subset ofE, and let{Ti}ri1 be a finite family of Lipschitzian pseudocontractive

self-mappings ofCsuch thatF:∩r_i₁FT_i_/∅. Let{x_n}be defined by

xnαnxn−1 1−αnTnxn, n≥1, 1.15

where Tn Tnmodr. If {αn}is chosen so that αn ∈ 0,1with lim sup_n_{→ ∞}αn < 1, then {xn} converges weakly to a common fixed point of the family{ T_i}r_i₁.

Here, we remark that the scheme in Theorem Z is implicit, and the convergence is weak

convergence.

More recently, Zegeye et al.23proved the following strong convergence of Ishikawa

iterative process for a common fixed point of finite family of Lipschitz pseudocontractive mappings.

Theorem ZSAsee23. LetCbe a nonempty, closed and convex subset of a real Hilbert space

H. LetTi :C → C, i1,2, . . . , N, be a finite family of Lipschitz pseudocontractive mappings with Lipschitzian constantsLi, fori1,2, . . . , N, respectively. Assume that the interior ofF:∩n_i1FTi

is nonempty. Let{x_n}be a sequence generated from an arbitraryx0∈Eby

yn1−βnxnβnTnxn,

xn1 1−αnxnαnTnyn, n≥1,

1.16

whereTn:TnmodNand{αn},{βn} ⊂0,1satisfying certain appropriate conditions. Then,{xn} converges strongly to a common fixed point of{T1, T2, . . . , T_N}.

From Theorem ZSA, we observe that the assumption that the interior of FT is

nonempty is severe restriction.

Motivated by Halpern20 and Zegeye et al. 23, it is our purpose, in this paper,

to prove strong convergence of Halpern-Ishikawa algorithm3.3to a common fixed point

of a finite family of Lipschitz pseudocontractive mappings. No compactness assumption is

imposed either on one of the mappings or onC. The assumption that interior ofFT is

nonempty is dispensed with. Moreover, computation of closed and convex set Cn for each

n ≥ 1 is not required. The results obtained in this paper improve and extend the results of

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2. Preliminaries

In what follows we will make use of the following lemmas.

Lemma 2.1. LetH be a real Hilbert space. Then for any givenx, y ∈ E, the following inequality

holds:

_x_y2_≤_x 2₂_{y, x}_y_. _2.1

Lemma 2.2see24. LetCbe a convex subset of a real Hilbert spaceH. Letx∈H. Thenx0P_Cx

if and only if

z−x0, x−x0 ≤0, ∀z∈C. 2.2

Lemma 2.3see25. Let{a_n}be a sequence of nonnegative real numbers satisfying the following

relation:

an1≤

1−βnanβnδn, n≥n0, 2.3

where{β_n} ⊂0,1and{δ_n} ⊂Rsatisfying the following conditions: lim_n_{→ ∞}β_n0,∞_n₁βn ∞, and lim sup_n_{→ ∞}δ_n≤0. Then, lim_n_{→ ∞}a_n0.

Lemma 2.4see18. LetHbe a real Hilbert space, letCbe a closed convex subset ofH, and let

T :C → Cbe a continuous pseudocontractive mapping; then

iFTis closed convex subset ofC;

ii I −Tis demiclosed at zero; that is, if{x_n}is a sequence inCsuch that x_n xand

Txn−xn → 0, asn → ∞, thenxTx.

Lemma 2.5see26. Let{an}be sequences of real numbers such that there exists a subsequence

{ni}of{n}such thatani < ani1for alli∈N. Then there exists a nondecreasing sequence{mk} ⊂N such thatmk → ∞,and the following properties are satisfied by all (suﬃciently large) numbersk∈N:

amk ≤amk1, ak≤amk1. 2.4

In fact,m_kmax{j≤k:a_j< a_j1}.

Lemma 2.6 see27. Let H be a real Hilbert space. Then for all xi ∈ H and αi ∈ 0,1 for

i1,2, . . . , nsuch thatα1α2· · ·αn1 the following equality holds:

α0x0α1x1· · ·αnxn 2

n

i0

αi xi 2−

0≤i,j≤n

αiαjxi−xj2. 2.5

3. Main Result

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Lemma 3.1. LetCbe a nonempty convex subset of a real Hilbert space H. Let Ti : C → C, i

1,2, . . . , N, be a finite family of Lipschitz pseudocontractive mappings with constantsL_i, respectively.

LetSθ1T1θ2T2· · ·θNTN, whereθ1θ2· · ·θN 1. ThenSis Lipschitz pseudocontractive

mapping onC.

Proof. Letx, y∈C. Then

Sx−Sy, x−yθ1T1x−T1y, x−y

θ2T2x−T2y, x−y· · ·θNTNx−TNy, x−y

≤θ1x−y2_θ2_x₋_y2_{· · ·}_θ

Nx−y2

x−y2_.

3.1

HenceSis pseudocontractive. Moreover, since

_Sx₋_Sy_θ1T1_θ2T2_{· · ·}_θ_N_T_N_x₋_θ1T1_θ2T2_{· · ·}_θ_N_T_N_y

≤θ1T1x−T1yθ2T2x−T2y· · ·θNTNx−TNy

≤Lx−y,

3.2

whereL:max{Li:i1,2, . . . , N}, we get thatSisL-Lipschitz. The proof is complete.

Let{Ti :i1,2, . . . , N}be a finite family of pseudocontractive mappings. The family

is said to satisfy conditionHifTix−x, Tjx−x ≥0, fori, j∈ {1,2, . . . , N}.

Theorem 3.2. LetCbe a nonempty, closed and convex subset of a real Hilbert spaceH. LetTi:C →

C, i1,2, . . . , Nbe a finite family of Lipschitz pseudocontractive mappings with Lipschitz constants

Li, respectively, satisfying conditionH. Assume thatF :Ni1FTiis nonempty. Let a sequence

{xn}be a sequence generated from an arbitraryx1w∈Cby

yn1−βnxnβnSnxn,

xn1αnw 1−αnγnSnyn1−γnxn,

3.3

whereSn:θn,1T1θn,2T2· · ·θn,NTN, for{θn,i} ⊆a, b⊂0,1such thatθn,1θn,2· · ·θn,N1, for alln≥1 and{α_n},{β_n},{γ_n} ⊂0,1satisfying the following conditions:i0≤α_n≤c <1, for

alln≥1 such that limn→ ∞αn0 andαn∞;ii0< α≤γn≤βn≤β <1/1L2 1, for

alln≥1, forL:max{Li:i1,2, . . . , N}. Then,{xn}converges strongly to a common fixed point

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Proof. LetpPFw. Then from3.3,Lemma 2.6,1.5, andLemma 3.1we have the following:

_x_n₁₋_p2 _α

nw 1−αnγnSnyn1−γnxn−p2

≤αnw−p2 1−αnγnSnyn1−γnxn−p2

αnw−p2 1−αn

γnSnyn−p2

1−γnxn−p2−γn1−γnSnyn−xn2

αnw−p2 1−αnγnSnyn−p2

≤αnw−p2 1−αnγn

yn−p2yn−Snyn2

1−αn1−γnxn−p2

−γn1−γn1−αnSnyn−xn2

αnw−p2 1−αnγnyn−p2 1−αnγnyn−Snyn2

1−αn1−γnxn−p2−γn1−γn1−αnSnyn−xn2

1−αn1−γnxn−p2−γn1−αn1−γnSnyn−xn2.

3.4

In addition, we have that

_y_n₋_S_n_y_n2

1−βnxn−SnynβnSnxn−Snyn2

1−βnxn−Snyn2βnSnxn−Snyn2

−βn1−βn xn−Snxn 2

≤1−β_nx_n−S_ny_n2β_nL2_x

n−yn2

1−βnxn−Snyn2β3nL2 xn−Snxn 2

1−βnxn−Snyn2βn L2β2nβn−1

xn−Snxn 2,

3.5

||yn−p||21−βnxn−pβnSnxn−p2

1−β_nx_n−p2β_nS_nx_n−p2−β_n1−β_n x_n−S_nx_n 2

≤1−βnxn−p2βnxn−p2 xn−Snxn 2

xn−p2βn2 xn−Snxn 2.

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Substituting3.5and3.6into3.4we obtain that

_x_n₁₋_p2_≤_α

nw−p2 1−αnγnxn−p2β2n xn−Snxn 2

1−α_nγ_n1−β_nx_n−S_ny_n2β_n L2β2_nβ_n−1

× xn−Snxn 2

1−α_n1−γ_nx_n−p2

−γn1−γn1−αnSnyn−xn2

1−αnγnβn2 xn−Snxn 2 1−αnγnβn L2βn2βn−1

× xn−Snxn 21−αn1−βnγn−1−αn1−γnγn

×xn−Snyn2

αnw−p2 1−αnxn−p2−1−αnγnβn 1−2βn−L2β2n

× xn−Snxn 2 1−αnγnγn−βnxn−Snyn2

αnw−p21−αnγn 1−αn1−γnxn−p2.

3.7

Since fromii, we have thatγn−βn≤0 and 1−2βn−L2β2n≥1−2β−L2β2>0 for alln≥1,

3.7implies that

_x_n₁₋_p2_≤_α

nw−p2 1−αnxn−p2

−1−α_nγ_nβ_n 1−2β−L2β2 x_n−S_nx_n 2

≤αnw−p2 1−αnxn−p2.

3.8

Thus, by induction,

_x_n₁₋_p2_≤

maxx1−p2,w−p2, ∀n≥1, 3.9

which implies that{xn}and hence{yn}are bounded.

Furthermore, from3.3,Lemma 2.1, and following the methods used in3.7we get

that

_x_n₁₋_p2 _α

nw−p 1−αnγnSnyn1−γnxn−p2

≤1−αnγnSnyn1−γnxn−p22αnw−p, xn1−p

1−αnγnSnyn−p2 1−αn1−γnxn−p2

−γn1−γn1−αnSnyn−xn22αnw−p, xn1−p

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≤1−α_nγ_ny_n−p2y_n−S_ny_n2 1−α_n 1−γ_nx_n−p2

−γn1−γn1−αnSnyn−xn22αnw−p, xn1−p

1−α_nγ_ny_n−p2 1−α_nγ_ny_n−S_ny_n2

≤1−α_nγ_nx_n−p2β_n2 x_n−S_nx_n 2 1−α_nγ_n

×1−β_nx_n−S_ny_n2β_n L2β2_nβ_n−1 x_n−S_nx_n 2

2αnw−p, xn1−p

≤1−αnxn−p22αnw−p, xn1−p

−1−α_nγ_nβ_n 1−2β_n−β_n2L2S_ny_n−x_n2

−1−cα2 1−2β−β2L2 S_nx_n−x_n 2

2α_nw−p, x_n1−p

.

3.10

On the other hand, usingLemma 2.6and conditionH, we get that

||xn−Snxn||2 xn−θn,1T1θn,2T2· · ·θn,NTNxn 2

θn,1xn−T1xn θn,2xn−T2xn · · ·θn,Nxn−TNxn 2

θn,1 xn−T1xn 2θn,2 xn−T2xn 2· · ·θn,N xn−TNxn 2

−

1≤i,j≤N

θn,iθn,jTixn−Tjxn2

−

1≤i,j≤N,i /j

θn,iθn,j

Tixn−xn 2xn−Tjxn2

θn,11−θn,2−θn,3− · · · −θn,N xn−T1xn 2

θn,21−θn,1−θn,3−θn,4− · · · −θn,N xn−T2xn 2. . .

θn,N1−θn,1−θn,2−θn,4− · · · −θn,N xn−TNxn 2

≥θn,1 xn−T1xn 2θn,2 xn−T2xn 2· · ·θn,N xn−TNxn 2.

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Thus, substituting3.11into3.10we obtain that

_x_n₁₋_p2_≤

1−αnxn−p22αnw−p, xn1−p

−1−cα2 1−2β−β2L2

×θn,11−θn,2−θn,3− · · · −θn,N xn−T1xn 2

θn,21−θn,1−θn,3−θn,4− · · · −θn,N xn−T2xn 2. . .

θn,N1−θn,1−θn,2−θn,4− · · · −θn,N−1 xn−TNxn 2

3.12

. 3.13

Now, we consider the following two cases.

Case 1. Suppose that there existsn0 ∈Nsuch that{||x_n−p||}is nonincreasing. Then, we get

that{||xn−p||}is convergent. Thus, from3.12and the fact thatαn → 0, asn → ∞, we

have that

xn−Tixn−→0, as n−→ ∞, 3.14

for eachi1,2, . . . , N. Letz_nγ_nS_ny_n 1−γ_nx_n. Then from3.3we obtain that

xn1−znαnw−zn−→0, as n−→ ∞. 3.15

Furthermore, from3.3and3.14we get that

_y_n₋_x_n_β_n_S_n_x_n₋_x_n_≤_S_n_x_n₋_x_n

≤θn,1 T1xn−xn θn,2 T2xn−xn · · ·θn,N TNxn−xn −→0,

3.16

asn → ∞, and hence3.16and the fact thatS_nisL-Lipschitz imply that

zn−xn γnSnyn−xnγnSnyn−SnxnγnSnxn−xn

≤γnLyn−xnγn Snxn−xn −→0, as n−→ ∞.

3.17

Now,3.15and3.17imply that

xn1−xn−→0, as n−→ ∞. 3.18

Moreover, since{xn}is bounded andEis reflexive, we choose a subsequence{xni1}

of{xn}such thatxni1 zand lim sup_n_{→ ∞}w−p, xn1−plimi→ ∞w−p, xni1−p. This

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for eachi1,2, . . . , N. Hence,z∈ ∩_iN₁FTi. Therefore, byLemma 2.2, we immediately obtain

that

lim sup

n−→∞

w−p, xn1−p

lim

i→ ∞

w−p, xni1−p

w−p, z−p≤0.

3.19

Then, since from3.13we have that

_x_n₁₋_p2_≤

1−α_nx_n−p22α_nw−p, x_n1−p

. 3.20

It follows from 3.20, 3.19, and Lemma 2.3 that ||xn − p|| → 0, as n → ∞.

Consequently,x_n → p.

Case 2. Suppose that there exists a subsequence{ni}of{n}such that

_x_n_i₋_p_<_x_n_i₁₋_p_, 3.21

for alli∈N. Then, byLemma 2.5, there exists a nondecreasing sequence{mk} ⊂Nsuch that

mk → ∞,||xmk−p|| ≤ ||xmk1−p||and||xk−p|| ≤ ||xmk1−p||for allk∈N. Now, from3.12

and the fact thatαn → 0, we get thatxmk −Tixmk → 0, ask → ∞, for eachi 1,2, . . . , N.

Thus, as in Case 1, we obtain thatxmk1−xmk → 0 and that

lim sup

k→ ∞

w−p, xmk1−p

≤0. _3.22

Now, from3.13we have that

_x_m_k₁₋_p2_≤

1−α_m_kx_m_k−p22α_m_kw−p, x_m_k1−p

, 3.23

and hence, since x_m_k−p 2≤ x_m_k1−p 2,3.23implies that

αmkxmk −p

2_≤_x

mk−p

2₋_x

mk1−p

2

2αmk

w−p, xmk1−p

≤2α_m_kw−p, x_m_k1−p

. 3.24

But noting thatα_m_k >0, we obtain that

_x_m_k₋_p2_≤

2w−p, xmk1−p

. 3.25

Then, from3.22we get that||xmk−p|| → 0, ask → ∞. This together with3.23gives

that||xmk1−p|| → 0, ask → ∞. But||xk−p|| ≤ ||xmk1−p||, for allk ∈N; thus we obtain

thatx_k → p. Therefore, from the previous two cases, we can conclude that{x_n}converges

strongly to an element ofF, and the proof is complete.

If, inTheorem 3.2, we consider single Lipschitz pseudocontractive mapping, then the

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Corollary 3.3. Let Cbe a nonempty, closed and convex subset of a real Hilbert space H. Let T :

C → Cbe a Lipschitz pseudocontractive mapping with Lipschitz constantsL. Assume thatFTis nonempty. Let a sequence{xn}be a sequence generated from an arbitraryx1 w∈Cby

yn1−βnxnβnTxn,

xn1αnw 1−αn

γnTyn1−γnxn,

3.26

where{αn},{βn} ⊂ 0,1satisfying the following conditions:i0 < αn ≤c <1, for alln≥1 such that limn→ ∞αn0 andαn∞;ii 0< α ≤γn ≤βn ≤β <1/1L2 1, for alln≥1. Then,{x_n}converges strongly to a fixed point ofTnearest tox1w.

Proof. PuttingSn :Tin3.3the scheme reduces to scheme3.26, and following the method

of proof ofTheorem 3.2we get thatsee,3.10

_x_n₁₋_p2_≤

1−α_nx_n−p22α_nw−p, x_n1−p

−1−αnγnβn 1−2β−β2L2

Txn−xn 2

−1−cγ_nβ_n 1−2β−β2L2 Tx_n−x_n 2

≤1−α_nx_n−p22α_nw−p, x_n1−p

.

3.27

Now, considering cases as in the proof ofTheorem 3.2we obtain the required result.

We now state and prove a convergence theorem for a common zero of finite family of monotone mappings.

Corollary 3.4. LetHbe a real Hilbert space. LetA_i :H → H, i 1,2, . . . , Nbe a finite family

of Lipschitz monotone mappings with Lipschitz constantsLi, respectively, satisfyingAix, Ajx ≥0, for alli, j∈ {1,2, . . . , N}.

Assume thatF:N_i₁NA_iis nonempty. Let a sequence{x_n}be generated from an arbitrary

x1∈Hby

ynxn−βnAnxn,

xn1αnw 1−αnxn−γnAnyn,

3.28

whereAn:θn,1A1θn,2A2· · ·θn,NAN, for{θn,i} ⊆a, b⊂0,1such thatθn,1θn,2· · ·θn,N

1, for alln≥1 and{α_n},{β_n},{γ_n} ⊂0,1satisfying the following conditions:i 0 < α_n≤c <1,

for alln≥0 such that limn→ ∞αn0 andαn∞;ii0< α≤γn≤βn≤β <1/1L2 1, for alln≥1, forL :max{1Li:i1,2, . . . , N}. Then,{xn}converges strongly to a common zero point of{A1, A2, . . . A_N}nearest tox1w.

Proof . Let Tix : I −Aix, fori 1,2, . . . , N. Then we get that every Ti for all i ∈ {1,

2, . . . , N}is Lipschitz pseudocontractive mapping with Lipschitz constantsLi: 1Liand

∩N

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we get that scheme3.28reduces to scheme3.3, and hence the conclusion follows from

Theorem 3.2.

If, inCorollary 3.4we consider a single Lipschitz monotone mapping, then we obtain

the following corollary.

Corollary 3.5. LetH be a real Hilbert space. LetA : H → H be Lipschitz monotone mappings

with Lipschitz constantL. Assume thatNAis nonempty. Let a sequence{xn}be generated from an arbitraryx1∈Hby

ynxn−βnAxn,

xn1αnw 1−αn

xn−γnAyn,

3.29

where{αn},{βn},{γn} ⊂0,1satisfying the following conditions:i0 < αn ≤c <1, for alln≥1 such that lim_n_{→ ∞}α_n 0 and α_n ∞;ii0 < α ≤ γ_n ≤ β_n ≤ β < 1/1L2 ₁_,

for alln≥1. Then,{xn}converges strongly to a zero point ofAnearest tox1w.

We now give examples of Lipschitz pseudocontractive mappings satisfying condition

H. LetX:RandC: −24,3⊂R. LetT1, T2:C → Cbe defined by

T1x:

x, x∈−24,0,

x−x3_{, x}_∈₀_,₃_,

T2x:

x, x∈−24,2,

3x−x2_{, x}_∈₂_,₃_.

3.30

Then we observe that FT1 −24,0,and FT2 −24,2, and hence common

fixed point ofT1 and T2 is−24,0 which is nonempty. Now, we show that T1 and T2 are

pseudocontractive mappings. But, since

A1x: I−T1x

0, x∈−24,0,

x3_{, x}_∈₀_,₃_,

A2x: I−T2x

0, x∈−24,2,

−2xx2_{, x}_∈₂_,₃

3.31

are monotone, we have thatT1andT2are pseudocontractive mappings.

Now, we show that T1 and T2 are Lipschitzian mappings. First, we show thatT1 is

Lipschitzian with constantL 28. Let C1 −24,0,C2 0,3. Ifx, y ∈ C1, then we have

that

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Ifx, y∈C2,then we have that

_T1x₋_T1y_x₋_x3₋ _y₋_y3_≤_x₋_y_x3₋_y3

x−yx−yx2_xy_y2

1x2xyy2x−y≤28x−y.

3.33

Ifx∈C1andy∈C2_,then we get that

_T1x₋_T1y_x₋ _y₋_y3_≤_x₋_y_y3_x₋_y_y2_y

≤x−yy2_y₋_x _y2₁_x₋_y

≤10x−y≤28x−y.

3.34

Therefore, from3.32,3.33, and3.34, we obtain thatT1is Lipschitz.

Next, we show thatT2is Lipschitz with constantL9.

LetD1 −24,2,D2 2,3. Ifx, y∈D1, then we have that

_T2x₋_T2y_x₋_y_≤₉_x₋_y_. _3.35

Ifx, y∈D2,then we get that

_T2x₋_T2y3x−x2− 3y−y2≤3x−yx2−y2

3x−yx−yxy

≤3xyx−y≤9x−y.

3.36

Ifx∈D1andy∈D2then we have that

_T2x₋_T2y_x₋ 3y−y2≤x−yy2₋₂_y _3.37

and forx∈0,2 3.37implies that

_T2x₋_T2y_≤_x₋_y_y2₋₂_y₋ _x2₋₂_x

≤x−y2x−yx−yxy

≤3xyx−y≤9x−y.

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Forx∈−24,0inequality3.37gives that

_T2x₋_T2y_≤_x₋_y_y₋2y,

≤x−yy−2y−x

1y−2x−y

≤2x−y≤9x−y.

3.39

Therefore, from3.35,3.36, and3.39we obtain thatT2 is Lipschitz. Furthermore,

we show thatT1andT2satisfy conditionH. Ifx∈D1,then we have thatT1x−x, T2x−x0,

and ifx ∈D2we get thatT1x−x, T2x−x x−x3₋_x,₃_x₋_x2₋_x ₋_x3_,₂_x₋_x2

−x3₂_x₋_x2_≥_{0. Therefore,}_T1_and_T1_{satisfy property}_H_.

Remark 3.6. Theorem 3.2 provides convergence sequence to a common fixed point of

finite family of Lipschitzian pseudocontractive mappings whereas Corollary 3.4 provides

convergence sequence to a common zero of finite family of monotone mappings in Hilbert

spaces. No compactness assumption is imposed either onT orC. This provides aﬃrmative

answer to the question raised.

Remark 3.7. Theorem 3.2improves Theorem I, Theorem 3.1 of Zhou15, Theorem 3.1 of Yao

et al.17, and Theorem 3.1 of Tang et al.16in the sense that either our convergence does

not require compactness ofTor computation ofCn1fromCnfor eachn≥1.

Remark 3.8. Theorem 3.2improves Theorems I and ZSA in the sense that our convergence is for a fixed point of a finite family of Lipschitz pseudocontractive mappings. The condition

that interior ofFTis nonempty is dispensed with.

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