An algorithm for a common fixed point of a family of pseudocontractive mappings

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R E S E A R C H

Open Access

An algorithm for a common ﬁxed point of a

family of pseudocontractive mappings

Habtu Zegeye

1

_{and Naseer Shahzad}

2*

*_{Correspondence:}

[email protected]

2_{Department of Mathematics, King}

Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia Full list of author information is available at the end of the article

Abstract

We introduce an iterative process which converges strongly to a common ﬁxed point of a family of Lipschitzian pseudocontractive mappings. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear operators.

MSC: 47H09; 47J20; 65J15

Keywords: ﬁxed points of mappings; monotone mappings; pseudocontractive mappings; strong convergence

1 Introduction

LetCbe a nonempty subset of a real Hilbert space H. A mappingT :C→H is called

pseudocontractiveif and only if

Tx–Ty≤ x–y+(I–T)x– (I–T)y, for allx,y∈C, (.)

andTis calledα-strictly pseudocontractive[] if and only if there existsα∈[, ) such that

Tx–Ty≤ x–y+α(I–T)x– (I–T)y, for allx,y∈C, (.)

Tis calledLipschitzianif and only if

Tx–Ty ≤Lx–y, for allx,y∈K. (.)

If in (.) we haveL≤, thenTis callednonexpansive. We note that inequalities (.) and (.) can be equivalently written as

x–y,Tx–Ty ≤ x–y, for allx,y∈C (.)

and

x–y,Tx–Ty ≤ x–y–λ(I–T)x– (I–T)y, (.)

for someλ> , respectively.

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We observe from (.), (.) and (.) that every nonexpansive mapping is anα-strictly pseudocontractive mapping and everyα-strictly pseudocontractive mapping is a pseudo-contractive mapping, and hence a class of pseudopseudo-contractive mappings is a more general class of mappings.

Furthermore, pseudocontractive mappings are related with the important class of non-linearmonotonemappings, where a mappingAwith domainD(A) and rangeR(A) inHis calledmonotoneif the inequality

x–y,Ax–Ay ≥, (.)

holds for everyx,y∈D(A). We note thatT is pseudocontractive if and only ifA:=I–T

is monotone, and hence a ﬁxed point of T, F(T) :={x∈D(T) :Tx=x}, is a zero ofA,

N(A) :={x∈D(A) :Ax= }. It is now well known (see,e.g., []) that ifAis monotone, then the solutions of the equationAx=  correspond to the equilibrium points of some evolution systems.

Finding a point in the intersection of fixed point sets of a family of nonexpansive map-pings is a task that occurs frequently in various areas of mathematical sciences and en-gineering. For example, the well-known convex feasibility problem reduces to finding a point in the intersection of fixed point sets of a family of nonexpansive mappings; see,e.g., [, ].

Consequently, considerable research efforts have been devoted to developing iterative methods for approximating a common fixed point (when it exists) for a family of nonex-pansive mappings andα-strictly pseudocontractive mappings. Bauschke [] was the first to introduce a Halpern-type iterative process (see,e.g., []) for approximating a common fixed point for a finite family ofNnonexpansive self-mappings. He proved the following theorem.

Theorem BSK (Bauschke [], Theorem .) Let K be a nonempty closed convex sub-set of a Hilbert space H, and let T,T, . . . ,TN be a ﬁnite family of nonexpansive

map-pings of K into itself with F:=N_i_=F(Ti)=∅and F=F(TNTN–· · ·T) =F(TTN· · ·T) =

F(TN–TN–· · ·TTN).Given points u,x∈C,let{xn}be generated by

xn+:=αn+u+ ( –αn+)Tn+xn, n≥, (.)

where Tn:=Tn(modN)and{αn} ⊂(, )satisﬁes

n≥|αn+r–αn|<∞.Then{xn}converges

strongly to PFu,where PF:H→F is the metric projection.

Various authors have studied iterative schemes similar to that of Theorem BSK in more general Banach spaces on the one hand, and using various conditions on the sequence{αn} on the other hand (see, for example, Colaoet al.[], Yao [], Takahashi and Takahashi [], Plubtieng and Punpaeng [], Cenget al.[] and the references therein). Many authors have also studied iterative methods for a family ofα-strictly pseudocontractive mappings (see,e.g., [–] and the references therein).

Our concern now is the following:Can we construct an iterative sequence for a common ﬁxed point of a ﬁnite family of pseudocontractive mappings?

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Hilbert spaces. LetCbe a closed convex subset ofE, and let{Ti}Ni=be a finite family of Lipschitzian pseudocontractive self-mappings ofCsubset ofEwhich is a real uniformly convex Banach space with a Frêchet differentiable norm andF:=N_i_=F(Ti)=∅. Let{xn} be defined by

xn=αnxn–+ ( –αn)Tnxn, n≥, (.)

whereTn=Tn(modN). He proved that{xn}converges weakly to a common ﬁxed point of the family{Ti}Ni=under certain conditions on the parameter{αn}.

We remark that the scheme in the theorem of Zhou [] isimplicitand the convergence isweak convergence.

Recently, Zegeyeet al.[] proved the following strong convergence of Ishikawa iterative process [] for a common ﬁxed point of a ﬁnite family of Lipschitz pseudocontractive mappings.

Theorem ZSA [] Let C be a nonempty, closed and convex subset of a real Hilbert space H.Let Ti:C→C,i= , , . . . ,N,be a ﬁnite family of Lipschitz pseudocontractive

mappings with Lipschitzian constants Li,for i= , , . . . ,N,respectively.Assume that the interior of F:=N_i_=F(Ti)is nonempty.Let{xn}be a sequence generated from an arbitrary

x∈E by ⎧ ⎨ ⎩

yn= ( –βn)xn+βnTnxn,

xn+= ( –αn)xn+αnTnyn, n≥,

(.)

where Tn:=Tn(modN) and{αn},{βn} ⊂(, )satisfy certain appropriate conditions.Then

{xn}converges strongly to a common ﬁxed point of{T,T, . . . ,TN}.

It is worth to mention that the assumption ‘interior of F(T)is nonempty’ in Theo-rem ZSA issevere restriction.

More recently, Daman and Zegeye [] proved the following strong convergence of a Halpern-type [] iterative process for a common fixed point of a finite family of Lipschitz pseudocontractive mappings under the assumption that the family satisfiescondition(H), that is, the family{Ti:i= , , . . . ,N}of pseudocontractive mappings is said to satisfy

con-dition(H) if and only ifTix–x,Tjx–x ≥ fori,j∈ {, , . . . ,N}. We can also see that condition (H) is againsevere restriction.

It is our purpose in this paper to introduce an iterative scheme which converges strongly to a common ﬁxed point of a family of Lipschitz pseudocontractive mappings. The as-sumption thatinterior of F(T)is nonemptyorcondition(H) is dispensed with. The results obtained in this paper improve and extend the results of Zhou [], Theorem ZSA, Daman and Zegeye [] and some other results in this direction.

2 Preliminaries

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

Lemma . Let H be a real Hilbert space.Then,for any given x,y∈E,the following in-equality holds:

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Lemma .[] Let C be a convex subset of a real Hilbert space H.Let x∈H.Then x=

PCx if and only if

z–x,x–x ≤, ∀z∈C.

Lemma .[] Let{an}be a sequence of nonnegative real numbers satisfying the following

relation:

an+≤( –αn)an+αnδn, n≥n,

where{αn} ⊂(, )and{δn} ⊂R satisfy the following conditions:limn→∞αn= ,

∞

n=αn=

∞,andlim sup_n_→∞δn≤.Thenlimn→∞an= .

Lemma .[] Let H be a real Hilbert space,C be a closed convex subset of H and T:

C→C be a continuous pseudocontractive mapping.Then

(i) F(T)is a closed convex subset ofC;

(ii) (I–T)is demiclosed at zero,i.e.,if{xn}is a sequence inCsuch thatxnxand

Txn–xn→,asn→ ∞,thenx=T(x).

Lemma .[] Let{an}be sequences of real numbers such that there exists a subsequence

{ni}of{n}such that ani<ani+ for all i∈N.Then there exists a nondecreasing sequence

{mk} ⊂N such that mk→ ∞and the following properties are satisﬁed by all(suﬃciently

large)numbers k∈N:

amk ≤amk+ and ak≤amk+.

In fact,mkis the largest number n in the set{, , . . . ,k}such that the condition an≤an+

holds.

Lemma .[] Let H be a real Hilbert space.Then,for all xi∈H andαi∈[, ]for

i= , , . . . ,n such thatα+α+· · ·+αn= ,the following equality holds:

αx+αx+· · ·+αnxn= n

i=

αixi–

≤i,j≤n

αiαjxi–xj.

3 Main result

Theorem . Let C be a nonempty,closed and convex subset of a real Hilbert space H.

Let T,T:C→C be Lipschitz pseudocontractive mappings with Lipschitz constants L

and L,respectively.Assume thatF=F(T)∩F(T)is nonempty.Let a sequence{xn}be

generated from an arbitrary x,u∈C by

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

zn= ( –cn)xn+cnTxn;

yn= ( –βn)xn+βnTxn;

xn+=αnu+ ( –αn)(θnxn+δnTyn+γnTzn), n≥,

(.)

where{δn},{θn},{γn} ⊂(a,b)⊂(, ),{αn} ⊂(,c)⊂(, )satisfy the following conditions: (i)θn+δn+γn= ; (ii)limn→∞αn= ,

αn=∞; (iii)δn+γn≤cn,βn≤β<√ 

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for L:=max{L,L}.Then{xn}converges strongly to a common ﬁxed point of T and T

nearest to u.

Proof We make use of some ideas of the paper []. Letp∈F. Then, from (.), (.) and

Lemma ., we have that

xn+–p =αnu+ ( –αn)(θnxn+δnTyn+γnTzn) –p 

≤αnu–p+ ( –αn)δn(Tyn–p) +θn(xn–p) +γn(Tzn–p) 

≤αnu–p+ ( –αn)

δnTyn–p+θnxn–p

+γnTzn–p

– ( –αn)δnθnTyn–xn

– ( –αn)θnγnTzn–xn,

and hence

xn+–p≤αnu–p+ ( –αn)δn

yn–p+yn–Tyn

+ ( –αn)θnxn–p+ ( –αn)γn

zn–p+zn–Tzn

– ( –αn)δnθnTyn–xn– ( –αn)θnγnTzn–xn

≤αnu–p+ ( –αn)δnyn–p+ ( –αn)δnyn–Tyn

+ ( –αn)θnxn–p+ ( –αn)γnzn–p

+ ( –αn)γnzn–Tzn– ( –αn)δnθnTyn–xn

– ( –αn)θnγnTzn–xn. (.)

In addition, from (.), Lemma . and (.), we get that

yn–p =( –βn)(xn–p) +βn(Txn–p) 

= ( –βn)xn–p+βnTxn–p

–βn( –βn)xn–Txn

≤( –βn)xn–p+βn

xn–p+xn–Txn

–βn( –βn)xn–Txn

=xn–p+βnxn–Txn. (.)

Similarly, we have that

zn–p=xn–p+cnxn–Txn. (.)

Furthermore, from (.) and Lemma ., we have that

yn–Tyn=( –βn)(xn–Tyn) +βn(Txn–Tyn) 

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≤( –βn)xn–Tyn+βnLxn–yn

–βn( –βn)xn–Txn

= ( –βn)xn–Tyn+βnLxn–Txn

= ( –βn)xn–Tyn–βn

 –Lβ_n–βnxn–Txn. (.)

Similarly, we obtain that

zn–Tzn= ( –cn)xn–Tzn–cn

 –Lc_n–cn

xn–Txn. (.)

Substituting (.), (.), (.) and (.) into (.), we obtain that

xn+–p≤αnu–p+ ( –αn)δn

xn–p+βnxn–Txn

+ ( –αn)δn

( –βn)xn–Tyn–βn

 –Lβ_n–βn

× xn–Txn

+ ( –αn)θnxn–p+ ( –αn)γn

xn–p

+c_nxn–Txn

+ ( –αn)γn

( –cn)xn–Tzn

–cn

xn–Txn

– ( –αn)δnθnTyn–xn

– ( –αn)θnγnTzn–xn

=αnu–p+ ( –αn)xn–p– ( –αn)δnβn

× –Lβ_n+ βn

xn–Txn

– ( –αn)γncn

 –Lc_n+ cn

xn–Txn

+ ( –αn)δn( –θn–βn)Tyn–xn

+ ( –αn)γn( –θn–cn)Tzn–xn,

and hence

xn+–p≤αnu–p+ ( –αn)xn–p– ( –αn)δnβn

× –Lβ_n+ βn

xn–Txn

– ( –αn)γncn

 –Lc_n+ cn

xn–Txn

+ ( –αn)δn(δn+γn–βn)Tyn–xn

+ ( –αn)γn(δn+γn–cn)Tzn–xn. (.)

Now, from (iii) of the hypothesis, we have that

 – βn–Lβ_n≥ – β–Lβ> , (.)

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and

(δn+γn) –βn≤, (δn+γn) –cn≤ for alln≥. (.)

Thus, inequality (.) implies that

xn+–p≤αnu–p+ ( –αn)xn–p. (.)

Thus, by induction,

xn+–p≤max

u–p,x–p

, ∀n≥,

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

Letx∗ =PF(u). Then, using (.), Lemma . and following the methods used to get (.), we obtain that

xn+–x∗ =αnu+ ( –αn)(θnxn+δnTyn+γnTzn) –x∗

=αn

u–x∗+ ( –αn)θnxn+δnTyn+γnTzn–x∗ 

≤( –αn)δnTyn+θnxn+γnTzn–x∗ 

+ αn

u–x∗,xn+–x∗

≤( –αn)δnTyn–x∗ 

+ ( –αn)θnxn–x∗ 

+ ( –αn)γnTzn–x∗– ( –αn)θnδnTyn–xn

– ( –αn)θnγnTzn–xn+ αn

and

xn+–x∗≤( –αn)δnyn–x∗+yn–Tyn

+ ( –αn)θnxn–x∗ 

+ ( –αn)γnzn–x∗ 

+zn–Tzn

– ( –αn)θnδnTyn–xn

– ( –αn)θnγnTzn–xn+ αn

≤( –αn)δnxn–x∗ 

+β_nxn–Txn

+ ( –αn)δn

( –βn)xn–Tyn–βn

 –Lβ_n–βn

× xn–Txn

+ ( –αn)θnxn–x∗ 

+ ( –αn)γnxn–x∗ 

+c_nxn–Txn

+ ( –αn)γn

( –cn)xn–Tzn–cn

× xn–Txn

– ( –αn)θnδnTyn–xn– ( –αn)θnγn

× Tzn–xn+ αn

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which implies that

xn+–x∗≤( –αn)xn–x∗– ( –αn)δnβn

 –Lβ_n– βn

× xn–Txn+ ( –αn)δn(δn+γn–βn)xn–Tyn

– ( –αn)δncn

 –Lc_n– cn

xn–Txn

+ ( –αn)γn(δn+γn–cn)xn–Tzn+ αn

(.)

≤( –αn)xn–x∗+ αn

. (.)

Now, we consider two cases.

Case . Suppose that there existsn∈Nsuch that{xn–x∗}is decreasing for alln≥n. Then, we get that{xn–x∗}is convergent. Thus, from (.), (.), (.) and (.), we have that

xn–Txn→, xn–Txn→ asn→ ∞. (.)

Furthermore, since{xn+} is a bounded subset ofH which is reﬂexive, we can choose a subsequence {xni+}of{xn+}such that xni+xandlim supn→∞u–x∗,xn+–x∗=

lim_i→∞u–x∗,xni+–x∗. Then, from (.) and Lemma ., we have thatx∈F(T) and

x∈F(T). Therefore, by Lemma ., we immediately obtain that

lim sup n→∞

= lim i→∞

u–x∗,xni+–x∗

=u–x∗,x–x∗≤. (.)

Then it follows from (.), (.) and Lemma . thatxn–x∗ → asn→ ∞. Conse-quently,xn→x∗=PF(u).

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

xni–x∗<xni+–x∗

for alli∈N. Then, by Lemma ., there exists a nondecreasing sequence{mk} ⊂Nsuch thatmk→ ∞, and

xmk–x∗≤xmk+–x∗ and xk–x∗≤xmk+–x∗ (.)

for allk∈N. Now, from (.), (.), (.) and (.), we get thatxmk–Txmk → and

xnk–Txnk→ ask→ ∞. Thus, like in Case , we obtain that

lim sup k→∞

u–x∗,xmk+–x

∗_≤_. _(.)

Now, from (.) we have that

xmk+–x∗ 

≤( –αmk)xmk–x∗ 

+ αmk

u–x∗,xmk+–x∗

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and hence (.) and (.) imply that

αmkxmk–x∗ 

≤xmk–x∗ 

–xmk+–x∗ 

+ αmk

u–x∗,xmk+–x∗

≤+αmk

u–x∗,xmk+–x

∗_.

But using the fact thatαmk>  and (.), we obtain that

lim sup k→∞

xmk–x∗ 

≤,

and hencexmk–x∗ → ask→ ∞. This together with (.) implies thatxmk+–x∗ →  ask→ ∞. Butxk–x∗ ≤ xmk+–x

∗ _{for all}_k_∈_N_{, thus we obtain that}_x

k→x∗. Therefore, from the above two cases, we can conclude that{xn}converges strongly to a common ﬁxed point ofTandTnearest tou. The proof is complete.

We note that the method of the proof of Theorem . provides a convergence theorem for a ﬁnite family of Lipschitzian pseudocontractive mappings. In fact, we have the fol-lowing theorem.

Theorem . Let C be a nonempty,closed and convex subset of a real Hilbert space H.

Let Ti:C→C,i= , , . . . ,N,be Lipschitz pseudocontractive mappings with Lipschitz

con-stants Li,i= , , . . . ,N,respectively.Assume thatF=N_i F(Ti)is nonempty.Let a sequence {xn}be generated from an arbitrary x,u∈C by

⎧ ⎨ ⎩

yni= ( –βn)xn+βnTixn, i= , , . . . ,N;

xn+=αnu+ ( –αn)(θnxn+ N

i=θniTiyni),

(.)

where{θni:i= , , , . . . ,N} ⊂(a,b)⊂(, ),{αn} ⊂(,c)⊂(, )satisfy the following

con-ditions: (i)θn+θn+· · ·+θnN= ; (ii)limn→∞αn= ,

αn=∞; (iii) N

i=θni≤βn≤β< 

√

+L_+,∀n≥,for L:=max{Li:i= , , . . . ,N}.Then{xn}converges strongly to a common ﬁxed point of Ti(i= , , . . . ,N)nearest to u.

If in Theorem . we assume thatT is nonexpansive, then we have thatT is Lipschitz pseudocontractive withL= , and hence we get the following corollary.

Corollary . Let C be a nonempty,closed and convex subset of a real Hilbert space H.Let

T,T:C→C be nonexpansive mappings.Assume thatF=F(T)∩F(T)is nonempty.

Let a sequence{xn}be generated from an arbitrary x,u∈C by ⎧

⎪ ⎪ ⎨ ⎪ ⎪ ⎩

zn= ( –cn)xn+cnTxn;

yn= ( –βn)xn+βnTxn;

xn+=αnu+ ( –αn)(θnxn+δnTyn+γnTzn),

(.)

where{δn},{θn},{γn} ⊂(a,b)⊂(, ),{αn} ⊂(,c)⊂(, )satisfy the following conditions: (i)δn+θn+γn= ; (ii)limn→∞αn= ,

αn=∞; (iii)δn+γn≤cn,βn≤β< (

√

 – ),∀n≥.

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

Corollary . Let H be a real Hilbert space.Let Ai:H→H,i= , ,be Lipschitz monotone

mappings with Lipschitz constants Land L,respectively.Assume that F:= 

i=N(Ai)is

nonempty.Let a sequence{xn}be generated from an arbitrary x,u∈H by ⎧

⎪ ⎪ ⎨ ⎪ ⎪ ⎩

zn=xn–cnAxn;

yn=xn–βnAxn;

xn+=αnu+ ( –αn)(θnxn+δn(I–A)yn+γn(I–A)zn),

(.)

where{δn},{θn},{γn} ⊂(a,b)⊂(, ),{αn} ⊂(,c)⊂(, )satisfy the following conditions: (i)δn+θn+γn= ; (ii)limn→∞αn= ,

αn=∞; (iii)δn+γn≤cn,βn≤β<√  +(+L)_+, ∀n≥,for L:=max{L,L}.Then{xn}converges strongly to a common zero point of Aand

Anearest to u.

Proof LetTix:= (I–Ai)xfori= , . Then we get that everyTifor alli∈ {, }is a Lipschitz pseudocontractive mapping with the Lipschitz constant L_i := ( +Li) and



i=F(Ti) = 

i=(Ai)=∅. Moreover, whenAiis replaced with (I–Ti), for eachi∈ {, }, then scheme (.) reduces to scheme (.), and hence the conclusion follows from Theorem ..

We may also have the following corollary for a ﬁnite family of monotone mappings.

Corollary . Let H be a real Hilbert space.Let Ai:H→H,i= , , . . . ,N,be Lipschitz

monotone mappings with Lipschitz constants Li,i= , , . . . ,N,respectively.Assume that

F:=N_i_=N(Ai)is nonempty.Let a sequence{xn}be generated from an arbitrary x,u∈H

by

⎧ ⎨ ⎩

yni=xn–βnAixn, i= , , . . . ,N;

xn+=αnu+ ( –αn)(θnxn+ N

i=θni(I–Ai)yni),

(.)

where{θni:i= , , , . . . ,N} ⊂(a,b)⊂(, ),{αn} ⊂(,c)⊂(, )satisfy the following

con-ditions: (i)θn+θn+· · ·+θnN= ; (ii)limn→∞αn= ,

αn=∞; (iii) N

i=θni≤βn≤β< 

√

+(+L)_+,∀n≥,for L:=max{Li:i= , , . . . ,N}.Then{xn}converges strongly to a com-mon zero point of Ai,i= , , . . . ,N,nearest to u.

If in Corollary . we consider a single Lipschitz monotone mapping, then we obtain the following corollary.

Corollary . Let H be a real Hilbert space.Let A:H→H be a Lipschitz monotone map-ping with Lipschitz constant L.Assume that F:=N(A)is nonempty.Let a sequence{xn}be

generated from an arbitrary x,u∈H by

⎧ ⎨ ⎩

yn=xn–βnAxn;

xn+=αnu+ ( –αn)(( –γn)xn+γn(I–A)yn),

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where {γn} ⊂ (a,b) ⊂ (, ), {αn} ⊂ (,c) ⊂ (, ) satisfy the following conditions: (i)limn→∞αn= ,

αn=∞; (ii)γn≤βn≤β< √ 

+(+L)_+,∀n≥.Then{xn}converges strongly to the zero point of A nearest to u.

4 Numerical example

Now, we give an example of a ﬁnite family of pseudocontractive mappings satisfying The-orem . and some numerical experiment result to explain the conclusion of the theThe-orem as follows.

Example . LetH=Rwith absolute value norm. LetK= [–, ] andT,T:K→Kbe deﬁned by

Tx:= ⎧ ⎨ ⎩

x+x_, _x_∈_{[–, ],}

x, x∈(, ] (.)

and

Tx:= ⎧ ⎨ ⎩

x, x∈[–,

],

x– (x–_)_, _x_∈₍ , ].

(.)

Clearly,F=F(T)∩F(T) = [, ]∩[–,_] = [,_], and forx,y∈Kwe have that

(I–T)x– (I–T)y,x–y

≥ (.)

and

(I–T)x– (I–T)y,x–y

≥, (.)

which show that both mappings are pseudocontractive. Next, we show that T is Lips-chitzian withL= . Ifx,y∈[–, ], then

|Tx–Ty|=x+x–y–y

=(x+y) + |x–y| ≤|x+y|. (.)

Ifx,y∈(, ], then

|Tx–Ty|=|x–y|.

Ifx∈[–, ] andy∈(, ], then

|Tx–Ty|=x+x–y

=x–y+x=x–y+x–y+y

=x–y+x–y+y

≤ |x+y+ | · |x–y|+|y–x|

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Ifx∈(, ] andy∈[–, ], then

|Tx–Ty|=x–

y+y

=x–y–y+x–x=x–y+ (x+y)(x–y) –x

=( +x+y)|x–y|+x

≤( +x+y)|x–y|+ (x–y)

≤| +x+y|+|x–y||x–y|

≤|x–y|. (.)

Thus, we get thatTis Lipschitzian pseudocontractive withL= . Similarly, we can show thatTis Lipschitzian pseudocontractive withL= .

Now, takingαn=_n_+ ,cn=βn=_ +_n_+ ,γn=δn= .[_ +_n_+ ] andθn=  – [_ + 

n+], we observe that the conditions of Theorem . are satisﬁed and scheme (.) pro-vides the data in Tables  and  and Figures  and .

(i) Whenu= . andx= –, we see that the sequence converges tox∗= . as shown in Table  and Figure .

(ii) Whenu= – andx= ., we see that the sequence converges tox∗=  as shown in Table  and Figure .

Table 1 Values of{xn}with initial valuesu= 0.6 andx0= –1

n 0 500 1,000 5,000 10,000 12,000 14,000 17,000

xn –1.0000 0.5639 0.5638 0.5387 0.5287 0.5264 0.5246 0.5225

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Table 2 Values of{xn}with initial valuesu= –1 andx0= 0.8

n 0 500 1,000 5,000 10,000 12,000 14,000 17,000

xn 0.8000 –0.4781 –0.3738 –0.1870 –0.1351 –0.1238 –0.1150 –0.1047

Figure 2 Figure of{xn}with initial valuesu= –1 andx0= 0.8.

Remark . Theorem . provides a convergence sequence to a common fixed point of two Lipschitzian pseudocontractive mappings, whereas Theorem . provides a conver-gence sequence to a common fixed point of a finite family of Lipschitzian pseudocontrac-tive mappings. In addition, Corollary . provides a convergence sequence to a common zero of two Lipschitzian monotone mappings, whereas Theorem . provides a conver-gence sequence to a common zero of a finite family of Lipschitzian monotone mappings.

Remark . Theorem . improves Theorem ZSA, Theorem . of Daman and Zegeye [] in the sense that our convergence does not require the assumption that interior of F(T)is nonemptyorcondition(H).

Remark . Theorem . improves Theorem . of Zhou [], Theorem . of Yaoet al.

[] and Theorem . of Tanget al.[] in the sense that our convergence requires neither compactness ofT nor computation of closed and convexCnofCfor eachn≥.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All the authors contributed equally. All authors read and approved the ﬁnal manuscript.

Author details

1_{Department of Mathematics, University of Botswana, Pvt. Bag 00704, Gaborone, Botswana.}2_{Department of}

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Acknowledgements

This article was funded by the Deanship of Scientiﬁc Research (DSR), King Abdulaziz University, Jeddah. The second author acknowledges with thanks DSR for ﬁnancial support.

Received: 8 April 2013 Accepted: 15 August 2013 Published: 5 September 2013 References

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