A new multi-step iteration for solving a fixed point problem of nonexpansive mappings

R E S E A R C H

Open Access

A new multi-step iteration for solving a fixed

point problem of nonexpansive mappings

Prasit Cholamjiak

*_{Correspondence:}

[email protected] School of Science, University of Phayao, Phayao, 56000, Thailand

Abstract

We introduce a new nonlinear mapping generated by a finite family of nonexpansive mappings. Weak and strong convergence theorems are also established in the setting of a Banach space.

MSC: 47H09; 47H10

Keywords: nonexpansive mapping; weak convergence; Banach space; fixed point;

strong convergence

1 Introduction

LetCbe a nonempty, closed and convex subset of a real Banach spaceE. LetT:C→C

be a nonlinear mapping. The fixed point set ofTis denoted byF(T), that is,F(T) ={x∈ C:x=Tx}. Recall that a mappingT is said to benonexpansive ifTx–Ty ≤ x–y

for allx,y∈C, and a mappingf :C→Cis called acontractionif there exists a constant

α∈(, ) such thatf(x) –f(y) ≤αx–yfor allx,y∈C. We useCto denote a class of

contractions with constantα.

Fixed point problems are now arising in a wide range of applications such as optimiza-tion, physics, engineering, economics and applied sciences. Many related problems can be cast as the problem of finding fixed points for nonlinear mappings. The interdisciplinary nature of fixed point problems is evident through a vast literature which includes a large body of mathematical and algorithmic developments.

In the literature, several types of iterations have been constructed and proposed in order to get convergence results for nonexpansive mappings in various settings. One classical iteration process is defined as follows:x∈Cand

xn+= ( –αn)xn+αnTxn, ∀n≥,

where{αn} ⊂(, ). This method was introduced in  by Mann [] and is known as the Mann iteration process. However, we note that it has only weak convergence in general; for instance, see [].

In , Ishikawa [] proposed the following two-step iteration:x∈Cand

yn= ( –βn)xn+βnTxn,

xn+= ( –αn)xn+αnTyn, ∀n≥,

where{αn}and{βn}are sequences in (, ). This method is often called theIshikawa iter-ation process.

Very recently, Agarwalet al.[] introduced a new iteration process as follows:x∈C

and

yn= ( –βn)xn+βnTxn,

xn+= ( –αn)Txn+αnTyn, ∀n≥,

where{αn}and{βn}are sequences in (, ). This method is called theS-iteration process.

The weak convergence was studied in [] for nonexpansive mappings. It was also shown in [] that the convergence rate of theS-iteration process is faster than the Picard and Mann iteration processes for contractive mappings.

Firstly, motivated by Agarwalet al.[], we have the aim to introduce and study a new mapping defined by the following definition.

Definition . Let C be a nonempty and convex subset of a real Banach space E.

Let T,T, . . . ,TN be a finite family of nonexpansive mappings of Cinto itself, and let λ,λ, . . . ,λN be real numbers such that ≤λi≤ for alli= , , . . . ,N. Define the

map-pingB:C→Cas follows:

U=λT+ ( –λ)I,

U=λTU+ ( –λ)T,

U=λTU+ ( –λ)T,

.. .

UN–=λN–TN–UN–+ ( –λN–)TN–,

B=UN=λNTNUN–+ ( –λN)TN–. (.)

Such a mappingBis called theB-mappinggenerated byT,T, . . . ,TNandλ,λ, . . . ,λN.

See [–] for the corresponding concept.

Secondly, using the definition above, we study weak convergence of the following Mann-type iteration process in a uniformly convex Banach space with a Fréchet differentiable norm or that satisfies Opial’s condition:x∈Cand

xn+= ( –αn)xn+αnBnxn, ∀n≥, (.)

whereBnis aB-mapping generated byT,T, . . . ,TNandλn,,λn,, . . . ,λn,N(see Section ).

Finally, we discuss strong convergence of the iteration scheme involving the modified viscosity approximation method [] defined as follows:x∈Cand

xn+=αnf(xn) +βnxn+γnBnxn, ∀n≥, (.)

where{αn},{βn}and{γn}are sequences in (, ), andf ∈C.

Throughout this paper, we use the notation:

• for weak convergence and→for strong convergence.

• ωω(xn) ={x:xnix}denotes the weakω-limit set of{xn}.

2 Preliminaries and lemmas

In this section, we begin by recalling some basic facts and lemmas which will be used in the sequel.

LetEbe a real Banach space and letU={x∈E:x= }be the unit sphere ofE. A Ba-nach spaceEis said to bestrictly convexif for anyx,y∈U,

x =y implies x+y 

< .

It is also said to beuniformly convexif for eachε∈(, ], there existsδ>  such that for anyx,y∈U,

x–y ≥ε implies x+y 

<  –δ.

It is known that a uniformly convex Banach space is reflexive and strictly convex. Define a functionδ: [, ]→[, ] called themodulus of convexityofEas follows:

δ(ε) =inf

 –x+y 

:x,y∈E,x=y= ,x–y ≥ε

.

ThenEis uniformly convex if and only ifδ(ε) >  for allε∈(, ]. A Banach spaceEis said to besmoothif the limit

lim t→

x+ty–x

t (.)

exists for allx,y∈U. The norm is said to beuniformly Gâteaux differentiableif fory∈U, the limit is attained uniformly forx∈U. It is said to beFréchet differentiableif forx∈U, the limit is attained uniformly fory∈U. It is said to beuniformly smoothoruniformly Fréchet differentiableif the limit (.) is attained uniformly forx,y∈U. Thenormalized duality mapping J:E→E∗_{is defined by}

J(x) =x∗∈E∗:x,x∗=x=x∗

for allx∈E. It is known thatEis smooth if and only if the duality mapping Jis single valued, and that ifEhas a uniformly Gâteaux differentiable norm,Jis uniformly norm-to-weak∗continuous on each bounded subset ofE. A Banach spaceEis said to satisfy Opial’s condition []. Ifx∈Eandxnx, then

lim sup

n→∞ xn–x<lim supn→∞ xn–y, ∀y∈E,x=y.

LetT:C→C. ThenI–T is demiclosed at  if for all sequence{xn}inC,xnqand

xn–Txn → implyq=Tq. It is known that ifEis uniformly convex,Cis nonempty

Lemma .[] Let E be a smooth Banach space.Then the following hold:

(i) x+y≥ x+ y,J(x)for allx,y∈E; (ii) x+y≤ x+ y,J(x+y)for allx,y∈E.

Lemma .[] In a strictly convex Banach space E,if

x=y=λx+ ( –λ)y

for all x,y∈E andλ∈(, ),then x=y.

Lemma .[] Let{xn}and{zn}be two sequences in a Banach space E such that

xn+=βnxn+ ( –βn)zn, n≥,

where{βn}satisfies the conditions <lim infn→∞βn≤lim supn→∞βn< . Iflim sup_n→∞(zn+–zn–xn+–xn)≤,thenzn–xn →as n→ ∞.

Lemma .[] Let E be a uniformly convex Banach space with a Fréchet differentiable norm.Let C be a closed and convex subset of E,and let{Sn}∞n=be a family of Ln-Lipschitzian self-mappings on C such that ∞_n=(Ln– ) <∞and F=

∞

n=F(Sn)=∅.For arbitrary x∈

C,define xn+=Snxnfor all n≥.Then,for every p,q∈F,limn→∞xn,J(p–q)exists,in particular,for all u,v∈ωω(xn)and p,q∈F,u–v,J(p–q)= .

Lemma .[] Let E be a reflexive and strictly convex Banach space with a uniformly Gâteaux differentiable norm,let C be a nonempty closed convex subset of E,let A:C→C be a continuous strongly pseudocontractive mapping with constant k∈[, ),and let T :

C→E be a continuous pseudocontractive mapping satisfying the weakly inward condition.

If T has a fixed point in C,then the path{xt}defined by

xt=tAxt+ ( –t)Txt

converges strongly to a fixed point q of T as t→,which is a unique solution of the varia-tional inequality

(I–A)q,J(q–p)≤, ∀p∈F(T).

Remark . Lemma . holds ifT:C→C is a nonexpansive mapping andA=f is a

contraction.

The following lemma gives us a nice property of real sequences.

Lemma .[] Assume that{an}is a sequence of nonnegative real numbers such that

an+≤( –cn)an+bn, ∀n≥,

where{cn}is a sequence in(, )and{bn}is a sequence such that (a) ∞_n=cn=∞;

(b) lim sup_n→∞bn

cn ≤or

∞

3 Weak convergence theorem

In this section, we give some properties concerning theB-mapping and then prove a weak convergence theorem for nonexpansive mappings.

Lemma . Let C be a nonempty,closed and convex subset of a strictly convex Banach space E. Let {Ti}Ni= be a finite family of nonexpansive mappings of C into itself such

that N_i=F(Ti) =∅, and let λ,λ, . . . ,λN be real numbers such that  <λi<  for all i= , , . . . ,N– and <λN ≤.Let B be the B-mapping generated by T,T, . . . ,TN and λ,λ, . . . ,λN.Then the following hold:

(i) F(B) =N_i=F(Ti); (ii) Bis nonexpansive.

Proof (i) SinceN_i=F(Ti)⊂F(B) is trivial, it suffices to show thatF(B)⊂

N

i=F(Ti). To

this end, letp∈F(B) andx∗∈N_i=F(Ti). Then we have

p–x∗=Bp–x∗=λN

TNUN–p–x∗

+ ( –λN)

TN–p–x∗

≤λNUN–p–x∗+ ( –λN)p–x∗

=λNλN–

TN–UN–p–x∗

+ ( –λN–)

TN–p–x∗+ ( –λN)p–x∗

≤λNλN–UN–p–x∗+ ( –λNλN–)p–x∗

=λNλN–λN–

TN–UN–p–x∗

+ ( –λN–)

TN–p–x∗

+ ( –λNλN–)p–x∗

≤λNλN–λN–UN–p–x∗+ ( –λNλN–λN–)p–x∗

.. .

=λNλN–· · ·λλ

TUp–x∗

+ ( –λ)

Tp–x∗

+ ( –λNλN–· · ·λ)p–x∗

≤λNλN–· · ·λTUp–x∗+ ( –λNλN–· · ·λ)p–x∗

≤λNλN–· · ·λUp–x∗+ ( –λNλN–· · ·λ)p–x∗

=λNλN–· · ·λλ

Tp–x∗

+ ( –λ)

p–x∗

+ ( –λNλN–· · ·λ)p–x∗

≤λNλN–· · ·λλTp–x∗+ ( –λNλN–· · ·λλ)p–x∗

≤λNλN–· · ·λλp–x∗+ ( –λNλN–· · ·λλ)p–x∗

=p–x∗. (.)

This shows that

p–x∗=λNλN–· · ·λλ

Tp–x∗

+ ( –λ)

p–x∗+ ( –λNλN–· · ·λ)p–x∗,

which turns out to be

p–x∗=λ

Tp–x∗

+ ( –λ)

Again by (.), we see thatp–x∗=Tp–x∗and thus

p–x∗=Tp–x∗=λ

Tp–x∗

+ ( –λ)

p–x∗.

Using Lemma ., we get thatTp=pand henceUp=p. Again by (.), we have

p–x∗=λNλN–· · ·λλ

TUp–x∗

+ ( –λ)

Tp–x∗

+ ( –λNλN–· · ·λ)p–x∗,

which implies that

_p–x∗_=λ



TUp–x∗

+ ( –λ)

Tp–x∗.

From (.) we see thatUp–x∗=TUp–x∗. SinceUp=pandTp=p,

p–x∗=Tp–x∗=λ

Tp–x∗

+ ( –λ)

p–x∗.

Using Lemma ., we get thatTp=pand henceUp=p.

By continuing this process, we can show thatTip=pandUip=pfor alli= , , . . . ,N– .

Finally, we obtain

p–TNp ≤ p–Bp+Bp–TNp

=p–Bp+ ( –λN)p–TNp,

which yields thatp=TNpsincep∈F(B). Hencep=Tp=Tp=· · ·=TNpand thusp∈

N

i=F(Ti).

(ii) The proof follows directly from (i).

Lemma . Let C be a nonempty closed convex subset of a real Banach space E.Let{Ti}Ni=

be a finite family of nonexpansive mappings of C into itself such thatN_i=F(Ti) =∅,and let B be the B-mapping generated by T,T, . . . ,TN andλ,λ, . . . ,λN.Let{λn,i}Ni=be a real

sequence in(, ).For every n∈N,let Bnbe the B-mapping generated by T,T, . . . ,TNand λn,,λn,, . . . ,λn,Nas follows:

Un,=λn,T+ ( –λn,)I,

Un,=λn,TU+ ( –λn,)T,

Un,=λn,TU+ ( –λn,)T,

.. .

Un,N–=λn,N–TN–UN–+ ( –λn,N–)TN–,

Bn=Un,N=λn,NTNUN–+ ( –λn,N)TN–.

Proof Let x∈C and Uk and Un,k be generated by T,T, . . . ,Tk and λ,λ, . . . ,λk, and T,T, . . . ,Tkandλn,,λn,, . . . ,λn,k, respectively. Then

Un,x–Ux=(λn,–λ)(Tx–x)≤ |λn,–λ|Tx–x.

Letk∈ {, , . . . ,N}andM=max{TkUk–x+Tk–x:k= , , . . . ,N}. Then

Un,kx–Ukx=λn,kTkUn,k–x+ ( –λn,k)Tk–x–λkTkUk–– ( –λk)Tk–x

=λn,kTkUn,k–x–λn,kTk–x–λkTkUk–+λkTk–x

≤λn,kTkUn,k–x–TkUk–x+|λn,k–λk|TkUk–x

+|λn,k–λk|Tk–x

≤ Un,k–x–Uk–x+|λn,k–λk|M.

It follows that

Bnx–Bx=Un,Nx–UNx

≤ Un,N–x–UN–x+|λn,N–λN|M

≤ Un,N–x–UN–x+|λn,N––λN–|M+|λn,N–λN|M

.. .

≤ Un,x–Ux+|λn,–λ|M+· · ·+|λn,N––λN–|M+|λn,N–λN|M

≤ |λn,–λ|Tx–x+|λn,–λ|M+· · ·+|λn,N––λN–|M

+|λn,N–λN|M.

Sinceλn,i→λiasn→ ∞(i= , , . . . ,N), we thus complete the proof.

Remark . It is easily seen that for alln∈N,Bnis nonexpansive.

Lemma . Let C be a nonempty closed convex subset of a real Banach space E.Let{Ti}Ni=

be a finite family of nonexpansive mappings of C into itself such thatN_i=F(Ti) =∅.Let

{λn,i}Ni=be a real sequence in(, ).For every n∈N,let Bnbe the B-mapping generated by T,T, . . . ,TN andλn,,λn,, . . . ,λn,N.

Iflimn→∞|λn+,i–λn,i|= for all i= , , . . . ,N,then lim

n→∞Bn+zn–Bnzn=  for each bounded sequence{zn}in C.

Proof Let{zn}be a bounded sequence inC. Forj∈ {, , . . . ,N– }and for someM> ,

we have

Un+,N–jzn–Un,N–jzn

–λn,N–jTN–jUn,N–j–zn– ( –λn,N–j)TN–j–zn

≤λn+,N–jTN–jUn+,N–j–zn–TN–jUn,N–j–zn

+|λn+,N–j–λn,N–j|TN–jUn,N–j–zn

+|λn+,N–j–λn,N–j|TN–j–zn

≤ Un+,N–j–zn–Un,N–j–zn+|λn+,N–j–λn,N–j|M.

Using the relation above, we can show that

Bn+zn–Bnzn=Un+,Nzn–Un,Nzn

≤M

N

j=

|λn+,j–λn,j|+|λn+,–λn,|

zn+Tzn

.

Sincelimn→∞|λn+,i–λn,i|=  for alli= , , . . . ,N, we obtain the desired result.

Using the concept ofB-mapping, we study weak convergence of the sequence generated by Mann-type iteration process (.).

Theorem . Let E be a uniformly convex Banach space having a Fréchet differentiable

norm or that satisfies Opial’s condition. Let C be a nonempty, closed and convex sub-set of E .Let{Ti}Ni= be a finite family of nonexpansive mappings of C into itself such that

N

i=F(Ti) =∅.Let{λn,i}iN= be a real sequence in(, )such thatλn,i→λi(i= , , . . . ,N). For every n∈N,let Bnbe the B-mapping generated by T,T, . . . ,TNandλn,,λn,, . . . ,λn,N. Let{αn}be a sequence in(, )satisfyinglim infn→∞αn( –αn) > .Let{xn}be generated by x∈C and

xn+= ( –αn)xn+αnBnxn, ∀n≥.

Then{xn}converges weakly to x∗∈

N

i=F(Ti).

Proof Letp∈N_i=F(Ti). Thenp=Bnpfor alln≥ and hence

xn+–p ≤( –αn)xn–p+αnBnxn–p ≤ xn–p.

It follows that{xn–p}is nonincreasing; consequently,limn→∞xn–pexists. Assume

xn–p> . SinceEis uniformly convex, it follows (see, for example, []) that

xn+–p ≤ xn–p

 – min{αn,  –αn}δE

xn–Bnxn

xn–p

,

which implies that

αn( –αn)xn–pδE

xn–Bnxn

xn–p

≤min{αn,  –αn}xn–pδE

xn–Bnxn

xn–p

≤  

xn–p–xn+–p

Sincelimn→∞xn–pexists andlim infn→∞αn( –αn) > , by the continuity ofδE, we

havelimn→∞xn–Bnxn= . Sinceλn,i→λi(i= , , . . . ,N), let the mappingB:C→Cbe

generated byT,T, . . . ,TNandλ,λ, . . . ,λN. Then, by Lemma ., we havelimn→∞Bnx– Bx=  for allx∈C. So we have

xn–Bxn ≤ xn–Bnxn+Bnxn–Bxn

≤ xn–Bnxn+ sup z∈{xn}

Bnz–Bz

→.

Since B is nonexpansive and E is uniformly convex, by the demiclosedness principle,

ωω(xn)⊂F(B). Moreover,F(B) =

N

i=F(Ti) by Lemma .(i).

We next show that ωω(xn) is a singleton. Indeed, suppose that x∗,y∗ ∈ωω(xn)⊂

N

i=F(Ti). DefineSn:C→Cby Snx= ( –αn)x+αnBnx, x∈C.

ThenSnis nonexpansive andx∗,y∗∈

∞

n=F(Sn). Using Lemma ., we havelimn→∞xn, J(x∗–y∗)exists. Suppose that{xnk}and{xmk}are subsequences of{xn}such thatxnkx∗

andxmky

∗_{. Then}

_x∗_–y∗

=x∗–y∗,Jx∗–y∗= lim k→∞

xnk–xmk,J

x∗–y∗= .

This shows thatx∗=y∗.

Assume thatEsatisfies Opial’s condition. Letx∗,y∗∈ωω(xn) and{xnk}and{xmk}be

sub-sequences of{xn}such thatxnkx∗andxmky∗. Ifx∗ =y∗, then

lim n→∞xn–x

∗_{= lim}

k→∞xnk–x

∗_{< lim}

k→∞xnk–y

∗_{= lim}

k→∞xmk–y

∗

< lim k→∞

xmk–x

∗_{= lim} n→∞xn–x

∗_,

which is a contradiction. It follows thatx∗=y∗. Thereforexnx∗∈

N

i=F(Ti) asn→ ∞.

This completes the proof.

4 Strong convergence theorem

In this section, we prove a strong convergence theorem for a finite family of nonexpansive mappings in Banach spaces.

Theorem . Let E be a strictly convex and reflexive Banach space having a uniformly

Gâteaux differentiable norm.Let C be a nonempty,closed and convex subset of E.Let{Ti}Ni=

be a finite family of nonexpansive mappings of C into itself such thatN_i=F(Ti) =∅.Let

{λn,i}Ni=be a real sequence in(, )such thatλn,i→λi(i= , , . . . ,N).For every n∈N,let Bnbe the B-mapping generated by T,T, . . . ,TN andλn,,λn,, . . . ,λn,N.Let{αn},{βn}and

{γn}be sequences in(, )which satisfy the conditions: (C) αn+βn+γn= ;

(C)  <lim infn→∞βn≤lim supn→∞βn< .

Let f ∈Cand define the sequence{xn}by x∈C and

xn+=αnf(xn) +βnxn+γnBnxn, ∀n≥.

Then{xn}converges strongly to q∈

N

i=F(Ti),where q is also the unique solution of the variational inequality

(I–f)(q),J(q–p)≤, ∀p∈ N

i=

F(Ti). (.)

Proof We divide the proof into the following steps. Step . We show that{xn}is bounded. Letp∈

N

i=F(Ti). Thenp=Bnpfor alln≥ and

hence, by the nonexpansiveness of{Bn}∞n=, we have

xn+–p=αn

f(xn) –p

+βn(xn–p) +γn(Bnxn–p)

≤αnf(xn) –p+βnxn–p+γnxn–p

≤αnf(xn) –f(p)+αnf(p) –p+ ( –αn)xn–p

≤αnαxn–p+αnf(p) –p+ ( –αn)xn–p

= –αn( –α)

xn–p+αnf(p) –p

≤max

xn–p,



 –αf(p) –p

.

By induction, we can conclude that{xn}is bounded. So are{f(xn)}and{Bnxn}.

Step . We show thatlimn→∞xn+–xn= . To this end, we definezn=xn+_––ββnnxn. From

(.) we have

zn+–zn=

αn+f(xn+) +γn+Bn+xn+

 –βn+

–αnf(xn) +γnBnxn  –βn

= αn+  –βn+

f(xn+) –Bnxn

+ αn

 –βn

Bnxn–f(xn)

+ γn+  –βn+

(Bn+xn+–Bnxn)

≤ αn+

 –βn+

M+ αn  –βn

M+Bn+xn+–Bnxn

≤

αn+

 –βn+

+ αn  –βn

M+Bn+xn+–Bn+xn

+Bn+xn–Bnxn

≤

αn+

 –βn+

+ αn  –βn

M+xn+–xn+Bn+xn–Bnxn

for someM> . It turns out that

zn+–zn–xn+–xn ≤

αn+

 –βn+

+ αn  –βn

From conditions (C), (C) and Lemma ., we have

lim sup n→∞

zn+–zn–xn+–xn

≤.

Lemma . yields thatzn–xn → and hence

xn+–xn= ( –βn)zn–xn →.

Step . We show thatlim_n→∞Bxn–xn= . Indeed, noting that

Bnxn–xn=



γn

(xn+–xn) +αn

xn–f(xn)

,

we have, by (C) and (C),

lim

n→∞Bnxn–xn= .

Let B:C→C be the B-mapping generated by T,T, . . . ,TN andλ,λ, . . . ,λN. So, by

Lemma ., we haveBnx→Bxfor allx∈C. It also follows that

Bxn–xn ≤ Bxn–Bnxn+Bnxn–xn

≤ sup

z∈{xn}

Bz–Bnz+Bnxn–xn

→.

Fort∈(, ), we define a contraction as follows:

Stx=tf(x) + ( –t)Bx.

Then there exists a unique pathxt∈Csuch that

xt=tf(xt) + ( –t)Bxt.

From Lemma ., we know that xt →q ast→, whereq∈F(B). Lemma .(i) also

yields thatq∈F(B) =_iN_=F(Ti). Moreover,qis the unique solution of variational

inequal-ity (.).

Step . We show thatlim sup_n→∞f(q) –q,J(xn–q) ≤. We see that

xt–xn= ( –t)(Bxt–xn) +t

f(xt) –xn

.

It follows, by Lemma .(ii) that

xt–xn≤( –t)Bxt–xn+ t

f(xt) –xn,J(xt–xn)

≤ – t+txt–xn+Bxn–xn



+ tf(xt) –xt,J(xt–xn)

which gives

f(xt) –xt,J(xn–xt)

≤( +t)xn–Bxn

t

xt–xn+xn–Bxn

+t

xt–xn

_.

So we have

lim sup n→∞

f(xt) –xt,J(xn–xt)

≤ t

M (.)

for someM> . SinceEhas a uniformly Gâteaux differentiable norm,Jis norm-to-weak∗ uniformly continuous on bounded subsets ofE. So we have

f(q) –q,J(xn–q) –J(xn–xt)

→ (.)

and

f(q) –f(xt) +xt–q,J(xn–xt)

→ (.)

ast→. On the other hand, we have

f(q) –q,J(xn–q)

=f(xt) –xt,J(xn–xt)

+f(q) –f(xt) +xt–q,J(xn–xt)

+f(q) –q,J(xn–q) –J(xn–xt)

. (.)

Sincelim sup_n→∞andlim sup_t→are interchangeable, using (.)-(.), we obtain

lim sup n→∞

f(q) –q,J(xn–q)

≤.

Step . We show thatxn→qasn→ ∞. In fact, we have

xn+–q=αn

f(xn) –q,J(xn+–q)

+βn

xn–q,J(xn+–q)

+γn

Bnxn–q,J(xn+–q)

≤αnαxn–qxn+–q+αn

f(q) –q,J(xn+–q)

+βnxn–qxn+–q+γnxn–qxn+–q

= –αn( –α)

xn–qxn+–q+αn

f(q) –q,J(xn+–q)

≤  

 –αn( –α)

xn–q+xn+–q

+αn

f(q) –q,J(xn+–q)

,

which implies that

xn+–q≤

 –αn( –α)

 +αn( –α)

xn–q+

αn

 +αn( –α)

f(q) –q,J(xn+–q)

=

 – αn( –α)  +αn( –α)

xn–q

+ αn  +αn( –α)

f(q) –q,J(xn+–q)

Putcn=_+ααnn(–(–αα)) andbn=

αn

+αn(–α)f(q) –q,J(xn+–q). So it is easy to check that{cn}is

a sequence in (, ) such that ∞_n=cn=∞andlim supn→∞bcnn≤. Hence, by Lemma .,

we conclude thatxn→qasn→ ∞. This completes the proof.

Competing interests

The author declares that he has no competing interests.

Acknowledgements

The author wishes to thank editor/referees for valuable suggestions and Professor Suthep Suantai for the guidance. This research was supported by the Thailand Research Fund, the Commission on Higher Education, and University of Phayao under Grant MRG5580016.

Received: 16 April 2013 Accepted: 5 July 2013 Published: 22 July 2013

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doi:10.1186/1687-1812-2013-198