A new general iterative algorithm with Meir-Keeler contractions for variational inequality problems in q-uniformly smooth Banach spaces

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R E S E A R C H

Open Access

A new general iterative algorithm with

Meir-Keeler contractions for variational

inequality problems in

q

-uniformly smooth

Banach spaces

Jinlin Guan, Changsong Hu

*

_{and Xin Wang}

*_{Correspondence:}

[email protected] School of Mathematics and Statistics, Hubei Normal University, Huangshi, 435002, China

Abstract

In this paper, we generalize the iterative scheme and extend the space studied in (Fixed Point Theory and Applications 2012:46, 2012). Further, we prove some strong convergence theorems of the new iterative scheme for variational inequality problems inq-uniformly smooth Banach spaces under very mild conditions. Our results improve and extend corresponding ones announced by many others.

MSC: 47H09; 47H10

Keywords: strong convergence; iterative algorithm; Meir-Keeler contractions; ﬁxed point;k-strict pseudo-contractions; Banach spaces

1 Introduction

Throughout this paper, we denote byXandX∗a real Banach space and the dual space of X, respectively. LetCbe a nonempty closed convex subset ofX.

The duality mappingJ:X→X∗ _{is deﬁned by} _{J(x) =}_{_x∗_∈_X∗_:_x,_x∗₌_x_,_x∗₌

x},∀x∈X. It is well known that ifXis smooth, thenJis single-valued, which is denoted byj. Letq>  be a real number. The generalized duality mappingJq:X→X

∗

is deﬁned by

Jq(x) =x∗∈X∗:x,x∗=xq,x∗=xq–,

for allx∈X, where·,· denotes the generalized duality pairing betweenXandX∗. In particular,J=Jis called the normalized duality mapping andJq(x) =xq–J(x) forx= .

It is well known that ifXis smooth, thenJqis single-valued, which is denoted byjq. Recall that a mappingf :C→Cis a contraction onCif there exists a constantα∈(, ) such that

f(x) –f(y)≤αx–y, ∀x,y∈C. (.)

A mappingW:C→Cis said to be nonexpansive if

W(x) –W(y)≤ x–y, ∀x,y∈C. (.)

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A mappingF:E→Cis said to beL-Lipschitzian if there exists a positive constantL such that

F(x) –F(y)≤Lx–y, ∀x,y∈C. (.)

A mappingF:E→Cis said to beη-strongly accretive if there existjq(x–y)∈Jq(x–y) andη>  such that

Fx–Fy,jq(x–y)≥ηx–yq, ∀x,y∈C. (.)

Without loss of generality, we can assume thatη∈(, ] andL∈[,∞).

Recall that ifCandDare nonempty subsets of a Banach spaceXsuch thatCis nonempty closed convex and D⊂C, then a mappingP:C→Dis sunny [] provided P(x+t(x– P(x))) =P(x) for allx∈Candt≥, wheneverx+t(x–P(x))∈C. A mappingP:C→Dis called a retraction ifPx=xfor allx∈D. Furthermore,Pis a sunny nonexpansive retraction fromContoDifPis a retraction fromContoDwhich is also sunny and nonexpansive. A subset DofC is called a sunny nonexpansive retraction ofC if there exists a sunny nonexpansive retraction fromContoD.

Let{Bn}be a family of mappings from a subsetCof a Banach spaceXinto itself with

∞

n=F(Bn)=∅. We say that{Bn}satisﬁes the AKTT-condition if for each bounded subset DofC,

∞

n=

sup

ω∈D

Bn+ω–Bnω<∞.

Proposition .(Banach []) Let(X,d)be a complete metric space,and let f be a contrac-tion on X,then f has a unique ﬁxed point.

Proposition .(Meir and Keeler []) Let(X,d)be a complete metric space,and letφbe a Meir-Keeler contraction(MKC,for short)on X,that is,for everyε> ,there existsδ>  such that d(x,y) <ε+δimplies d(φ(x),φ(y)) <εfor all x,y∈X.Thenφhas a unique ﬁxed point.

This proposition is one of generalizations of Proposition ., because the contractions are Meir-Keeler contractions.

Proposition .[] Let C be a closed convex subset of a smooth Banach space X.LetC be a nonempty subset of C.Let QC:C→C be a retraction,and let J be the normalized duality mapping on X.Then the following are equivalent:

(i) QCis sunny and nonexpansive.

(ii) QCx–QCy_≤_x_–_y,_J(QCx_–_QCy)_,_∀_x,_y_∈_C_.

(iii) x–QCx,J(y–QCx) ≤,∀x∈C,y∈C.

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dynamics and it is experiencing an explosive growth in both theory and applications. Sev-eral numerical methods have been developed for solving variational inequalities and re-lated optimization problems; see [–] and the references therein.

LetC be a nonempty closed convex subset of a real Hilbert spaceH. Recall that the classical variational inequality is to ﬁnd anx∗such that

Fx∗,x–x∗≥, ∀x∈C, (.)

where F :C→C is a nonlinear mapping. The set of solutions of (.) is denoted by

VI(F,C).

In , Yaoet al.[] modiﬁed Mann’s iterative scheme by using the viscosity approx-imation method which was introduced by Moudaﬁ []. More precisely, they introduced and studied the following iterative algorithm:

⎧ ⎪ ⎨ ⎪ ⎩

x=x∈C,

yn=βnxn+ ( –βn)Txn,

xn+=αnf(xn) + ( –αn)yn, n≥,

(.)

where T is a nonexpansive mapping ofK into itself andf is a contraction onK. They obtained a strong convergence theorem under some mild restrictions on the parameters. Zhou [] and Qinet al.[] modiﬁed normal Mann’s iterative process (.) fork-strictly pseudo-contractions to have strong convergence in Hilbert spaces. Qinet al.[] intro-duced the following iterative algorithm scheme:

⎧ ⎪ ⎨ ⎪ ⎩

x=x∈K,

yn=PK[βnxn+ ( –βn)Txn],

xn+=αnf(xn) + ( –αnA)yn, n≥,

(.)

whereT is ak-strictly pseudo-contraction,f is a contraction andAis a strong positive linear bounded operator,PK is the metric projection. They proved, under certain appro-priate assumptions on the sequences{αn}and{βn}, that{xn}deﬁned by (.) converges strongly to a ﬁxed point of thek-strictly pseudo-contraction, which solves some varia-tional inequality.

Very recently, Songet al.[] introduced the following iteration process:

⎧ ⎪ ⎨ ⎪ ⎩

x=x∈C,

yn=PC[βnxn+ ( –βn)∞_i_=μ(_in)Tixn],

xn+=αnφ(xn) +γnxn+ (( –γn)I–αnF)yn, n≥,

(.)

whereTi is aki-strictly pseudo-contraction,φ is an MKC contraction andF:C→Cis anL-Lipschitzian andη-strongly monotone mapping in a Hilbert space,PCis the metric projection. Under certain appropriate assumptions on the sequences{αn},{βn},{γn}and

{μn_i}, the sequence{xn}defined by (.) converges strongly to a common fixed point of an infinite family ofki-strictly pseudo-contractions, which solves some variational inequality.

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Question  Can the projectionPC in Song [] be changed to the sunny nonexpansive retractionQCand be put to other place of the iteration process?

Question  Can we extend the iterative scheme of algorithm (.) to a more general iter-ative scheme?

Question  Can we remove the very strict conditionC+C⊂Cwhich is necessary in

Lemma . and Theorem . of Song []?

The purpose of this paper is to give aﬃrmative answers to these questions mentioned above. In this paper we study a new general iterative scheme as follows:

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

x=x∈C,

yn=βnxn+ ( –βn)∞_i_=μ(_in)Tixn, xn+=QC[αnγ φ(xn) +γn(αI+ ( –α)

_∞

i=μ (n)

i Ti)xn+ (( –γn)I–αnF)yn], n≥,

(.)

whereTiis aλi-strictly pseudo-contraction,φ is an MKC contraction,QC is the sunny nonexpansive retraction andF:X→Cis anL-Lipschitzian andη-strongly accretive map-ping in aq-uniformly smooth Banach space. Under some suitable assumptions on the se-quences{αn},{βn},{γn}and{μ(in)}, the sequence{xn}defined by (.) converges strongly to a common fixed point of an infinite family ofλi-strictly pseudo-contractions, which solves some variational inequality.

2 Preliminaries

In this section, we ﬁrst recall some notations.Tis said to be aλ-strict pseudo-contraction in the terminology of Browder and Petryshyn [] if there exists a constantλ∈[, ) such that

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

for everyx,y∈Cand for somejq(x–y)∈Jq(x–y). It is clear that (.) is equivalent to the following:

(I–T)x– (I–T)y,jq(x–y)≥λ(I–T)x– (I–T)yq. (.)

A Banach spaceXis said to be strictly convex if wheneverxandyare not collinear, then

x+y<x+y.

Then the modulus of convexity ofXis deﬁned by

δX() =inf

 – 

x+y:x,y ≤,x–y ≥

for all∈[, ].Xis said to be uniformly convex ifδX() =  andδX() >  for all  <≤, and ifδX()≥cp withp≥, thenXis said to bep-uniformly convex. A Hilbert space His -uniformly convex, whileLp_is_max_{_{p, }_}_{-uniformly convex for every}_p_{> . Let}_ρ

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[,∞)→[,∞) be the modulus of smoothness ofXdeﬁned by

ρX(t) =sup

 

x+y+x–y–  :x∈S(X),y ≤t

.

A Banach spaceXis said to be uniformly smooth ifρX(t)

t → ast→. A Banach spaceX is said to beq-uniformly smooth if there exists a ﬁxed constantc>  such thatρX(t)≤ctq withq> . A typical example of uniformly smooth Banach spaces isLp_{, where}_p_{> . More} precisely,Lp_is_min_{_{p, }_}_{-uniformly smooth for every}_p_{> .}

The norm of a Banach spaceXis said to be Gâteaux diﬀerentiable if the limit

lim

t→

x+ty–x

t (.)

exists for allx,yon the unit sphereS(X) ={x∈X:x= }. If, for eachy∈S(X), the limit (.) is uniformly attained forx∈S(X), then the norm ofXis said to be uniformly Gâteaux diﬀerentiable. The norm ofXis said to be Fréchet diﬀerentiable if, for eachx∈S(X), the limit (.) is attained uniformly fory∈S(X).

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

Lemma .[] Letφbe an MKC on a convex subset C of a Banach space X.Then,for eachε> ,there exists r∈(, )such that

x–y ≥εimpliesφx–φy ≤rx–y, ∀x,y∈C.

Lemma .[] Let X be a real q-uniformly smooth Banach space,then there exists a constant Cq> such that

x+yq≤ xq+qy,Jq(x)+Cqyq for all x,y∈X.

Lemma .[] Let{αn}be a sequence of nonnegative numbers satisfying the property

αn+≤( –γn)αn+bn+γncn, n≥, where{γn},{bn},{cn}satisfy the restrictions:

(i) lim sup_n_→∞γn= ,

∞

n=γn=∞;

(ii) bn≥,∞_n_=bn<∞;

(iii) lim sup_n_→∞cn≤.Thenlimn→∞αn= .

Lemma .[] Let C be a nonempty convex subset of a real q-uniformly smooth Banach space X,and let T:C→C be aλ-strict pseudo-contraction.Forα∈(, ),we deﬁne Tαx= ( –α)x+αTx.Then,asα∈(,μ],μ=min{, (_Cqqλ)q}_,_T

α:C→C is nonexpansive such that F(Tα) =F(T).

Lemma .[] Let q> ,then the following inequality holds:

ab≤  qa

q₊q–  q b

q q–

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Lemma . Let F be an L-Lipschitzian andη-strongly accretive operator on a nonempty closed convex subset C of a real q-uniformly smooth Banach space X with <ηq≤and  <t< (_Cqqη_Lq)



q–_._{Then G}_{= (I}_–_{tF) :}_C→_{X is a contraction with contraction coeﬃcient}

τt=  –_q(qtη–CqLqtq).

Proof From the deﬁnition ofη-strongly accretive andL-Lipschitzian operator, we have

Gx–Gyq=x–y–t(Fx–Fy)q

≤ x–yq–qtFx–Fy,jq(x–y)+CqtqFx–Fyq

≤ x–yq–qtηx–yq+CqLqtqx–yq

= –qtη–CqLqtqx–yq.

Therefore, we have

Gx–Gy ≤ –qtη–CqLqtq

 q_x_–_y

≤

 – q

qtη–CqLqtq

x–y

for allx,y∈C. From  <ηq≤ and  <t< (_CqLqηq) 

q–_{, we have  <  –} 

q(qtη–CqLqtq) <  and

Gx–Gy ≤τtx–y,

whereτt=  –_q(qtη–CqLqtq)∈(, ). Hence,Gis a contraction with contraction

coeﬃ-cientτt. This completes the proof.

Lemma .([], Demiclosedness principle) Let C be a nonempty closed convex subset of a reﬂexive Banach space X which satisﬁes Opial’s condition,and suppose that T:C→X is nonexpansive.Then the mapping I–T is demiclosed at zero,that is,xnx,xn–Txn→ implies x=Tx.

Lemma . Let C be a closed convex subset of a smooth Banach space X.Let C be a

nonempty subset of C.Let QC:C→C be a retraction,and let j,jqbe the normalized du-ality mapping and generalized dudu-ality mapping on X,respectively.Then the following are equivalent:

(i) QCis sunny and nonexpansive.

(ii) QCx–QCy≤ x–y,j(QCx–QCy),∀x,y∈C.

(iii) x–QCx,j(y–QCx) ≤,∀x∈C,y∈C.

(iv) x–QCx,jq(y–QCx) ≤,∀x∈C,y∈C.

Proof From Proposition ., we have (i)⇔(ii)⇔(iii). We need only to prove (iii)⇔(iv). Indeed, if y–QCx= , it follows from the fact jq(x) =xq–_{j(x) that} _x_–_QCx,_j(y_–

QCx) ≤⇔ x–QCx,jq(y–QCx) ≤,∀x∈C,y∈C.

Ify–QCx= , thenx–QCx,j(y–QCx)=x–QCx,jq(y–QCx)= ,∀x∈C,y∈C. This

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Lemma .[] Suppose that{Bn}satisﬁes the AKTT-condition,then for each bounded subset D of C:

(i) {Bn}converges strongly to some point inCfor eachx∈C;

(ii) Furthermore,if the mappingB:C→Cis deﬁned byBx=limn→∞Bnxfor allx∈D,

thenlimn→∞supω∈DBω–Bnω= .

Lemma . Let C be a closed convex subset of a reflexive Banach space X which admits a weakly sequentially continuous duality mapping jq from X to X∗. Let S:C→C be a nonexpansive mapping with F(S)=∅andφbe an MKC on C.Suppose that F:C→X is anη-strongly accretive and L-Lipschitzian mapping with coefficient andη>γ > .Then the sequence{xt}defined by xt=QC[tγ φ(xt) + ( –tF)Sxt]converges strongly as t→to a fixed pointx of S,which solves the variational inequality

(F–γ φ)x,jq(x–z)≤, ∀z∈F(S). (.)

Proof The definition of{xt}is a good definition. Indeed, from the definition of MKC, we can see that an MKC is also a nonexpansive mapping. Consider a mappingLtonCdefined by

Ltx=QCtγ φ(x) + (I–tF)Sx, x∈C.

It is easy to see thatLtis a contraction when  <t< (q_Cq(η–_Lγq)) 

q–_{. Indeed, by Lemmas . and}

., we have

Ltx–Lty=QCtγ φ(x) + (I–tF)Sx–QCtγ φ(y) + (I–tF)Sy

≤tγ φ(x) + (I–tF)Sx–tγ φ(y) – (I–tF)Sy

≤tγφ(x) –φ(y)+(I–tF)Sx– (I–tF)Sy

≤tγφ(x) –φ(y)+τtSx–Sy

≤tγx–y+τtx–y

≤θtx–y,

whereθt=tγ+τt∈(, ). HenceLthas a unique ﬁxed point, denoted byxt, which uniquely solves the ﬁxed point equation

xt=QCtγ φ(xt) + (I–tF)Sxt. (.)

Next we show the uniqueness of a solution of the variational inequality (.). Suppose thatx∈F(S) andxˆ∈F(S) are solutions to (.), then, without loss of generality, we may assume that there is a numberεsuch thatˆx–x ≥ε. Then, by Lemma ., there is a numberr∈(, ) such thatφxˆ–φx ≤rˆx–x. From (.) we have

(F–γ φ)x,jq(x–x)ˆ ≤, (.)

(F–γ φ)x,ˆ jq(xˆ–x)

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Adding up (.) and (.), we obtain

(F–γ φ)xˆ– (F–γ φ)x,jq(xˆ–x)≤. (.)

Meanwhile, we notice that

(F–γ φ)xˆ– (F–γ φ)x,jq(xˆ–x)

=Fxˆ–Fx,jq(xˆ–x)

–γφ(x) –ˆ φ(x),jq(xˆ–x)

≥ηˆx–xq–γφ(x) –ˆ φ(x)ˆx–xq–

≥ηˆx–xq–γrˆx–xˆx–xq–

≥ηˆx–xq–γrˆx–xq

= (η–γr)ˆx–xq

≥(η–γr)ε

> .

Thusxˆ=xand the uniqueness is proved. Below, we usexto denote the unique solution of (.).

First, we prove that{xt}is bounded.

Assume that  <t<η_L–γ for∀z∈F(S), ﬁxedεfor eacht.

Case . (xt–z<ε). In this case, we can see easily that{xt}is bounded.

Case . (xt–z ≥ε). In this case, by Lemma ., there is a numberr∈(, ) such that

φ(xt) –φ(p)<rxt–p, then we have

xt–z=QCtγ φ(xt) + (I–tF)Sxt–z

=tγ φ(xt) + (I–tF)Sxt–z

=tγ φ(xt) –Fz+ (I–tF)Sxt– (I–tF)z

≤tγ φ(xt) –p+τtxt–z

≤tγ φ(xt) –γ φ(z)+tγ φ(z) –Fz+τtxt–z

≤tγrxt–z+tφ(z) –Fz+τtxt–z,

which impliesxt–z ≤ γ φ_η_–(z_γ)–z. Thus{xt}is bounded. Then, we prove thatxt→x(x∈F(S)) ast→.

SinceXis reﬂexive and{xt}is bounded, there exists a subsequence{xtn}of{xt}such that xtnx∗. Settingyt=tγ φ(xt) + (I–tF)Sxt, we obtainxt=QCyt.

We claimxtn–x∗ →. It follows from Lemma . that

yt–QCyt,jqx∗–QCyt≤, (.)

then we have

xtm–x∗q=QCytm–ytm,jqxtm–x∗+ytm–x∗,jqxtm–x∗

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=(I–tmF)Sxtm– (I–tmF)x∗,jqxtm–x∗+tmγ φxtm–Fx∗,jqxtm–x∗

≤τmxtm–x∗ q

+tmγ φxtm–Fx∗,jqxtm–x∗,

which implies that

xtm–x∗q≤ tm  –τm

γ φxtm–Fx∗,jqxtm–x∗. (.)

Sincetm→ asm→ ∞, by (.) we obtain thatxtm→x∗. Hence, we havextn→x∗. Now, we prove thatx∗solves the variational inequality (.).

Since

xt=QCyt=QCyt–yt+tγ φ(xt) + (I–tF)Sxt, (.)

we get that

(F–γ φ)xt=

t(QCyt–yt) – 

t(I–S)xt+ (Fxt–FSxt). (.)

Notice that

(I–S)xt– (I–S)z,jq(xt–z)

=xt–z,jq(xt–z)

–Sxt–Sz,jq(xt–z)

≥ xt–zq–Sxt–Szxt–zq–

≥ xt–zq–xt–zq

= .

Then, forz∈F(S),

(F–γ φ)xt,jq(xt–z)=  t

QCyt–yt,jq(xt–z)– t

(I–S)xt,jq(xt–z)

+Fxt–FSxt,jq(xt–z)

=  t

QCyt–yt,jq(xt–z)

– t

(I–S)xt– (I–S)z,jq(xt–z)

+Fxt–FSxt,jq(xt–z)

≤Fxt–F(Sxt),jq(xt–z)

≤Mxt–Sxt, (.)

whereM=sup_n_≥_{Lxt–zq–}<∞. Notice xt–Sxt=t

γ φ(xt) –FSxt

.

Thus, we have

xt–Sxt→ ast→.

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Hencex=x∗by uniqueness. Thus, we have shown that every cluster point of{xt}(att→)

equalsx, thereforext→xast→.

Lemma . Let X be a q-uniformly smooth Banach space,and let C be a nonempty convex subset of X.Assume that Ti:C→C is a countable family ofλi-strict pseudo-contractions for some≤λi< andinf{λi:i∈N}> such thatF:=

∞

i=F(Ti)=∅.Assume that{μi} is a positive sequence such that∞_i_=μi= .Then

∞

i=μiTi:C→C is aλ-strict

pseudo-contraction withλ=inf{λi:i∈N}and F(

∞

i=μiTi) =

∞

i=F(Ti).

Proof LetHnx=μTx+μTx+· · ·+μnTnx, where∞i=μi= . ThenHn:C→Xis a

λ-strict pseudo-contraction withλ=min{λi: ≤i≤n}. Step . We ﬁrstly prove the case ofn= .

(I–H)x– (I–H)y,jq(x–y)

=μ(I–T)x+μ(I–T)x–μ(I–T)y–μ(I–T)y,jq(x–y)

=μ

(I–T)x– (I–T)y,jq(x–y)

+μ

(I–T)x– (I–T)y,jq(x–y)

≥μλ(I–T)x– (I–T)y

q

+μλ(I–T)x– (I–T)y

q

≥λμ(I–T)x– (I–T)y

q

+μ(I–T)x– (I–T)y

q

=λ(I–H)x– (I–H)yq,

whereλ=min{λi:i= , }, which shows thatH:C→Cis aλ-strict pseudo-contraction.

Using the same means, our proof method can easily carry over to the general finite case. Step . We prove the infinite case. From the definition ofλ-strict pseudo-contraction, we have

(I–Tn)x– (I–Tn)y,jq(x–y)≥λ(I–Tn)x– (I–Tn)yq, (.)

then we obtain

(I–Tn)x– (I–Tn)y≤



λ



q–

x–y. (.)

Takingp∈F(Tn), it follows from (.) that

(I–Tn)x=(I–Tn)x– (I–Tn)p≤



λ



q–

x–p. (.)

Thus, for∀x∈X, if∞_i_=F(Ti)=∅withμi>  and

∞

i=μi= , then

∞

i=μiTistrongly

converges. Let

Hx=

∞

i=

(11)

then we obtain

Hx=

∞

i=

μiTix= lim

n→∞

n

i=

μiTix= lim

n→∞



n

i=μi n i= μiTix. Therefore

(I–H)x– (I–H)y,jq(x–y)

= lim

n→∞

I–_n i=μi

n

i=

μiTi

x+

I–_n i=μi

n

i=

μiTi

y,jq(x–y)

= lim

n→∞ 

n

i=μi n

i=

μi

(I–Ti)x– (I–Ti)y,jq(x–y)

≥ lim

n→∞



n

i=μi n

i=

μiλ(I–Ti)x– (I–Ti)y q

≥λ lim

n→∞

I–_n i=μi

n

i=

μiTi

x–

I–_n i=μi

n

i=

μiTi

y q

=λ(I–H)x– (I–H)yq.

Thus,His aλ-strict pseudo-contraction. Step . We proveF(∞_i_=μiTi) =∞_i_=F(Ti).

Letx=∞_i_=μiTix, then, forp∈∞_i_=F(Ti), we obtain

x–pq=x–p,jq(x–p)

=

_∞

i=

μiTix–p,jq(x–p)

= ∞ i= μi

Tix–p,jq(x–p)

≤ x–pq–λ

∞

i=

μix–Tixq,

whereλ=inf{λi:i∈N}. Thus, we obtainx=Tix, it follows thatx∈

∞

i=F(Ti). 3 Main results

Lemma . Let C be a nonempty closed convex subset of a q-uniformly smooth Banach

space X.Let QCbe the sunny nonexpansive retraction from X onto C,and letφbe an MKC on C.Let F:C→X beη-strongly accretive and L-Lipschitzian with <γ <η,and let Ti: C→C be aλi-strictly pseudo-contractive non-self-mapping such thatF:=∞_i_=F(Ti)=∅. Assumeλ=inf{λi:i∈N}> .Let{xn}be a sequence of C generated by(.).We assume that the following parameters are satisﬁed:

(i)  <αn< ,∞i=αn=∞,limn→∞αn= ,∞i=|αn+–αn|<∞;

(ii)  <  – (_Cqqλ)q ≤_β n< ,

_∞

i=|βn+–βn|<∞;

(12)

(iv)  <γn<a< ,

∞

i=|γn+–γn|<∞,αn+βnL+αγn++βn>αn+L+βnγn+;

(v)  –b≤α< ,b:=min{, (_Cqqλ)q}_. Thenlimn→∞xn+–xn= .

Proof LetBn=

∞

i=μ (n)

i Ti, by Lemma . we obtain that for eachn≥,Bnis aλ-strict pseudo-contraction onCandF(Bn) =∞_i_=F(Ti). Further, we can get that

∞

n=

Bn+x–Bnx=

∞

n=

∞

i=

μ(_in+)–μ_i(n)Tix<∞,

thus{Bn}∞_n_= satisﬁes the AKTT-condition. LetWn=αI+ ( –α)∞_i_=μ(_in)Ti, whereα∈

[ –b, ),b:=min{, (q_Cqλ)q}_{. From Lemma . and Lemma . we have that}_W

nis a non-expansive mapping andF(Wn) =∞_i_=F(Ti) =F, then the iterative algorithm (.) can be rewritten as follows:

⎧ ⎪ ⎨ ⎪ ⎩

x=x∈C,

yn=βnxn+ ( –βn)Bnxn,

xn+=QC[αnγ φ(xn) +γnWnxn+ (( –γn)I–αnF)yn], n≥.

(.)

We divide the rest of the proof into two parts.

Step . We will prove that the sequence{xn}is bounded. Letting

Lnx=βnx+ ( –βn)Bnx,

from Lemma . and condition (ii), we get thatLn:C→Cis nonexpansive. Taking a point p∈∞_i_=F(Ti), we haveLnp=pandp∈F(Wn). Therefore, we obtain

yn–p=Lnxn–p ≤ xn–p.

From the deﬁnition of MKC and Lemma ., for anyε> , there is a numberrε∈(, ), if

xn–p<ε, thenφ(xn) –φ(p)<ε; ifxn–p ≥ε, thenφ(xn) –φ(p) ≤rεxn–p. It follows from (.) and Lemma . that

xn+–p

=QC

αnγ φ(xn) +γnWnxn+

( –γn)I–αnF

yn

–p

≤αnγ φ(xn) +γnWnxn+

( –γn)I–αnFyn–p

=αn

γ φ(xn) –Fp+γn(Wnxn–p) +( –γn)I–αnFyn–( –γn)I–αnFp

≤

 –γn–

αnη–

CqLqαqn q( –γn)q–

xn–p+γnxn–p+αnγ φ(xn) –Fp

≤

 –

αnη–

CqLqαnq q( –γn)q–

xn–p+αnγmax

rxn–p,ε+αnγ φ(p) –Fp

=max

 –

αnη–

CqLq_αq n q( –γn)q–

(13)

 –

αnη–

CqLqαnq q( –γn)q–

xn–p+αnγ ε+αnγ φ(p) –Fp

=max

 –

αnη–

CqLqαnq

q( –γn)q––αnγr

xn–p+αnγ φ(p) –Fp,

 –

αnη–

CqLq_αq n q( –γn)q–

xn–p+αnγ ε+αnγ φ(p) –Fp

.

By induction, we obtain

xn+–p

≤max

 –αn

η– CqL q_αq–

n q( –γn)q–

–γr

xn–p

+αn

n q( –γn)q–

–γr

γ φ(p) –Fp

η– CqLqα q– n q(–γn)q– –γr

,

 –αn

n q( –γn)q–

xn–p

+αn

n q( –γn)q–

γ ε+γ φ(p) –Fp

η– CqLqα q– n q(–γn)q–

.

Hence, we obtain

xn–p ≤max

x–p,M

, n≥,

whereMis a constant such that

M=max

sup

n≥

γ φ(p) –Fp

η– CqLqα q– n q(–γn)q– –γr

,sup

γ ε+γ φ(p) –Fp

η– CqLqα q– n q(–γn)q–

,

which implies that{xn}is bounded, so are{yn}and{Lnxn}. Step . We claim thatxn+–xn → asn→ ∞. From (.) we have

xn+–xn+

=QCαn+γ φ(xn+) +γn+Wn+xn++

( –γn+)I–αn+F

Ln+xn+

–QC

Lnxn

≤αn+γ φ(xn+) +γn+Wn+xn++

( –γn+)I–αn+F

Ln+xn+

–αnγ φ(xn) –γnWnxn–

Lnxn

=( –γn+)I–αn+F

Ln+xn+–

( –γn)I–αnFLnxn

+αn+γ φ(xn+) –αnγ φ(xn)

+ (γn+Wn+xn+–γnWnxn)

≤( –γn+)I–αn+F

Ln+xn+–

( –γn+)I–αn+F

Ln+xn

+( –γn+)I–αn+F

Ln+xn–

(14)

+αn+γ φ(xn+) –αnγ φ(xn)+γn+Wn+xn+–γnWnxn

≤

 –γn+–αn+

n+

q( –γn)q–

xn+–xn

+ ( –γn+)Ln+xn–Lnxn+|γn+–γn|Lnxn

+αn+FLn+xn–FLnxn+|αn+–αn|FLnxn+αn+γ φ(xn+) –αn+γ φ(xn)

+αn+γ φ(xn) –αnγ φ(xn)+γn+Wn+xn+–γn+Wn+xn

+γn+Wn+xn–γnWnxn

≤

 –γn+–αn+

n+

q( –γn)q–

xn+–xn

+ ( –γn+)Ln+xn–Lnxn+|γn+–γn|Lnxn

+αn+FLn+xn–FLnxn+|αn+–αn|FLnxn+αn+γxn+–xn

+|αn+–αn|γ φ(xn)+γn+xn+–xn

+γn+Wn+xn–Wnxn+|γn+–γn|Wnxn

≤

 –αn+

n+

q( –γn)q––γ

xn+–xn

+ ( –γn+)Ln+xn–Lnxn+γn+Wn+xn–Wnxn

+|γn+–γn|

Lnxn+Wnxn+αn+LLn+xn–Lnxn

+|αn+–αn|

FLnxn+γ φ(xn)

=

 –αn+

n+

q( –γn)q––γ

xn+–xn

+|αn+–αn|

FLnxn+γ φ(xn)

+|γn+–γn|

Lnxn+Wnxn

+ ( +αn+L–γn+)Ln+xn–Lnxn

+γn+Wn+xn–Wnxn. (.)

Next, we estimateLn+xn–LnxnandWn+xn–Wnxn. Notice that

Ln+xn–Lnxn=βn+xn+ ( –βn+)Bn+xn

–βnxn+ ( –βn)Bnxn

≤ |βn+–βn|xn–Bn+xn+ ( –βn)Bn+xn–Bnxn, (.)

Wn+xn–Wnxn=αxn+Bn+xn–αxn+ ( –α)Bnxn

= ( –α)Bn+xn–Bnxn. (.) Substituting (.) and (.) into (.) and using condition (iv), we have

xn+–xn+ ≤

 –αn+

n+

q( –γn)q– –γ

xn+–xn+M

+ – (αn+βnL+αγn++βn–αn+L–βnγn+)

(15)

≤

 –αn+

n+

q( –γn)q––γ

xn+–xn+M

+Bn+xn–Bnxn, (.) whereMis an appropriate constant such that

M≥ xn–Bn+xn+FLnxn+Lnxn+γφ(xn)+Wnxn for alln. Since{Bn}satisﬁes the AKTT-condition, we get that

∞

i=

Bn+xn–Bnxn<∞.

Noticing conditions (i), (iii) and (iv) and applying Lemma . to (.), we obtain

lim

n→∞xn+–xn= . (.)

This completes the proof.

Lemma . Let C be a nonempty closed convex subset of a q-uniformly smooth Banach

space X.Let QCbe the sunny nonexpansive retraction from X onto C,and letφbe an MKC on C.Let F:C→X beη-strongly accretive and L-Lipschitzian with <γ <η,and let Ti: C→C be aλi-strictly pseudo-contractive non-self-mapping such thatF:=∞_i_=F(Ti)=∅. Assumeλ=inf{λi:i∈N}> .Let{xn}be a sequence of C generated by(.).We assume that the parameters{αn},{βn},{μi(n)}and{γn}satisfy the conditions(i), (ii), (iii), (iv), (v)in Lemma.and(vi)limn→∞βn=α.Then{xn}converges strongly tox∈F,which solves the following variational inequality:

γ φ(x) –Fx,jq(p–x)

≤, ∀p∈

∞

i=

F(Ti).

Proof The proof of the lemma will be split into three parts.

Step . We will prove thatlimn→∞Txn–xn= , whereT:C→Cis deﬁned byTx=

αx+ ( –α)BxandBx=limn→∞Bnx. From (.) we have

Lnxn–xn+=QC

Lnxn

–Lnxn

≤αnγ φ(xn) +γnWnxn+

Lnxn–Lnxn

=αnγ φ(xn) –αnFLnxn+γnWnxn–γnLnxn

≤αnγ φ(xn) –FLnxn+γnWnxn–Lnxn

=αnγ φ(xn) –FLnxn+γnαxn+ ( –α)Bnxn–βnxn– ( –βn)Bnxn

=αnγ φ(xn) –FLnxn+γn|βn–α|xn–Bnxn.

Using conditions (i) and (vi), we obtain

lim

(16)

Since{Bn}∞_n_=satisﬁes the AKTT-condition, from Lemma . we can obtainlimn→∞Bnx–

Bx= . Furthermore, notice that

Bx–By,jq(x–y)= lim

n→∞

Bnx–Bny,jq(x–y)

≤ lim

n→∞

Bnx–Bnyq–λ(I–Bn)x– (I–Bn)yq

=Bx–Byq–λ(I–B)x– (I–B)yq,

therefore, we deduce thatB:C→Cis aλ-strict pseudo-contraction. Applying Lemma ., we obtain thatTis nonexpansive withF(T) =F(B). Notice that

Txn–xn

≤ Lnxn–Txn+Lnxn–xn++xn+–xn

=βnxn+ ( –βn)Bnxn–αxn– ( –α)Bxn+Lnxn–xn++xn+–xn

=(βn–α)(xn–Bnxn) + ( –α)(Bnxn–Bxn)+Lnxn–xn++xn+–xn

≤ |βn–α|xn–Bnxn+ ( –α)Bnxn–Bxn+Lnxn–xn++xn+–xn.

Using (.), (.) and (vi), we obtain

lim

n→∞Txn–xn= . (.)

Step . We will show that

lim sup

n→∞

γ φ(x) –Fx,jq(xn–x)≤, (.)

wherex=limt→xtwithxtbeing the ﬁxed point of the contraction

x→QCtγ φ(x) + ( –tF)Tx.

From the above, we know thatx∈F=F(T), then we take a subsequence{xnk}of{xn}and assume thatxn_k ω, whereω∈F(T). Since the Banach spaceXhas a weakly sequentially continuous generalized duality mappingjq:X→X∗, by using Lemma ., . and (.), we have

lim sup

n→∞

γ φ(x) –Fx,jq(xn–x)

=lim sup

k→∞

γ φ(x) –Fx,jq(xnk–x)

=γ φ(x) –Fx,jq(ω–x)

≤.

Step . We will prove thatlim_n_→∞xn–x= . By contradiction, there is a numberεsuch that

lim sup

n→∞

(17)

First, let

zn=αnγ φ(xn) +γnWnxn+

( –γn)I–αnFyn. (.)

Now, we will obtain the contradiction from two cases.

Case . Fixε (ε<ε), if for somen>N∈Nsuch thatxn–x ≥ε–ε, and for the

othern>N∈Nsuch thatxn–x<ε–ε.

Let

Mn=qγ φ(x) –Fx,jq(xn+–x) (ε–ε)q

.

From (.) we know lim sup_n_→∞Mn≤, thus there are two numbershandN. When

n>N, we haveMn≤h, whereh=min{η– CqL q_αq–

n q(–γn)q– –

γ}. We extract a numbern>N

satisfyingxn–x<ε–ε, then from Lemma . and (.), we have

xn+–xq

=QCzn–zn,jq(xn+–x)

+zn–x,jq(xn+–x)

≤zn–x,jq(xn+–x)

=αnγ φ(xn) +γnWnxn+

( –γn)I–αnF

yn–x,jq(xn+–x)

=( –γn)I–αnF

yn–( –γn)I–αnF

x

+αn

γ φ(xn) –Fx

+γn(Wnxn–x),jq(xn+–x)

=( –γn)I–αnF

yn–

( –γn)I–αnF

x,jq(xn+–x)

+αn

γ φ(xn) –γ φ(x)

,jq(xn+–x)

+αn

γ φ(x) –Fx,jq(xn+–x)

+γn

Wnxn–x,jq(xn+–x)

≤( –γn)I–αnF

yn–

( –γn)I–αnF

xxn+–xq–

+αnγxn–xxn+–xq–

+αn

γ φ(x) –Fx,jq(xn+–x)

+γnxn–xxn+–xq–

≤

 –γn–αn

n q( –γn)q–

yn–xxn+–xq–

+αn

γ φ(x) –Fx,jq(xn+–x)

+γnxn–xxn+–xq–

≤

 –γn–αn

η– CqL

q_αq– n q( –γn)q–

xn–xxn+–xq–

+αn

γ φ(x) –Fx,jq(xn+–x)

+γnxn–xxn+–xq–

=

 –αn

n q( –γn)q–

–γ

xn–xxn+–xq–

+αn

γ φ(x) –Fx,jq(xn+–x)

(18)

≤

q

 –αn

η– CqL

q_αq–

n q( –γn)q–

–γ

xn–xq

+q– 

q xn+–x q₊_α

n

γ φ(x) –Fx,jq(xn+–x)

≤

q

 –αn

η– CqL

q_αq–

n q( –γn)q–

–γ

(ε–ε)q+

q– 

q xn+–x q

+αn

γ φ(x) –Fx,jq(xn+–x)

,

which implies that

xn+–xq<

 –αn

n q( –γn)q–

–γ

(ε–ε)q

+qαn

γ φ(x) –Fx,jq(xn+–x)

=

 –αn

η– CqL

q_αq– n q( –γn)q–

–γ–Mn

(ε–ε)q

≤(ε–ε)q.

Hence, we have

xn+–x<ε–ε.

In the same way, we obtain

xn–x<ε–ε, ∀n≥n.

It contradictslim sup_n_→∞xn–x ≥ε.

Case . Fixε(ε<ε), ifxn–x ≥ε–εfor alln>N∈N. In this case, from Lemma .

there exists a numberr∈(, ) such that

φ(xn) –φ(x)≤rxn–x, n≥N.

From (.) we have

xn+–xq

=QCzn–zn,jq(xn+–x)

+zn–x,jq(xn+–x)

≤zn–x,jq(xn+–x)

=αnγ φ(xn) +γnWnxn+

( –γn)I–αnFyn–x,jq(xn+–x)

=( –γn)I–αnFyn–( –γn)I–αnFx

+αn

γ φ(xn) –Fx+γn(Wnxn–x),jq(xn+–x)

=( –γn)I–αnF

yn–

x,jq(xn+–x)

+αn

γ φ(xn) –γ φ(x),jq(xn+–x)

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+αn

γ φ(x) –Fx,jq(xn+–x)

+γn

Wnxn–x,jq(xn+–x)

≤( –γn)I–αnFyn–( –γn)I–αnFxxn+–xq–

+αnγrxn–xxn+–xq–

+αn

+γnxn–xxn+–xq–

≤

 –γn–αn

n q( –γn)q–

yn–xxn+–xq–

+αn

≤

 –γn–αn

n q( –γn)q–

xn–xxn+–xq–

+αn

=

 –αn

n q( –γn)q––γr

xn–xxn+–xq–

+αn

γ φ(x) –Fx,jq(xn+–x)

≤

q

 –αn

n q( –γn)q–

–γr

xn–xq

+q– 

q xn+–x q₊_α

n

,

which implies that

xn+–xq<

 –αn

n q( –γn)q––γr

xn–xq

+qαn

. (.)

Putan=αn(η–CqLqα q– n

q(–γn)q––γr) andcn=

qγ φ(x)–Fx,jq(xn+–x)

η–CqLqα

q– n q(–γn)q––γr

. Applying Lemma . to (.), we

obtainxn→xasn→ ∞, which contradictsxn–x ≥ε–ε. Thuslimn→∞xn–x= .

This completes the proof.

Theorem . Let C be a nonempty closed subset of a q-uniformly smooth Banach space X. Let QC be the sunny nonexpansive retraction from X onto C,and letφ be an MKC on C.Let F:C→X be anη-strongly accretive L-Lipschitzian and linear mapping with <

γ <η,and let Ti:C→C be aλi-strictly pseudo-contractive non-self-mapping such that

F:=∞_i_=F(Ti)=∅.Assumeλ=inf{λi:i∈N}> .Let{xn}be a sequence of C generated by (.).We assume that the following parameters are satisﬁed:

(i)  <αn< ,

∞

i=αn=∞,limn→∞αn= ,

∞

i=|αn+–αn|<∞;

(ii)  <  – (_Cqqλ)q ≤_β n< ,

∞

i=|βn+–βn|<∞;

(iii) ∞_i_=μ_i(n)= ,∞_n_=_i∞_=|μ(_in+)–μ(_in)|<∞; (iv)  <γn<a< ,

∞

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(v)  –b≤α< ,b:=min{, (_Cqqλ)q}_;

(vi) limn→∞βn=α.

Then{xn}generated by(.)converges strongly tox∈F,which solves the following varia-tional inequality:

γ φ(x) –Fx,jq(p–x)

≤, ∀p∈

∞

i=

F(Ti).

Proof Combining the proof of Lemma . with Lemma ., we can obtain the

conclu-sion.

Remark . Compared with Theorem . of Song [], our results are diﬀerent from those in the following aspects:

(i) Theorem . improves and extends Theorem . of Song []. Especially, our results extend the above results from a Hilbert space to a more generalq-uniformly smooth and uniformly convex Banach space.

(ii) We change the metric projectionPCin Song [] into the sunny nonexpansive retractionQCand put it to the other place of the iteration process so that our iteration process is better deﬁned.

(iii) We generalize the iteration process so that our iteration process is more general. (iv) We remove the very strict conditionC+C⊂Cin Lemma . and Theorem . of

Song [], and it is worth stressing that the strict condition is also very necessary in Qinet al.[] and Caiet al.[].

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors read and approved the ﬁnal manuscript.

Received: 20 June 2013 Accepted: 23 August 2013 Published:09 Nov 2013 References

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Cite this article as:Guan et al.:A new general iterative algorithm with Meir-Keeler contractions for variational