Volume 2010, Article ID 354202,19pages doi:10.1155/2010/354202
Research Article
Strong Convergence Theorems of a New General
Iterative Process with Meir-Keeler Contractions for
a Countable Family of
λ
i
-Strict Pseudocontractions
in
q
-Uniformly Smooth Banach Spaces
Yanlai Song and Changsong Hu
Department of Mathematics, Hubei Normal University, Huangshi 435002, China
Correspondence should be addressed to Yanlai Song,[email protected]
Received 9 August 2010; Revised 2 October 2010; Accepted 14 November 2010
Academic Editor: Mohamed Amine Khamsi
Copyrightq2010 Y. Song and C. Hu. 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.
We introduce a new iterative scheme with Meir-Keeler contractions for strict pseudocontractions inq-uniformly smooth Banach spaces. We also discuss the strong convergence theorems for the new iterative scheme inq-uniformly smooth Banach space. Our results improve and extend the corresponding results announced by many others.
1. Introduction
Throughout this paper, we denote byEandE∗a real Banach space and the dual space ofE, respectively. LetCbe a subset ofE, and lrt T be a non-self-mapping ofC. We useFTto denote the set of fixed points ofT.
The norm of a Banach spaceEis said to be Gˆateaux differentiable if the limit
lim
t→0
xty− x
t 1.1
LetρE:0,1 → 0,1be the modulus of smoothness ofEdefined by
ρEt sup
1
2xyx−y−1 :x∈SE, y≤t
. 1.2
A Banach spaceEis said to be uniformly smooth ifρEt/t → 0 ast → 0. Letq >1.
A Banach spaceEis said to beq-uniformly smooth, if there exists a fixed constantc >0 such thatρEt ≤ ctq. It is well known thatEis uniformly smooth if and only if the norm ofEis
uniformly Fr´echet differentiable. IfEisq-uniformly smooth, thenq ≤ 2 andEis uniformly smooth, and hence the norm ofEis uniformly Fr´echet differentiable, in particular, the norm of
Eis Fr´echet differentiable. Typical examples of both uniformly convex and uniformly smooth Banach spaces areLp, wherep >1. More precisely,Lpis min{p,2}-uniformly smooth for every
p >1.
By a gauge we mean a continuous strictly increasing functionϕdefinedR : 0,∞
such that ϕ0 0 and limr→ ∞ϕr ∞. We associate with a gauge ϕ a generally
multivaluedduality mapJϕ:E → E∗defined by
Jϕx
x∗∈E∗:x, x∗ xϕx, x∗ϕx. 1.3
In particular, the duality mapping with gauge functionϕt tq−1 denoted byJ
q, is referred
to the generalized duality mapping. The duality mapping with gauge functionϕt t
denoted byJ, is referred to the normalized duality mapping. Browder1initiated the study
Jϕ. Set fort≥0
Φt
t
0
ϕrdr. 1.4
Then it is known thatJϕxis the subdifferential of the convex functionΦ · atx. It is well
known that ifEis smooth, thenJqis single valued, which is denoted byjq.
The duality mappingJq is said to be weakly sequentially continuous if the duality
mapping Jq is single valued and for any{xn} ∈ E with xn x, Jqxn J∗ qx. Every
lp 1 < p < ∞ space has a weakly sequentially continuous duality map with the gauge
ϕt tp−1. Gossez and Lami Dozo2proved that a space with a weakly continuous duality
mapping satisfies Opial’s condition. Conversely, if a space satisfies Opial’s condition and has a uniformly Gˆateaux differentiable norm, then it has a weakly continuous duality mapping. We already know that inq-uniformly smooth Banach space, there exists a constantCq > 0
such that
xyq
≤ xqqy, Jqx
Cqyq, 1.5
for allx, y∈E.
Recall that a mappingT is said to be nonexpansive, if
T is said to be a λ-strict pseudocontraction in the terminology of Browder and Petryshyn3, if there exists a constantλ >0 such that
Tx−Ty, jq
x−y≤x−yq−λI−Tx−I−Tyq, 1.7
for everyx,y, andCfor somejqx−y∈Jqx−y. It is clear that1.7is equivalent to the
following:
I−Tx−I−Ty, jq
x−y≥λI−Tx−I−Tyq. 1.8
The following famous theorem is referred to as the Banach contraction principle.
Theorem 1.1Banach4. LetX, dbe a complete metric space and letfbe a contraction onX, that is, there existsr ∈ 0,1such thatdfx, fy ≤ rdx, yfor allx, y ∈ X. Thenf has a unique fixed point.
Theorem 1.2Meir and Keeler5. LetX, dbe a complete metric space and letφbe a Meir-Keeler contraction (MKC, for short) onX, that is, for everyε >0, there existsδ >0such thatdx, y< εδ
impliesdφx, φy< εfor allx, y∈X. Thenφhas a unique fixed point.
This theorem is one of generalizations ofTheorem 1.1, because contractions are Meir-Keeler contractions.
In a smooth Banach space, we define an operatorAis strongly positive if there exists a constantγ >0 with the property
Ax, Jx ≥γx2, aI−bA sup
x≤1
{|aI−bAx, Jx |:a∈0,1, b∈0,1}, 1.9
whereIis the identity mapping andJis the normalized duality mapping.
Attempts to modify the normal Mann’s iteration method for nonexpansive mappings and λ-strictly pseudocontractions so that strong convergence is guaranteed have recently been made; see, for example,6–11and the references therein.
Kim and Xu6introduced the following iteration process:
x1x∈C,
yn βnxn
1−βn
Txn,
xn1 αnu 1−αnyn, n≥0,
1.10
whereT is a nonexpansive mapping ofCinto itselfu∈Cis a given point. They proved the sequence{xn}defined by1.10converges strongly to a fixed point ofT, provided the control
Hu and Cai12introduced the following iteration process:
x1x∈C,
ynPC
βnxn
1−βn
N
i1
ηinTixn
,
xn1αnγfxn γnxn
1−γn
I−αnA
yn, n≥1.
1.11
whereTiis non-self-λi-strictly pseudocontraction,fis a contraction andAis a strong positive
linear bounded operator in Banach space. They have proved, under certain appropriate assumptions on the sequences {αn},{γn}, and{βn}, that {xn} defined by1.11converges
strongly to a common fixed point of a finite family ofλi-strictly pseudocontractions, which
solves some variational inequality.
Question 1. Can Theorem 3.1 of Zhou 8, Theorem 2.2 of Hu and Cai 12 and so on be extended from finiteλi-strictly pseudocontraction to infiniteλi-strictly pseudocontraction?
Question 2. We know that the Meir-Keeler contractionMKC, for shortis more general than the contraction. What happens if the contraction is replaced by the Meir-Keeler contraction?
The purpose of this paper is to give the affirmative answers to these questions mentioned above. In this paper we study a general iterative scheme as follows:
x1x∈C,
ynPC
βnxn
1−βn
∞
i1
ηinTixn
,
xn1αnγφxn γnxn
1−γn
I−αnA
yn, n≥1,
1.12
whereTnis non-selfλn-strictly pseudocontraction,φis a MKC contraction andAis a strong
positive linear bounded operator in Banach space. Under certain appropriate assumptions on the sequences{αn},{βn},{γn}, and{μni}, that{xn}defined by1.12converges strongly to a
common fixed point of an infinite family ofλi-strictly pseudocontractions, which solves some
variational inequality.
2. Preliminaries
In order to prove our main results, we need the following lemmas.
Lemma 2.1see13. Let{xn},{zn}be bounded sequences in a Banach space Eand {βn} be a
sequence in0,1which satisfies the following condition:0 <lim infn→ ∞βn ≤lim supn→ ∞βn <1.
Suppose thatxn1 1−βnxnβnznfor alln≥0andlim supn→ ∞zn1−zn − xn1−xn≤0.
Lemma 2.2see Xu14. Assume that{αn}is a sequence of nonnegative real numbers such that
αn1≤1−γnαnδn, whereγnis a sequence in (0, 1) and{δn}is a sequence inRsuch that
i∞n1γn∞,
iilim supn→ ∞δn/γn≤0or∞n1|δn|<∞.
Thenlimn→ ∞αn0.
Lemma 2.3see15demiclosedness principle. LetCbe a nonempty closed convex subset of a reflexive Banach spaceEwhich satisfies Opial’s condition, and supposeT :C → Eis nonexpansive. Then the mappingI−T is demiclosed at zero, that is,xn x,xn−Txn → 0impliesxTx.
Lemma 2.4see16, Lemmas 3.1, 3.3. LetEbe real smooth and strictly convex Banach space, andCbe a nonempty closed convex subset ofEwhich is also a sunny nonexpansive retraction ofE. Assume thatT :C → Eis a nonexpansive mapping andP is a sunny nonexpansive retraction ofE
ontoC, thenFT FP T.
Lemma 2.5see17, Lemma 2.2. LetCbe a nonempty convex subset of a realq-uniformly smooth Banach spaceEandT : C → Cbe aλ-strict pseudocontraction. For α ∈ 0,1, we defineTαx
1−αxαTx. Then, asα∈0, μ,μmin{1,{qλ/Cq}1/q−1},Tα:C → Cis nonexpansive such
thatFTα FT.
Lemma 2.6see12, Remark 2.6. WhenTis non-self-mapping, theLemma 2.5also holds.
Lemma 2.7see12, Lemma 2.8. Assume thatAis a strongly positive linear bounded operator on a smooth Banach spaceEwith coefficientγ >0and0< ρ≤ A−1. Then,
I−ρA≤1−ργ. 2.1
Lemma 2.8see18, Lemma 2.3. Letφbe an MKC on a convex subsetCof a Banach spaceE. Then for eachε >0, there existsr∈0,1such that
x−y≥εimplies φx−φy≤rx−y ∀x, y∈C. 2.2
Lemma 2.9. LetCbe a closed convex subset of a reflexive Banach space Ewhich admits a weakly sequentially continuous duality mappingJqfromEtoE∗. LetT :C → Cbe a nonexpansive mapping
with FT/∅ and φ : C → Cbe a MKC, A is strongly positive linear bounded operator with coefficientγ >0. Assume that0< γ < γ. Then the sequence{xt}define byxttγφxt 1−tATxt
converges strongly ast → 0to a fixed pointxofT which solves the variational inequality:
A−γφx, J qx−z
≤0, z∈FT. 2.3
Proof. The definition of{xt}is well definition. Indeed, from the definition of MKC, we can
see MKC is also a nonexpansive mapping. Consider a mappingStonCdefined by
It is easy to see thatStis a contraction. Indeed, byLemma 2.8, we have
Stx−Sty≤tγφx−φ
yI−tATx−Ty
≤tγφx−φy1−tγx−y
≤tγx−y1−tγx−y
≤1−tγ−γx−y.
2.5
Hence,St has a unique fixed point, denoted byxt, which uniquely solves the fixed point
equation
xttγφxt I−tATxt. 2.6
We next show the uniqueness of a solution of the variational inequality2.3. Suppose bothx ∈FTandx∈ FTare solutions to2.3, not lost generality, we may assume there is a number ε such that x−x ≥ ε. Then by Lemma 2.8, there is a number r such that
φx−φx ≤rx−x. From2.3, we know
A−γφx, J qx−x
≤0,
A−γφx, J qx−x
≤0. 2.7
Adding up2.7gets
A−γφx−A−γφx, J qx−x
≤0. 2.8
Noticing that
A−γφx−A−γφx, J qx−x
Ax−x, Jqx−x −γ
φx−φx, J qx−x
≥γx−xq−γφx−φxx−xq−1
≥γx−xq−γrx−xq
≥γ−γrx−xq
≥γ−γrεq
>0.
2.9
Thereforexxand the uniqueness is proved. Below, we usexto denote the unique solution of2.3.
We observe that{xt}is bounded. Indeed, we may assume, with no loss of generality,
t <A−1, for allp∈FT, fixedε1, for eacht∈0,1.
Case 2xt−p ≥ε1. In this case, by Lemmas2.7and2.8, there is a numberr1such that φxt−φ
p< r1xt−p,
xt−ptγφxt I−tATxt−p tγφxt−Ap
I−tATxt−p
≤tγφxt−Ap
1−tγxt−p
≤tγφxt−γφ
pγφp−Ap1−tγxt−p
≤tγr1xt−ptγφ
p−Ap1−tγxt−p,
2.10
therefore,xt−p ≤ γφp−Ap/γ−γr1. This implies the{xt}is bounded.
To prove thatxt → xx∈FTast → 0.
Since{xt}is bounded andEis reflexive, there exists a subsequence{xtn}of{xt}such
thatxtn x∗. Byxt−Txt tγφxt−ATxt. We havextn −Txtn → 0, astn → 0. SinceE
satisfies Opial’s condition, it follows fromLemma 2.3thatx∗∈FT. We claim
xtn−x
∗ −→0. 2.11
By contradiction, there is a numberε0and a subsequence{xtm}of{xtn}such thatxtm−x∗ ≥ ε0. FromLemma 2.8, there is a numberrε0 >0 such thatφxtm−φx∗ ≤rε0xtm −x∗, we
write
xtm−x∗tm
γφxtm−Ax∗
I−tmATxtm−x∗, 2.12
to derive that
xtm−x∗
q
tmγφxtm−Ax∗, Jqxtm−x∗ I−tmATxtm−x∗, Jqxtm−x∗
≤tm
γφxtm−Ax
∗, J
qxtm−x
∗1−t
mγ
xtm−x
∗q. 2.13
It follows that
xtm−x
∗q≤ 1
γ
γφxtm−Ax
∗, J
qxtm −x
∗
1
γ
γφxtm−γφx∗, Jqxtm−x∗
γφx∗−Ax∗, Jqxtm−x∗
≤ 1
γ
γrε0xtm−x
∗qγφx∗−Ax∗, J
qxtm −x
∗.
2.14
Therefore,
xtm−x∗
q≤
γφx∗−Ax∗, Jqxtm−x∗
γ−γrε0
Using that the duality mapJqis single valued and weakly sequentially continuous fromEto
E∗, by2.15, we get thatxtm → x∗. It is a contradiction. Hence, we havextn → x∗.
We next prove thatx∗solves the variational inequality2.3. Since
xttγφxt I−tATxt, 2.16
we derive that
A−γφxt−1
tI−tAI−Txt. 2.17
Notice
I−Txt−I−Tz, Jqxt−z
≥ xt−zq− Txt−Tzxt−zq−1
≥ xt−zq− xt−zq
0.
2.18
It follows that, forz∈FT,
A−γφxt, Jqxt−z
−1
t
I−tAI−Txt, Jqxt−z
−1
tI−Txt−I−Tz, Jqxt−z
AI−Txt, Jqxt−z
≤AI−Txt, Jqxt−z
.
2.19
Now replacingtin2.19withtnand lettingn → ∞, noticingI−Txtn → I−Tx∗0
forx∗∈FT, we obtainA−γφx∗, Jqx∗−z ≤0. That is,x∗∈FTis a solution of2.3;
Hencexx∗by uniqueness. In a summary, we have shown that each cluster point of{xt}at
t → 0equalsx, therefore,xt → xast → 0.
Lemma 2.10see, e.g., Mitrinovi´c19, page 63. Letq >1. Then the following inequality holds:
ab≤ 1 qa
qq−1
q b
q/q−1, 2.20
for arbitrary positive real numbersa, b.
Lemma 2.11. Let E be a q-uniformly smooth Banach space which admits a weakly sequentially continuous duality mapping Jq from E to E∗ and C be a nonempty convex subset of E. Assume
that Ti : C → E is a countable family of λi-strict pseudocontraction for some0 < λi < 1 and
inf{λi:i∈N}>0such thatF∞i1FTi/∅. Assume that{ηi}∞i1is a positive sequence such that ∞
i1ηi 1. Then∞i1ηiTi :C → Eis aλ-strict pseudocontraction withλ inf{λi :i∈N}and
Proof. Let
Gnxη1T1xη2T2x· · ·ηnTnx 2.21
andni1ηi 1. Then,Gn :C → Eis aλi-strict pseudocontraction withλ min{λi : 1≤i≤
n}. Indeed, we can firstly see the case ofn2.
I−G2x−I−G2y, Jq
x−y
η1I−T1xη2I−T2x−η1I−T1y−η2I−T2y, Jq
x−y
η1
I−T1x−I−T1y, Jq
x−yη2
I−T2x−I−T2y, Jq
x−y
≥η1λ1I−T1x−I−T1yqη2λ2I−T2x−I−T2yq
≥λη1I−T1x−I−T1yqη2I−T2x−I−T2yq
≥λI−G2x−I−G2yq,
2.22
which shows thatG2 :C → Eis aλ-strict pseudocontraction withλmin{λi :i1,2}. By
the same way, our proof method easily carries over to the general finite case.
Next, we prove the infinite case. From the definition ofλ-strict pseudocontraction, we know
I−Tnx−I−Tny, Jq
x−y≥λI−Tnx−I−Tnyq. 2.23
Hence, we can get
I−Tnx−I−Tny≤1
λ
1/q−1
x−y. 2.24
Takingp∈FTn, from2.24, we have
I−TnxI−Tnx−I−Tnp≤
1
λ
1/q−1
x−p. 2.25
Consquently, for allx∈E, ifF ∞i1FTi/∅,ηi >0i∈Nand∞i1ηi 1, then∞i1ηiTi
strongly converges. Let
Tx
∞
i1
ηiTix, 2.26
we have
Tx
∞
i1
ηiTix lim n→ ∞
n
i1
ηiTix lim n→ ∞
1 n
i1ηi n
i1
Hence,
I−Tx−I−Ty, Jq
x−y
lim
n→ ∞
I−n1
i1ηi n
i1
ηiTi
x
I−n1
i1ηi n
i1
ηiTi
y, Jq
x−y
lim
n→ ∞
1 n
i1ηi n
i1
ηi
I−Tix−I−Tiy, Jq
x−y
≥ lim
n→ ∞
1 n
i1ηi n
i1
ηiλI−Tix−I−Tiyq
≥λlim
n→ ∞
I−n1 i1ηi
n
i1
ηiTi
x−
I−n1 i1ηi
n
i1
ηiTi
y q
λI−Tx−I−Tyq.
2.28
So, we getTisλ-strict pseudocontraction.
Finally, we showF∞i1ηiTi F. Suppose thatx∞i1ηiTix, it is sufficient to show
thatx∈F. Indeed, forp∈F, we have x−pq
x−p, Jq
x−p
∞
i1
ηiTix−p, Jq
x−p
∞
i1
ηi
Tix−p, Jq
x−p
≤x−pq−λ
∞
i1
ηix−Tixq,
2.29
whereλinf{λi:i∈N}. Hence,xTixfor eachi∈N, this means thatx∈F.
3. Main Results
Lemma 3.1. Let E be a realq-uniformly smooth, strictly convex Banach space andCbe a closed convex subset ofEsuch thatC±C ⊂ C. LetCbe also a sunny nonexpansive retraction ofE. Let
φ : C → Cbe a MKC. LetA : C → Cbe a strongly positive linear bounded operator with the coefficient γ > 0such that0 < γ < γ andTi : C → Ebeλi-strictly pseudo-contractive
non-self-mapping such thatF ∞i1FTi/∅. Letλ inf {λi : i ∈ N} > 0. Let{xn}be a sequence ofC
generated by1.12with the sequences{αn},{βn}and{γn}in0,1, assume for eachn,{ηin}be an
infinity sequence of positive number such that∞i1ηin 1for allnandηin > 0. The following control conditions are satisfied
i∞i1αn∞,limn→ ∞αn 0,
iiilimn→ ∞βn1−βn 0,limn→ ∞∞i1|ηin1−ηni|0,
iv0<lim infn→ ∞γn≤lim supn→ ∞γn<1.
Then,limn→ ∞xn1−xn0.
Proof. Write, for each n ≥ 0, Bn
∞
i1ηinTi. By Lemma 2.11, each Bn is a λ-strict
pseudocontraction onCandFBn F for allnand the algorithm1.12can be rewritten
as
x1x∈C,
ynPC
βnxn
1−βn
Bnxn
,
xn1αnγφxn γnxn
1−γn
I−αnA
yn, n≥1.
3.1
The rest of the proof will now be split into two parts.
Step 1. First, we show that sequences{xn}and{yn}are bounded. Define a mapping
Lnx:PC
βnx
1−βn
Bnx
. 3.2
Then, from the control condition ii, Lemmas 2.5 and 2.6, we obtain Ln : C → C is
nonexpansive. Taking a pointp∈F, byLemma 2.4, we can getLnpp. Hence, we have
yn−pLnxn−p≤xn−p. 3.3
From definition of MKC andLemma 2.8, for each ε > 0 there is a number rε ∈ 0,1, if
xn−z< εthenφxn−φz< ε; Ifxn−z ≥εthenφxn−φz ≤rεxn−z. It follow
3.1
xn1−pαnγφxn γnxn 1−γn
I−αnA
yn−p
αn
γφxn−Ap
γn
xn−p
1−γn
I−αnA
yn−p
≤1−γn−αnγxn−pγnxn−pαnγφxn−Ap
≤1−αnγxn−pαnγmax
rεxn−p, ε
αnγφ
p−Ap
max1−αnγxn−pαnγrεxn−pαnγφ
p−Ap,
1−αnγxn−pαnγεαnγφ
p−Ap
max1−αnγαnγrεxn−pαnγφ
p−Ap,1−αnγxn−p
αnγεαnγφ
p−Ap
max1−αnγ−αnγrεxn−pαnγφ
p−Ap,1−αnγxn−p
αnγεαnγφ
p−Ap.
By induction, we have
xn−p≤maxx0−p,γφp−Ap γ−γrε ,
γεγφp−Ap γ
, n≥1, 3.5
which gives that the sequence{xn}is bounded, so are{yn}and{Lnxn}.
Step 2. In this part, we shall claim thatxn1−xn → 0, asn → ∞. From3.1, we get
xn1αnγφxn γnxn
1−γn
I−αnA
Lnxn. 3.6
Define
xn11−γn
lnγnxn, ∀n≥0, 3.7
where
ln xn1−γnxn
1−γn . 3.8
It follows that
ln1−ln
αn1γφxn1 γn1xn11−γn1I−αn1ALn1xn1−γn1xn1
1−γn1
−αnγφxn γnxn
1−γn
I−αnA
Lnxn−γnxn
1−γn
αn1
γφxn1−ALn1xn1
1−γn1 −
αn
γφxn−ALnxn
1−γn Ln1xn1−Lnxn,
3.9
which yields that
ln1−ln ≤
αn1γφxn1−ALn1xn1
1−γn1
αnγφxn−ALnxn
1−γn Ln1xn1−Lnxn
≤ αn1γφxn1−ALn1xn1
1−γn1
αnγφxn−ALnxn
1−γn Ln1xn1−Ln1xn
Ln1xn−Lnxn
≤ αn1γφxn1−ALn1xn1
1−γn1
αnγφxn−ALnxn
1−γn xn1−xn
Ln1xn−Lnxn.
Next, we estimateLn1xn−Lnxn. Notice that
Ln1xn−LnxnPC
βn1xn
1−βn1Bn1xn
−PC
βnxn
1−βn
Bnxn
≤βn1xn
1−βn1
Bn1xn
−βnxn
1−βn
Bnxn
≤βn1−βnxn−Bn1xn
1−βn
Bn1xn−Bnxn
≤βn1−βnxn−Bn1xn
1−βn
∞
i−1
ηin1−ηinTixn.
3.11
Substituting3.11into3.10, we have
ln1−ln ≤
αn1γφxn1−ALn1xn1
1−γn1
αnγφxn−ALnxn
1−γn xn1−xn
βn1−βnxn−Bn1xn
1−βn
∞
i1
ηin1−ηinTixn.
3.12
Hence, we have
ln1−ln − xn1−xn ≤
αn1γφxn1−ALn1xn1
1−γn1
αnγφxn−ALnxn
1−γn
xn−Bn1xnβn1−βn
1−βn
∞
i1
ηin1−ηinTixn.
3.13
Observing conditions i, iii, iv, and the boundedness of {xn}, {yn}, {fxn}, {Tnxn},
{Tnyn}it follows that
lim sup
n→ ∞
{ln1−ln − xn1−xn} ≤0. 3.14
Thus byLemma 2.1, we have limn→ ∞ln−xn0.
From3.7, we have
xn1−xn
1−γn
ln−xn. 3.15
Therefore,
lim
n→ ∞xn1−xn0. 3.16
Theorem 3.2. Let E be a real q-uniformly smooth, strictly convex Banach space which admits a weakly sequentially continuous duality mappingJqfromEtoE∗andCbe a closed convex subset of
a MKC. LetA : C → Cbe a strongly positive linear bounded operator with the coefficientγ > 0
such that0 < γ < γ and Ti : C → Ebe λi-strictly pseudo-contractive non-self-mapping such
that F ∞i1FTi/∅. Let λ inf{λi : i ∈ N} > 0. Let{xn} be a sequence of C generated
by1.12with the sequences{αn},{βn}and{γn}in 0,1, assume for eachn,Σ∞i1ηin 1for all
nand ηin > 0 for all i ∈ N. They satisfy the conditions (i), (ii), (iii), (iv) ofLemma 3.1 and (v)
limn→ ∞βnα,limn→ ∞i∞1|ηni −ηi|0and
∞
i1ηi1. Then{xn}converges strongly tox∈F,
which also solves the following variational inequality
γφx−Ax, J q
p−x≤0, ∀p∈F. 3.17
Proof. From3.1, we obtain
Lnxn−xn ≤ xn−xn1xn1−Lnxn
xn−xn1αnγφxn γnxn−Lnxn−αnALnxn
≤ xn−xn1αnγφxnALnxn
γnxn−Lnxn.
3.18
SoLnxn−xn ≤ 1/1−γnxn−xn1αnγφxnALnxn, which together with the
conditioni,ivandLemma 3.1implies
lim
n→ ∞Lnxn−xn0. 3.19
Define B ∞i1ηiTi, then B : C → E is a λ-strict pseudocontraction such thatFB
∞
i1FTi F byLemma 2.11, furthermoreBnx → Bxasn → ∞for all x ∈ C. Defines
T :C → Eby
Txαx 1−αBx. 3.20
Then, T is nonexpansive with FT FBbyLemma 2.5. It follows fromLemma 2.4that
FPCT FT F. Notice that
PCTxn−xn ≤ xn−LnxnLnxn−PCTxn
≤ xn−Lnxnβnxn
1−βn
Bnxn−αxn 1−αBxn
≤ xn−Lnxnβn−α
xn−Bnxn 1−αBnxn−Bxn
≤ xn−Lnxn
βn−a
xn−Bnxn 1−αBnxn−Bxn
3.21
which combines with3.19yielding that
lim
Next, we show that
lim sup
n→ ∞
γφx−Ax, J qxn−x
≤0, 3.23
wherexlimt→0xtwithxtbeing the fixed point of the contraction
x−→tγφx 1−tAPCTx. 3.24
To see this, we take a subsequence{xnk}of{xn}such that
lim sup
n→ ∞
γφx−Ax, J xn−x
lim
k→ ∞
γφx−Ax, J xnk−x
. 3.25
We may also assume thatxnk q. Note thatq ∈FTin virtue ofLemma 2.3and3.22. It
follow from theLemma 2.9andJqis weak weakly sequentially continuous duality mapping
that
lim sup
n→ ∞
γφx−Ax, J qxn−x
lim
k→ ∞
γφx−Ax, J qxnk−x
γφx−Ax, J q
q−x≤0.
3.26
Hence, we have
lim sup
n→ ∞
γφx−Ax, J qxn−x
≤0. 3.27
Finally, We showxn−x → 0. By contradiction, there is a numberε0such that
lim sup
n→ ∞
xn−x ≥ε0. 3.28
Case 1. Fixedε1ε1 < ε0, if for somen≥N∈Nsuch thatxn−x ≥ε0−ε1, and for the other
n≥N∈Nsuch thatxn−x< ε0−ε1.
Let
Mn
qγφx−Ax, J xn1−x ε0−ε1q
From3.23, we know lim supn→ ∞Mn ≤ 0. Hence, there is a numberN, whenn > N, we
haveMn≤γ−γ. We extract a numbern0≥Nstastifyingxn0−x< ε0−ε1, then we estimate
xn01−x.
xn01−x
q
αn0γφxn0 γn0xn0
1−γn0
I−αn0A
yn0−x q
1−γn0
I−αn0A
yn0−x αn0
γφxn0−Ax
γn0xn0−x q
1−γn0
I−αn0A
yn0−x
αn0
γφxn0−Ax
γn0xn0−x, Jqxn01−x
1−γn0
I−αn0A
yn0−x
, Jqxn01−x
αn0
γφxn0−Ax
, Jqxn01−x
γn0xn0−x, Jqxn01−x
1−γn0
I−αn0A
yn0−x
, Jqxn01−x αn0γ
φxn0−φx, Jqxn01−x
αn0γφx−Ax, J qxn01−x
γn0xn0−x, Jqxn01−x
≤1−γn0−αn0γ
xn0−xxn01−x q−1
αn0γφxn0−φxxn01−x q−1
αn0γφx−Ax, J qxn01−x γn0xn0−xxn01−x q−1
<1−αn0
γ−γε0−ε1xn01−x q−1
αn0
γφx−Ax, J qxn01−x
≤ 1
q
1−αn0
γ−γqε0−ε1q
q−1
q xn01−x q
αn0
γφx−Ax, J qxn01−x
by Lemma 2.10,
3.30
which implies that
xn01−x q
<1−αn0
γ−γqε0−ε1qqαn0
γφx−Ax, J qxn01−x
<1−αn0
γ−γε0−ε1qqαn0
γφx−Ax, J qxn01−x
1−αn0
γ−γ−Mn
ε0−ε1q
≤ε0−ε1q.
3.31
Hence, we have
xn01−x< ε0−ε1. 3.32
In the same way, we can get
xn−x< ε0−ε1, ∀n≥n0. 3.33
Case 2. Fixedε1ε1 < ε0, ifxn−x ≥ ε0−ε1for alln≥N ∈N, fromLemma 2.8, there is a
numberr,0< r <1such that
φxn−φx≤rxn−x, n≥N. 3.34
It follow3.1that
xn1−xqαnγφxn γnxn
1−γn
I−αnA
yn−xq
1−γnI−αnAyn−x αnγφxn−Ax γnxn−xq
1−γn
I−αnA
yn−x
αn
γφxn−Ax
γnxn−x, Jqxn1−x 1−γn
I−αnA
yn−x
, Jqxn1−xαn
γφxn−Ax
, Jqxn1−x γnxn−x, Jqxn1−x
1−γn
I−αnA
yn−x
, Jqxn1−xαn
γφxn−φx
, Jqxn1−x αn
γφx−Ax, Jqxn1−x
γnxn−x, Jqxn1−x
≤1−γn−αnγ
xn−xxn1−xq−1αnγrxn−xxn1−xq−1
αnγφx−Ax, J qxn1−x γnxn−xxn1−xq−1
1−αn
γ−γrxn−xxn1−xq−1αn
γφx−Ax, J qxn1−x
≤1−αn
γ−γr1
qxn−x
q q−1
q xn1−x
q
αnγφx
−Ax, J qxn1−x by Lemma 2.10,
3.35
which implies that
xn1−xq≤1−αn
γ−γrxn−xqqαn
γφx−Ax, J qxn1−x. 3.36
ApplyLemma 2.2to3.36to concludexn → xasn → ∞. It contradict thexn−x ≥ε0−ε1.
This completes the proof.
Corollary 3.3. LetDbe a closed convex subset of a Hilbert spaceHsuch thatD±D⊂Dandf∈D
with the coefficient0 < α < 1. LetA :C → Cbe a strongly positive linear bounded operator with the coefficientγ >0such that0< γ < γ andTi :C → Ebeλi-strictly pseudo-contractive
non-self-mapping such thatF ∞i1FTi/∅. Letλ inf {λi : i ∈ N} > 0. Let{xn}be a sequence ofC
generated by1.12with the sequences{αn},{βn}, and{γn}in0,1, assume for eachn,Σi∞1ηin1
limn→ ∞βnα,limn→ ∞i∞1|ηin−ηi|0and
∞
i1ηi 1. Then{xn}converges strongly tox∈F,
which also solves the following variational inequality
γφx−Ax, p −x≤0, ∀p∈F. 3.37
Remark 3.4. We conclude the paper with the following observations.
iTheorem 3.2improve and extends Theorem 3.1 of Zhang and Su17, Theorem 1 of Yao et al.11, and Theorem 2.2 of Cai and Hu12.Corollary 3.3also improve and extend Theorem 2.1 of Choa et al.20, Theorem 2.1 of Jung21, Theorem 2.1 of Qin et al.22and includes those results as special cases. Especially, Our results extends above results form contractions to more general Meir-Keeler contraction
MKC, for short. Our iterative scheme studied in present paper can be viewed as a refinement and modification of the iterative methods in12,13,17,22. On the other hand, our iterative schemes concern an infinite countable family ofλi-strict
pseudocontractions mappings, in this respect, they can be viewed as an another improvement.
iiThe advantage of the results in this paper is that less restrictions on the parameters
{αn}, {βn}, {γn} and {ηni} are imposed. Our results unify many recent results
including the results in12,17,22.
iiiIt is worth noting that we obtained two strong convergence results concerning an infinite countable family of λi-strict pseudocontractions mappings. Our result is
new and the proofs are simple and different from those in11,12,17,19–25.
Acknowledgments
The authors are extremely grateful to the referee and the editor for their useful comments and suggestions which helped to improve this paper. This work was supported by the National Science Foundation of China under Grant no. 10771175.
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