Meir-Keeler-Type Contraction for a Finite Family of Nonexpansive Semigroups in Banach Spaces

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Volume 2012, Article ID 720192,14pages doi:10.1155/2012/720192

Research Article

Implicit and Explicit Iterations with

Meir-Keeler-Type Contraction for a Finite Family of Nonexpansive Semigroups in Banach Spaces

Jiancai Huang,

¹

Shenghua Wang,

²

and Yeol Je Cho

³

1School of Control and Computer Engineering, North China Electric Power University, Baoding 071003, China

2School of Mathematics and Physics, North China Electric Power University, Baoding 071003, China

3Department of Mathematics Education and the RINS, Gyeongsang National University, Jinju 660-701, Republic of Korea

Correspondence should be addressed to Yeol Je Cho,[email protected] Received 31 December 2011; Accepted 28 January 2012

Academic Editor: Rudong Chen

Copyrightq2012 Jiancai Huang et al. 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 an implicit and explicit iterative schemes for a finite family of nonexpansive semigroups with the Meir-Keeler-type contraction in a Banach space. Then we prove the strong convergence for the implicit and explicit iterative schemes. Our results extend and improve some recent ones in literatures.

1. Introduction

LetCbe a nonempty subset of a Banach spaceEandT :C → Cbe a mapping. We callT nonexpansive ifTx−Ty ≤ x−yfor allx, y∈E. The set of all fixed points ofTis denoted by FixT, that is, FixT {x∈C:xTx}.

One parameter familyT{Tt:t≥0}is said to a semigroup of nonexpansive mappings or nonexpansive semigroup onCif the following conditions are satisfied:

1T0xxfor allx∈C;

2Tst TsTtfor alls, t≥0;

3for eacht≥0,Ttx−Tty ≤ x−yfor allx, y∈C;

4for eachx∈C, the mappingT·xfromR, whereRdenotes the set of all nonnegative reals, intoCis continuous.

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We denote by FixT the set of all common fixed points of semigroup T, that is, FixT {x∈C:Ttxx, 0≤t <∞}andNby the set of natural numbers.

Now, we recall some recent work on nonexpansive semigroup in literatures. In 1, Shioji and Takahashi introduced the following implicit iteration for a nonexpansive semigroup in a Hilbert space:

xnαnx 1−αn1 tn

_t_n

0

Tsxnds, ∀n∈N, 1.1

where{αn} ⊂ 0,1and {tn} ⊂ 0,∞. Under the certain conditions on{αn}and{tn}, they proved that the sequence{xn}defined by1.1converges strongly to an element in FixT.

In2, Suzuki introduced the following implicit iteration for a nonexpansive semigroup in a Hilbert space:

xnαnu 1−αnTtnxn, ∀n∈N, 1.2

where{αn} ⊂0,1and{tn} ⊂0,∞. Under the conditions that limn→ ∞tnlimn→ ∞αn/tn 0, he proved that{xn}defined by1.2converges strongly to an element of FixT. Later on, Xu3extended the iteration1.2to a uniformly convex Banach space that admits a weakly sequentially continuous duality mapping. Song and Xu4also extended the iteration1.2 to a reflexive and strictly convex Banach space.

In 2007, Chen and He 5 studied the following implicit and explicit viscosity approximation processes for a nonexpansive semigroup in a reflexive Banach space admitting a weakly sequentially continuous duality mapping:

xnαnfxn 1−αnTtnxn, y_n1βnf

yn

1−βn

Ttnyn, ∀n∈N, 1.3

wherefis a contraction,{αn} ⊂0,1and{tn} ⊂0,∞. They proved the strong convergence for the above iterations under some certain conditions on the control sequences.

Recently, Chen et al.6introduced the following implicit and explicit iterations for nonexpansive semigroups in a reflexive Banach space admitting a weakly sequentially continuous duality mapping:

ynαnxn 1−αnTtnxn, xnβnfxn

1−βn

yn, ∀n∈N, 1.4

ynαnxn 1−αnTtnxn, xn1βnfxn

1−βn

yn, ∀n∈N, 1.5

wheref is a contraction,{αn} ⊂0,1and{tn} ⊂ 0,∞. They proved that{xn}defined by 1.4and1.5converges strongly to an elementqof FixT, which is the unique solution of the following variation inequality problem:

f−I , j

x−q

≤0, ∀x∈FixT. 1.6

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For more convergence theorems on implicit and explicit iterations for nonexpansive semigroups, refer to7–13.

In this paper, we introduce an implicit and explicit iterative process by a generalized contraction for a finite family of nonexpansive semigroups in a Banach space. Then we prove the strong convergence for the iterations and our results extend the corresponding ones of Suzuki2, Xu3, Chen and He5, and Chen et al.6.

2. Preliminaries

LetEbe a Banach space andE^∗the duality space ofE. We denote the normalized mapping fromEto 2^E^∗byJdefined by

Jx

j∈E^∗: x, jx

x²j , ∀x∈E, 2.1 where·,·denotes the generalized duality pairing. For anyx, y ∈ Ewithjx ∈ Jxand jxy∈Jxy, it is well known that the following inequality holds:

x²2 y, jx

≤xy²≤ x²2 y, j

xy

. 2.2

The dual mappingJis called weakly sequentially continuous ifJis single valued, and {xn} x∈E, wheredenotes the weak convergence, thenJxnweakly star converges to Jx 14–16. A Banach spaceEis called to satisfy Opial’s condition17if for any sequence {xn}inE,xn x,

lim sup

n→ ∞ xn−x<lim sup

n→ ∞

xn−y, ∀y∈Ewithx /y. 2.3

It is known that ifEadmits a weakly sequentially continuous duality mappingJ, thenEis smooth and satisfies Opial’s condition14.

A functionψ :R → Ris said to be anL-function ifψ0 0,ψt>0 for anyt >0, and for everyt >0 ands >0, there existsu > ssuch thatψt≤s, for allt∈s, u. This implies thatψt< tfor allt >0.

Letf :C → Cbe a mapping.fis said to be aψ, L-contraction if there exists aL-func- tionψ:R → Rsuch thatfx−fy< ψx−yfor allx, y∈Cwithx /y. Obviously, ifψt ktfor allt >0, wherek∈0,1, thenfis a contraction.fis called a Meir-Keeler-type mapping if for each >0, there existsδ>0 such that for allx, y∈C, if <x−y< δ, thenfx−fy< .

In this paper, we always assume that ψt is continuous, strictly increasing and lim_{t→ ∞}ηt ∞, whereηt t−ψt, is strictly increasing and onto.

The following lemmas will be used in next section.

Lemma 2.1see18. LetX, dbe a metric space andf :X → X be a mapping. The following assertions are equivalent:

ifis a Meir-Keeler-type mapping,

iithere exists anL-functionψ:R → Rsuch thatfis aψ, L-contraction.

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Lemma 2.2see19. LetEbe a Banach space andCbe a convex subset ofE. LetT :C → Cbe a nonexpansive mapping andfbe aψ, L-contraction. Then the following assertions hold:

iT◦fis aψ, L-contraction onCand has a unique fixed point inC;

iifor eachα∈0,1, the mappingx→αfx 1−αTxis of Meir-Keeler-type and it has a unique fixed point inC.

Lemma 2.3see20. LetEbe a Banach space andCbe a convex subset ofE. Letf:C → Cbe a Meir-Keeler-type contraction. Then for each >0 there existsr ∈0,1such that, for eachx, y∈C withx−y ≥,fx−fy ≤rx−y.

Lemma 2.4 see21. Let C be a closed convex subset of a strictly convex Banach space E. Let Tm :C → Cbe a nonexpansive mapping for each 1≤m≤r, whereris some integer. Suppose that

∩^r_m1FixTmis nonempty. Let{λn}be a sequence of positive numbers with_r

n1λn 1. Then the mappingS:C → Cdefined by

Sx^r

m1

λmTmx, ∀x∈C, 2.4

is well defined, nonexpansive and FixS ∩^r_m1FixTmholds.

Lemma 2.5see22. Assume that{αn}is a sequence of nonnegative real numbers such that

α_n1≤ 1−γn

αnδn, n∈N, 2.5

where{γn}is a sequence in0,1and{δn}is a sequence inRsuch that ilimn→ ∞γn0;

ii_∞

n1γn∞;

iiilim sup_n_{→ ∞}δn/γn≤0 or_∞

n1|δn|<∞.

Then limn→ ∞αn0.

3. Main Results

In this section, by a generalized contraction mapping we mean a Meir-Keeler-type mapping orψ, L- contraction. In the rest of the paper we suppose thatψ from the definition of the ψ, L-contraction is continuous, strictly increasing andηtis strictly increasing and onto, whereηt t−ψt, for allt∈R. As a consequence, we have theηtis a bijection onR. Theorem 3.1. LetCbe a nonempty closed convex subset of a reflexive Banach spaceEwhich admits a weakly sequentially continuous duality mappingJfromEintoE^∗. For everyi1, . . . , NN≥1, letTi{Tit:t≥0}be a semigroup of nonexpansive mappings onCsuch thatF∩^N_i1FixTi/∅ andf : C → Cbe a generalized contraction onC. Let{αn},{βn} ⊂ 0,1and{tn} ⊂ 0,∞be

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the sequences satisfying limn→ ∞tnlimn→ ∞αn/tn 0 and lim sup_n_{→ ∞}βn <1. Let{xn}be a se- quence generated by

xnαnfxn

1−αn

N N

i1

yin,

yinβnxn 1−βn

Titnxn, i1, . . . , N.

3.1

Then{xn}converges strongly to a pointx^∗ ∈ F, which is the unique solution to the following varia- tional inequality:

f−I

x^∗, jx−x^∗

≤0, ∀x∈ F. 3.2

Proof. First, we show that the sequence {xn} generated by3.1is well defined. For every n∈Nandi1, . . . , N, letUinβnI 1−βnTitnand defineWn:C → Cby

Wnxαnfx 1−αnGnx, ∀x∈C, 3.3 whereGnx 1/N_N

i1Uinx. SinceUinis nonexpansive,Gnis nonexpansive. ByLemma 2.2 we see thatWnis a Meir-Keeler-type contraction for eachn∈N. Hence, eachWnhas a unique fixed point, denoted asxn, which uniquely solves the fixed point equation3.3. Hence{xn} generated by3.1is well defined.

Now we prove that{xn}generated by3.1is bounded. For anyp∈ F, we have yin−p≤βnxn−p

1−βnTitnxn−p≤xn−p. 3.4 Using3.4, we get

xn−p²

αnfxn 1−αn

N N

i1

yin−p, j

xn−p αn

fxn−f p

, j

xn−p αn

f p

−p, j

xn−p 1−αn

N N

i1

yin−p, j

xn−p

≤αnψxn−pxn−pαnf p

−pxn−p 1−αn

N N

i1

yin−pxn−p αnψxn−pxn−pαnf

p

−pxn−p 1−αnxn−p²

3.5

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and hence

xn−p≤ψxn−pf p

−p, 3.6

which implies that

ηxn−pxn−p−ψxn−p≤f p

−p. 3.7

Hence

xn−p≤η⁻¹f p

−p. 3.8

This shows that{xn}is bounded, and so are{Titnxn},{fxn}and{yin}.

SinceEis reflexivity and{xn} is bounded, there exists a subsequence{xnj} ⊂ {xn} such thatxnj x^∗for somex^∗∈Casj → ∞. Now we prove thatx^∗∈ F. For any fixedt >0, we have

N i1

xnj−Titx^∗≤^N

i1

⎡

⎣^t/tⁿⁱ⁻¹

k0

Ti

k1tnj

xnj−Ti

ktnj

xnj

Ti

t tnj

tnj

xnj−Ti

t tnj

tnj

x^∗

Ti

t tnj

tnj

xnj−Titx^∗

≤^N

i1

t tnj

Ti

tnj

xnj−xnj

xnj−x^∗ Ti

t−

t tnj

tnj

xnj−x^∗

≤^N

i1

t tnj

Ti

tnj

xnj−xnj

xnj−x^∗max

Tisx^∗−x^∗: 0≤s≤tnj

≤ Nαnj

t/tnj

1−αnj

1−βnj

xnj−f xnj

Nxnj−x^∗

^N

i1

max

Tisx^∗−x^∗: 0≤s≤tnj

≤ Nt

1−αnj

1−βnj

αnj

tnj

xnj−f xnj

Nxnj−x^∗

^N

i1

max

Tisx^∗−x^∗: 0≤s≤tnj .

3.9

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By hypothesis on{tn},{αn},{βn}, we have

jlim→ ∞

Nt 1−αnj

1−βnj

αnj

tnj

0. 3.10

Further, from3.9we get

lim sup

j→ ∞

N i1

xnj−Titx^∗≤lim sup

j→ ∞ Nxnj−x^∗. 3.11

SinceEadmits a weakly sequentially duality mapping, we see thatEsatisfies Opial’s condition. Thus ifx^∗∈ F, we have/

lim sup

j→ ∞ Nxnj−x^∗<lim sup

j→ ∞

N i1

xnj−Tix^∗. 3.12

This contradicts3.11. Sox^∗∈ F.

In3.5, replacingpwithx^∗andnwithnj, we see that xnj−x^∗²αnj

f xnj

−fx^∗, j

xnj−x^∗ αnj

fx^∗−x^∗, j

xnj−x^∗ 1−αnj

N N

i1

yinj−x^∗, j

xnj−x^∗

≤αnjψxnj−x^∗xnj−x^∗αnj

fx^∗−x^∗, j

xnj−x^∗ 1−αnj

N N

i1

yinj−x^∗xnj−x^∗

≤αnjψxnj−x^∗xnj−x^∗αnj

fx^∗−x^∗, j

xnj−x^∗

1−αnj

xn−p²,

3.13

which implies that xnj−x^∗

ψxnj−x^∗

−xnj−x^∗

≤

fx^∗−x^∗, j

xnj−x^∗

. 3.14

Now we prove that{xn}is relatively sequentially compact. Sincejis weakly sequentially continuous, we have

jlim→ ∞

xnj−x^∗

ψxnj−x^∗

−xnj−x^∗

≤0, 3.15

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which implies that

jlim→ ∞

xnj−x^∗0, or lim

j→ ∞

ψxnj−x^∗

−xnj−x^∗

0. 3.16

If limj→ ∞xnj −x^∗ 0, then {xn}is relatively sequentially compact. If limj→ ∞ψxnj − x^∗−xnj−x^∗ 0, we have limj→ ∞xnj−x^∗limj→ ∞ψxnj−x^∗. Sinceψis continuous, lim_j_{→ ∞}xnj−x^∗ψlimj→ ∞xnj−x^∗. By the definition ofψ, we conclude that limj→ ∞xnj− x^∗0, which implies that{xn}is relatively sequentially compact.

Next, we prove thatx^∗is the solution to3.2. Indeed, for anyx∈ F, we have

xn−x² αn

fxn−xnxn−x

, jxn−x

1−αn

N N

i1

yin−x, jxn−x αn

fxn−xn, jxn−x αn

xn−x, jxn−x 1−αn

N N

i1

βn

xn−x, jxn−x

1−βn

Titnxn−x, jxn−x^∗

≤αn

fxn−xn, jxn−x

αnxn−x² 1−αn

N N

i1

βnxn−x² 1−βn

Titnxn−xxn−x

≤αn

fxn−xn, jxn−x

αnxn−x² 1−αn

N N

i1

βnxn−x² 1−βn

xn−x² αn

fxn−xn, jxn−x

xn−x².

3.17

Therefore,

fxn−xn, jx−xn

≤0. 3.18

Sincexnj x^∗andjis weakly sequentially continuous, we have fx^∗−x^∗, jx−x^∗

lim

j→ ∞

f

xnj

−xnj, j x−xnj

≤0. 3.19

This shows thatx^∗is the solution of the variational inequality3.2.

Finally, we prove that x^∗ is the unique solution of the variational inequality 3.2.

Assume thatx∈ Fwithx /x^∗is another solution of3.2. Then there exists >0 such that x−x^∗> . ByLemma 2.3there existsr ∈0,1such thatfx −fx^∗ ≤rx−x^∗. Since bothxandx^∗are the solution of3.2, we have

fx^∗−x^∗, jx−x^∗

≤0,

fx −x, jx ^∗−x

≤0. 3.20

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Adding the above inequalities, we get

0<1−r² <1−rx−x^∗²≤ I−f

x^∗− I−f

x

, jx^∗−x ≤0, 3.21

which is a contradiction. Therefore, we must havexx^∗, which implies thatx^∗is the unique solution of3.2.

In a similar way it can be shown that each cluster point of sequence{xn}is equal tox^∗. Therefore, the entire sequence{xn}converges strongly tox^∗. This completes the proof.

If lettingβn0 for alln∈NinTheorem 3.1, then we get the following.

Corollary 3.2. LetCbe a nonempty closed convex subset of a reflexive Banach spaceEwhich admits a weakly sequentially continuous duality mappingJfromEintoE^∗. For everyi1, . . . , N(N≥1), letTi{Tit:t≥0}be a semigroup of nonexpansive mappings onCsuch thatF∩^N_i1FixTi/∅ andf :C → Cbe a generalized contraction onC. Let{αn} ⊂0,1and{tn} ⊂0,∞be sequences satisfying lim_{n→ ∞}tnlim_n_{→ ∞}α_n/tn 0. Let{xn}be a sequence generated by

xnαnfxn

1−αn

N N i1

Titnxn. 3.22

Then{xn}converges strongly to a pointx^∗ ∈ F, which is the unique solution to the following varia- tional inequality:

f−I

x^∗, jx−x^∗

≤0, ∀x∈ F. 3.23

Theorem 3.3. LetCbe a nonempty closed convex subset of a reflexive and strictly convex Banach spaceEwhich admits a weakly sequentially continuous duality mappingJfromEintoE^∗. For every i1,· · · , NN≥ 1, letTi {Tit:t≥0}be a semigroup of nonexpansive mappings onCsuch thatF∩^N_i1FixTi/∅andf :C → Cbe a generalized contraction onC. Let{αn},{βn} ⊂0,1 and{tn} ⊂ 0,∞be the sequences satisfying lim

n→ ∞tn lim

n→ ∞β_n/tn 0. Let{xn}be a sequence generated

yin αnxn 1−αnTitnxn, i1, . . . , N,

x_n1βnfxn

1−βn

N N

i1

yin, ∀n∈N. 3.24

Then{xn}converges strongly to a pointx^∗∈ F, which is the unique solution of variational inequality 3.2.

Proof. Letp∈ FandMmax{x1−p, η⁻¹fp−p}. Now we show by induction that xn−p≤M, ∀n∈N. 3.25

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It is obvious that3.25holds forn1. Suppose that3.25holds for somenk, wherek >1.

Observe that

yik−pαk

xk−p

1−αk

Titkxk−p

≤αkxk−p 1−αkTitkxk−p≤xk−p. 3.26 Now, by using3.24and3.26, we have

xk1−p βk

fxk−p

1−βk

N N

i1

yik−p

≤βkfxk−f

pβkf p

−p1−βk

N N

i1

yik−p

≤βkψxk−pβkf p

−p1−βk

N N

i1

xk−p βkψxk−pβkf

p

−p

1−βkxk−p βkψxk−pβkη

η⁻¹f p

−p

1−βkxk−p

≤βkψM βkηM 1−βk

M βkψM βk

M−ψM

1−βk

MM.

3.27

By induction we conclude that3.25holds for alln∈N. Therefore,{xn}is bounded and so are{fxn},{yin},{Titnxn}.

For eachi1, . . . , Nandn∈N, define the mappingUtn 1/N_N

i1Sitn, where Sitn αnI 1−αnTitn. Then we rewrite the sequence3.24to

xn1βnfxn

1−βn

Utnxn. 3.28

Obviously, each Utnis nonexpansive. Since{xn} is bounded andE is reflexive, we may assume that some subsequence{xnj}of{xn}converges weakly top. Next we show thatp∈ F.

Putxj xnj,βjβnj, andtj tnjfor eachj∈N. Fixt >0. By3.28we have

xj−Utp

t/tj−1 k0

U

k1tj

xj−U ktj

xj

U

t tj

tj

xj−U

t tj

tj

p

U

t tj

tj

p−Utp

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≤ t

tj

U tj

xj−xj1xj1−p U

t−

t tj

tj

p−p

t tj

βjU

tj

xj−f

xjxj1−p U

t−

t tj

tj

p−p

≤ tβj

tj

U tj

xj−f

xjxj1−pmaxUsp−p: 0≤s≤tj

. 3.29

So, for allj∈N, we have

lim sup

j→ ∞

xj−Utp≤lim sup

j→ ∞

xj1−plim sup

j→ ∞

xj−p. 3.30

SinceEhas a weakly sequentially continuous duality mapping satisfying Opials’ condition, this impliesp Utp. By Lemma 2.4, we have FixUt ∩^N_i1FixTitfor each t > 0. Therefore,p ∈ F. In view of the variational inequality3.2and the assumption that duality mappingJis weakly sequentially continuous, we conclude that

lim sup

n→ ∞

f−I q, j

xn1−q lim

j→ ∞

f−I q, j

xnj1−q

I−f q, j

p−q

≤0.

3.31

Finally, we prove thatxn → qasn → ∞. Suppose thatxn−q0. Then there exists >0 and subsequence{xnj}of{xn}such thatxnj−q ≥for allj∈N. Putxjxnj,βj βnj

andtj tnj. ByLemma 2.3one hasfxj−fq ≤ rxj−qfor allj ∈N. Now, from2.2 and3.28we have

x_j1−q²1−βnUtjxj−q βnfxj−q²

≤ 1−βj

₂U tj

xj−q²2βj

f xj

−q, j

xj1−q

≤ 1−βj

₂xj−q²2βn

f xj

−f q

, j

xj1−q 2βj

f q

−q, j

xj1−q

≤ 1−βj

₂xj−q²2βjrxj−qx_j1−q2βn

f q

−q, j

x_j1−q

≤ 1−βj

2xj−q²βjrxj−q²xj1−q² 2βj

f q

−q, j

xj1−q

1−βj

2βjrxj−q²βjrx_j1−q²2βj

f q

−q, j

x_j1−q .

3.32

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It follows that

xj1≤ 1−2−rβjβ²_j

1−βjr xj−q² 2βj

1−βjr f

q

−q, j

xj1−q

≤ 1−βjr−21−rβj

1−βjr xj−q² 2βj

1−βjr f

q

−q, j

x_j1−q β²_jM

1− 21−rβj

1−βjr

xj−q² 2βj

1−βjr f

q

−q, j

x_j1−q β²_jM

≤

1−21−rβjxj−q²βj

2 1−r

f q

−q, j

xj1−q

βjM ,

3.33

whereMis a constant.

Letγj21−rβjandδjβj2/1−rfq−q, jxj1−qβjM. It follows from 3.33that

xj1−q≤

1−γjxj−qδj. 3.34

It is easy to see thatγj → 0,_∞

j1γj∞andnoting3.28

lim sup

j→ ∞

δj

γj lim sup 1 1−r²

f q

−q, j

xj1−q

M 21−rβj,

lim sup

n→ ∞

1 1−r²

f q

−q, j

xj1−q

≤0.

3.35

UsingLemma 2.5, we conclude thatxj−q → 0 asj → ∞. It is a contradiction. Therefore, xn → qasn → ∞. This completes the proof.

If lettingαn0 for alln∈NinTheorem 3.3, then we get the following.

Corollary 3.4. LetCbe a nonempty closed convex subset of a reflexive and strictly convex Banach spaceEwhich admits a weakly sequentially continuous duality mappingJfromEintoE^∗. For every i1, . . . , NN ≥ 1, letTi {Tit:t≥0}be a semigroup of nonexpansive mappings onCsuch thatF ∩^N_i1FixTi/∅andf :C → Cbe a generalized contraction onC. Let{βn} ⊂0,1and {tn} ⊂ 0,∞be sequences satisfying lim_{n→ ∞}tn limn→ ∞βn/tn 0. Let{xn} be a sequence generated

x_n1βnfxn

1−βn

N N

i1

Titnxn, ∀n∈N. 3.36

Then{xn}converges strongly to a pointx^∗∈ F, which is the unique solution of variational inequality 3.2.

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Remark 3.5. Theorem 3.1andCorollary 3.2extend the corresponding ones of Suzuki2, Xu 3, and Chen and He 5 from one nonexpansive semigroup to a finite family of nonexpansive semigroups. ButTheorem 3.3andCorollary 3.4are not the extension of Theorem 3.2 of Chen and He5since Banach space inTheorem 3.3and Corollary 3.4is required to be strictly convex. But if lettingN 1 inTheorem 3.3and Corollary 3.4, we can remove the restriction on strict convexity and hence they extend Theorem 3.2 of Chen and He5from a contraction to a generalized contraction.

Remark 3.6. OurTheorem 3.1extends and improves Theorems 3.2 and 4.2 of Song and Xu4 from a nonexpansive semigroup to a finite family of nonexpansive semigroups and a contraction to a generalized contraction. Our conditions on the control sequences are diﬀerent with ones of Song and Xu4.

Acknowledgment

This work was supported by the Basic Science Research Program through the National Research Foundation of KoreaNRFfunded by the Ministry of Education, Science, and Tech- nologyGrant Number: 2011-0021821.

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Meir-Keeler-Type Contraction for a Finite Family of Nonexpansive Semigroups in Banach Spaces

Research Article

Implicit and Explicit Iterations with

Meir-Keeler-Type Contraction for a Finite Family of Nonexpansive Semigroups in Banach Spaces

Jiancai Huang,

Shenghua Wang,

and Yeol Je Cho

1. Introduction

2. Preliminaries

3. Main Results

Acknowledgment

References

Submit your manuscripts at http://www.hindawi.com

Mathematics

Complex Analysis

Optimization

Combinatorics

Function Spaces

The Scientific World Journal

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