Báo cáo hóa học: " Research Article A New General Iterative Method for a Finite Family of Nonexpansive Mappings in Hilbert Spaces Urailuk Singthong1 and Suthep Suantai1, 2" potx

Volume 2010, Article ID 262691, 12 pagesdoi:10.1155/2010/262691 Research Article A New General Iterative Method for a Finite Family of Nonexpansive Mappings in Hilbert Spaces Urailuk Sin

Trang 1

Volume 2010, Article ID 262691, 12 pages

doi:10.1155/2010/262691

Research Article

A New General Iterative Method for a Finite Family

of Nonexpansive Mappings in Hilbert Spaces

Urailuk Singthong1 and Suthep Suantai1, 2

1 Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand

2 PERDO National Centre of Excellence in Mathematics, Faculty of Science, Mahidol University,

Bangkok 10400, Thailand

Correspondence should be addressed to Suthep Suantai,scmti005@chiangmai.ac.th

Received 10 February 2010; Revised 21 June 2010; Accepted 15 July 2010

Academic Editor: Massimo Furi

Copyrightq 2010 U Singthong and S Suantai 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 general iterative method by using theK-mapping for finding a common fixed

point of a finite family of nonexpansive mappings in the framework of Hilbert spaces A strong convergence theorem of the purposed iterative method is established under some certain control conditions Our results improve and extend the results announced by many others

1 Introduction

LetH be a real Hilbert space, and let C be a nonempty closed convex subset of H A mapping

T of C into itself is called nonexpansive if Tx − Ty ≤ x − y for all x, y ∈ C A point x ∈ C is

called a fixed point ofT provided that Tx x We denote by FT the set of fixed points of T

i.e., FT {x ∈ H : Tx x} Recall that a self-mapping f : C → C is a contraction on C, if

there exists a constantα ∈ 0, 1 such that fx − fy ≤ αx − y for all x, y ∈ C A bounded

linear operatorA on H is called strongly positive with coeﬃcient γ if there is a constant γ > 0

with the property

Ax, x ≥ γx2, ∀x ∈ H. 1.1

In 1953, Mann1 introduced a well-known classical iteration to approximate a fixed point of

a nonexpansive mapping This iteration is defined as

x n1 α n x n 1 − α n Tx n , n ≥ 0, 1.2

Trang 2

where the initial guessx0is taken inC arbitrarily, and the sequence {α n}∞n0is in the interval

0, 1 But Mann’s iteration process has only weak convergence, even in a Hilbert space

setting In general for example, Reich2 showed that if E is a uniformly convex Banach space

and has a Frehet diﬀerentiable norm and if the sequence {αn} is such that Σ∞

n1 α n 1 − α n ∞, then the sequence {x n} generated by process 1.2 converges weakly to a point in FT.

Therefore, many authors try to modify Mann’s iteration process to have strong convergence

In 2005, Kim and Xu3 introduced the following iteration process:

x0 x ∈ C arbitrarily chosen,

y n β n x n1− β n

Tx n ,

x n1 α n u 1 − α n y n

1.3

They proved in a uniformly smooth Banach space that the sequence {x n} defined by 1.3 converges strongly to a fixed point ofT under some appropriate conditions on {α n } and {β n}

In 2008, Yao et al 4 alsomodified Mann’s iterative scheme 1.2 to get a strong convergence theorem

Let{T i}N

i1be a finite family of nonexpansive mappings withF :N

n1 FT i / ∅ There

are many authors introduced iterative method for finding an element ofF which is an optimal

point for the minimization problem Forn > N, T nis understood asT n mod Nwith the mod function taking values in{1, 2, , N} Let u be a fixed element of H.

In 2003, Xu5 proved that the sequence {x n} generated by

x n1 1 − n AT n1 x n n1 u 1.4 converges strongly to the solution of the quadratic minimization problem

min

x∈F

1

2Ax, x − x, u, 1.5 under suitable hypotheses on nand under the additional hypothesis

F FT1T2· · · T N FT N T1· · · T N−1 · · · FT2T3· · · T N T1. 1.6

In 1999, Atsushiba and Takahashi6 defined the mapping W nas follows:

U n,0 I,

U n,1 γ n,1 T11− γ n,1I,

U n,2 γ n,2 T2U n,11− γ n,2I,

U n,3 γ n,3 T3U n,21− γ n,3

I,

U n,N−1 γ n,N−1 T N − 1U n,N−21− γ n,N−1I,

W n U n,N γ n,N T N U n,N−11− γ n,N

I,

1.7

Trang 3

where{γ n,i}N i ⊆ 0, 1 This mapping is called the W-mapping generated by T1, T2, , T Nand

γ n,1 , γ n,2 , , γ n,N

In 2000, Takahashi and Shimoji7 proved that if X is strictly convex Banach space,

thenFW n N i1 FT i , where 0 < λ n,i < 1, i 1, 2, , N.

In 2007,Shang et al.8 introduced a composite iteration scheme as follows:

y n β n x n1− β n

W n x n ,

x n1 α n γfx n I − α n Ay n ,

1.8

wheref ∈Cis a contraction, andA is a linear bounded operator.

Note that the iterative scheme1.8 is not well-defined, because x n n ≥ 1 may not lie

inC, so W n x nis not defined However, ifC H, the iterative scheme 1.8 is well-defined and Theorem 2.1 8 is obtained In the case C / H, we have to modify the iterative scheme

1.8 in order to make it well-defined

In 2009, Kangtunyakarn and Suantai 9 introduced a new mapping, called

K-mapping, for finding a common fixed point of a finite family of nonexpansive mappings For

a finite family of nonexpansive mappings{T i}N

i1and sequence{γ n,i}N

i in0, 1, the mapping

K n:C → C is defined as follows:

U n,1 γ n,1 T11− γ n,1

I,

U n,2 γ n,2 T2U n,11− γ n,2

U n,1 ,

U n,3 γ n,3 T3U n,21− γ n,3U n,2 ,

U n,N−1 γ n,N−1 T N − 1U n,N−21− γ n,N−1

U n,N−2 ,

K n U n,N γ n,N T N U n,N−11− γ n,NU n,N−1

1.9

The mappingK n is called the K-mapping generated by T1, , T Nandγ n,1 , γ n,2 , , γ n,N

In this paper, motivated by Kim and Xu3, Marino and Xu 10, Xu 5, Yao et al 4, andShang et al.8, we introduce a composite iterative scheme as follows:

y n β n x n1− β nK n x n ,

x n1 P C

α n γfx n I − α n Ay n

,

1.10

wheref ∈C is a contraction, andA is a bounded linear operator We prove, under certain

appropriate conditions on the sequences{α n } and {β n } that {x n} defined by 1.10 converges strongly to a common fixed point of the finite family of nonexpansive mappings{T i}N i1, which solves a variational inequaility problem

Trang 4

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

Lemma 1.1 For all x, y ∈ H, there holds the inequality

x y2

≤ x2 2y, x y, x, y ∈ H. 1.11

Lemma 1.2 see 11 Let {x n } and {z n } be bounded sequences in a Banach space X, and let {β n}

be a sequence in 0, 1 with 0 < lim inf n → ∞ β n≤ lim supn → ∞ β n < 1 Suppose that

x n1 β n x n1− β n

for all integer n ≥ 0, and

lim sup

n → ∞ z n1 − z n − x n1 − x n ≤ 0. 1.13

Then lim n → ∞ x n − z n 0.

Lemma 1.3 see 5 Assume that {a n } is a sequence of nonnegative real numbers such that a n1≤

1 − γ n a n δ n n ≥ 0, where {γ n } ⊂ 0, 1 and {δ n } is a sequence in R such that

i ∞

n1 γ n ∞,

ii lim supn → ∞ δ n /γ n ≤ 0 or ∞

n1 |δ n | < ∞.

Then lim n → ∞ a n 0.

Lemma 1.4 see 10 Let A be a strongly positive linear bounded operator on a Hilbert space H

with coeﬃcient γ and 0 < ρ ≤ A−1 Then I − ρA ≤ 1 − ργ.

Lemma 1.5 see 10 Let H be a Hilbert space Let A be a strongly positive linear bounded operator

with coeﬃcient γ > 0 Assume that 0 < γ < γ/α Let T : C → C be a nonexpansive mapping with a fixed point x t ∈ C of the contraction C x → tγfx 1 − tATx Then x t converges strongly as

t → 0 to a fixed point x of T, which solves the variational inequality

A − γfx, z − x≥ 0, z ∈ FT. 1.14

Lemma 1.6 see 1 Demiclosedness principle Assume that T is nonexpansive self-mapping of

closed convex subset C of a Hilbert space H If T has a fixed point, then I − T is demiclosed That is, whenever {x n } is a sequence in C weakly converging to some x ∈ C and the sequence {I − Tx n}

strongly converges to some y, it follows that I − Tx y Here, I is identity mapping of H.

Lemma 1.7 see 9 Let C be a nonempty closed convex subset of a strictly convex Banach space.

Let {T i}N i1 be a finite family of nonexpansive mappings of C into itself withN i1 FT i / ∅, and let

λ1, , λ N be real numbers such that 0 < λ i < 1 for every i 1, , N − 1 and 0 < λ N ≤ 1 Let K be

the K-mapping of C into itself generated by T1, , T N and λ1, , λ N Then FK N

i1 FT i .

Trang 5

By using the same argument as in9, Lemma 2.10, we obtain the following lemma.

Lemma 1.8 Let C be a nonempty closed convex subset of Banach space Let {T i}N

i1 be a finite family of nonexpanxive mappings of C into itself and {λ n,i}N i1 sequences in 0, 1 such that λ n,i → λ i , as n →

∞, i 1, 2, , N Moreover, for every n ∈ N, let K and K n be the K -mappings generated by

T1, T2, , T N and λ1, λ2, , λ N , and T1, T2, , T N and λ n,1 , λ n,2 , , λ n,N , respectively Then, for every bounded sequence x n ∈ C, one has lim n → ∞ K n x n − Kx n 0.

Let H be real Hilbert space with inner product ·, ·, C a nonempty closed convex

subset ofH Recall that the metric nearest point projection P C from a real Hilbert spaceH

to a closed convex subsetC of H is defined as follows Given that x ∈ H, P C x is the only

point inC with the property x − P C x inf{x − y : y ∈ C} BelowLemma 1.9can be found

in any standard functional analysis book

Lemma 1.9 Let C be a closed convex subset of a real Hilbert space H Given that x ∈ H and y ∈ C

then

i y P C x if and only if the inequality x − y, y − z ≥ 0 for all z ∈ C,

ii P C is nonexpansive,

iii x − y, P C x − P C y ≥ P C x − P C y2for all x, y ∈ H,

iv x − P C x, P C x − y ≥ 0 for all x ∈ H and y ∈ C.

2 Main Result

In this section, we prove strong convergence of the sequences{x n} defined by the iteration scheme1.10

Theorem 2.1 Let H be a Hilbert space, C a closed convex nonempty subset of H Let A be a strongly

positive linear bounded operator with coeﬃcient γ > 0, and let f ∈c·Let {T i}N i1 be a finite family of nonexpansive mappings of C into itself, and let K n be defined by1.9 Assume that 0 < γ < γ/α and

F N

i1 FT i / ∅ Let x0 ∈ C, given that {α n}∞

n0 and {β n}∞

n0 are sequences in 0, 1, and suppose

that the following conditions are satisfied:

C1 α n → 0;

C2 ∞

n0 α n ∞;

C3 0 < lim inf n → ∞ β n≤ lim supn → ∞ β n < 1;

C4 ∞n1 |γ n,i − γ n−1,i | < ∞, for all i 1, 2, , N and {γ n,i}N i1 ⊂ a, b, where 0 < a ≤ b < 1;

C5 ∞n1 |α n1 − α n | < ∞;

C6 ∞

n1 |β n1 − β n | < ∞.

If {x n}∞

n1 is the composite process defined by1.10, then {x n}∞

n1 converges strongly to q ∈ F, which also solves the following variational inequality:

γfq− Aq, p − q≤ 0, p ∈ F. 2.1

Trang 6

Proof First, we observe that {x n}∞n0is bounded Indeed, take a pointu ∈ F, and notice that

y n − u ≤ β n x n − u 1− β nK n x n − u ≤ x n − u. 2.2

Sinceα n → 0, we may assume that α n ≤ A−1 for all n ByLemma 1.4, we haveI − α n A ≤

1− α n γ for all n.

It follows that

x n1 − u P C

− P C u

≤α n

γfx n − Au I − α n Ay n − u

≤ α n γfx n − Au 1 − α n γ y n − u

≤ α n γfx n − γfu α n γfu − Au 1 − α n γ y n − u

≤ αγα n x n − u α n γfu − Au 1 − α n γx n − u

1−γ − γαα n

x n − u α n γfu − Au

1−γ − γαα n

x n − u γ − γαα n γfu − Au

γ − γα

≤ max x n − u, γfu − Au

γ − γα

.

2.3

By simple inductions, we have

x n − u ≤ max x0− u, γfu − Au

γ − γα

Therefore {x n } is bounded, so are {y n } and {fx n } Since K n is nonexpansive and y n

β n x n 1 − β n K n x n, we also have

y n1 − y n ≤β n1 x n11− β n1K n1 x n1−β n x n1− β nK n x n

β n1 x n1 − β n1 x n β n1 x n − β n x n1− β n1

K n1 x n1 − K n1 x n

1− β n1K n1 x n − K n x n 1− β n1K n x n−1− β nK n x n

≤ β n1 x n1 − x n β n1 − β n x n 1− β n1K n1 x n1 − K n1 x n

1− β n1

K n1 x n − K n x n β n − β n1 K n x n

≤ β n1 x n1 − x n β n1 − β n x n 1− β n1x n1 − x n

1− β n1

x n1 − x n β n1 − β n x n 1− β n1

K n1 x n − K n x n βn − β n1 K n x n .

2.5

Trang 7

By using the inequalities2.6 and 2.11 of 9, Lemma 2.11, we can conclude that

K n x n−1 − K n−1 x n−1 ≤ M N

whereM sup{ N j2 T j U n,j−1 x n U n,j−1 x n T1x n x n}

By2.5 and 2.6, we have

x n1 − x n P C

−P C

α n−1 γfx n−1 I − α n−1 Ay n−1

≤I − α n Ay n − y n−1

− α n − α n−1 Ay n−1

γα n

fx n − fx n−1 γα n − α n−1 fx n−1

≤1− α n γ y n − y n−1 |α n − α n−1|Ayn−1

 γαα n x n − x n−1 γ|α n − α n−1|fxn−1

≤1− α n γx n − x n−1 β n − β n−1 x n−1

1− β n K n x n−1 − K n−1 x n−1 β n−1 − β n K n−1 x n−1

 |α n − α n−1|Ayn−1 γαα n x n − x n−1 γ|α n − α n−1|fxn−1

≤1− α n γx n − x n−1 β n − β n−1 x n−1

1− β n K n x n−1 − K n−1 x n−1 β n−1 − β n K n−1 x n−1

 |α n − α n−1 |Ay n−1 γαα n x n − x n−1 γ|α n − α n−1 |fx n−1

1−γ − γαα nx n − x n−1 L β n−1 − β n M|α n − α n−1|

1− β n M N

j1 γ n,j − γ n−1,j ,

2.7

whereL sup{x n−1 K n−1 x n−1 : n ∈ N}, M max{Ay n−1 γfx n−1} Since ∞

n1 |α n−

we obtainx n1 − x n → 0 It follows that

x n1 − y n P C

− P Cy n

≤α n γfx n I − α n Ay n − y n

α n γfx n Ay n . 2.8

Trang 8

Sinceα n → 0 and {fx n }, {Ay n } are bounded, we have x n1 − y n → 0 as n → ∞ Since

x n − y n ≤ x n − x n1 x n1 − y n , 2.9

it implies thatx n − y n → 0 as n → ∞.

On the other hand, we have

K n x n − x n ≤x n − y n y n − K n x n x n − y n β n x n − K n x n , 2.10

which implies that1 − β n K n x n − x n ≤ x n − y n

From conditionC3 and x n − y n → 0 as n → ∞, we obtain

K n x n − x n → 0. 2.11

ByC4, we have limn → ∞ γ n,i γ i ∈ a, b for all i 1, 2, , N Let K be the K-mapping

generated byT1, , T Nandγ1, , γ N Next, we show that

lim sup

n → ∞

γfq− Aq, x n − q≤ 0, 2.12

whereq lim t → 0 x twithx tbeing the fixed point of the contractionx → tγfx I − tAKx.

Thus, x t solves the fixed point equation x t tγfx t I − tAKx t By Lemma 1.5 and

Lemma 1.7, we haveq ∈ F and γfq − Aq, p − q ≥ 0 for all p ∈ F It follows by 2.11 and Lemma 1.8thatKx n − x n → 0 Thus, we have x t − x n I − tAKx t − x n

tγfx t − Ax n It follows fromLemma 1.1that for 0< t < A−1,

x t − x n2I − tAKx t − x n tγfx t − Ax n2

≤1− γt2Kx t − x n2 2tγfx t − Ax n , x t − x n

≤1− γt2

Kx t − Kx n2 2Kx t − K n x n Kx n − x n Kx n − x n2

 2tγfx t − Ax t , x t − x n Ax t − Ax n , x t − x n

≤1− 2γt γt2

x t − x n2 f n t 2tγfx t − Ax t , x t − x n

 2tAx t − Ax n , x t − x n ,

2.13

where

f n t 2x t − x n x n − Kx n x n − Kx n −→ 0, as n → 0. 2.14

Trang 9

It follows that

Ax t − γfx t , x t − x n

≤

−2γt γt2

2t

x t − x n2 1

2t f n t Ax t − Ax n , x t − x n

≤−2 γt

2

γx t − x n2 1

≤

−1 γt 2

Ax t − Ax n , x t − x n 1

≤ γt

2Ax t − Ax n , x t − x n 1

2t f n t.

2.15 Lettingn → ∞ in 2.15 and 2.14, we get

lim sup

n → ∞

Ax t − γfx t , x t − x n

≤ t

whereM0 > 0 is a constant such that M0 ≥ γAx t − Ax n , x t − x n for all t ∈ 0, 1 and n ≥ 1.

Takingt → 0 in 2.16, we have

lim sup

t → 0 lim sup

n → ∞

Ax t − γfx t , x t − x n≤ 0. 2.17

On the other hand, one has

γfq− Aq, x n − qγfq− Aq, x n − q−γfq− Aq, x n − x t

γfq− Aq, x n − x t−γfq− Ax t , x n − x t

γfq− Ax t , x n − x t−γfx t − Ax t , x n − x t

γfx t − Ax t , x n − x t

.

γfq− Aq, x t − qAx t − Aq, x n − x t

γfq− γfx t , x n − x t

≤γfq − Aqx t − q Ax t − q γαx t − qx n − x t

γfq − Aqx t − q A γαx t − qx n − x t

.

2.18

It follows that

lim sup

n → ∞

γfq− Aq, x n − q≤γfq − Aqx t − q A γαx t − qlim sup

n → ∞ x n − x t

lim sup

n → ∞

γfx t − Ax t , x n − x t.

2.19

Trang 10

Therefore, from 2.17 and limt → 0 x t − q 0, we have

lim sup

n → ∞ γfq− Aq, x n − q ≤ lim sup

t → 0

lim sup

n → ∞ γfq− Aq, x n − q

≤ lim sup

t → 0 lim sup

n → ∞

≤ 0. 2.20

Hence 2.12 holds Finally, we prove that x n → q By using 2.2 and together with the Schwarz inequality, we have

x n1 − q2P C

− PC

q2

≤α n

γfx n − Aq I − α n Ay n − q2

I − α n Ay n − q2 α2

n γfx n − Aq2

 2α n

I − α n Ay n − q, γfx n − Aq

≤1− α n γ2y n − q2 α2

n γfx n − Aq2

 2α n

y n − q, γfx n − Aq− 2α2

n

Ay n − q, γfx n − Aq

≤1− α n γ2x n − q2 α2

n γfx n − Aq2

 2α n

y n − q, γfx n − γfq 2α n

y n − q, γfq− Aq

− 2α2

n

≤1− α n γ2x n − q2 α2

n γfx n − Aq2

 2α n y n − qγfx n − γfq 2α n

− 2α2

n

≤1− α n γ2x n − q2 α2

n γfx n − Aq2

 2γαα n y n − qx n − q 2α n

− 2α2

n

≤1− α n γ2x n − q2 α2

n γfx n − Aq2

 2γαα n x n − q2 2α n

− 2α2

n

≤

1− α n γ2 2γαα nx

n − q2 2α ny n − q, γfx n − Aq

 α2

n γfx n − Aq2 2α2

n Ay n − qγfx n − Aq

1− 2γ − γαα n x n − q2

 α n2

α nγfx

n − Aq2 2Ayn − qγfx n − Aq γ2x n − q2

.

2.21

Trang 7

By using the inequalities2.6 and 2.11 of 9, Lemma 2.11, we can conclude that

K... n1

K n1 x n − K n x n β n − β n1 K n x n... n1

Định dạng
Số trang	12
Dung lượng	510,16 KB