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Volume 2011, Article ID 681214, 7 pagesdoi:10.1155/2011/681214 Research Article Strong Convergence Theorems by Shrinking Qiao-Li Dong,1, 2 Songnian He,1, 2 and Fang Su3 1 College of Scie

Volume 2011, Article ID 681214, 7 pages

doi:10.1155/2011/681214

Research Article

Strong Convergence Theorems by Shrinking

Qiao-Li Dong,1, 2 Songnian He,1, 2 and Fang Su3

1 College of Science, Civil Aviation University of China, Tianjin 300300, China

2 Tianjin Key Laboratory for Advanced Signal Processing, Civil Aviation University of China,

Tianjin 300300, China

3 Department of Mathematics and Systems Science, National University of Defense Technology,

Changsha 410073, China

Correspondence should be addressed to Qiao-Li Dong,dongqiaoli@ymail.com

Received 9 December 2010; Accepted 2 February 2011

Academic Editor: S Al-Homidan

Copyrightq 2011 Qiao-Li Dong 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 prove a strong convergence theorem by a shrinking projection method for the class ofT mappings Using this theorem, we get a new result We also describe a shrinking projection method for a nonexpansive mapping on Hilbert spaces, which is the same as that of Takahashi et al.2008

1 Introduction

LetH be a real Hilbert space with inner product ·, · and norm  · , and let C be a nonempty

Tx − Ty ≤ x − y for all x, y ∈ H The set of fixed points of T is FixT : {x ∈ H : Tx  x}.

T : H → H is said to be quasi-nonexpansive if FixT is nonempty and Tx − p ≤

x − p for all x ∈ H and p ∈ FixT.

Givenx, y ∈ H, let

Hx, y:z ∈ H :z − y, x − y≤ 0 1.1

cutter if T ∈ T  {T : H → H | domT  H and FixT ⊂ Hx, Tx, for all x ∈ H}

Remark 1.1 The class T is fundamental because it contains several types of operators commonly found in various areas of applied mathematics and in particular in approximation and optimization theorysee 1 for details

Combettes 2, Bauschke, and Combettes 1 studied properties of the class T mappings and presented several algorithms They introduced an abstract Haugazeau method

in1 as follows: starting x0∈ H,

x n1  P Hx0,x n∩Hxn ,T n x n x0. 1.2

quasi-nonexpansive, one can easily obtain hybrid methods introduced by Nakajo and Takahashi

nonexpan-sive mappingsT n:C → C Let x0 ∈ H, C1 C, x1 P C1x0, and

y n  α n  1 − α n T n x n ,

C n1z ∈ C n:y n − z ≤ x n − z,

x n1  P C n1 x0, n  1, 2, ,

1.3

K of H.

projection method for a nonexpansive mapping on Hilbert spaces, which is the same as presented by Takahashi et al.4

We will use the following notations:

1  for weak convergence and → for strong convergence;

2 ω w x n   {x : ∃x n j  x} denotes the weak ω-limit of {x n}

Lemma 1.2 see 1 Let H be a Hilbert space Let I be the identity operator of H.

i If dom T  H, then 2T − I is quasi-nonexpansive if and only if T ∈ T.

ii If T ∈ T, then λI  1 − λT ∈ T, for all λ ∈ 0, 1.

Definition 1.3 Let T n ∈ T for each n The sequence {T n} is called to be coherent if, for every

∞



n0

v n1 − v n2< ∞,

∞



n0

v n − T n v n2< ∞,

⇒ ω w v n ⊂ ∞

n0

Definition 1.4 T is called demiclosed at y ∈ H if Tx  y whenever {x n } ⊂ H, x n  x and

Tx n → y.

Next lemma shows that nonexpansive mappings are demeiclosed at 0

Lemma 1.5 Goebel and Kirk 5 Let C be a closed convex subset of a real Hilbert space H, and

let T : C → C be a nonexpansive mapping such that FixT / ∅ If a sequence {x n } in C is such that

x n  z and x n − Tx n → 0, then z  Tz.

Lemma 1.6 see 6 Let K be a closed convex subset of H Let {x n } be a sequence in H and u ∈ H.

Let q  P K u If x n is such that ω w x n  ⊂ K and satisfies the condition

then x n → q.

Lemma 1.7 Goebel and Kirk 5 Let K be a closed convex subset of real Hilbert space H, and let

P K be the (metric or nearest point) projection from H onto K (i.e., for x ∈ H, P K x is the only point

in K such that x − P K x  inf{x − z : z ∈ K}) Given x ∈ H and z ∈ K, then z  P K x if and only if there holds the relation



2 Main Results

and prove strong convergence theorem

Theorem 2.1 Let T n ∈ T for each n such that F : ∞n1FixTn  / ∅ Suppose that the sequence {T n}

is coherent Let x0∈ H For C1 H and x1 x0, define a sequence {x n } as follows:

x n1  P C n1 x0, n  1, 2, ,

Then, {x n } converges strongly to PFx0.

Proof We first show by induction that F ⊂ C n for all n ∈ N.F ⊂ C1 is obvious Suppose

F ⊂ C kfor somek ∈ N Note that, by the definition of T k ∈ T, we always have F ⊂ FixT k ⊂

Hx k , T k x k, that is,

From the definition ofC k1andF ⊂ C k, we obtainF ⊂ C k1 This implies that

It is obvious thatC1 H is closed and convex So, from the definition, C nis closed and convex for alln ∈ N So we get that {x n} is well defined

Sincex nis the projection ofx0ontoC nwhich containsF, we have

Takingy  PFx0 ∈ F, we get

The last inequality ensures that{x0− x n } is bounded From x n  P C n x0andx n1  P C n1 x0∈

C n1 ⊂ C n, usingLemma 1.7, we get

It follows that

x0− x n12 x0− x n  − x n1 − x n2

 x0− x n2− 2x0− x n , x n1 − x n   x n1 − x n2

≥ x0− x n2 x n1 − x n2

≥ x0− x n2.

2.7

Thus{x n − x0} is increasing Since {x n − x0} is bounded, limn → ∞ x n − x0 exists From

2.7, it follows that

n1 x n1 − x n2< ∞ On the other hand, by x n1  P C n1 x0∈ C n1, we have

Hence,

x n1 − x n2 x n1 − T n x n  − x n − T n x n2

 x n1 − T n x n2− 2x n1 − T n x n , x n − T n x n   x n − T n x n2

≥ x n1 − T n x n2 x n − T n x n2.

2.10

n1 x n − T n x n2 < ∞ Since the sequence {T n} is coherent, we have

ω w x n ⊂ F FromLemma 1.6and2.5, the result holds

Remark 2.2 We take C1 H so that F ⊂ C1is satisfied

Theorem 2.3 Let T n ∈ T for each n such that F : ∞n1FixTn  / ∅ Suppose that the sequence {T n}

is coherent Let x0∈ H For C1 H and x1 x0, define a sequence {x n } as follows:

y n  α n x n  1 − α n T n x n ,

C n1z ∈ C n:

z − y n , x n − y n

≤ 0,

x n1  P C n1 x0, n  1, 2, ,

2.11

where 0 ≤ α n ≤ a < 1 Then, {x n } converges strongly to PFx0.

Proof Set S n  α n I  1 − α n T n ByLemma 1.2ii, we have that S n ∈ T From x n − S n x n 

1 − α n x n − T n x n , it follows that 1 − ax n − T n x n  ≤ x n − S n x n  ≤ x n − T n x n which implies that the sequence{S n } is coherent It is obvious that FixS n   FixT n , for all n ∈ N.

n1FixSn  ∞

n1FixTn UsingTheorem 2.1, we get the desired result

3 Deduced Results

Hilbert space

Theorem 3.1 Let T ∈ T such that FixT / ∅ and satisfying that I −T is demiclosed at 0 Let x0∈ H.

For C1 H and x1 x0, define a sequence {x n } as follows:

y n  α n x n  1 − α n Tx n ,

C n1z ∈ C n:z − y n , x n − y n ≤ 0,

x n1  P C n1 x0, n  1, 2, ,

3.1

where 0 ≤ α n ≤ a < 1 Then, {x n } converges strongly to PFixTx0.

Proof Let T n  T in 2.11 for all n ∈ N Following the proof ofTheorem 2.1, we can easily get2.5 and∞

n1 x n − Tx n2 < ∞ By 2.5, we obtain that {x n } is bounded and ω w x n is nonempty For any x ∈ ω w x n , there exists a subsequence {x n j } of the sequence {x n} such thatx n j  x From∞n1 x n − Tx n2 < ∞, it follows that x n − Tx n  → 0 Since I − T is

and2.5 implies that x n → PFixTx0

Theorem 3.2 Let H be a Hilbert space Let S be a quasi-nonexpansive mapping on H such that

FixS / ∅ and satisfying that I − S is demiclosed at 0 Let x0∈ H For C1 H and x1 x0, define a sequence {x n } as follows:

u n  α n x n  1 − α n Sx n ,

C n1  {z ∈ C n:z − u n  ≤ x n − z},

x n1  P C n1 x0, n  1, 2, ,

3.2

where 0 ≤ α n ≤ a < 1 Then, {x n } converges strongly to PFixSx0.

Proof ByLemma 1.2i, S  I/2 ∈ T Substitute T in 3.1 by S  I/2 Then y n  1 

α n /2x n  1 − α n /2Sx n Setu n  2y n − x n  α n x n  1 − α n Sx n, theny n  u n  x n /2 So,

we have

C n1z ∈ C n:

z − y n , x n − y n

≤ 0

 {z ∈ C n:2z − x n  u n , x n − u n ≤ 0}

 {z ∈ C n:z − u n  ≤ x n − z}.

3.3

the result by usingTheorem 3.1

Theorem 3.2, we have following corollary

Corollary 3.3 Let H be a Hilbert space Let S be a nonexpansive mapping H such that FixS / ∅.

Let x0∈ H For C1 H and x1 x0, define a sequence {x n } as follows:

u n  α n x n  1 − α n Sx n ,

C n1  {z ∈ C n: z − u n  ≤ x n − z},

x n1  P C n1 x0, n  1, 2, ,

3.4

where 0 ≤ α n ≤ a < 1 Then, {x n } converges strongly to PFixSx0.

Remark 3.4. Corollary 3.3is a special case of Theorem 4.1 in4 when C1 H.

Acknowledgments

The authors would like to express their thanks to the referee for the valuable comments and suggestions for improving this paper This paper is supported by Research Funds of Civil

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Acknowledgments

The authors would like to express their thanks to the referee for the valuable comments and suggestions for improving this paper This paper is supported by Research. .. < Then, {x n } converges strongly to PFixSx0.

Proof...

1 H H Bauschke and P L Combettes, “A weak-to -strong convergence principle for Fej´er-monotone

methods in Hilbert spaces,” Mathematics of Operations Research, vol 26, no 2, pp 248–264,