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Volume 2009, Article ID 261932, 14 pagesdoi:10.1155/2009/261932 Research Article Strong Convergence of Monotone Hybrid Method for Maximal Monotone Operators and Hemirelatively Nonexpansi

Volume 2009, Article ID 261932, 14 pages

doi:10.1155/2009/261932

Research Article

Strong Convergence of Monotone

Hybrid Method for Maximal Monotone Operators and Hemirelatively Nonexpansive Mappings

Chakkrid Klin-eam and Suthep Suantai

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

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

Received 20 May 2009; Accepted 21 September 2009

Recommended by Wataru Takahashi

We prove strong convergence theorems for finding a common element of the zero point set of a maximal monotone operator and the fixed point set of a hemirelatively nonexpansive mapping

in a Banach space by using monotone hybrid iteration method By using these results, we obtain new convergence results for resolvents of maximal monotone operators and hemirelatively nonexpansive mappings in a Banach space

Copyrightq 2009 C Klin-eam 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

1 Introduction

Let E be a real Banach space and let E∗be the dual space of E Let A be a maximal monotone operator from E to E∗ It is well known that many problems in nonlinear analysis and

optimization can be formulated as follows Find a point u ∈ E satisfying

We denote by A−10 the set of all points u ∈ C such that 0 ∈ Au Such a problem

contains numerous problems in economics, optimization, and physics and is connected with a variational inequality problem It is well known that the variational inequalities are equivalent to the fixed point problems There are many authors who studied the problem of finding a common element of the fixed point of nonlinear mappings and the set of solutions

of a variational inequality in the framework of Hilbert spaces see; for instance,1 11 and the reference therein

A well-known method to solve problem 1.1 is called the proximal point algorithm:

x0∈ E and

x n1  J r n x n , n  0, 1, 2, 3, , 1.2

where {r n } ⊂ 0, ∞ and J r n are the resovents of A Many researchers have studies this

algorithm in a Hilbert space; see, for instance,12–15 and in a Banach space; see, for instance,

16,17

In 2005, Matsushita and Takahashi 18 proposed the following hybrid iteration method it is also called the CQ method with generalized projection for relatively

nonexpansive mapping T in a Banach space E: x0 x ∈ C chosen arbitrarily,

u n  J−1α n Jx n  1 − α n JTx n ,

C nz ∈ C : φz, u n  ≤ φz, x n,

Q n  {z ∈ C : x n − z, Jx − Jx n  ≥ 0},

x n1 ΠC n ∩Q n x,

1.3

where J is the duality mapping on E, {α n } ⊂ 0, 1 They proved that {x n} generated by 1.3

converges strongly to a fixed point of T under condition that lim sup n → ∞ α n < 1.

In 2008, Su et al.19 modified the CQ method 1.3 for approximation a fixed point of

a closed hemi-relatively nonexpansive mapping in a Banach space Their method is known

as the monotone hybrid method defined as the following x0 x ∈ C chosen arbitrarily, then

x1 x ∈ C, C−1 Q−1 C,

u n  J−1α n Jx n  1 − α n JTx n ,

C nz ∈ C n−1 ∩ Q n−1 : φz, u n  ≤ φz, x n,

Q n  {z ∈ C n−1 ∩ Q n−1:x n − z, Jx − Jx n  ≥ 0},

x n1 ΠC n ∩Q n x,

1.4

where J is the duality mapping on E, {α n } ⊂ 0, 1 They proved that {x n} generated by 1.4

converges strongly to a fixed point of T under condition that lim sup n → ∞ α n < 1.

Note that the hybrid method iteration method presented by Matsushita and Takahashi

18 can be used for relatively nonexpansive mapping, but it cannot be used for hemi-relatively nonexpansive mapping

Very recently, Inoue et al.20 proved the following strong convergence theorem for finding a common element of the zero point set of a maximal monotone operator and the fixed point set of a relatively nonexpansive mapping by using the hybrid method

Theorem 1.1 Inoue et al 20 Let E be a uniformly convex and uniformly smooth Banach space

and let C be a nonempty closed convex subset of E Let A ⊂ E × E∗be a monotone operator satisfying

DA ⊂ C and let J r  J  rA−1J for all r > 0 Let T : C → C be a relatively nonexpansive mapping such that FT ∩ A−10/ ∅ Let {x n } be a sequence generated by x0 x ∈ C and

u n  J−1α n Jx n  1 − α n JTJ r n x n ,

C nz ∈ C : φz, u n  ≤ φz, x n,

Q n  {z ∈ C : x n − z, Jx − Jx n  ≥ 0},

x n1 ΠC n ∩Q n x

1.5

for all n ∈ N ∪ {0}, where J is the duality mapping on E, {α n } ⊂ 0, 1 and {r n } ⊂ a, ∞ for some

a > 0 If lim inf n → ∞ 1 − α n  > 0, then {x n } converges strongly to Π FT∩A−1 0x0, whereΠFT∩A−1 0is the generalized projection from C onto FT ∩ A−10.

Employing the ideas of Inoue et al 20 and Su et al 19, we modify iterations

1.4 and 1.5 to obtain strong convergence theorems for finding a common element of the zero point set of a maximal monotone operator and the fixed point set of a hemi-relatively nonexpansive mapping in a Banach space Using these results, we obtain new convergence results for resolvents of maximal monotone operators and hemi-relatively nonexpansive mappings in a Banach space The results of this paper modify and improve the results of Inoue et al.20, and some others

2 Preliminaries

Throughout this paper, all linear spaces are real LetN and R be the sets of all positive integers

and real numbers, respectively Let E be a Banach space and let E∗be the dual space of E For

a sequence{x n } of E and a point x ∈ E, the weak convergence of {x n } to x and the strong

convergence of{x n } to x are denoted by x n  x and x n → x, respectively.

Let E be a Banach space Then the duality mapping J from E into 2 E∗is defined by

Jx x∗∈ E∗:x, x∗  x2 x∗2

Let SE be the unit sphere centered at the origin of E Then the space E is said to be

smooth if the limit

lim

t → 0

x  ty − x

exists for all x, y ∈ SE It is also said to be uniformly smooth if the limit exists uniformly

in x, y ∈ SE A Banach space E is said to be strictly convex if x  y/2 < 1 whenever

x, y ∈ SE and x / y It is said to be uniformly convex if for each  ∈ 0, 2, there exists δ > 0

such thatx  y/2 < 1 − δ whenever x, y ∈ SE and x − y ≥  We know the following

see, 21:

i if E in smooth, then J is single valued;

ii if E is reflexive, then J is onto;

iii if E is strictly convex, then J is one to one;

iv if E is strictly convex, then J is strictly monotone;

v if E is uniformly smooth, then J is uniformly norm-to-norm continuous on each bounded subset of E.

Let E be a smooth strictly convex and reflexive Banach space and let C be a closed convex subset of E Throughout this paper, define the function φ : E × E → R by

φ

y, x

Observe that, in a Hilbert space H,2.3 reduces to φx, y  x − y2, for all x, y ∈ H It is obvious from the definition of the function φ that for all x, y ∈ E,

1 x − y2≤ φx, y ≤ x  y2,

2 φx, y  φx, z  φz, y  2x − z, Jz − Jy,

3 φx, y  x, Jx − Jy  y − x, Jy ≤ xJx − Jy  y − xy.

Following Alber 22, the generalized projection ΠC from E onto C is a map that assigns to an arbitrary point x ∈ E the minimum point of the functional φy, x; that is,

ΠC x  x, where x is the solution to the minimization problem

φx, x  min

y, x

Existence and uniqueness of the operatorΠC follow from the properties of the functional

φy, x and strict monotonicity of the mapping J In a Hilbert space, Π C is the metric

projection of H onto C.

Let C be a closed convex subset of a Banach space E, and let T be a mapping from C into itself We use FT to denote the set of fixed points of T; that is, FT  {x ∈ C : x  Tx} Recall that a self-mapping T : C → C is hemi-relatively nonexpansive if FT / ∅ and φu, Tx ≤ φu, x for all x ∈ C and u ∈ FT.

A point u ∈ C is said to be an asymptotic fixed point of T if C contains a sequence {x n } which converges weakly to u and lim n → ∞ x n − Tx n  0 We denote the set of all

asymptotic fixed points of T by FT A hemi-relative nonexpansive mapping T : C → C

is said to be relatively nonexpansive if FT  FT / ∅ The asymptotic behavior of a relatively

nonexpansive mapping was studied in23

Recall that an operator T in a Banach space is call closed, if x n → x and Tx n → y, then

Tx  y.

We need the following lemmas for the proof of our main results

Lemma 2.1 Kamimura and Takahashi 13 Let E be a uniformly convex and smooth Banach

space and let {x n } and {y n } be two sequences in E such that either {x n } or {y n } is bounded If

limn → ∞ φx n , y n   0, then lim n → ∞ x n − y n   0.

Lemma 2.2 Matsushita and Takahashi 18 Let C be a closed convex subset of a smooth, strictly

convex, and reflexive Banach space E and let T be a relatively hemi-nonexpansive mapping from C into itself Then FT is closed and convex.

Lemma 2.3 Alber 22, Kamimura and Takahashi 13 Let C be a closed convex subset of a

smooth, strictly convex, and reflexive Banach space, x ∈ E and let z ∈ C Then, z  Π C x if and only

if y − z, Jx − Jz ≤ 0 for all y ∈ C.

Lemma 2.4 Alber 22, Kamimura and Takahashi 13 Let C be a closed convex subset of a

smooth, strictly convex, and reflexive Banach space Then

φ

x, Π C y

 φΠC y, y

≤ φx, y

, ∀x ∈ C, y ∈ E. 2.5

Let E be a smooth, strictly convex, and reflexive Banach space, and let A be a set-valued mapping from E to E∗with graph GA  {x, x∗ : x∗ ∈ Ax}, domain DA  {z ∈

E : Az / ∅}, and range RA  ∪{Az : z ∈ DA} We denote a set-valued operator A from E

to E∗by A ⊂ E × E∗.A is said to be monotone of x − y, x∗− y∗ ≥ 0, for all x, x∗, y, y∗ ∈ A.

A monotone operator A ⊂ E × E∗is said to be maximal monotone if its graph is not properly

contained in the graph of any other monotone operator It is known that a monotone mapping

A is maximal if and only if for x, x∗ ∈ E × E∗, x − y, x∗− y∗ ≥ 0 for every y, y∗ ∈ GA implies that x∗ ∈ Ax We know that if A is a maximal monotone operator, then A−10 {z ∈

DA : 0 ∈ Az} is closed and convex; see 19 for more details The following result is well known

Lemma 2.5 Rockafellar 24 Let E be a smooth, strictly convex, and reflexive Banach space and

let A ⊂ E × E∗be a monotone operator Then A is maximal if and only if RJ  rA  E∗for all r > 0.

Let E be a smooth, strictly convex, and reflexive Banach space, let C be a nonempty closed convex subset of E, and let A ⊂ E × E∗be a monotone operator satisfying

DA ⊂ C ⊂ J−1

r>0 RJ  rA



Then we can define the resolvent J r : C → DA by

J r x  {z ∈ DA : Jx ∈ Jz  rAz}, ∀x ∈ C. 2.7

We know that J r x consists of one point For r > 0, the Yosida approximation A r : C → E∗is

defined by A r x  Jx − JJ r x/r for all x ∈ C.

Lemma 2.6 Kohsaka and Takahashi 25 Let E be a smooth, strictly convex, and reflexive Banach

space, let C be a nonempty closed convex subset of E, and let A ⊂ E × E∗ be a monotone operator satisfying

DA ⊂ C ⊂ J−1

r>0 RJ  rA



Let r > 0 and let J r and A r be the resolvent and the Yosida approximation of A, respectively Then, the following hold:

i φu, J r x  φJ r x, x ≤ φu, x, for all x ∈ C, u ∈ A−10;

ii J r x, A r x ∈ A, for all x ∈ C;

iii FJ r   A−10.

Lemma 2.7 Kamimura and Takahashi 13 Let E be a uniformly convex and smooth Banach

space and let r > 0 Then there exists a strictly increasing, continuous and convex function g :

0, 2r → 0, ∞ such that g0  0 and

for all x, y ∈ B r 0, where B r 0  {z ∈ E : z ≤ r}.

3 Main Results

In this section, we prove a strong convergence theorem for finding a common element of the zero point set of a maximal monotone operator and the fixed point set of a hemi-relatively nonexpansive mapping in a Banach space by using the monotone hybrid iteration method

Theorem 3.1 Let E be a uniformly convex and uniformly smooth Banach space and let C be a

nonempty closed convex subset of E Let A ⊂ E × E∗be a monotone operator satisfying DA ⊂ C and let J r  J  rA−1J for all r > 0 Let T : C → C be a closed hemi-relatively nonexpansive mapping such that FT ∩ A−10/ ∅ Let {x n } be a sequence generated by

x0 x ∈ C, C−1 Q−1 C,

u n  J−1α n Jx n  1 − α n JTJ r n x n ,

C nz ∈ C n−1 ∩ Q n−1 : φz, u n  ≤ φz, x n,

Q n  {z ∈ C n−1 ∩ Q n−1:x n − z, Jx − Jx n  ≥ 0},

x n1 ΠC n ∩Q n xl

3.1

for all n ∈ N ∪ {0}, where J is the duality mapping on E, {α n } ⊂ 0, 1 and {r n } ⊂ a, ∞ for some

a > 0 If lim inf n → ∞ 1 − α n  > 0, then {x n } converges strongly to Π FT∩A−1 0x0, whereΠFT∩A−1 0is the generalized projection from C onto FT ∩ A−10.

Proof We first show that C n and Q n are closed and convex for each n≥ 0 From the definition

of C n and Q n , it is obvious that C n is closed and Q n is closed and convex for each n≥ 0 Next,

we prove that C nis convex

Since

φz, u n  ≤ φz, x n 3.2

is equivalent to

which is affine in z, and hence Cn is convex So, C n ∩ Q n is a closed and convex subset of E for all n ≥ 0 Let u ∈ FT ∩ A−10 Put y n  J r n x n for all n ≥ 0 Since T and J r n are hemi-relatively nonexpansive mappings, we have

φu, u n   φu, J−1

α n Jx n  1 − α n JTy n

 u2− 2 u, α n Jx n  1 − α n JTy n α n Jx n  1 − α n JTy n2

≤ u2− 2α n u, Jx n  − 21 − α n u, JTy n  α nxn2 1 − α nTyn2

 α n φu, x n   1 − α n φu, Ty n

≤ α n φu, x n   1 − α n φu, y n

 α n φu, x n   1 − α n φu, J r n x n

≤ α n φu, x n   1 − α n φu, x n

 φu, x n .

3.4

So, u ∈ C n for all n ≥ 0, which implies that FT∩A−10⊂ C n Next, we show that FT∩A−10⊂

Q n for all n ≥ 0 We prove that by induction For k  0, we have FT ∩ A−10 ⊂ C  Q−1

Assume that FT ∩ A−10 ⊂ Q k−1 for some k ≥ 0 Because x k is the projection of x0 onto

x k − z, Jx0− Jx k  ≥ 0, ∀z ∈ C k−1 ∩ Q k−1 3.5

Since FT ∩ A−10⊂ C k−1 ∩ Q k−1, we have

This together with definition of Q n implies that F T∩A−10⊂ Q k and hence FT∩A−10⊂ Q n

for all n ≥ 0 So, we have that FT∩A−10⊂ C n ∩Q n for all n ≥ 0 This implies that {x n} is well

defined From definition of Q n we have x n ΠQ n x0 So, from x n1 ΠC n ∩Q n x0∈ C n ∩Q n ⊂ Q n,

we have

Therefore,{φx n , x0} is nondecreasing It follows fromLemma 2.4and x n ΠQ n x0that

φx n , x0  φΠQ x0, x0



≤ φu, x0 − φu, Π Q x0



for all u ∈ FT ∩ A−1 ⊂ Q n Therefore,{φx n , x0} is bounded Moreover, by definition of φ,

we know that{x n } and {J r n x n }  {y n } are bounded So, the limit of {φx n , x0} exists From

x n ΠQ n x0, we have that for any positive integer,

φx nk , x n   φx nk , Π Q n x0



≤ φx nk , x0 − φΠQ n x0, x0



 φx nk , x0 − φx n , x0 3.9

This implies that limn → ∞ φx nk , x n   0 Since {x n } is bounded, there exists r > 0 such that {x n } ⊂ B r0 UsingLemma 2.7, we have, for m, n with m > n,

gx m − x n  ≤ φx m , x n  ≤ φx m , x0 − φx n , x0, 3.10

where g : 0, 2r → 0, ∞ is a continuous, strictly increasing, and convex function with

g0  0 Then the properties of the function g yield that {x n } is a Cauchy sequence in C So there exists w ∈ C such that x n → w In view of x n1 ΠC n ∩Q n x0 ∈ C n and definition of C n,

we also have

It follows that limn → ∞ φx n1 , u n  limn → ∞ φx n1 , x n   0 Since E is uniformly convex and

smooth, we have fromLemma 2.1that

lim

n → ∞ x n1 − x n  lim

So, we have limn → ∞ x n −u n   0 Since J is uniformly norm-to-norm continuous on bounded

sets, we have

lim

n → ∞ Jx n1 − Jx n  lim

n → ∞ Jx n1 − Ju n  lim

On the other hand, we have

Jx n1 − Ju n Jx n1 − α n Jx n − 1 − α n JTy n

α n Jx n1 − Jx n   1 − α nJx n1 − JTy n

1 − α nJx n1 − JTy n

− α n Jx n − Jx n1

≥ 1 − α nJxn1 − JTy n  − α n Jx n − Jx n1 .

3.14

This follows

Jx n1 − JTy n ≤ 1

1− α n Jx n1 − Ju n   α n Jx n − Jx n1 . 3.15 From3.13 and lim infn → ∞ 1 − α n  > 0, we obtain that lim n → ∞ Jx n1 − JTy n   0.

Since J−1is uniformly norm-to-norm continuous on bounded sets, we have

lim

From

x n − Ty n  ≤ x n − x n1 x n1 − Ty n , 3.17

we have

lim

From3.4, we have

φ

u, y n

1− α n



φu, u n  − α n φu, x n. 3.19

Using y n  J r n x nandLemma 2.6, we have

φ

y n , x n

 φJ r n x n , x n  ≤ φu, x n  − φu, J r n x n   φu, x n  − φu, y n

It follows that

φ

y n , x n

≤ φu, x n  − φu, y n

≤ φu, x n − 1

1− α n



φu, u n  − α n φu, x n

1− α n



φu, x n  − φu, u n

1− α n



x n2− u n2− 2u, Jx n − Ju n

1− α n

x

n2− u n2  2|u, Jx

n − Ju n|

1− α n |x n  − u n |x n   u n   2uJx n − Ju n

1− α n x n − u n x n   u n   2uJx n − Ju n .

3.21

From3.13 and limn → ∞ x n − u n   0, we have lim n → ∞ φy n , x n   0.

Since E is uniformly convex and smooth, we have fromLemma 2.1that

lim

From limn → ∞ x n − Ty n   0, we have

lim

Since x n → w and lim n → ∞ x n − y n   0, we have y n → w Since T is a closed operator and

y n → w, w is a fixed point of T Next, we show w ∈ A−10 Since J is uniformly norm-to-norm

continuous on bounded sets, from3.22 we have

lim

From r n ≥ a, we have

lim

n → ∞

1

Therefore, we have

lim

n → ∞ A r n x n  lim

n → ∞

1

Forp, p∗ ∈ A, from the monotonicity of A, we have p − y n , p∗− A r n x n  ≥ 0 for all n ≥ 0 Letting n → ∞, we get p − w, p∗ ≥ 0 From the maximality of A, we have w ∈ A−10 Finally,

we prove that w ΠFT∩A−1 0x0 FromLemma 2.4, we have

φ

w, Π FT∩A−1 0x0



 φΠFT∩A−1 0x0, x0



Since x n1 ΠC n ∩Q n x0and w ∈ FT ∩ A−10⊂ C n ∩ Q n , we get fromLemma 2.4that

φ

ΠFT∩A−1 0x0, x n1

 φx n1 , x0 ≤ φΠFT∩A−1 0x0, x0

By the definition of φ, it follows that φw, x0 ≤ φΠ FT∩A−1 0x0, x0 and φw, x0 ≥

φΠ FT∩A−1 0x0, x0, whence φw, x0  φΠ FT∩A−1 0x0, x0 Therefore, it follows from the uniqueness of theΠFT∩A−1 0x0that w ΠFT∩A−1 0x0

Let r > and let J r and A r be the resolvent and the Yosida approximation of A, respectively Then,... the monotone hybrid iteration method

Theorem 3.1 Let E be a uniformly convex and uniformly smooth Banach space and let C be a

nonempty closed convex subset of E