Convergence Theorems of Modified Ishikawa Iterative Scheme for Two Nonexpansive Semigroups

(1)

Volume 2010, Article ID 914702,12pages doi:10.1155/2010/914702

Research Article

Convergence Theorems of

Modified Ishikawa Iterative Scheme for

Two Nonexpansive Semigroups

Kriengsak Wattanawitoon and Poom Kumam

Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), Bangmod, Thrungkru, Bangkok 10140, Thailand

Correspondence should be addressed to Poom Kumam,[email protected]

Received 26 September 2009; Accepted 24 November 2009

Academic Editor: Tomonari Suzuki

Copyrightq2010 K. Wattanawitoon and P. Kumam. 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 convergence theorems of modified Ishikawa iterative sequence for two nonexpansive semigroups in Hilbert spaces by the two hybrid methods. Our results improve and extend the corresponding results announced by Saejung2008and some others.

1. Introduction

LetCbe a subset of real Hilbert spaces Hwith the inner product·,·and the norm · .

T :C → Cis called a nonexpansive mapping if

_Tx₋_Ty_≤_x₋_y _∀_{x, y}_∈_C. _1.1

We denote byFTthe set of fixed points ofT, that is,FT {x∈C:xTx}.

Let{Tt:t≥0}be a family of mappings from a subsetCofHinto itself. We call it a nonexpansive semigroup onCif the following conditions are satisfied:

iT0xxfor allx∈C;

iiTst TsTtfor alls, t≥0;

iiifor eachx∈Cthe mappingt→Ttxis continuous;

(2)

The Mann’s iterative algorithm was introduced by Mann1in 1953. This iterative process is now known as Mann’s iterative process, which is defined as

xn1αnxn 1−αnTxn, n≥0, 1.2

where the initial guessx0is taken inCarbitrarily and the sequence{αn}∞_n0is in the interval

0,1.

In 1967, Halpern2first introduced the following iterative scheme:

x0u∈C chosen arbitrarily,

xn1αnu 1−αnTxn,

1.3

see also Browder3. He pointed out that the conditions lim_n_{→ ∞}α_n0 and∞_n₁αn ∞are

necessary in the sence that, if the iteration1.3converges to a fixed point ofT, then these conditions must be satisfied.

On the other hand, in 2002, Suzuki4was the first to introduce the following implicit iteration process in Hilbert spaces:

xnαnu 1−αnTtnxn, n≥1, 1.4

for the nonexpansive semigroup. In 2005, Xu5established a Banach space version of the sequence1.4of Suzuki4.

In 2007, Chen and He 6 studied the viscosity approximation process for a nonexpansive semigroup and prove another strong convergence theorem for a nonexpansive semigroup in Banach spaces, which is defined by

xn1αnfxn 1−αnTtnxn, ∀n∈N, 1.5

wheref:C → Cis a fixed contractive mapping.

Recently He and Chen7is proved a strong convergence theorem for nonexpansive semigroups in Hilbert spaces by hybrid method in the mathematical programming. Very recently, Saejung8proved a convergence theorem by the new iterative method introduced by Takahashi et al.9without Bochner integrals for a nonexpansive semigroup{Tt:t≥0} withF:∞_t₀FTt/∅in Hilbert spaces:

x0∈H taken arbitrary,

C1C,

x1PC1x0,

ynαnxn 1−αnTtnxn,

Cn1

z∈Cn:yn−z≤ xn−z,

xn1 PCn1x0,

1.6

(3)

In 1974, Ishikawa10introduced a new iterative scheme, which is defined recursively by

ynβnxn1−βnTxn,

xn1αnxn 1−αnTyn,

1.7

where the initial guessx0is taken inCarbitrarily and the sequences{αn}and{βn}are in the

interval0,1.

In this paper, motivated by the iterative sequences1.6given by Saejung in8and Ishikawa10, we introduce the modified Ishikawa iterative scheme for two nonexpansive semigroups in Hilbert spaces. Further, we obtain strong convergence theorems by using the hybrid methods. This result extends and improves the result of Saejung8and some others.

2. Preliminaries

This section collects some lemmas which will be used in the proofs for the main results in the next section.

It is known that every Hilbert spaceHsatisfies the Opial’s condition11, that is,

lim inf

n→ ∞ xn−x<lim infn→ ∞ xn−y, ∀y∈X, y /x. 2.1

Recall that the metricnearest pointprojectionPCfrom a Hilbert spaceHto a closed

convex subsetCofHis defined as follows. Givenx∈H, PCxis the only point inCwith the

property

x−PCxinfx−y:y∈C. 2.2

PCxis characterized as follows.

Lemma 2.1. LetHbe a real Hilbert space,Ca closed convex subset ofH. Givenx∈Handy∈C. ThenyPCxif and only if there holds the inequality

x−y, y−z≥0, ∀z∈C. 2.3

Lemma 2.2. There holds the identity in a Hilbert spaceH

_λx 1−λy2λx2 1−λy2−λ1−λx−y2 2.4

(4)

Lemma 2.3see12, Lemma 1. Let{tn}be a real sequence and letτ be a real number such that

lim inf_nt_n ≤τ≤lim sup_nt_n. Suppose that either of the following holds:

ilim sup_ntn1−tn≤0or

iilim inf_nt_n1−tn≥0,

thenτ is a cluster point of{tn}. Moreover, forε > 0, k, m ∈ N, there exists m0 ≥ m such that |tj−τ|< εfor every integerjwithm0≤j≤m0k.

3. Main Results

3.1. The Shrinking Projection Method

In this section, we prove strong convergence of an iterative sequence generated by the shrinking hybrid projection method in mathematical programming.

Theorem 3.1. LetC be a closed convex subset of a real Hilbert spaceH. Let{Tt : t ≥ 0}and {St : t ≥ 0}be nonexpansive semigroups onCwith a nonempty common fixed point setF, that is,F : ∞_t₀FTt∩∞_t₀FSt/∅. Let{αn} ⊂ 0, a ⊂ 0,1,{βn} ⊂ b, c ⊂ 0,1and {tn}be the sequences such thatlim infn→ ∞tn 0,lim supn→ ∞tn >0,andlimn→ ∞tn1−tn 0. Suppose that{x_n}is a sequence generated by the following iterative scheme:

x0∈H taken arbitrary,

C1C,

x1PC1x0,

znβnxn1−βnTtnxn,

ynαnxn 1−αnStnzn,

Cn1

u∈Cn:yn−u≤ xn−u,

xn1 PCn1x0,

3.1

then{xn}converges strongly toPFx0.

Proof. We first show thatC_n1is closed and convex for eachn≥0. From the definition ofCn1

it is obvious thatCn1 is closed for eachn ≥ 0. We show thatCn1is convex for any n≥ 0.

Since

_y_n₋_u_≤_x_n₋_u_⇐⇒₂_x_n₋_y_n_{, u}_≤_x_n2₋_y

(5)

and henceCn1is convex. Next we show thatF⊂Cn1for alln≥0. Letp∈F, then we have

_z_n₋_p_β_n_x_n

1−βnTtnxn−p ≤βnxn−p1−βnTtnxn−p ≤βnxn−p1−βnxn−p ≤xn−p,

3.3

_y_n₋_p_α_n_x_n ₁₋_α_n_S_t_n_z_n₋_p

≤αnxn−p 1−αnStnzn−p ≤αnxn−p 1−αnzn−p.

3.4

Substituting3.3into3.4, we have

_y_n₋_p_≤_x_n₋_p_. 3.5

This means thatp ∈ Cn1 for all n ≥ 0. Thus,{xn}is well defined. Sincexn PCnx0and

xn1∈Cn1 ⊂Cn, we get

x0−xn, xn−xn1 ≥0 ∀n∈N. 3.6

Consequently,

0≤ x0−xn, xn−xn1

x0−xn, xn−x0x0−xn1

−xn−x0, xn−x0x0−xn, x0−xn1

≤ −xn−x02x0−xnx0−xn1,

3.7

forn∈N. This implies that

x0−xn ≤ x0−xn1 ∀n∈N. 3.8

Therefore,{x0−xn}is nondecreasing. FromxnPCnx0, we also havex0−xn, xn−p ≥0, for allp∈Cn.

SinceF⊆C_n, we get

(6)

Thus, forp∈F, we obtain

0≤ x0−xn, xn−p

−xn−x0, xn−x0 x0−xn, x0−p

≤ −xn−x02x0−xnx0−p.

3.10

Thus,x_n−x0 ≤ x0−p, for allp∈Fandn∈N. Then limn→ ∞xn−x0exists and{xn}is

bounded.

Next, we show thatxn1−xn → 0 asn → ∞. From3.6we have

xn−xn12xn−x0x0−xn12

xn−x022xn−x0, x0−xn1x0−xn12

xn−x022xn−x0, x0−xnxn−xn1x0−xn12

xn−x02−2x0−xn, x0−xn −2x0−xn, xn−xn1x0−xn12

≤ xn−x02−2xn−x02x0−xn12

−xn−x02x0−xn12.

3.11

Since lim_n_{→ ∞}x_n−x0exists, then

lim

n→ ∞xn−xn10. 3.12

Further, as in the proof of8, page 3, we have{x_n}which is a Cauchy sequence. So, we have

xn z.By definition ofyn, we have

_y_n₋_x_n ₁₋_α_n_S_t_n_z_n₋_x_n_. _3.13

Sincex_n1∈Cn1and3.12, we obtain

Stnzn−xn ₁₋1_α n

_y_n₋_x_n

≤ 1

1−αn

_y_n₋_x_n₁_x_n₁₋_x_n

≤ 1

1−α_nxn−xn1xn1−xn

≤ 2

1−αnxn−xn1 −→0 asn−→ ∞.

3.14

(7)

For p ∈ F, we have xn −p ≤ xn −Stnzn ≤ Stnzn− p. This implies that

0≤ x_n−p − z_n−p ≤ x_n−St_nz_n → 0 and hencex_n−p2− z_n−p2 → 0.Moreover, since

_z_n₋_p2_β

nxn−p21−βnTtnxn−p2−βn1−βnxn−Ttnxn2, 3.15

we have

bcxn−Ttnxn2≤βn1−βnxn−Ttnxn2

≤βnxn−p21−βnTtnxn−p2−zn−p2

≤xn−p2−zn−p2 −→0.

3.16

And sinceStnis a nonexpansive mapping, we obtain

xn−Stnxn ≤ xn−StnznStnzn−Stnxn, ≤ xn−Stnznzn−xn.

3.17

Sincez_n−x_n 1−β_nTt_nx_n−x_n → 0 andx_n−St_nz_n → 0, we obtain

lim

n→ ∞xn−Stnxn0. 3.18

As in the proof of12, Theorem 4, byLemma 2.3, we can choose a sequence{t_n_k}of positive real numbers such that

tnk −→0, 1

tnk

xnk−Ttnkxnk −→0, ask−→ ∞. 3.19

In similar way, we also have

tnk −→0, 1

tnk

(8)

Next, we show thatz∈F. To see this, we fixt >0,

xnk−Ttz ≤ t/tnk−1

j0

_T

jtnk

xnk−T

j1t_n_kx_n_k

T

_t

tnk

xnk−T

_t

tnk

zT

_t

tnk

z−Ttz

≤

_t

tnk

xnk−Ttnkxnkxnk−z

T

t−

_t

tnk

z−z

≤ _tt nk

xnk−Ttnkxnkxnk−zsup{Tsz−z: 0≤s≤tnk}.

3.21

Asxnk → zand3.19, we obtainxnk → Ttzand soTtzz.Similarly, we haveStzz. Thusz∈F.

Finally, we show thatzPFx0.SinceF ⊂Cn1andxn1PCn1x0,

xn1−x0 ≤q−x0 ∀n∈N, q∈F. 3.22

Butx_n → zasn → ∞, we have

z−x0 ≤q−x0 ∀q∈F. 3.23

HencezPFx0as required. This completes the proof.

Corollary 3.2. Let C be a closed convex subset of a real Hilbert space H. Let {Tt : t ≥ 0} be nonexpansive semigroups onC with a nonempty common fixed point set F, that is, F :

_∞

t0FTt/∅. Let{αn} ⊂ 0, a ⊂ 0,1,{βn} ⊂ b, c ⊂ 0,1and{tn}be the sequences such thatlim inf_n_{→ ∞}t_n 0,lim sup_n_{→ ∞}t_n > 0,andlim_n_{→ ∞}t_n1−tn 0. Suppose that {xn} is a sequence iteratively generated by the following iterative scheme:

C1C,

x1PC1x0,

ynαnxn 1−αnTtnzn,

Cn1

xn1 PCn1x0,

3.24

(9)

Corollary 3.3see8, Theorem 2.1. LetCbe a closed convex subset of a real Hilbert spaceH. Let {Tt : t ≥ 0}be a nonexpansive semigroups on Cwith a nonempty common fixed point set

F, that is, F : ∞_t₀FTt_/∅. Let {αn} ⊂ 0, a ⊂ 0,1and {tn} be the sequences such that

lim infn→ ∞tn0,lim sup_n_{→ ∞}tn>0,andlimn→ ∞tn1−tn 0. Suppose that{xn}is a sequence iteratively generated by the following iterative scheme:

C1C,

x1PC1x0,

zn αnxn 1−αnTtnxn,

Cn1

xn1 PCn1x0,

3.25

thenxn → PFx0.

Proof. IfStn Ttnfor alln ∈ NandTt I for everyt > 0 inTheorem 3.1then3.1

reduced to3.25. By usingTheorem 3.1, we get the following conclusion.

3.2. The CQ Hybrid Method

In this section, we consider the modified Ishikawa iterative scheme computing by the CQ hybrid method 13–15. We use the same idea as Saejung’s Theorem 2.2 in 8 and our

Theorem 3.1to obtain the following result and the proof is omitted.

Theorem 3.4. LetC be a closed convex subset of a real Hilbert spaceH. Let{Tt : t ≥ 0}and {St : t ≥ 0}be nonexpansive semigroups onCwith a nonempty common fixed point setF, that is,F : ∞_t₀FTt∩∞_t₀FSt/∅. Let{αn} ⊂ 0, a ⊂ 0,1,{βn} ⊂ b, c ⊂ 0,1and {tn}be the sequences such thatlim infn→ ∞tn 0,lim supn→ ∞tn >0,andlimn→ ∞tn1−tn 0. Suppose that{x_n}is a sequence generated by the following iterative scheme:

ynαnxn 1−αnStnzn,

Cnu∈C:yn−u≤ xn−u,

Qn{u∈C:xn−x0, u−xn ≥0},

xn1PCn∩Qnx0,

3.26

Proof. First, we show that bothC_n and Q_n are closed and convex, and C_n ∩Q_n_/∅ for all

n ∈ N∪ {0}. It follows easily from the definition thatCn and Qn are just intersection ofC

and the half-spaces see also 9. As in the proof of the preceding theorem, we haveF ⊂

(10)

we havep∈Ck∩Qk. In particular,xk1−x0, p−xk1 ≥0,that is,p∈Qk1. It follows from

the induction thatF⊂Q_nfor alln∈N∪ {0}. This proves the claim. Next, we show thatxn−Ttnxn → 0,andxn−Stnxn → 0.

We first claim thatxn1−xn → 0.Indeed, asxn1∈QnandxnPQnx0,

xn−x0 ≤ xn1−x0 ∀n∈N. 3.27

For fixedz∈F. It follows fromF ⊂Qnfor alln∈Nthat

xn−x0 ≤ z−x0 ∀n∈N. 3.28

This implies that sequence{xn}is bounded and

lim

n→ ∞xn−x0exists. 3.29

Notice that

xn1−xn, xn−x0 ≥0. 3.30

This implies that

xn1−xn2 xn1−x02−2xn1−xn, xn−x0 − x0−xn2

≤ xn1−x02− xn−x02−→0.

3.31

By using the same argument of Saejung 8, Theorem 2.2, page 6 and in the proof of

Theorem 3.1, we haveTtnxn−xn → 0 andStnxn−xn → 0. And we can choose a

subsequence{n_k}of{n}such thatx_n_k z∈C,t_n_k → 0,1/t_n_kx_n_k −Tt_n_kx_n_k → 0 and

1/tnkxnk−Stnkxnk → 0 ask → ∞. From3.21, we obtain

lim sup

k→ ∞ xnk −Ttz ≤

lim sup

k→ ∞ xnk−z,

lim sup

k→ ∞ xnk−Stz ≤lim supk→ ∞ xnk−z.

3.32

By the Opial’s condition ofH, we havezTtzandzStzfor allt >0, that is,z∈F. We note that

x0−PFx0 ≤ x0−z ≤lim inf

k→ ∞ x0−xnk ≤lim sup_k_{→ ∞} x0−xnk ≤ x0−PFx0. 3.33

This implies that

lim

(11)

Therefore,

xnk −→PFx0 z, ask−→ ∞. 3.35

Hence the whole sequence must converge to PFx0 z, as required. This completes the

proof.

Corollary 3.5. Let C be a closed convex subset of a real Hilbert space H. Let {Tt : t ≥ 0}

be nonexpansive semigroups on C with a nonempty common fixed point set F, that is, F :

_∞

t0FTt/∅. Let{αn} ⊂0, a⊂0,1,{βn} ⊂b, c⊂0,1and{tn}be the sequences such that

lim infn→ ∞tn0,lim sup_n_{→ ∞}tn>0,andlimn→ ∞tn1−tn 0. Suppose that{xn}is a sequence iteratively generated by the following iterative scheme:

ynαnxn 1−αnTtnzn,

Qn{u∈C:xn−x0, u−xn ≥0},

xn1PCn∩Qnx0,

3.36

Proof. IfSt_n Tt_nfor alln∈N∪ {0}, inTheorem 3.4then3.26reduced to3.36. So, we obtain the result immediately.

We also deduce the following corollary.

Corollary 3.6see8, Theorem 2.2. LetCbe a closed convex subset of a real Hilbert spaceH. Let {Tt : t ≥ 0}be a nonexpansive semigroups on Cwith a nonempty common fixed point set

F, that is, F : ∞_t₀FTt/∅. Let {αn} ⊂ 0, a ⊂ 0,1and {tn} be the sequences such that

lim infn→ ∞tn0,lim supn→ ∞tn >0andlimn→ ∞tn1−tn 0. Suppose that{xn}is a sequence iteratively generated by the following iterative scheme:

zn αnxn 1−αnTtnxn,

Qn{u∈C:xn−x0, u−xn ≥0},

xn1PCn∩Qnx0,

3.37

(12)

Acknowledgments

The authors would like to thank the editors and the anonymous referees for their valuable suggestions which help to improve this paper. This research was supported by the Computational Science and Engineering Research Cluster, King Mongkut’s University of Technology Thonburi KMUTT National Research Universities under CSEC Project no. E01008.

References

1 W. R. Mann, “Mean value methods in iteration,”Proceedings of the American Mathematical Society, vol. 4, pp. 506–510, 1953.

2 B. Halpern, “Fixed points of nonexpanding maps,”Bulletin of the American Mathematical Society, vol. 73, pp. 957–961, 1967.

3 F. E. Browder, “Fixed-point theorems for noncompact mappings in Hilbert space,”Proceedings of the National Academy of Sciences of the United States of America, vol. 53, pp. 1272–1276, 1965.

4 T. Suzuki, “On strong convergence to common fixed points of nonexpansive semigroups in Hilbert spaces,”Proceedings of the American Mathematical Society, vol. 131, no. 7, pp. 2133–2136, 2002.

5 H.-K. Xu, “A strong convergence theorem for contraction semigroups in Banach spaces,”Bulletin of the Australian Mathematical Society, vol. 72, no. 3, pp. 371–379, 2005.

6 R. Chen and H. He, “Viscosity approximation of common fixed points of nonexpansive semigroups in Banach space,”Applied Mathematics Letters, vol. 20, no. 7, pp. 751–757, 2007.

7 H. He and R. Chen, “Strong convergence theorems of the CQ method for nonexpansive semigroups,”

Fixed Point Theory and Applications, vol. 2007, Article ID 59735, 8 pages, 2007.

8 S. Saejung, “Strong convergence theorems for nonexpansive semigroups without Bochner integrals,”

Fixed Point Theory and Applications, vol. 2008, Article ID 745010, 7 pages, 2008.

9 W. Takahashi, Y. Takeuchi, and R. Kubota, “Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 341, no. 1, pp. 276–286, 2008.

10 S. Ishikawa, “Fixed points by a new iteration method,”Proceedings of the American Mathematical Society, vol. 44, pp. 147–150, 1974.

11 Z. Opial, “Weak convergence of the sequence of successive approximations for nonexpansive mappings,”Bulletin of the American Mathematical Society, vol. 73, pp. 591–597, 1967.

12 T. Suzuki, “Strong convergence of Krasnoselskii and Mann’s type sequences for one-parameter non-expansive semigroups without Bochner integrals,”Journal of Mathematical Analysis and Applications, vol. 305, no. 1, pp. 227–239, 2005.

13 Y. Haugazeau,Sur les In´equations variationnelles et la minimisation de fonctionnelles convexes, Ph.D. thesis, Universit´e Paris, Paris, France, 1968.

14 K. Nakajo and W. Takahashi, “Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups,”Journal of Mathematical Analysis and Applications, vol. 279, no. 2, pp. 372– 379, 2003.