Hybrid Iterative Methods for Convex Feasibility Problems and Fixed Point Problems of Relatively Nonexpansive Mappings in Banach Spaces

Volume 2008, Article ID 583082,19pages doi:10.1155/2008/583082

Research Article

Hybrid Iterative Methods for Convex Feasibility Problems and Fixed Point Problems of Relatively Nonexpansive Mappings in Banach Spaces

Somyot Plubtieng and Kasamsuk Ungchittrakool

Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand

Correspondence should be addressed to Somyot Plubtieng,[email protected] Received 2 July 2008; Accepted 23 December 2008

Recommended by Hichem Ben-El-Mechaiekh

The convex feasibility problemCFPof finding a point in the nonempty intersectionN i1Ciis considered, whereN1 is an integer and theCi’s are assumed to be convex closed subsets of a Banach spaceE. By using hybrid iterative methods, we prove theorems on the strong convergence to a common fixed point for a finite family of relatively nonexpansive mappings. Then, we apply our results for solving convex feasibility problems in Banach spaces.

Copyrightq2008 S. Plubtieng and K. Ungchittrakool. 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

We are concerned with the convex feasibility problemCFP

finding an x∈^N

i1

C_i, 1.1

whereN1 is an integer, andC₁, . . . , C_Nare intersecting closed convex subsets of a Banach spaceE. This problem is a frequently appearing problem in diverse areas of mathematical and physical sciences. There is a considerable investigation onCFPin the framework of Hilbert spaces which captures applications in various disciplines such as image restoration 1–4, computer tomography5, and radiation theraphy treatment planning6. In computer tomography with limited data, in which an unknown image has to be reconstructed from a priori knowledge and from measured results, each piece of information gives a constraint which in turn, gives rise to a convex set Ci to which the unknown image should belong see7. The advantage of a Hilbert spaceHis that thenearest pointprojectionPK onto a closed convex subsetK of H is nonexpansivei.e.,PKx−P_Ky x−y, x, y ∈ H.

So projection methods have dominated in the iterative approaches toCFPin Hilbert spaces;

see 6, 8–11 and the references therein. In 1993, Kitahara and Takahashi 12 deal with the convex feasibility problem by convex combinations of sunny nonexpansive retractions in uniformly convex Banach spaces see also Takahashi and Tamura 13, O’Hara et al.

14, and Chang et al.15 for the previous results on this subject. It is known that if C is a nonempty closed convex subset of a smooth, reflexive, and strictly convex Banach space E, then the generalized projection ΠC see, Alber 16 or Kamimura and Takahashi 17 from E ontoC is relatively nonexpansive, whereas the metric projection P_C from E onto C is not generally nonexpansive. Our purpose in the present paper is to obtain an analogous result for a finite family of relatively nonexpansive mappings in Banach spaces.

This notion was originally introduced by Butnariu et al. 18. Recently, Matsushita and Takahashi 19 reformulated the definition of the notion and obtained weak and strong convergence theorems to approximate a fixed point of a single relatively nonexpansive mapping. Motivated by Nakajo and Takahashi20, Matsushita and Takahashi21studied the strong convergence of the sequence{xn}generated by

x₀ x∈C, y_n J⁻¹

α_nJx_n 1−α_n

JTx_n , H_n

z∈C:φ z, y_n

φ

z, x_n , W_n

z∈C:

x_n−z, Jx−Jx_n 0 , x_n1 ΠHn∩Wnx, n0,1,2, . . . ,

1.2

whereJis the duality mapping onE,{αn} ⊂0,1,T is a relatively nonexpansive mapping fromCinto itself, andΠFT·is the generalized projection fromContoFT.

Very recently, Plubtieng and Ungchittrakool22studied the strong convergence to a common fixed point of two relatively nonexpansive mappings of the sequence{xn}generated by

x₀x∈C, y_nJ⁻¹

α_nJx_n 1−α_n

Jz_n , z_nJ⁻¹

β¹_n Jx_nβ²_n JTx_nβ_n³JSx_n , H_n

z∈C:φ z, y_n

φ

z, x_n , W_n

z∈C:

x_n−z, Jx−Jx_n 0 , x_n1PHn∩Wnx, n0,1,2, . . . ,

1.3

whereJ is the duality mapping on E, andP_F·is the generalized projection from Conto F:FS∩FT.

We note that the block iterative method is a method which often used by many authors to solve the convex feasibility problem CFP see, 23, 24, etc.. In 2008, Plubtieng and Ungchittrakool25established strong convergence theorems of block iterative methods for a finite family of relatively nonexpansive mappings in a Banach space by using the hybrid method in mathematical programming. In this paper, we introduce the following iterative

process by using the shrinking method proposed, whose studied by Takahashi et al.26, which is different from the method in25. Let C be a closed convex subset ofE and for each i 1,2, . . . , N, let Ti : C → C be a relatively nonexpansive mapping such that F:_N

i1FTi/∅. Define{xn}in the two following ways:

x0∈E, C1C, x1 ΠC1x0, ynJ⁻¹

αnJxn 1−αn

Jzn , znJ⁻¹

β¹_n Jxn^N

i1

βⁱ¹_n JTixn

, C_n1

z∈C_n:φ z, y_n

φ

z, x_n , x_n1 ΠCn1x0, n0,1,2, . . . ,

1.4

and

x₀∈C, y_nJ⁻¹

α_nJx_n 1−α_n

Jz_n , z_nJ⁻¹

β¹_n Jx_n^N

i1

βⁱ¹_n JT_ix_n

, H_n

z∈C:φ z, y_n

φ

z, x_n , W_n

z∈C:

x_n−z, Jx₀−Jx_n 0 , x_n1 ΠHn∩Wnx₀, n0,1,2, . . . ,

1.5

where{αn},{βⁱn } ⊂0,1, _N1

i1 βⁱ_n 1 satisfy some appropriate conditions.

We will prove that both iterations1.4and1.5converge strongly to a common fixed point of_N

i1FTi. Using this results, we also discuss the convex feasibility problem in Banach spaces. Moreover, we apply our results to the problem of finding a common zero of a finite family of maximal monotone operators and equilibrium problems.

Throughout the paper, we will use the notations:

i → for strong convergence andfor weak convergence;

iiωwxn {x:∃xnr x}denotes the weakω-limit set of{xn}.

2. Preliminaries

LetEbe a real Banach space with norm·and letE^∗be the dual ofE. Denote by ·,·the duality product. The normalized duality mappingJfromEtoE^∗is defined by

Jx

x^∗∈E^∗:

x, x^∗ x²x^∗2

2.1 forx∈E.

A Banach spaceEis said to be strictly convex ifxy/2 <1 for allx, y ∈Ewith xy1 andx /y. It is also said to be uniformly convex if lim_{n→ ∞}xn−y_n0 for any two sequences{xn}, {yn}inEsuch thatxnyn 1 and lim_{n→ ∞}xnyn/21. Let U{x∈E:x1}be the unit sphere ofE. Then the Banach spaceEis said to be smooth provided that

limt→0

xty − x

t 2.2

exists for eachx, y∈U. It is also said to be uniformly smooth if the limit is attained uniformly for x, y ∈ U. It is well known that ^p and L^p 1 < p < ∞ are uniformly convex and uniformly smooth; see Cioranescu 27 or Diestel28. We know that ifE is smooth, then the duality mappingJis single valued. It is also known that ifEis uniformly smooth, then J is uniformly norm-to-norm continuous on each bounded subset ofE. Some properties of the duality mapping have been given in27,29,30. A Banach spaceEis said to have the Kadec-Klee property if a sequence{xn}of Esatisfying thatxn x ∈ Eandxn → x, thenxn → x. It is known that ifEis uniformly convex, thenEhas the Kadec-Klee property;

see27,30for more details. LetEbe a smooth Banach space. The functionφ:E×E → Ris defined by

φy, x y²−2 y, Jxx² 2.3

for allx, y∈E. It is obvious from the definition of the functionφthat 1 y − x²φy, xyx²,

2φx, y φx, z φz, y 2 x−z, Jz−Jy,

3φx, y x, Jx−Jy y−x, JyxJx−Jyy−xy,

for allx, y, z∈E. LetEbe a strictly convex, smooth, and reflexive Banach space, and letJbe the duality mapping fromEintoE^∗. ThenJ⁻¹is also single-valued, one-to-one, and surjective, and it is the duality mapping fromE^∗ into E. We make use of the following mapping V studied in Alber16:

V x, x^∗

x²−2

x, x^∗ x^∗2 2.4

for allx∈Eandx^∗ ∈E^∗. In other words,Vx, x^∗ φx, J⁻¹x^∗for allx∈Eandx^∗ ∈E^∗. For eachx∈E, the mappingVx,·:E^∗ → Ris a continuous and convex function fromE^∗ intoR.

LetCbe a nonempty closed convex subset of a smooth, strictly convex, and reflexive Banach spaceE, for anyx∈E, there exists a pointx0∈Csuch thatφx0, x min_y∈Cφy, x.

The mappingΠC:E → Cdefined byΠCxx₀is called the generalized projection16,17,31.

The following are well-known results. For example, see16,17,31.

This section collects some definitions and lemmas which will be used in the proofs for the main results in the next section. Some of them are known; others are not hard to derive.

Lemma 2.1see27,30,32. IfEis a strictly convex and smooth Banach space, then forx, y∈E, φy, x 0 if and only ifxy.

Proof. It is sufficient to show that ifφy, x 0 thenx y. From1, we havex y.

This implies y, Jx y² Jx². From the definition ofJ, we haveJx Jy. SinceJ is one-to-one, we havexy.

Lemma 2.2 Kamimura and Takahashi17. LetEbe a uniformly convex and smooth Banach space and let{yn},{zn}be two sequences ofE. Ifφyn, zn → 0 and either{yn}or{zn}is bounded, theny_n−z_n → 0.

LetCbe a nonempty closed convex subset of a smooth, strictly convex, and reflexive Banach spaceE, letTbe a mapping fromCinto itself, and letFTbe the set of all fixed points ofT. Then a pointp ∈ Cis said to be an asymptotic fixed point ofT see Reich33if there exists a sequence{xn}inCconverging weakly topand limn→ ∞xn−Txn 0. We denote the set of all asymptotic fixed points ofTbyFT and we say thatTis a relatively nonexpansive mapping if the following conditions are satisfied:

R1FTis nonempty;

R2φu, Txφu, xfor allu∈FTandx∈C;

R3FT FT.

Lemma 2.3 Alber 16, Alber and Reich 31, Kamimura and Takahashi 17. LetC be a nonempty closed convex subset of a smooth Banach spaceE, letx∈E, and letx₀∈C. Then,x₀ ΠCx if and only if x0−y, Jx−Jx₀0 for ally∈C.

Lemma 2.4 Alber 16, Alber and Reich31, Kamimura and Takahashi 17. Let E be a reflexive, strictly convex and smooth Banach space, letCbe a nonempty closed convex subset ofEand letx∈E. Thenφy,ΠCx φΠCx, xφy, xfor ally∈C.

Lemma 2.5. LetEbe a uniformly convex Banach space and letBr0 {x∈E:xr}be a closed ball ofE. Then there exists a continuous strictly increasing convex functiong :0,∞ → 0,∞with g0 0 such that

N i1

ωⁱxi

2

^N

i1

ωⁱxi²−ω^jω^kgxj−xk, for anyj, k∈ {1,2, . . . , N}, 2.5 where{xi}^N_i1⊂B_r0and{ωⁱ}^N_i1 ⊂0,1with_N

i1ωⁱ1.

Proof. It sufficient to show that

N

i1

ωⁱxi

2

^N

i1

ωⁱxi²−ω¹ω²gx1−x2. 2.6

It is obvious that2.6holds forN 1,2 see34for more details. Next, we assume that 2.6is true forN−1. It remains to show that2.6holds forN. We observe that

N i1

ωⁱxi

2

ω^NxN

1−ω^N_N−1

i1

ωⁱ 1−ω^Nxi

2

ω^Nx_N²

1−ω^N

N−1

i1

ωⁱ 1−ω^Nx_i

2

ω^Nx_N²

1−ω^N_N−1

i1

ωⁱ

1−ω^Nxi²− ω¹ω²

1−ω^N2gx₁−x₂ ^N

i1

ωⁱx_i²− ω¹ω²

1−ω^Ngx₁−x₂

^N

i1

ωⁱx_i²−ω¹ω²gx₁−x₂.

2.7

This completes the proof.

Lemma 2.6. LetCbe a closed convex subset of a smooth Banach spaceEand letx, y ∈E. Then the setK:{v∈C:φv, yφv, x}is closed and convex.

Proof. As a matter of fact, the defining inequality inKis equivalent to the inequality

v,2Jx−Jy x²− y². 2.8

This inequality is affine invand hence the setKis closed and convex.

3. Main result

In this section, we prove strong convergence theorems for finding a common fixed point of a finite family of relatively nonexpansive mappings in Banach spaces by using the hybrid method in mathematical programming.

Theorem 3.1. LetE be a uniformly convex and uniformly smooth Banach space, and let Cbe a nonempty closed convex subset ofE. Let{Ti}^N_i1be a finite family of relatively nonexpansive mappings fromCinto itself such thatF:_N

i1FTiis nonempty and letx0∈E. ForC1Candx1 ΠC1x0, define a sequence{xn}ofCas follows:

ynJ⁻¹

αnJxn 1−αn

Jzn , znJ⁻¹

β¹_n Jxn^N

i1

βⁱ¹_n JTixn

, C_n1

z∈Cn:φ z, yn

φ

z, xn , x_n1 ΠC_n1x0, n0,1,2, . . . ,

3.1

where{αn},{βnⁱ} ⊂0,1satisfy the following conditions:

i0α_n<1 for alln∈N∪ {0}and lim sup_{n→ ∞}α_n<1, ii0βⁱ_n 1 for alli1,2, . . . , N1,_N1

i1 βⁱ_n 1 for alln∈N∪ {0}. If either alim infn→ ∞β¹_n βⁱ¹_n >0 for alli1,2, . . . , Nor

blim_{n→ ∞}β¹_n 0 and lim inf_{n→ ∞}β^k1_n β^l1_n >0 for alli /j,k, l1,2, . . . , N.

Then the sequence{xn}converges strongly toΠFx₀, whereΠF is the generalized projection fromE ontoF.

Proof. We first show by induction thatF ⊂C_nfor alln∈N.F ⊂ C₁is obvious. Suppose that F⊂Ckfor somek∈N. Then, we have, foru∈F ⊂Ck,

φ u, yk

φ u, J⁻¹

αkJxk 1−αk

Jzk V

u, αkJxk 1−αk

Jzk αkV

u, Jxk

1−αk V

u, Jzk αkφ

u, xk

1−αk φ

u, zk ,

φ u, zk

V

u, β¹_k Jxk^N

i1

βⁱ¹_k JTixk

β¹_k V u, Jxk

^N

i1

βⁱ¹_k V

u, JTixk

φ

u, xk .

3.2

It follow that

φ u, y_k

φ

u, x_k

3.3 and henceu∈C_k1. This implies thatF ⊂ Cnfor alln ∈N. Next, we show thatCnis closed and convex for alln∈N. Obvious thatC₁Cis closed and convex. Suppose thatC_kis closed and convex for somek∈N. Forz∈C_k, we note byLemma 2.6thatC_k1is closed and convex.

Then for anyn ∈ N,Cn is closed and convex. This implies that{xn}is well-defined. From x_n ΠCnx₀, we have

φ x_n, x₀

φ

u, x₀

−φ u, x_n

φ

u, x₀

∀u∈C_n. 3.4

In particular, letu∈F, we have φ

xn, x₀

φ

u, x₀

∀n∈N. 3.5

Thereforeφxn, x₀ is bounded and hence{xn} is bounded by 1. Fromx_n ΠCnx₀ and x_n1∈C_n1 ⊂C_n, we have

φ xn, x₀

min

y∈Cn

φ y, x₀

φ

x_n1, x₀

∀n∈N. 3.6

Therefore{φxn, x0}is nondecreasing. So there exists the limit ofφxn, x0. ByLemma 2.4, we have

φ

x_n1, xn φ

x_n1,ΠCnx0

φ x_n1, x0

−φ

ΠCnx0, x0

φ x_n1, x0

−φ xn, x0

. 3.7

for eachn∈N. This implies that limn→ ∞φx_n1, x_n 0. Sincex_n1∈C_n1it follows from the definition ofC_n1that

φ

x_n1, y_n

φ

x_n1, x_n

∀n∈N. 3.8

Lettingn → ∞, we have lim_{n→ ∞}φx_n1, y_n 0. ByLemma 2.2, we obtain

nlim→ ∞x_n1−yn lim

n→ ∞x_n1−xn0. 3.9

SinceJis uniformly norm-to-norm continuous on bounded sets, we have

nlim→ ∞Jx_n1−Jy_n lim

n→ ∞Jx_n1−Jx_n0. 3.10

SinceJx_n1−Jy_nJx_n1−α_nJx_n−1−α_nJzn1−α_nJx_n1−Jz_n −α_nJxn−Jx_n1 for eachn∈N∪ {0}, we get that

Jx_n1−Jzn 1

1−α_nJx_n1−JynαnJxn−Jx_n1

1

1−α_nJx_n1−Jy_nJx_n−Jx_n1.

3.11

From 3.10 and lim sup_{n→ ∞}α_n < 1, we have lim_{n→ ∞}Jx_n1−Jz_n 0.Since J⁻¹ is also uniformly norm-to-norm continuous on bounded sets, it follows that

nlim→ ∞x_n1−z_n lim

n→ ∞J⁻¹ Jx_n1

−J⁻¹

Jz_n0. 3.12

Fromxn−z_nxn−x_n1x_n1−z_n, we have limn→ ∞xn−z_n0.

Next, we show thatxn−Tixn → 0 for alli1,2, . . . , N. Since{xn}is bounded and φp, Tix_nφp, xnfor alli1,2, . . . , N, wherep∈F. We also obtain that{Jxn}and{JTix_n} are bounded for alli1,2, . . . , N. Then there existsr >0 such that{Jxn},{JTixn} ⊂Br0for alli1,2, . . . , N. ThereforeLemma 2.5is applicable. Assume thataholds, we observe that

φ p, z_n

p²−2

p, β_n¹Jx_n^N

i1

βⁱ¹_n JT_ix_n

β¹_n Jx_n^N

i1

βⁱ¹_n JT_ix_n

2

p²−2β¹_n

p, Jx_n ^N

i1

βⁱ¹_n

p, JT_ix_n β¹_n x_n^2N

i1

βⁱ¹_n T_ix_n²

−β¹_n βⁱ¹_n gJx_n−JT_ix_n β¹_n

p²−2

p, Jx_n x_n^{2 N}

i1

βⁱ¹_n

p²2

p, JT_ix_n T_ix_n²

−β¹_n βⁱ¹_n gJx_n−JT_ix_n β¹n φ

p, xn ^N

i1

βnⁱ¹φ p, Tixn

−βn¹βⁱ¹n gJxn−JTixn

φ

p, xn

−β_n¹βⁱ¹_n gJxn−JTixn

3.13

and hence

β¹_n βⁱ¹_n gJxn−JTixnφ p, xn

−φ p, zn 2

p, zn−xn xnznxn−zn 2pz_n−x_nx_nz_nx_n−z_n

−→0,

3.14

whereg :0,∞ → 0,∞is a continuous strictly increasing convex function withg0 0 in Lemma 2.5. Bya, we have limn→ ∞gJxn−JTixn 0 and then limn→ ∞Jxn−JTixn0 for alli1,2, . . . , N. SinceJ⁻¹is also uniformly norm-to-norm continuous on bounded sets, we obtain

nlim→ ∞xn−Tixn lim

n→ ∞J⁻¹ Jxn

−J⁻¹

JTixn0, 3.15

for alli1,2, . . . , N. Ifbholds, we get

φ p, z_n

p²−2

p, β_n¹Jx_n^N

i1

βⁱ¹_n JT_ix_n

β¹_n Jx_n^N

i1

βⁱ¹_n JT_ix_n

2

p²−2β¹_n

p, Jx_n ^N

i1

βⁱ¹_n

p, JT_ix_n β¹_n x_n^2N

i1

βⁱ¹_n T_ix_n²

−β^k1_n β^l1_n gJT_kx_n−JT_lx_n β¹_n

p²−2

p, Jxn xn^{2 N}

i1

βⁱ¹_n

p²2

p, JTixn Tixn²

−β^k1_n β^l1_n gJTkxn−JTlxn β¹n φ

p, xn ^N

i1

βnⁱ¹φ p, Tixn

−βn^k1βn^l1gJTkxn−JTlxn

φ

p, x_n

−β_n^k1β^l1_n gJT_kx_n−JT_lx_n

3.16

and hence

β^k1_n β^l1_n gJT_kx_n−JT_lx_nφ p, x_n

−φ p, z_n 2

p, z_n−x_n x_nz_nx_n−z_n 2pz_n−x_nx_nz_nx_n−z_n

−→0.

3.17

Then by the same argument above, we have limn→ ∞Tkxn−Tlxn0 for allk, l1,2, . . . , N.

Next, we observe that

φTkxn, zn V

Tkxn, β_n¹Jxn^N

i1

βⁱ¹_n JTixn

β¹_n V

Tkxn, Jxn ^N

i1

βⁱ¹_n V

Tkxn, JTixn

β¹_n φ

T_kx_n, x_n ^N

i1

βⁱ¹_n φ

T_kx_n, T_ix_n

−→0.

3.18

asβ¹_n → 0. By Lemma 2.2, we have lim_{n→ ∞}Tkx_n −z_n 0 for allk 1,2, . . . , N, and hence

T_ix_n−x_nT_ix_n−z_nz_n−x_n−→0 asn−→ ∞, 3.19 for alli1,2, . . . , N. Thenω_wxn⊂_N

i1FT i _N

i1FTi F.

Finally, we show thatxn → ΠFx0. Let{xnk}be a subsequence of{xn}such thatxnk v∈ω_wxn⊂F. Putw: ΠFx₀∈F⊂C_nk, we observe that

φ xnk, x₀

φ

ΠC_nkx₀, x₀ min

y∈C_nkφ y, x₀

φ

w, x₀ min

z∈F φ z, x₀

φ

v, x₀

. 3.20

Sinceφ·, x0is weakly lower semicontinuous, we obtain φ

v, x₀

lim inf

k→ ∞ φ x_nk, x₀

lim sup

k→ ∞

φ x_nk, x₀

φ

w, x₀

φ

v, x₀

. 3.21

This implies that v w and lim_{k→ ∞}xnk w and then the Kadec-Klee property ofE yieldsxnk → w. Since{xnk}is an arbitrary,xn → w. This completes the proof.

Corollary 3.2. LetE be a uniformly convex and uniformly smooth Banach space, and let C be a nonempty closed convex subset ofE. Let{Ωi}^N_i1be a finite family of nonempty closed convex subset of Csuch thatΩ:_N

i1Ωiis nonempty and letx0∈E. ForC1Candx1 ΠC1x0, define a sequence {xn}ofCas follows:

y_nJ⁻¹

α_nJx_n 1−α_n

Jz_n ,

z_nJ⁻¹

β¹_n Jx_n^N

i1

βⁱ¹_n JΠ_Ωix_n

,

C_n1

z∈Cn:φ z, yn

φ

z, xn , x_n1 ΠCn1x₀, n0,1,2, . . . ,

3.22

where{αn},{βnⁱ} ⊂0,1satisfy the following conditions:

i0αn<1 for alln∈N∪ {0}and lim sup_{n→ ∞}αn<1, ii0βⁱ_n 1 for alli1,2, . . . , N1,_N1

i1 βⁱ_n 1 for alln∈N∪ {0}. If either alim infn→ ∞β¹_n βⁱ¹_n >0 for alli1,2, . . . , Nor

blim_{n→ ∞}β¹_n 0 and lim inf_{n→ ∞}β^k1_n β^l1_n >0 for alli /j,k, l1,2, . . . , N.

Then the sequence{xn}converges strongly toΠΩx0, whereΠΩ is the generalized projection fromE ontoΩ.

Theorem 3.3. LetE be a uniformly convex and uniformly smooth Banach space, and let Cbe a nonempty closed convex subset ofE. Let{Ti}^N_i1be a finite family of relatively nonexpansive mappings fromCinto itself such thatF:_N

i1FTiis nonempty. Let a sequence{xn}defined by x₀∈C,

y_nJ⁻¹

α_nJx_n 1−α_n

Jz_n , znJ⁻¹

β¹n Jxn^N

i1

βnⁱ¹JTixn

, H_n

z∈C:φ z, y_n

φ

z, x_n , W_n

z∈C:

x_n−z, Jx₀−Jx_n 0 , x_n1 ΠHn∩Wnx0, n0,1,2, . . . ,

3.23

where{αn},{βnⁱ} ⊂0,1satisfy the following conditions:

i0αn<1 for alln∈N∪ {0}and lim sup_{n→ ∞}αn<1, ii0βⁱ_n 1 for alli1,2, . . . , N1,_N1

i1 βⁱ_n 1 for alln∈N∪ {0}. If either alim infn→ ∞β¹_n βⁱ¹_n >0 for alli1,2, . . . , Nor

blim_{n→ ∞}β¹_n 0 and lim inf_{n→ ∞}β^k1_n β^l1_n >0 for alli /j,k, l1,2, . . . , N.

Then the sequence{xn}converges strongly toΠFx0, whereΠF is the generalized projection fromE ontoF.

Proof. From the definition ofHnandWn, it is obviousHnandWnare closed and convex for eachn ∈N∪ {0}. Next, we show thatF ⊂Hn∩Wnfor eachn∈ N∪ {0}. Letu ∈F and let n∈N∪ {0}. Then, as in the proof ofTheorem 3.1, we have

φ u, zn

φ

u, xn

3.24 for alln∈N∪ {0}, and thenφu, ynφu, xn. Thus, we haveu∈Hn. Therefore we obtain F ⊂ H_n for eachn ∈ N∪ {0}. We note by21, Proposion 2.4that eachFTiis closed and convex and so is F. Using the same argument presented in the proof of21, Theorem 3.1;

page 261-262, we haveF ⊂Hn∩Wnfor eachn∈N∪ {0},{xn}is well defined and bounded, and

nlim→ ∞x_n1−yn lim

n→ ∞x_n1−xn0. 3.25

SinceJis uniformly norm-to-norm continuous on bounded sets, we have

nlim→ ∞Jx_n1−Jyn lim

n→ ∞Jx_n1−Jxn0. 3.26

As in the proof ofTheorem 3.1, we also have that Jx_n1−Jzn 1

1−α_nJx_n1−JynαnJxn−Jx_n1

1

1−α_nJx_n1−Jy_nJx_n−Jx_n1.

3.27

From 3.26 and lim sup_{n→ ∞}αn < 1, we have limn→ ∞Jxn1−Jzn 0.Since J⁻¹ is also uniformly norm-to-norm continuous on bounded sets, we obtain

nlim→ ∞x_n1−zn lim

n→ ∞J⁻¹ Jx_n1

−J⁻¹

Jzn0. 3.28

Fromxn−znxn−xn1xn1−znwe have limn→ ∞xn−zn0. By the same argument as in the proof ofTheorem 3.1, we have{xn}converges strongly toΠFx₀.

Corollary 3.4. LetE be a uniformly convex and uniformly smooth Banach space, and let C be a nonempty closed convex subset ofE. Let{Ωi}^N_i1be a finite family of nonempty closed convex subset ofCsuch thatΩ:_N

i1Ωiis nonempty. Let a sequence{xn}defined by x₀∈C,

y_nJ⁻¹

α_nJx_n 1−α_n

Jz_n , z_nJ⁻¹

β¹_n Jx_n^N

i1

βⁱ¹_n JΠ_Ωix_n

, Hn

z∈C:φ z, yn

φ

z, xn , Wn

z∈C:

xn−z, Jx₀−Jxn 0 , x_n1 ΠHn∩Wnx₀, n0,1,2, . . . ,

3.29

where{αn},{βnⁱ} ⊂0,1satisfy the following conditions:

i0αn<1 for alln∈N∪ {0}and lim sup_{n→ ∞}αn<1, ii0βⁱ_n 1 for alli1,2, . . . , N1,_N1

i1 βⁱ_n 1 for alln∈N∪ {0}. If either alim inf_{n→ ∞}β¹_n βⁱ¹_n >0 for alli1,2, . . . , Nor

blimn→ ∞β¹n 0 and lim infn→ ∞β^k1n β^l1n >0 for alli /j,k, l1,2, . . . , N.

Then the sequence{xn}converges strongly toΠΩx0, whereΠΩ is the generalized projection fromE ontoΩ.

IfN2,T₁T andT₂S, thenTheorem 3.3reduces to the following corollary.

Corollary 3.5Plubtieng and Ungchittrakool22, Theorem 3.1. LetEbe a uniformly convex and uniformly smooth Banach space, and letCbe a nonempty closed convex subset ofE. LetSandT

be two relatively nonexpansive mappings fromCinto itself withF :FS∩FTis nonempty. Let a sequence{xn}be defined by

x₀x∈C, y_nJ⁻¹

α_nJx_n 1−α_n

Jz_n , z_nJ⁻¹

β¹_n Jx_nβ²_n JTx_nβ_n³JSx_n , H_n

z∈C:φ z, y_n

φ

z, x_n , Wn

z∈C:

xn−z, Jx−Jxn 0 , x_n1PHn∩Wnx, n0,1,2, . . . ,

3.30

with the following restrictions:

i0αn<1 for alln∈N∪ {0}and lim sup_{n→ ∞}αn<1,

ii0 β_n¹, β²_n , β³_n 1,β¹_n β²_n β³_n 1 for alln∈ N∪ {0}, limn→ ∞β_n¹ 0 and lim infn→ ∞β²_n β_n³>0.

Then the sequence{xn}converges strongly toPFx, wherePFis the generalized projection fromConto F.

4. Applications

4.1. Maximal monotone operators

LetAbe a multivalued operator fromEtoE^∗ with domainDA {z ∈ E : Az /∅}and rangeRA ∪{Az:z∈DA}. An operatorAis said to be monotone if x1−x2, y₁−y20 for eachxi ∈ DAandyi ∈ Axi,i 1,2. A monotone operatorAis said to be maximal if its graphGA {x, y : y ∈ Ax} is not properly contained in the graph of any other monotone operator. We know that ifAis a maximal monotone operator, thenA⁻¹0is closed and convex. LetE be a reflexive, strictly convex and smooth Banach space, and letAbe a monotone operator fromEtoE^∗, we known from Rockafellar35thatAis maximal if and only ifRJrA E^∗for allr >0. LetJ_r :E → DAdefined byJ_r JrA⁻¹Jand such aJr is called the resolvent ofA. We know thatJr is a relatively nonexpansive; see21and A⁻¹0 FJrfor allr >0; see30,32for more details.

Theorem 4.1. LetEbe a uniformly convex and uniformly smooth Banach space. LetA_i ⊂E×E^∗be a maximal monotone operator for eachi1,2, . . . , Nsuch thatΛ:_N

i1A⁻¹_i 0is nonempty and let x0∈E. ForC1 E, define a sequence{xn}as follows:

ynJ⁻¹

αnJxn 1−αn

Jzn , znJ⁻¹

β¹_n Jxn^N

i1

βⁱ¹_n JJ_r^A_iⁱxn

, C_n1

z∈Cn:φ z, yn

φ

z, xn , x_n1 ΠC_n1x0, n0,1,2, . . . ,

4.1