Strong Convergence Theorems for an Infinite Family of Equilibrium Problems and Fixed Point Problems for an Infinite Family of Asymptotically Strict Pseudocontractions

Volume 2011, Article ID 859032,15pages doi:10.1155/2011/859032

Research Article

Strong Convergence Theorems for an Infinite Family of Equilibrium Problems and Fixed Point Problems for an Infinite Family of Asymptotically Strict Pseudocontractions

Shenghua Wang,

Shin Min Kang,

and Young Chel Kwun

1School of Applied Mathematics and Physics, North China Electric Power University, Baoding 071003, China

2Department of Mathematics and RINS, Gyeongsang National University, Jinju 660-701, Republic of Korea

3Department of Mathematics, Dong-A University, Pusan 614-714, Republic of Korea

Correspondence should be addressed to Young Chel Kwun,[email protected] Received 12 October 2010; Accepted 29 January 2011

Academic Editor: Jong Kim

Copyrightq2011 Shenghua Wang 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 for an infinite family of asymptotically strict pseudo- contractions and an infinite family of equilibrium problems in a Hilbert space. Our proof is simple and different from those of others, and the main results extend and improve those of many others.

1. Introduction

LetCbe a closed convex subset of a Hilbert spaceH. LetS :C → Hbe a mapping and if there exists an elementx∈ Csuch thatx Sx, thenxis called a fixed point ofS. The set of fixed points ofSis denoted byFS. Recall that

1Sis called nonexpansive if

Sx−Sy≤x−y, ∀x, y∈C, 1.1 2S is called asymptotically nonexpansive1if there exists a sequence {kn} ⊂ 1,∞

withk_n → 1 such that

Sⁿx−Sⁿy≤k_nx−y, ∀x, y∈C, n≥1, 1.2

3S is called to be a κ-strict pseudo-contraction 2if there exists a constant κwith 0≤κ <1 such that

Sx−Sy²≤x−y²κx−y

−

Sx−Sy², ∀x, y∈C, 1.3

4Sis called an asymptoticallyκ-strict pseudo-contraction3,4if there exists a constant κwith 0≤κ <1 and a sequence{γn} ⊂0,∞with lim_{n→ ∞}γn0 such that

Sⁿx−Sⁿy²≤

1γnx−y²κx−y

−

Sⁿx−Sⁿy², ∀x, y∈C, n≥1. 1.4 It is clear that every asymptotically nonexpansive mapping is an asymptotically 0- strict pseudo-contraction and everyκ-strict pseudo-contraction is an asymptoticallyκ-strict pseudo-contraction with γn 0 for all n ≥ 1. Moreover, every asymptotically κ-strict pseudo-contraction with sequence {γ_n} is uniformly L-Lispchitzian, where L sup{κ 1γn1−κ/1−κ : n ≥ 1}and the fixed point set of asymptoticallyκ-strict pseudo- contraction is closed and convex; see3, Proposition 2.6.

Let Φ be a bifunction from C×C to ^Ê, where ^Ê is the set of real numbers. The equilibrium problem for Φ : C×C → ^Ê is to find x ∈ C such thatΦx, y ≥ 0 for all y∈C. The set of such solutions is denoted by EPΦ.

In 2007, S. Takahashi and W. Takahashi 5first introduced an iterative scheme by the viscosity approximation method for finding a common element of the set of solutions of the equilibrium problem and the set of fixed points of a nonexpansive mapping in a Hilbert spaceHand proved a strong convergence theorem which is connected with Combettes and Hirstoaga’s result6 and Wittmann’s result 7. More precisely, they gave the following theorem.

Theorem 1.1see5. LetCbe a nonempty closed convex subset ofH. LetΦbe a bifunction from C×Cto^Êsatisfying the following assumptions:

A1 Φx, x 0 for allx∈C;

A2 Φis monotone, that is,Φx, y Φy, x≤0 for allx, y∈C;

A3for allx, y, z∈C,

limt↓0 Φ

tz 1−tx, y

≤Φ x, y

; 1.5

A4for allx∈C,y→Φx, yis convex and lower semicontinuous.

LetS :C → Hbe a nonexpansive mapping such thatFS∩EPΦ/∅,f :H → Hbe a contraction and{x_n},{u_n}be the sequences generated by

x1 ∈H, Φ

u_n, y 1

r_ny−u_n, u_n−x_n ≥0, ∀y∈C, x_n1α_nfx_n 1−α_nSu_n, ∀n≥1,

1.6

where{α_n} ⊂0,1and{r_n} ⊂0,∞satisfy the following conditions:

nlim→ ∞αn0, ^∞

n1

αn ∞, ^∞

n1

|αn1−αn|<∞,

lim inf

n→ ∞ rn>0, ^∞

n1

|rn1−rn|<∞.

1.7

Then, the sequences {xn} and {un} converge strongly to z ∈ FS ∩ EPΦ, where z PFS∩EPΦfz.

In8, Tada and Takahashi proposed a hybrid algorithm to find a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of an equilibrium problem and proved the following strong convergence theorem.

Theorem 1.2see8. LetCbe a nonempty closed convex subset of a Hilbert spaceH. LetΦbe a bifunction fromC×C → ^ÊsatisfyingA1–A4and letSbe a nonexpansive mapping ofCintoH such thatFS∩EPΦ/∅. Let{xn}and{un}be sequences generated byx1x∈Hand

u_n∈C such thatΦ u_n, y

1 rn

y−u_n, u_n−x_n

≥0, ∀y∈C, w_n 1−α_nx_nα_nSu_n,

Cn{z∈H:wn−z ≤ xn−z}, Dn{z∈H:xn−z, x−xn ≥0},

x_n1P_Cn∩Dnx, ∀n≥1,

1.8

where{α_n} ⊂a,1for somea∈ 0,1and{r_n} ⊂ 0,∞satisfies lim inf_{n→ ∞}r_n > 0. Then{x_n} converges strongly toPFS∩EPΦx.

Many methods have been proposed to solve the equilibrium problems and fixed point problems; see9–13.

Recently, Kim and Xu 3 proposed a hybrid algorithm for finding a fixed point of an asymptoticallyκ-strict pseudo-contraction and proved a strong convergence theorem in a Hilbert space.

Theorem 1.3see3. LetCbe a closed convex subset of a Hilbert spaceH. LetT :C → Cbe an asymptoticallyκ-strict pseudo-contraction for some 0 ≤ κ <1. Assume thatFTis nonempty and bounded. Let{xn}be the sequence generated by the following algorithm:

x0∈C chosen arbitrarily, ynαnxn 1−αnTⁿxn,

C_n z∈H:y_n−z≤ xn−z² κ−α_n1−α_nx_n−Tⁿx_n²θ_n , D_n{z∈H:x_n−z, x0−x_n ≥0},

xn1PCn∩Dnx0, ∀n≥1,

1.9

where

θ_n Δ²_n1−α_nγ_n−→0 n−→ ∞, Δ_nsup{x_n−z:z∈FT}<∞. 1.10 Assume that the control sequence {αn} is chosen such that lim sup_{n→ ∞}αn < 1−κ. Then {xn} converges strongly toPFTx0.

In this paper, motivated by3,8, we propose a new algorithm for finding a common element of the set of fixed points of an infinite family of asymptotically strict pseudo- contractions and the set of solutions of an infinite family of equilibrium problems and prove a strong convergence theorem. Our proof is simple and different from those of others, and the main results extend and improve those Kim and Xu3, Tada and Takahashi8, and many others.

2. Preliminaries

LetHbe a Hilbert space, and letCbe a nonempty closed convex subset ofH. It is well known that, for allx, y∈Candt∈0,1,

tx 1−ty²tx² 1−ty²−t1−tx−y, 2.1 and hence

tx 1−ty²≤tx² 1−ty², 2.2 which implies that

n i1

t_ix_i

2

≤ⁿ

i1

t_ix_i² 2.3

for all{x_i} ⊂Hand{t_i} ⊂0,1with_n

i1t_i 1.

For anyx∈H, there exists a unique nearest point inC, denoted byPCx, such that zPCx⇐⇒

x−z, z−y

≥0, ∀y∈C. 2.4

LetIdenote the identity operator ofH, and let{xn}be a sequence in a Hilbert space Handx ∈H. Throughout the rest of the paper,x_n → xdenotes the strong convergence of {x_n}tox.

We need the following lemmas for our main results in this paper.

Lemma 2.1see14. Let C be a nonempty closed convex subset of a Hilbert spaceH. LetΦbe a bifunction fromC×Cto^ÊsatisfyingA1–A4. Letr >0 andx∈H. Then there existsz∈Csuch that

Φ z, y

1 r

y−z, z−x

≥0, ∀y∈C. 2.5

Lemma 2.2see6. Let C be a nonempty closed convex subset of a Hilbert spaceH. Let Φbe a bifunction fromC×Cto^Ê satisfying A1–A4. For anyr > 0 and x ∈ H, define a mapping T_r :H → Cas follows:

T_rx

z∈C:Φ z, y

1 r

y−z, z−x

≥0, ∀y∈C

, ∀x∈H. 2.6

Then the following hold:

1T_r is single-valued,

2T_r is firmly nonexpansive, that is, for anyx, y∈H, T_rx−T_ry²≤

T_rx−T_ry, x−y

, 2.7

3FT_r EPΦ, and 4EPΦis closed and convex.

3. Main Results

Now, we are ready to give our main results.

Lemma 3.1. LetCbe a nonempty closed convex subset of a Hilbert spaceH. LetT :C → Cbe an asymptoticallyκ-strict pseudo-contraction with sequence{γ_n} ⊂0,∞such thatFT/∅. Assume that{β_n} ⊂κ,1and define a mappingS_n β_nI 1−β_nTⁿfor eachn≥ 1. Then the following hold:

Snx−Sny²≤

1γnx−y², ∀x, y∈C,

Snx−x² ≤γnx−x^∗22x−Snx, x−x^∗, ∀x∈C, x^∗ ∈FT. 3.1

Proof. For allx, y∈C, we have Snx−Sny²βn

x−y

1−βn

Tⁿx−Tⁿy ² βnx−y²

1−βn

Tⁿx−Tⁿy²−βn 1−βn

I−Tⁿx−I−Tⁿy²

≤β_nx−y²

1−β_n 1γ_n

x−y²κI−Tⁿx−I−Tⁿy²

−β_n 1−β_n

I−Tⁿx−I−Tⁿy² βnx−y²

1−βn 1γn

x−y²

1−βn κ−βn

I−Tⁿx−I−Tⁿy²

≤β_nx−y²

1−β_n 1γ_n

x−y²

≤ 1γn

x−y².

3.2 By this result, for allx∈Candx^∗∈FT, we have

1γn

x−x^∗2≥ Snx−Snx^∗2Snx−xx−x^∗2

S_nx−x²x−x^∗22S_nx−x, x−x^∗, 3.3 and hence

S_nx−x²≤γ_nx−x^∗22x−S_nx, x−x^∗. 3.4 This completes the proof.

Lemma 3.2. Let C be a nonempty closed subset of a Hilbert space H. Let T : C → C be an asymptotically κ-strict pseudo-contraction with sequence {γn} ⊂ 0,∞ satisfying γn → 0 as n → ∞. Let {zn} be a sequence in C such that zn −zn1 → 0 and zn −Tⁿzn → 0 as n → ∞. Thenz_n−Tz_n → 0 asn → ∞.

Proof. The proof method of this lemma is mainly from 15, Lemma 2.7. Since T is an asymptoticallyκ-strict pseudo-contraction, we obtain from3, Proposition 2.6that

Tⁿ¹zn−Tⁿ¹zn1≤Lzn−zn1, 3.5 whereLsup{κ

1γn1−κ/1−κ:n≥1}. Note thatzn−zn1 → 0, which implies thatTⁿ¹z_n−Tⁿ¹z_n1 → 0, and observe that

z_n−Tz_n ≤ z_n−z_n1z_n1−Tⁿ¹z_n1Tⁿ¹z_n1−Tⁿ¹z_nTⁿ¹z_n−Tz_n

≤1Lz_n−z_n1z_n1−Tⁿ¹z_n1Tⁿ¹z_n−Tz_n. 3.6

SinceT is uniformly Lipschitzian,Tis uniformly continuous. So we have

Tⁿ¹z_n−Tz_n−→0 asn−→ ∞. 3.7

It follows fromzn−zn1 → 0 andzn−Tⁿzn → 0 asn → ∞that lim_{n→ ∞}zn−Tzn0.

This completes the proof.

LetHbe a Hilbert space, and, letCbe a nonempty closed and convex subset ofH. Let {Φ_n}be a countable family of bifunctions fromC×Cto^ÊsatisfyingA1–A4and let{r_n} be a real number sequence inr,∞withr >0. Define

Trix

z∈C:Φi z, y

1 r_i

y−z, z−x

≥0, ∀y∈C

, ∀x∈H. 3.8

Lemma 2.2 shows that every Tri i ≥ 1 is a firmly nonexpansive mapping and hence nonexpansive andFT_ri EPΦ_i.

Theorem 3.3. LetCbe a nonempty closed convex subset of a Hilbert spaceH. Let{Ti} :C → C be an infinite family of asymptoticallyκi-strict pseudocontractions with the sequence{γi,n} ⊂0,∞ satisfying γ_i,n → 0 as n → ∞for each i ≥ 1 andγ_1,n ≥ γ_i,n for each i ≥ 1 andn ≥ 1. Let {Φn}be a countable family of bifunctions fromC×Cto^ÊsatisfyingA1–A4. Assume thatΩ _∞

i1FTi∩EPΦiis nonempty and bounded. Setα01 andθ01. Assume that{αi}is a strictly decreasing sequence in0, afor some 0< a <1,{θ_n}is a strictly decreasing sequence in0,1,{β_i,n} is a sequence inκi, κwith 0< κi < κ <1 for eachi≥1, and{rn}is a sequence inr,∞withr >0.

The sequence{xn}is generated byx1x∈Cand

znθnxnⁿ

i1

θi−1−θiTrixn,

wnαnxnⁿ

i1

αi−1−αi

βi,nI

1−βi,n T_iⁿ

zn,

C_n{v∈C:w_n−v ≤ x_n−vλ_n}, D_nⁿ

j1

C_j,

x_n1 P_Dnx, ∀n≥1,

3.9

where{T_ri}is defined by3.8and

λn 1−αnγ1,nΔn−→0 n−→ ∞, Δnsup{xn−v:v∈Ω}. 3.10

Then{x_n}converges strongly toP_Ωx.

Proof. We show first that the sequence{xn}is well defined. Obviously,Cn is closed for all n≥1. Since

w_n−v ≤ x_n−vλ_n 3.11

is equivalent to

w_n−x_n²2w_n−x_n, x_n−z ≤λ_n, 3.12 C_nis convex for alln≥1. SoD_nn

j1C_jis also closed and convex for alln≥1.

For eachn≥1 andi≥1, putSi,nβi,nI 1−βi,nT_iⁿ. Letp∈Ω. Note thatθ01,{θn} is strictly decreasing and eachTriis firmly nonexpansive. Hence we have

zn−p≤θnxn−pⁿ

i1

θi−1−θiTrixn−p

≤θnxn−pⁿ

i1

θi−1−θixn−p

≤θ_nx_n−p 1−θ_nx_n−p xn−p, ∀n≥1.

3.13

Sinceα01 and{αn}is strictly decreasing, by3.13and Lemma3.1, we have wn−p≤αnxn−pⁿ

i1

αi−1−αiSi,nzn−p

≤α_nx_n−pⁿ

i1

α_i−1−α_i

1γ_i,nz_n−p

≤α_nx_n−pⁿ

i1

α_i−1−α_i

1γ_1,nx_n−p

≤x_n−pλ_n.

3.14

So we havep ∈C_nand hencep∈D_{n n}

j1C_j for alln≥ 1. This shows thatΩ ⊂D_nfor all n≥1. This implies that the sequence{xn}is well defined.

SinceΩis a nonempty closed convex subset ofH, there exists a uniquez^∗ ∈Ωsuch that

z^∗P_Ωx. 3.15

Fromxn1PDnx, we have

xn1−x ≤ z−x, ∀z∈Dn. 3.16

Sincez^∗∈Ω⊂Dn, we have

xn1−x ≤ z^∗−x, ∀n≥1. 3.17

Therefore,{xn}is bounded. From3.13and3.14,{zn}and{wn}are also bounded.

Fromx_n1P_DnxandD_n1⊂D_n, one sees thatx_n2P_Dn1x∈D_n1⊂D_nfor alln≥1.

It follows that

xn1−x ≤ xn2−x, ∀n≥1. 3.18

Since{xn}is bounded, the sequence{x−xn}is bounded and nondecreasing. So there exists c∈^Êsuch that

c lim

n→ ∞x−xn. 3.19

Sincex_n1 P_Dnx∈D_n,x_n2P_Dn1x∈D_n1⊂D_nandx_n1x_n2/2∈D_n, we have x−xn1²≤x− xn1xn2

2 ²

1

2x−xn1 1

2x−xn2 ² 1

2x−xn1²1

2x−xn2²− 1

4xn1−xn2².

3.20

So we get

1

4xn1−xn2²≤ 1

2x−xn2²−1

2x−xn1². 3.21

Since lim_{n→ ∞}x−xn1lim_{n→ ∞}x−xn2c, we obtain

nlim→ ∞x_n1−x_n20, 3.22

that is,

nlim→ ∞xn−xn10. 3.23

Now, for eachl≥1, from3.23we get

xnl−xn ≤ xnl−xnl−1· · ·xn1−xn

−→0 asn−→ ∞. 3.24

This implies that there exists an elementx∈Csuch thatxn → xasn → ∞.

Next we show thatx∈_∞

i1FT_iandx∈_∞

i1EPΦ_i. Fromxn1∈Cn, we have

xn−wn ≤ xn−xn1xn1−wn

≤2xn−xn1λn. 3.25

By3.10and3.23, we obtain

nlim→ ∞xn−wn0. 3.26

Forp∈Ω, we have, from Lemma2.2,

Trixn−p²Trixn−Trip²

≤

Trixn−Trip, xn−p

T_rix_n−p, x_n−p 1

2

Trixn−p²xn−p²− xn−Trixn² ,

3.27

and hence

Trixn−p²≤xn−p²− xn−Trixn², ∀i≥1. 3.28

Therefore

z_n−p²≤θ_nx_n−p²ⁿ

i1

θ_i−1−θ_iT_rix_n−p²

≤θ_nx_n−p²ⁿ

i1

θ_i−1−θ_ix_n−p²− xn−T_rix_n² xn−p²−ⁿ

i1

θi−1−θixn−Trixn².

3.29

By3.29and Lemma3.1, we have w_n−p²≤α_nx_n−p²ⁿ

i1

α_i−1−α_iS_i,nz_n−p²

≤αnxn−p²ⁿ

i1

αi−1−αi

1γ1,n₂zn−p² α_nx_n−p² 1−α_n

1γ_1,n2z_n−p²

≤α_nx_n−p² 1−α_n

1γ_1,n2

x_n−p²−ⁿ

i1

θ_i−1−θ_ix_n−T_rix_n²

xn−p² 1−αn

2γ1,nγ_1,n² xn−p²

−1−α_n

1γ_1,n2ⁿ

i1

θ_i−1−θ_ix_n−T_rix_n²,

3.30

and hence

1−α_n

1γ_1,n2n i1

θ_i−1−θ_ix_n−T_rix_n²

≤xn−p²−wn−p² 1−αn

2γ1,nγ_1,n² xn−p²

≤ xn−wnxn−pwn−p 1−αn

2γ1,nγ_1,n² xn−p².

3.31

This shows that

1−αn

1γ1,n2θi−1−θixn−Trixn²

≤ xn−wnxn−pwn−p 1−α_n

2γ_1,nγ_1,n² x_n−p², ∀i≥1.

3.32

Since{α_n} ⊂0, awith 0< a <1,γ_1,n → 0,{θ_n}is strictly decreasing andx_n−w_n → 0, we get

nlim→ ∞xn−Trixn0, ∀i≥1. 3.33

LetMn sup_i≥1{xn−Trixn}for eachn≥1. ThenMn → 0 asn → ∞. Hence, from3.33, one has

x_n−z_n ≤ⁿ

i1

θ_i−1−θ_iT_rix_n−x_n

≤ⁿ

i1

θi−1−θiMn 1−θnMn

−→0.

3.34

From3.26and3.34, we obtain

z_n−w_n ≤ z_n−x_nx_n−w_n −→0. 3.35

Noting that

n i1

αi−1−αizn−Si,nzn αnxn 1−αnzn−wn

αnxn−wn 1−αnzn−wn,

3.36

we have

n i1

α_i−1−α_i

z_n−S_i,nz_n, z_n−p αn

xn−wn, zn−p

1−αn

zn−wn, zn−p .

3.37

By Lemma3.1, we have

zn−Si,nzn²≤γi,nzn−p²2

zn−Si,nzn, zn−p

≤γ1,nzn−p²2

zn−Si,nzn, zn−p

. 3.38

Therefore, combining this inequality with3.37, we get n

i1

αi−1−αizn−Si,nzn²

≤γ1,n1−αnzn−p²2αn

xn−wn, zn−p 21−αn

zn−wn, zn−p ,

3.39

and hencenoting thatαi−1> αifor eachi≥1 zn−Si,nzn²≤ γ1,n1−αn

α_i−1−α_i zn−p² 2αn

α_i−1−α_i

xn−wn, zn−p

21−αn α_i−1−α_i

zn−wn, zn−p .

3.40

From3.26,3.35and lim_{n→ ∞}γ_1,n0,we have

nlim→ ∞z_n−S_i,nz_n0, ∀i≥1. 3.41

From the definition ofSi,nand3.41, we havenoting that{βi,n} ⊂κi, κ⊂0,1 z_n−T_iⁿz_n≤ 1

1−βi,nz_n−S_i,nz_n −→0, ∀i≥1. 3.42 We next show3.42implies that

nlim→ ∞zn−Tizn0, ∀i≥1. 3.43

As a matter of fact, from3.23and3.34we have

zn−zn1 ≤ zn−xnxn−xn1xn1−zn1

−→0. 3.44

Now,3.42,3.44, and Lemma3.2imply3.43.

Since eachTiis uniformly continuous andzn → xasn → ∞, one getx ∈FTifor eachi≥1 and hencex∈_∞

i1FT_i. Now we showx∈_∞

i1EPΦ_i.

Since everyTriis nonexpansive, from3.33andxn → x, we have x∈FTriand hence

x∈_∞

i1FT_ri. Lemma2.2shows thatx∈_∞

i1EPΦ_i.

Finally, we prove thatxP_Ωx. Fromx_n1 P_Dnx, one sees

xn1−z, x−xn1 ≥0, ∀z∈Dn. 3.45 SinceΩ⊂Dnfor alln≥1, one arrives at

xn1−z, x−xn1 ≥0, ∀z∈Ω. 3.46

Taking the limit for above inequality, we get

x−z, x−x ≥ 0, ∀z∈Ω. 3.47 HencexP_Ωx. This completes the proof.

As direct consequences of Theorem3.3, we can obtain the following corollaries.

Corollary 3.4. Let C be a nonempty closed convex subset of a Hilbert space H. Let {Φ_n} be a countable family of bifunctions from:C×Cto^ÊsatisfyingA1–A4. Assume thatΩ _∞

i1EPΦi is nonempty and bounded. Let{rn}be a sequence inr,∞withr >0. Setθ01. The sequence{xn} is generated byx1 x∈Cand

z_nθ_nx_nⁿ

i1

θ_i−1−θ_iT_rix_n,

Cn{v∈C:zn−v ≤ xn−v}, Dnⁿ

j1

Cj,

xn1 PDnx, ∀n≥1,

3.48

where{Tri}is defined by3.8and{θn}is a strictly decreasing sequence in0,1. Then{xn}converges strongly toPΩx.

Proof. Putting T_i I for all i ≥ 1 and α_n 0 for all n ≥ 1 in Theorem 3.3, we obtain Corollary3.4.

Corollary 3.5. LetCbe a nonempty closed subset of a Hilbert spaceH. LetT be an asymptotically κ-strict pseudo-contraction with sequence{γ_n} ⊂0,∞satisfyingγ_n → 0 asn → ∞andFT/∅.

Let{xn}and{un}be sequences generated byx1x∈Hand znθnxn 1−θnPCxn, wnαnxn 1−αn

βnI 1−βn

Tⁿ zn, C_n{v∈C:w_n−v ≤ x_n−v},

D_nⁿ

j1

C_j,

x_n1P_Dnx, ∀n≥1,

3.49

where{θn} ⊂0,1,{αn} ⊂ 0, awith 0 < a <1, and{βn} ⊂ κ, κwithκ < κ <1. Then{xn} converges strongly toP_FTx.

Proof. Put Φix, y 0 for all x, y ∈ C and set rn 1 for all n ≥ 1 in Theorem 3.3.

By Lemma2.2, we have Trixn PCxn for each i ≥ 1. Hence, by Theorem3.3, we obtain Corollary3.5.

Remark 3.6. Our algorithms are of interest because the sequence{xn}in Theorem3.3is very different from the known manner. The proof is simple and different from those of others. The main results extend and improve those of Kim and Xu3, Tada and Takahashi8, and many others.

Remark 3.7. Putα01,θ01,κ3/4,r1,γi,n1/4ⁱⁿ,κi 1/41/3i,αn 1/1n, θ_n1/41/8n,β_i,n1/41/3i 1/8nfor alli≥1 and alln≥1,r01, andr_n11/n.

Then these control sequences satisfy all the conditions of Theorem3.3.

Acknowledgments

The authors thank the referees for useful comments and suggestions. This study was supported by research funds from Dong-A University.

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