Strong convergence theorems for strongly nonexpansive sequences

Strong convergence theorems for strongly nonexpansive sequences

Applied Mathematics and Computation 217 (2011) 7537–7545 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homep...

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Applied Mathematics and Computation 217 (2011) 7537–7545

Contents lists available at ScienceDirect

Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

Strong convergence theorems for strongly nonexpansive sequences Koji Aoyama ⇑, Yasunori Kimura Department of Economics, Chiba University, Yayoi-cho, Inage-ku, Chiba-shi, Chiba 263-8522, Japan Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, O-okayama, Meguro-ku, Tokyo 152-8552, Japan

a r t i c l e

i n f o

Keywords: Nonexpansive mapping Strongly nonexpansive sequence Fixed point Strong convergence

a b s t r a c t The aim of this paper is to prove a strong convergence theorem for a pair of sequences of nonexpansive mappings in a Hilbert space, where one of them is a strongly nonexpansive sequence, and provide some applications of the theorem. Ó 2011 Elsevier Inc. All rights reserved.

1. Introduction In order to approximate a common fixed point of a pair of sequences of nonexpansive mappings, we consider the following iterative sequence: u, x1 2 C and

xnþ1 ¼ bn xn þ ð1  bn ÞSn ðan u þ ð1  an ÞT n xn Þ

ð1:1Þ

for n 2 N, where C is a nonempty closed convex subset of a Hilbert space H, {Sn} and {Tn} are sequences of nonexpansive selfmappings of C, and {an} and {bn} are sequences in [0, 1]. In particular, we focus on the case where {Sn} or {Tn} is a strongly nonexpansive sequence [4,5]. Then we prove that the iterative sequence converges strongly to the nearest point of the set of common fixed points of {Sn} and {Tn} from u under some assumptions. We also discuss some applications of our results. The iterative sequence defined by (1.1) was studied in many papers. For example, Takahashi, Takahashi, and Toyoda [25] used this iteration in order to solve the fixed point problem for a nonexpansive mapping and the zero point problem for a monotone operator. Yao and Yao [27] used the iteration to solve the fixed point problem for a nonexpansive mapping and the variational inequality problem for an inverse-strongly monotone mapping. Our main result (Theorem 3.1) is a generalization of these results. The paper is organized as follows: Some preliminaries are described in Section 2. The main result and its proof are given in Section 3. Some examples of strongly nonexpansive sequences which satisfy the assumptions of the main result are listed in Section 4. In the final section, by using our main result, we prove a strong convergence theorem for a finite family of quasinonexpansive mappings. 2. Preliminaries Throughout the paper, H denotes a real Hilbert space, h, i the inner product of H, kk the norm of H, I the identity mapping on H, and N the set of positive integers. Strong convergence of a sequence {xn} in H to x is denoted by xn ? x and weak convergence by xn N x. The following inequality holds for all x, y 2 H:

kx þ yk2 6 kxk2 þ 2hy; x þ yi:

⇑ Corresponding author at: Department of Economics, Chiba University, Yayoi-cho, Inage-ku, Chiba-shi, Chiba 263-8522, Japan. E-mail addresses: [email protected] (K. Aoyama), [email protected] (Y. Kimura). 0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.01.092

ð2:1Þ

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Let C be a nonempty closed convex subset of H and T : C ? H a mapping. The set of fixed points of T is denoted by F(T). A mapping T is said to be nonexpansive if kTx  Tyk 6 kx  yk for all x, y 2 C. It is known that F(T) is closed and convex if T is nonexpansive. The metric projection of H onto C is denoted by PC, that is, PC(x) 2 C and kPC(x)  xk 6 ky  xk for all x 2 H and y 2 C. It is known that PC is nonexpansive and

hy  P C ðxÞ; x  PC ðxÞi 6 0

ð2:2Þ

for all x 2 H and y 2 C. Let {Tn} be a sequence of mappings of C into H. The set of common fixed points of {Tn} is denoted by F({Tn}), that is, T FðfT n gÞ ¼ 1 n¼1 FðT n Þ. A sequence {zn} in C is said to be an approximate fixed point sequence of {Tn} if zn  Tnzn ? 0. The set of all bounded approximate fixed point sequences of {Tn} is denoted by e F ðfT n gÞ; see [4]. It is clear that e F ðfT n gÞ is nonempty if {Tn} has a common fixed point. A sequence {Tn} is said to be a strongly nonexpansive sequence if each Tn is nonexpansive and

xn  yn  ðT n xn  T n yn Þ ! 0 whenever {xn} and {yn} are sequences in C such that {xn  yn} is bounded and kxn  ynk  kTnxn  Tn ynk ? 0. A sequence {Tn} having a common fixed point is said to satisfy the condition (Z) if every weak cluster point of {xn} is a common fixed point whenever fxn g 2 e F ðfT n gÞ. A sequence {Tn} of nonexpansive mappings of C into H is said to satisfy the condition (R) if

lim sup kT nþ1 y  T n yk ¼ 0

n!1 y2D

for every nonempty bounded subset D of C; see [2]. We need the following lemmas: Lemma 2.1. Let H be a Hilbert space, C a nonempty subset of H, and {Sn} and {Tn} sequences of nonexpansive self-mappings of C. Suppose that {Sn} and {Tn} satisfy the condition (R) and fT n y : n 2 N; y 2 Dg is bounded for any bounded subset D of C. Then {SnTn} satisfies the condition (R). Proof. The nonexpansiveness of Sn and Tn implies that each SnTn is nonexpansive and

kSnþ1 T nþ1 y  Sn T n yk 6 kSnþ1 T nþ1 y  Sn T nþ1 yk þ kSn T nþ1 y  Sn T n yk 6 kSnþ1 T nþ1 y  Sn T nþ1 yk þ kT nþ1 y  T n yk

ð2:3Þ

for all y 2 C. Let D be a nonempty bounded subset of C. Since {Sn} and {Tn} satisfy the condition (R) and D0 ¼ fT n y : n 2 N; y 2 Dg is bounded, it follows from (2.3) that

sup kSnþ1 T nþ1 y  Sn T n yk 6 sup kSnþ1 z  Sn zk þ sup kT nþ1 y  T n yk ! 0: y2D

z2D0

y2D

Therefore, {SnTn} satisfies the condition (R). h Lemma 2.2 ([4, Theorem 3.7, 3.9], [5, Theorem 3.3]). Let H be a Hilbert space, C a nonempty subset of H, and {Sn} and {Tn} sequences of nonexpansive self-mappings of C. Suppose that {Sn} or {Tn} is a strongly nonexpansive sequence and e F ðfT n gÞ is nonempty. Then e F ðfSn gÞ \ e F ðfT n gÞ ¼ e F ðfSn T n gÞ. F ðfSn gÞ \ e Lemma 2.3 [21, Lemma 2.2]. Let {xn} and {yn} be bounded sequences in a Banach space and {bn} a sequence in [0, 1]. Suppose that xnþ1 ¼ ð1  bn Þxn þ bn yn for every n 2 N, 0 < lim inf n!1 bn 6 lim supn!1 bn < 1, and

lim supðkynþ1  yn k  kxnþ1  xn kÞ 6 0:

n!1

Then xn  yn ? 0. Lemma 2.4 [3, Lemma 2.3]. Let {tn} be a sequence of nonnegative real numbers, {an} a sequence in [0, 1], and {cn} a sequence of P real numbers. Suppose that tnþ1 6 ð1  an Þtn þ an cn for all n 2 N, lim supn?1 cn 6 0, and 1 n¼1 an ¼ 1. Then tn ? 0. Lemma 2.5 [13, Lemma 3]. Let C be a nonempty closed convex subset of a strictly convex Banach space E. Let {Sj} be a sequence of P nonexpansive mappings of C into E and {cj} a sequence of positive real numbers such that 1 j¼1 cj ¼ 1. If F({Sj}) is nonempty, then a P1 mapping Q ¼ j¼1 cj Sj is well defined and nonexpansive, and moreover, F(Q) = F({Sj}). 3. Strong convergence theorem We prove the following strong convergence theorem:

K. Aoyama, Y. Kimura / Applied Mathematics and Computation 217 (2011) 7537–7545

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Theorem 3.1. Let H be a Hilbert space, C a nonempty closed convex subset of H, and {Sn} and {Tn} sequences of nonexpansive selfmappings of C. Suppose that F = F({Sn}) \ F({Tn}) is nonempty, both {Sn} and {Tn} satisfy the conditions (R) and (Z), and {Sn} or {Tn} is a strongly nonexpansive sequence. Let {an} and {bn} be sequences in [0, 1] such that

an ! 0;

1 X n¼1

an ¼ 1 and 0 < lim inf bn 6 lim sup bn < 1: n!1

ð3:1Þ

n!1

Let x, u 2 C and let {xn} be a sequence in C defined by x1 ¼ x 2 C and

xnþ1 ¼ bn xn þ ð1  bn ÞSn ðan u þ ð1  an ÞT n xn Þ

ð3:2Þ

for n 2 N. Then {xn} converges strongly to PF(u). First we prove needed lemmas; then we will prove Theorem 3.1. Lemma 3.2. {xn}, {Tnxn}, and fSn ðan u þ ð1  an ÞT n xn Þg are bounded. Proof. Let z 2 F and put yn ¼ an u þ ð1  an ÞT n xn for n 2 N. Since Sn z ¼ z, T n z ¼ z, and both Sn and Tn are nonexpansive, we have

kSn yn  zk 6 kyn  zk 6 an ku  zk þ ð1  an ÞkT n xn  zk 6 an ku  zk þ ð1  an Þkxn  zk

ð3:3Þ

and hence

kxnþ1  zk 6 bn kxn  zk þ ð1  bn ÞkSn yn  zk 6 ð1  an ð1  bn ÞÞkxn  zk þ an ð1  bn Þku  zk: Thus, by induction on n,

kT n xn  zk 6 kxn  zk 6 max fkx1  zk; ku  zkg for every n 2 N. This shows that {xn} and {Tn xn} are bounded, and moreover, {Snyn} is also bounded by (3.3). h F ðfSn gÞ and e F ðfAn T n gÞ ¼ e F ðfT n gÞ, where An ¼ an u þ ð1  an ÞI for n 2 N. Lemma 3.3. e F ðfSn An gÞ ¼ e Proof. We first show the former equality. Let {zn} be a bounded sequence in C. Since Sn is nonexpansive and an ? 0, it holds that

kSn An zn  Sn zn k 6 kAn zn  zn k ¼ an ku  zn k ! 0: Thus if fzn g 2 e F ðfSn An gÞ, then

kzn  Sn zn k 6 kzn  Sn An zn k þ kSn An zn  Sn zn k ! 0 and hence fzn g 2 e F ðfT n gÞ. On the other hand, if fzn g 2 e F ðfSn gÞ, then

kzn  Sn An zn k 6 kzn  Sn zn k þ kSn zn  Sn An zn k ! 0 and hence fzn g 2 e F ðfSn An gÞ. We next show the latter equality. Let {zn} be a bounded sequence in C and z 2 F. Since

kT n zn k 6 kT n zn  zk þ kzk 6 kzn  zk þ kzk; we know that {Tnzn} is also bounded and hence

kAn T n zn  T n zn k ¼ an ku  T n zn k ! 0: Thus if fzn g 2 e F ðfAn T n gÞ, then

kzn  T n zn k 6 kzn  An T n zn k þ kAn T n zn  T n zn k ! 0 and therefore fzn g 2 e F ðfT n gÞ. On the other hand, if fzn g 2 e F ðfT n gÞ, then

kzn  An T n zn k 6 kzn  T n zn k þ kT n zn  An T n zn k and hence fzn g 2 e F ðfAn T n gÞ. h Lemma 3.4. The following hold: (1) (2) (3)

e F ðfSn gÞ \ e F ðfT n gÞ; F ðfSn An T n gÞ ¼ e {SnAnTn} satisfies the condition (R); fxn g 2 e F ðfSn gÞ \ e F ðfT n gÞ, where An ¼ an u þ ð1  an ÞI for n 2 N.

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Proof. We first show (1). Since F is nonempty, Lemma 3.3 implies that

e F ðfSn An gÞ \ e F ðfT n gÞ ¼ e F ðfSn gÞ \ e F ðfT n gÞ – ;

ð3:4Þ

e F ðfSn gÞ \ e F ðfAn T n gÞ ¼ e F ðfSn gÞ \ e F ðfT n gÞ – ;:

ð3:5Þ

and

Suppose that {Sn} is a strongly nonexpansive sequence. By (3.5) and Lemma 2.2, we have

e F ðfSn An T n gÞ ¼ e F ðfSn gÞ \ e F ðfAn T n gÞ ¼ e F ðfSn gÞ \ e F ðfT n gÞ: On the other hand, suppose that {Tn} is a strongly nonexpansive sequence. By (3.4) and Lemma 2.2, we have

e F ðfSn An T n gÞ ¼ e F ðfSn An gÞ \ e F ðfT n gÞ ¼ e F ðfSn gÞ \ e F ðfT n gÞ: Therefore, (1) holds. We next show (2). It is not difficult to check that {An} satisfies the condition (R) and fAn y : n 2 N; y 2 Dg is bounded for any nonempty bounded subset D of C. Thus Lemma 2.1 implies that {SnAn} satisfies the condition (R). Similarly, since F is nonempty, fT n y : n 2 N; y 2 Dg is bounded for any nonempty bounded subset D of C. Lemma 2.1 implies that {SnAnTn} satisfies the condition (R). We finally show (3). By Lemma 3.2, there is a bounded subset D of C such that xn 2 D for every n 2 N. Put U n ¼ Sn An T n for n 2 N. Since Un is nonexpansive and {Un} satisfies the condition (R) by (2), we have

kU nþ1 xnþ1  U n xn k 6 kU nþ1 xnþ1  U n xnþ1 k þ kU n xnþ1  U n xn k 6 sup kU nþ1 y  U n yk þ kxnþ1  xn k y2D

and hence

lim supðkU nþ1 xnþ1  U n xn k  kxnþ1  xn kÞ 6 lim sup kU nþ1 y  U n yk ¼ 0: n!0 y2D

n!1

Thus Lemmas 3.2, 2.3, and (1) imply that

F ðfU n gÞ ¼ e F ðfSn gÞ \ e F ðfT n gÞ:  fxn g 2 e Lemma 3.5. lim supn?1hu  PF(u),Tnxn  PF(u)i 6 0. Proof. By Lemma 3.2, {xn} is bounded and thus there exist a point

v 2 C and a subsequence fxn g of {xn} such that i



 lim sup hu  PF ðuÞ; xn  PF ðuÞi ¼ lim u  PF ðuÞ; xni  PF ðuÞ

ð3:6Þ

i!1

n!1

and xni * v . Since {Sn} and {Tn} satisfy the condition (Z), Lemma 3.4 implies that v 2 F. Therefore it follows from Tnxn  xn ? 0, (3.6) and (2.2) that

lim sup hu  PF ðuÞ; T n xn  P F ðuÞi ¼ lim sup ðhu  PF ðuÞ; T n xn  xn i þ hu  PF ðuÞ; xn  PF ðuÞiÞ n!1 n!1   ¼ lim sup u  PF ðuÞ; xni  PF ðuÞ ¼ hu  PF ðuÞ; v  PF ðuÞi 6 0:



i!1

Proof (Proof of Theorem 3.1). Put w ¼ PF ðuÞ and yn ¼ an u þ ð1  an ÞT n xn for n 2 N. Since w 2 F, and both Sn and Tn are nonexpansive, it follows from (2.1) that

kSn yn  wk2 6 kyn  wk2 ¼ kð1  an ÞðT n xn  wÞ þ an ðu  wÞk2 6 ð1  an Þ2 kT n xn  wk2 þ 2han ðu  wÞ; yn  wi 6 ð1  an Þkxn  wk2 þ 2an hu  w; yn  wi: Thus this inequality and the convexity of k  k2 imply that

  kxnþ1  wk2 6 bn kxn  wk2 þ ð1  bn ÞkSn yn  wk2 6 bn kxn  wk2 þ ð1  bn Þ ð1  an Þkxn  wk2 þ 2an hu  w; yn  wi 6 ð1  an ð1  bn ÞÞkxn  wk2 þ 2an ð1  bn Þhu  w; yn  wi for every n 2 N. Lemmas 3.2 and 3.5 show that

lim suphu  w; yn  wi ¼ lim supðhu  w; T n xn  wi þ an hu  w; u  T n xn iÞ ¼ lim suphu  w; T n xn  wi 6 0: n!1

Since

P1

n¼1

n!1

an ð1  bn Þ ¼ 1, it follows from Lemma 2.4 that xn ! w ¼ PF ðuÞ. h

n

K. Aoyama, Y. Kimura / Applied Mathematics and Computation 217 (2011) 7537–7545

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4. Examples In this section, we provide some sequences of mappings which satisfy the assumptions of Theorem 3.1. Throughout this section, C is a nonempty closed convex subset of H. Example 4.1. Let T : C ? C be a nonexpansive mapping with a fixed point and {Tn} a sequence defined by T n ¼ T for n 2 N. Then {Tn} satisfies the conditions (R) and (Z). Moreover, if T is strongly nonexpansive [14], then {Tn} is a strongly nonexpansive sequence. Proof. It is clear that FðfT n gÞ ¼ FðTÞ and {Tn} satisfies the conditions (R). It is known that I  T is demiclosed, that is, z 2 F(T) whenever {xn} is a sequence in C such that xn  Txn ? 0 and xn N z; see [15]. Therefore, {Tn} satisfies the conditions (Z). It is also clear that if T is strongly nonexpansive, then {Tn} is a strongly nonexpansive sequence; see [4, Example 3.1]. h Let B be a maximal monotone operator on H and q > 0. It is known that ðI þ qBÞ1 is a single-valued firmly nonexpansive mapping of H into {x 2 H : Bx – ;}; see [23] for more details. Example 4.2. Let B be a maximal monotone operator on H with a zero point, {qn} a sequence of positive real numbers, and {Tn} a sequence defined by T n ¼ ðI þ qn BÞ1 for n 2 N. Then {Tn} is a strongly nonexpansive sequence. Moreover, if qn+1  qn ? 0 and inf n qn > 0, then {Tn} satisfies the conditions (R) and (Z). Proof. It is known that FðT n Þ ¼ B1 ð0Þ for every N; see, for example, [23]. Thus FðfT n gÞ ¼ B1 ð0Þ is nonempty. Since each Tn is firmly nonexpansive, {Tn} is a strongly nonexpansive sequence; see [4, Example 3.2]. We next show that {Tn} satisfies the condition (R). Let D be a bounded subset of H and suppose that qn+1  qn ? 0 and inf n qn > 0. It is known that

kT nþ1 y  T n yk2 6

 j qnþ1  qn j  2 2 kT nþ1 y  yk  kT n y  yk 

qnþ1

ð4:1Þ

holds for all y 2 D and n 2 N; see [11, Lemma 4.4] and [10, Corollary 3.2]. Since Tn is nonexpansive and F({Tn}) is nonempty, fT n y : y 2 D; n 2 Ng is bounded and hence

 n o   M ¼ sup kT nþ1 y  yk2  kT n y  yk2  : y 2 D; n 2 N < 1;

so that

sup kT nþ1 y  T n yk2 6 y2D

j qnþ1  qn j M!0 inf n qn

as n ? 1. Therefore, {Tn} satisfies the condition (R). For the condition (Z), see, for example, [4, Lemma 5.1], [7, Lemma 2.1] and [9, Lemma 2.4]. h A mapping A : C ? H is said to be inverse-strongly monotone if there is a constant a > 0 such that hx  y, Ax  Ayi P akAx  Ayk2 for all x, y 2 C. Such a mapping A is called an a-inverse-strongly monotone mapping. It is known that I  kA is nonexpansive if 0 < k < 2a and A is an a-inverse-strongly monotone mapping; see [26]. The solution set of the variational inequality problem for a mapping A : C ? H is denoted by VI(C, A), that is, VI(C, A) = {x 2 C : h y  x, Axi P 0 for all y 2 C}. Example 4.3. Let a > 0 and let A : C ? H be an a-inverse-strongly monotone mapping such that VI(C, A) is nonempty. Let {kn} be a sequence of positive real numbers such that

0 < inf kn 6 sup kn < 2a and knþ1  kn ! 0 n

ð4:2Þ

n

and {Tn} a sequence of mappings defined by T n ¼ PC ðI  kn AÞ for n 2 N. Then {Tn} is a strongly nonexpansive sequence that satisfies the conditions (R) and (Z). Proof. It is known that FðT n Þ ¼ VIðC; AÞ for every n 2 N. Thus FðfT n gÞ ¼ VIðC; AÞ is nonempty. It is also known that {Tn} is a strongly nonexpansive sequence; see [4, Examples 3.2 and 3.3] and [4, Theorem 3.4]. For the conditions (R) and (Z), see [2, Example 8]. h Example 4.4. Let a > 0 and let A : C ? H be an a-inverse-strongly monotone mapping and B a maximal monotone operator on H. Suppose that (A + B)10 is nonempty. Let {kn} be a sequence of positive real numbers satisfying (4.2) and {Tn} a sequence of mappings defined by T n ¼ ðI þ kn BÞ1 ðI  kn AÞ for n 2 N. Then {Tn} is a strongly nonexpansive sequence that satisfies the conditions (R) and (Z).

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Proof. It is known that FðT n Þ ¼ ðA þ BÞ1 ð0Þ for every n 2 N; see [4, Lemma 5.8]. Thus FðfT n gÞ ¼ ðA þ BÞ1 ð0Þ is nonempty. It is also known that {Tn} is a strongly nonexpansive sequence; see the proof of [4, Theorem 5.9]. As in the proofs of Examples 4.2 and 4.3, we can prove that {Tn} satisfies the condition (R). For the condition (Z), see [1,4,6,8]. h Example 4.5. Let {Rn} be a sequence of nonexpansive mappings of C into itself having a common fixed point and {ln} a sequence in [0, 1]. For each n 2 N, a W-mapping [22] Tn generated by Rn, Rn1, . . . , R1 and ln, ln1, . . . , l1 is defined as follows:

U n;n ¼ ln Rn þ ð1  ln ÞI; U n;n1 ¼ ln1 Rn1 U n;n þ ð1  ln1 ÞI; ... U n;k ¼ lk Rk U n;kþ1 þ ð1  lk ÞI; ... U n;2 ¼ l2 R2 U n;3 þ ð1  l2 ÞI; T n ¼ U n;1 ¼ l1 R1 U n;2 þ ð1  l1 ÞI: See also [17,20,24]. If 0 < l1 6 1 and 0 < ln 6 b for n P 2 and 0 < b < 1, then {Tn} satisfies the conditions (R) and (Z). Proof. It is known [12] that FðfT n gÞ ¼ FðfRn gÞ and thus there exists z 2 F({Tn}). Let D be a nonempty bounded subset of C. Following the proof of [20, Lemma 3.2], we have

kT nþ1 y  T n yk ¼ k





l1 R1 U nþ1;2 y þ ð1  l1 Þy  ðl1 R1 U n;2 y þ ð1  l1 ÞyÞk ¼ l1 kR1 U nþ1;2 y  R1 U n;2 yk

6 l1 kU nþ1;2 y  U n;2 yk 6 l1 l2 kU nþ1;3 y  U n;3 yk 6 . . . 6

n Y

!

lk kU nþ1;nþ1 y  yk ¼

k¼1 n

nþ1 Y

!

lk kRnþ1 y  yk

k¼1

n

6 b ðkRnþ1 y  zk þ kz  ykÞ 6 2b ky  zk for any y 2 D. Thus we have n

0 6 lim sup kT nþ1 y  T n yk 6 2 lim b sup ky  zk ¼ 0 n!1 y2D

n!1

y2D

and hence {Tn} satisfies the condition (R). For the condition (Z), see [18, Theorem 5.1] and [5]. h Example 4.6. Let {Rn} be a sequence of nonexpansive mappings of C into H having a common fixed point, and P fcn;k : n; k 2 N; k 6 ng a family of nonnegative real numbers such that nk¼1 cn;k ¼ 1 for every n 2 N. For each n 2 N, define Pn a mapping Tn : C ? H by T n ¼ k¼1 cn;k Rk . Then the following hold. P (1) If limn!1 nk¼1 j cn;k  cnþ1;k j¼ 0, then {Tn} satisfies the condition (R); P1 Pn (2) if n¼1 k¼1 j cn;k  cnþ1;k j< 1 and limn!1 cn;k ¼ ck > 0 for every k 2 N , then {Tn} satisfies the condition (Z).

Proof. We first prove (1). Let D be a nonempty bounded subset of C. Using Lemma 2.5, we have

FðfT n gÞ ¼

1 \

FðT n Þ ¼

n¼1

1 \ n \

FðRk Þ ¼

n¼1 k¼1

1 \

FðRn Þ ¼ FðfRn gÞ – ;:

n¼1

Thus, letting z0 2 F({Rn}), for every n 2 N, we have that Rn is nonexpansive and

sup kRn yk 6 sup kRn y  z0 k þ kz0 k 6 sup ky  z0 k þ kz0 k 6 M; y2D

y2D

y2D

where M ¼ supy2D kyk þ 2kz0 k. Since

T nþ1  T n ¼

nþ1 X

cnþ1;k Rk 

k¼1

¼

n X k¼1

n X k¼1

cn;k 

n X k¼1

cn;k Rk ¼ cnþ1;nþ1 Rnþ1 þ !

cnþ1;k Rnþ1 þ

n  X



cnþ1;k  cn;k Rk ¼ 1 

k¼1

n X

!

cnþ1;k Rnþ1 þ

k¼1

n  X k¼1

n  X

n  n     X X cnþ1;k  cn;k Rk ¼ cn;k  cnþ1;k Rnþ1 þ cnþ1;k  cn;k Rk ;

k¼1

k¼1

k¼1



cnþ1;k  cn;k Rk

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we have

sup kT nþ1 y  T n yk 6 y2D

n X

j cn;k  cnþ1;k j sup kRnþ1 yk þ y2D

k¼1

n X

j cnþ1;k  cn;k j sup kRk yk 6 2M

k¼1

y2D

n X

j cnþ1;k  cn;k j

k¼1

for n 2 N. When n ? 1, we have that {Tn} satisfies the condition (R). P P1 Let us prove (2). From the argument in [3, pp. 2357–2358], we have that 1 k¼1 ck ¼ 1 and by putting T ¼ k¼1 ck Rk , we have

FðTÞ ¼ FðfRn gÞ ¼ FðfT n gÞ and

lim sup kT n y  Tyk ¼ 0

n!1 y2D

for each bounded subset D of C. Let z be a weak cluster point of a bounded sequence {xn} such that xn  Tnxn ? 0. Then there exists a subsequence fxni g of {xn} which converges weakly to z. Since fxni g is bounded, we have that limi!1 kT ni xni  Txni k ¼ 0 and therefore

kxni  Txni k 6 kxni  T ni xni k þ kT ni xni  Txni k ! 0 as i ? 1. By Lemma 2.5, T is nonexpansive so that I  T is demiclosed. Thus we have z 2 FðTÞ ¼ FðfT n gÞ. Hence {Tn} satisfies the condition (Z), which is the desired result. h

5. Deduced results In this section, we provide three results which are deduced from Theorem 3.1. The following two corollaries are direct consequences of Theorem 3.1. Corollary 5.1 (Takahashi, Takahashi, and Toyoda [25]). Let H be a Hilbert space, C a nonempty closed convex subset of H, a a positive real number, A : C ? H an a-inverse-strongly monotone mapping, B a maximal monotone operator on H such that the domain is included in C, and S : C ? C a nonexpansive mapping. Suppose that F ¼ FðSÞ \ ðA þ BÞ 1ð0Þ is nonempty. Let {an} and {bn} be sequences in [0, 1] and {kn} a sequence of positive real numbers. Suppose that (3.1) and (4.2) hold. Let x, u 2 C and let {xn} be a sequence defined by x1 ¼ x 2 C and

  xnþ1 ¼ bn xn þ ð1  bn ÞS an u þ ð1  an ÞðI þ kn BÞ1 ðI  kn AÞxn

ð5:1Þ

for n 2 N. Then {xn} converges strongly to PF(u). Proof. Let {Sn} and {Tn} be sequences of mappings defined by Sn ¼ S and T n ¼ ðI þ kn BÞ1 ðI  kn AÞ for n 2 N. Then Examples 4.1 and 4.4 show that

FðfSn gÞ \ FðfT n gÞ ¼ FðSÞ \ ðA þ BÞ1 ð0Þ ¼ F; both {Sn} and {Tn} satisfy the conditions (R) and (Z), and {Tn} is a strongly nonexpansive sequence. Moreover, it is clear that (5.1) coincides with (3.2). Therefore, Theorem 3.1 implies the conclusion. h Corollary 5.2 (Yao and Yao [27]). Let H be a Hilbert space, C a nonempty closed convex subset of H, a a positive real number, A : C ? H an a-inverse-strongly monotone mapping, and S : C ? C a nonexpansive mapping. Suppose that F ¼ FðSÞ \ VIðC; AÞ is nonempty. Let {an} and {bn} be sequences in [0, 1] and {kn} a sequence of positive real numbers. Suppose that (3.1) and (4.2) hold. Let x, u 2 C and let {xn} be a sequence defined by x1 ¼ x 2 C and

xnþ1 ¼ an u þ bn xn þ ð1  an  bn ÞSPC ðI  kn AÞPC ðI  kn AÞxn

ð5:2Þ

for n 2 N. Then {xn} converges strongly to PF(u). Proof. Let {Sn} and {Tn} be sequences of mappings defined by Sn ¼ I and T n ¼ SP C ðI  kn AÞPC ðI  kn AÞ for n 2 N. Then Example 4.1 and [2, Example 8 and Remark 9] show that

FðfSn gÞ \ FðfT n gÞ ¼ FðSÞ \ VIðC; AÞ ¼ F both {Sn} and {Tn} satisfy the conditions (R) and (Z), and {Sn} is a strongly nonexpansive sequence. By assumption, without loss of generality, we may assume that bn – 1 for every n 2 N. Thus it follows from (5.2) and (3.1) that

xnþ1 ¼ bn xn þ ð1  bn ÞSn ðcn u þ ð1  cn ÞT n xn Þ;

cn ? 0, and

P1

c ¼ 1, where cn ¼ an =ð1  bn Þ for n 2 N. Therefore, Theorem 3.1 implies the conclusion. h

n¼1 n

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K. Aoyama, Y. Kimura / Applied Mathematics and Computation 217 (2011) 7537–7545

A mapping Q : C ? H is called a quasinonexpansive mapping if F(Q) is nonempty and kQx  zk 6 kx  zk for all x 2 C and z 2 F(Q). It is known that F(Q) is closed and convex if Q : C ? H is a quasinonexpansive mapping. We say that Q satisfies the condition (⁄) if x0 2 F(Q) whenever both sequences {xn} in C and {Qxn} converge strongly to x0; see [16]. Motivated by the method of [19, Theorem 3.3], we obtain the following result. See also [3]. Theorem 5.3. Let H be a Hilbert space and C a nonempty closed convex subset of H and u 2 H. Let {Q1, Q2, . . . , Qm} be a finite family T of quasinonexpansive mappings of C into H such that Qj satisfies the condition (⁄) for each j ¼ 1; 2; . . . ; m. Let F ¼ m j¼1 FðQ j Þ and suppose that F is nonempty. Let x, u 2 C and let {xn} be a sequence defined by x1 ¼ x 2 C and



C jn ¼ z 2 H : kQ j xn  zk 6 kxn  zk for m X 1 P j for k ¼ 1; 2; . . . ; n; Rk ¼ m j¼1 C k xnþ1 ¼ bn xn þ ð1  bn ÞPC an u þ ð1  an Þ

j ¼ 1; 2; . . . ; m;

n nþ1 X 1 R k xn n k¼1 kðk þ 1Þ

!

for n 2 N, where {an} and {bn} are sequences in [0, 1] with (3.1). Then {xn} converges strongly to PF(u). Proof. Fix j 2 {1, 2, . . . , m}. Since C jn is a closed half space of H, it is obviously closed, convex, and nonempty. Therefore the metric projection PC j onto C jn is well defined. Further, since Qj is quasinonexpansive, it follows that kQj xn  zk 6 kxn  zk n for every z 2 F and thus we have ; – F  C jn for all n 2 N. Letting

Tn ¼

n nþ1 X 1 Rk n k¼1 kðk þ 1Þ

for each n 2 N, we have

xnþ1 ¼ bn xn þ ð1  bn ÞPC ðan u þ ð1  an ÞT n xn Þ for every n 2 N. Since Rn and Tn are defined as convex combinations of nonexpansive mappings, by Lemma 2.5 we have that T Tm j Tn Rn and Tn are also nonexpansive and FðRn Þ ¼ m j¼1 FðP C jn Þ ¼ j¼1 C n and FðT n Þ ¼ k¼1 FðRk Þ. We show that {Tn} satisfies the conditions (R) and (Z). Indeed, letting

cn;k ¼

nþ1 nkðk þ 1Þ

for n 2 N and k 2 {1, 2, . . . ,n}, we have that n X

j cn;k  cnþ1;k j¼

lim cn;k ¼

n!1

c

k¼1 n;k

n X

n X

k¼1

k¼1

k¼1

Therefore we obtain

Pn

ðcn;k  cnþ1;k Þ ¼

P1 Pn n¼1

k¼1

¼ 1 for every n 2 N. Further, since cnþ1;k < cn;k , we have that

cn;k 

n X

cnþ1;k ¼ cnþ1;nþ1 ¼

k¼1

1 ðn þ 1Þ2

:

j cn;k  cnþ1;k j< 1. It also holds that

1 >0 kðk þ 1Þ

for every k 2 N. From Example 4.6, {Tn} satisfies the conditions (R) and (Z). By Theorem 3.1, we have that {xn} converges strongly to x0 ¼ P F 0 ðuÞ, where

F 0 ¼ FðfT n gÞ \ C ¼ FðfT n gÞ ¼

1 \ k¼1

FðRk Þ ¼

1 \ m \ k¼1 j¼1

C jk ¼

m \ 1 \

C jn :

j¼1 n¼1

Then, since x0 2 F 0  C jn for all j ¼ 1; 2; . . . ; m and n 2 N, we have that

kQ j xn  x0 k 6 kxn  x0 k ! 0 as n ? 1. This implies that {Qjxn} converges strongly to x0 as n ? 1 for each j ¼ 1; 2; . . . ; m. Consequently, we have that T x0 2 F(Qj) for each j ¼ 1; 2; . . . ; m by using the condition (⁄) for Qj. Since x0 2 F ¼ m j¼1 FðQ j Þ  F 0 and x0 ¼ P F 0 ðuÞ, we obtain that x0 = PF(u), which completes the proof. h Remark 5.4. Let v 2 H be a nonzero vector, q a real number, and L ¼ fz 2 H : hv ; zi 6 qg a closed half space of H. It is easy to see that we can express the metric projection PL by

PL ðxÞ ¼ x 

max f0; hv ; xi  qg kv k2

v

K. Aoyama, Y. Kimura / Applied Mathematics and Computation 217 (2011) 7537–7545

7545

for every x 2 H. Since C jn in the previous theorem can be given by

o

n C jn ¼ z 2 H : kQ j xn  zk 6 kxn  zk ¼ z 2 H : kQ j xn  zk2 6 kxn  zk2 n o ¼ z 2 H : hxn  Q j xn ; zi 6 ðkxn k2  kQ j xn k2 Þ=2 ;

we have

PC j ðxn Þ ¼ xn  k

n   o max 0; hxk  Q j xk ; xn i  kxk k2  kQ j xk k2 =2 kxk  Q j xk k2

ðxk  Q j xk Þ

if xk – Qjxk and PC j ðxn Þ ¼ xn if xk ¼ Q j xk for every n 2 N, k ¼ 1; 2; . . . ; n, and j ¼ 1; 2; . . . ; m. From this fact, we can calculate k P C j ðxn Þ directly from known vectors. Moreover, if a closed convex set C is sufficiently simple and PC is easily calculated, then k a sequence {xn} also can be calculated easily by known vectors. Acknowledgements The authors thank anonymous referees for their valuable comments and suggestions. The authors are supported by Grant-in-Aid for Scientific Research No. 22540175 from Japan Society for the Promotion of Science. References [1] K. Aoyama, Asymptotic fixed points of sequences of quasi-nonexpansive type mappings, Banach and function spaces III, Yokohama Publ., Yokohama, in press. [2] K. Aoyama, An iterative method for fixed point problems for sequences of nonexpansive mappings, in: Fixed Point Theory and Applications, Yokohama Publ., Yokohama, 2010, pp. 1–7. [3] K. Aoyama, Y. Kimura, W. Takahashi, M. 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