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Viscosity approximation process for a sequence of quasinonexpansive mappings

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Abstract

We study the viscosity approximation method due to Moudafi for a fixed point problem of quasinonexpansive mappings in a Hilbert space. First, we establish a strong convergence theorem for a sequence of quasinonexpansive mappings. Then we employ our result to approximate a solution of the variational inequality problem over the common fixed point set of the sequence of quasinonexpansive mappings.

MSC:47H09, 47H10, 41A65.

1 Introduction

Let C be a nonempty closed convex subset of a Hilbert space. This paper is devoted to the study of strong convergence of a sequence { x n } in C defined by an arbitrary point x 1 C and

x n + 1 = α n f n ( x n )+(1 α n ) T n x n
(1.1)

for nN, where α n is a real number in [0,1], f n is a contraction-like mapping on C, and T n is a quasinonexpansive mapping on C. This iterative method (1.1) is called the viscosity approximation method [1]. In Section 3, we establish that, under some appropriate assumptions, the sequence { x n } converges strongly to a certain common fixed point of { T n } by using the technique developed in [2]. Then, in Section 4, we apply our result to approximate a solution of a variational inequality problem over the common fixed point set of { T n }.

The viscosity approximation method (1.1) is based on the study of Moudafi [1], who considered a fixed point problem of a single nonexpansive mapping and proved strong convergence of sequences generated by the method. After that, Xu [3] extended Moudafi’s results [1] in the framework of Hilbert spaces and Banach spaces; Suzuki [4] gave simple proofs of Xu’s results [3]; Aoyama and Kimura [5] investigated a relationship between viscosity approximation methods and Halpern [6] type iterative methods for a sequence of nonexpansive mappings.

On the other hand, Maingé [7] adopted the viscosity approximation method for a fixed point problem of a single quasinonexpansive mapping; Wongchan and Saejung [8] extended Maingé’s result [7]. Our main result (Theorem 3.1) is a generalization of Wongchan and Saejung’s result [8] and is closely related to the study in [5]. Moreover, it is also applicable to an approximation method, which is called the hybrid steepest descent method [9, 10], for a variational inequality problem over the common fixed point set of a sequence of quasinonexpansive mappings.

2 Preliminaries

Throughout the present paper, H denotes a real Hilbert space, , the inner product of H, the norm of H, C a nonempty closed convex subset of H, I the identity mapping on H, the set of real numbers, and the set of positive integers. Strong convergence of a sequence { x n } in H to xH is denoted by x n x and weak convergence by x n x.

Let T:CH be a mapping. The set of fixed points of T is denoted by Fix(T). A mapping T is said to be quasinonexpansive if Fix(T) and Txpxp for all xC and pFix(T); T is said to be nonexpansive if TxTyxy for all x,yC; T is said to be strongly quasinonexpansive if T is quasinonexpansive and T x n x n 0 whenever { x n } is a bounded sequence in C and x n pT x n p0 for some point pFix(T); T is demiclosed at 0 if Tp=0 whenever { x n } is a sequence in C such that x n p and T x n 0. We know that if T:CH is quasinonexpansive, then Fix(T) is closed and convex; see [[11], Theorem 1].

It is known that, for each xH, there exists a unique point x 0 C such that

x x 0 =min { x y : y C } .

Such a point x 0 is denoted by P C (x) and P C is called the metric projection of H onto C. It is known that the metric projection P C is nonexpansive; see [12].

Let f:CC be a mapping, F a nonempty subset of C, and θ a real number in [0,1). A mapping f is said to be a θ-contraction with respect to F if f(x)f(z)θxz for all xC and zF; f is said to be a θ-contraction if f is a θ-contraction with respect to C. By definition, it is easy to check the following results.

Lemma 2.1 Let F be a nonempty subset of C and f:CC a θ-contraction with respect to F, where 0θ<1. If F is closed and convex, then P F f is a θ-contraction on F, where P F is the metric projection of H onto F.

Lemma 2.2 Let f:CC be a θ-contraction, where 0θ<1 and T:CC a quasinonexpansive mapping. Then fT is a θ-contraction with respect to Fix(T).

Let D be a nonempty subset of C. A sequence { f n } of mappings of C into H is said to be stable on D [5] if { f n (z):nN} is a singleton for every zD. It is clear that if { f n } is stable on D, then f n (z)= f 1 (z) for all nN and zD.

A function τ:NN is said to be eventually increasing [2] if lim n τ(n)= and τ(n)τ(n+1) for all nN. By definition, we easily obtain the following.

Lemma 2.3 Let τ:NN be an eventually increasing function and { ξ n } a sequence of real numbers such that ξ n ξ. Then ξ τ ( n ) ξ.

The following is a direct consequence of [[13], Lemma 3.1].

Lemma 2.4 ([[2], Lemma 3.4])

Let { ξ n } be a sequence of nonnegative real numbers which is not convergent. Then there exist NN and an eventually increasing function τ:NN such that ξ τ ( n ) ξ τ ( n ) + 1 for all nN and ξ n ξ τ ( n ) + 1 for all nN.

Under the assumptions of Lemma 2.4, we cannot choose a strictly increasing function τ; see [[2], Example 3.3].

Let { T n } be a sequence of mappings of C into H such that F= n = 1 Fix( T n ) is nonempty. Then

  • { T n } is said to be strongly quasinonexpansive type if each T n is quasinonexpansive and T n x n x n 0 whenever { x n } is a bounded sequence in C and

    x n p T n x n p0

for some point pF;

  • { T n } is said to satisfy the condition (Z) [2, 1416] if every weak cluster point of { x n } belongs to F whenever { x n } is a bounded sequence in C such that T n x n x n 0.

Remark 2.5 Since β n α n 0 if and only if β n 2 α n 2 0 for all bounded sequences { α n } and { β n } in [0,), { T n } is strongly quasinonexpansive type if and only if it is a strongly relatively nonexpansive sequence in the sense of [2, 17]. See also [18, 19].

We know several examples of strongly quasinonexpansive type sequences satisfying the condition (Z); see [17] and Example 4.5 in Section 4.

The following lemma follows from [[2], Lemma 3.5] and Remark 2.5.

Lemma 2.6 Let { T n } be a sequence of mappings of C into H such that F= n = 1 Fix( T n ) is nonempty, τ:NN an eventually increasing function, and { z n } a bounded sequence in C such that z n p T τ ( n ) z n p0 for some pF. If { T n } is strongly quasinonexpansive type, then T τ ( n ) z n z n 0.

In order to prove our main result in Section 3, we need the following lemmas.

Lemma 2.7 ([[2], Lemma 3.6])

Let { T n } be a sequence of mappings of C into H such that F= n = 1 Fix( T n ) is nonempty, τ:NN an eventually increasing function, and { z n } a bounded sequence in C such that T τ ( n ) z n z n 0. Suppose that { T n } satisfies the condition (Z). Then every weak cluster point of { z n } belongs to F.

Lemma 2.8 ([[2], Lemma 3.7])

Let { T n } be a sequence of mappings of C into H, F a nonempty closed convex subset of H, { z n } a bounded sequence in C such that T n z n z n 0, and uH. Suppose that every weak cluster point of { z n } belongs to F. Then

lim sup n T n z n w,uw0,

where w= P F (u).

The following lemma is well known; see [20, 21].

Lemma 2.9 Let { ξ n } be a sequence of nonnegative real numbers, { δ n } a sequence of real numbers, and { β n } a sequence in [0,1]. Suppose that ξ n + 1 (1 β n ) ξ n + β n δ n for every nN, lim sup n δ n 0, and n = 1 β n =. Then ξ n 0.

3 Strong convergence of a viscosity approximation process

In this section, we prove the following strong convergence theorem.

Theorem 3.1 Let H be a real Hilbert space, C a nonempty closed convex subset of H, { S n } a sequence of mappings of C into C such that F= n = 1 Fix( S n ) is nonempty, { α n } a sequence in (0,1] such that α n 0 and n = 1 α n =, and { f n } a sequence of mappings of C into C such that each f n is a θ-contraction with respect to F and { f n } is stable on F, where 0θ<1. Let { x n } be a sequence defined by x 1 C and

x n + 1 = α n f n ( x n )+(1 α n ) S n x n
(3.1)

for nN. Suppose that { S n } is strongly quasinonexpansive type and satisfies the condition (Z). Then { x n } converges strongly to wF, where w is the unique fixed point of a contraction P F f 1 .

Note that Lemma 2.1 implies that P F f 1 is a θ-contraction on F and hence it has a unique fixed point on F.

First, we show some lemmas; then we prove Theorem 3.1. In the rest of this section, we set

β n = α n ( 1 + ( 1 2 θ ) ( 1 α n ) )

and

γ n = α n 2 f n ( x n ) w 2 +2 α n (1 α n ) S n x n w , f 1 ( w ) w

for nN.

Lemma 3.2 { x n }, { S n x n }, and { f n ( x n )} are bounded, and moreover,

x n + 1 w α n f n ( x n ) w + S n x n w
(3.2)

and

x n + 1 w 2 (1 β n ) x n w 2 + γ n
(3.3)

hold for every nN.

Proof Since f n is a θ-contraction with respect to F, S n is quasinonexpansive, wFFix( S n ), and { f n } is stable on F, it follows that

x n + 1 w α n f n ( x n ) w + ( 1 α n ) S n x n w α n ( f n ( x n ) f n ( w ) + f n ( w ) w ) + ( 1 α n ) S n x n w ( 1 α n ( 1 θ ) ) x n w + α n ( 1 θ ) f 1 ( w ) w 1 θ
(3.4)

for every nN. Thus, by induction on n, we have

S n x n w x n wmax { x 1 w , f 1 ( w ) w / ( 1 θ ) } .

Therefore, it turns out that { x n } and { S n x n } are bounded, and moreover, { f n ( x n )} is also bounded.

Equation (3.2) follows from (3.4).

Next, we show (3.3). By assumption, it follows that

S n x n w , f n ( x n ) w S n x n w f n ( x n ) f n ( w ) + S n x n w , f n ( w ) w θ x n w 2 + S n x n w , f 1 ( w ) w ,

and thus

x n + 1 w 2 = α n 2 f n ( x n ) w 2 + ( 1 α n ) 2 S n x n w 2 + 2 α n ( 1 α n ) S n x n w , f n ( x n ) w α n 2 f n ( x n ) w 2 + ( ( 1 α n ) 2 + 2 α n ( 1 α n ) θ ) x n w 2 + 2 α n ( 1 α n ) S n x n w , f 1 ( w ) w = ( 1 β n ) x n w 2 + γ n
(3.5)

for every nN. Therefore, (3.3) holds. □

Lemma 3.3 The following hold:

  • 0< β n 1 for every nN;

  • 2 α n (1 α n )/ β n 1/(1θ);

  • α n 2 f n ( x n ) w 2 / β n 0;

  • n = 1 β n =.

Proof Since 0< α n 1 and 1<12θ1, we know that

0< α n 2 = α n ( 1 + ( 1 ) ( 1 α n ) ) β n α n ( 1 + ( 1 α n ) ) = α n (2 α n )1.

It follows from α n 0 that 2 α n (1 α n )/ β n 1/(1θ).

Since { f n ( x n )} is bounded by Lemma 3.2 and

α n 2 β n = α n 1 + ( 1 2 θ ) ( 1 α n ) 0,

it follows that α n 2 f n ( x n ) w 2 / β n 0.

Finally, we prove n = 1 β n =. Suppose that 12θ0. Then it is clear that β n α n for every nN. Thus, n = 1 β n n = 1 α n =. Next, we suppose that 12θ<0. Then it is clear that β n >2(1θ) α n for every nN. Thus, n = 1 β n 2(1θ) n = 1 α n =. This completes the proof. □

Lemma 3.4 { x n w} is convergent.

Proof We assume, in order to obtain a contraction, that { x n w} is not convergent. Then Lemma 2.4 implies that there exist NN and an eventually increasing function τ:NN such that

x τ ( n ) w x τ ( n ) + 1 w
(3.6)

for every nN and

x n w x τ ( n ) + 1 w
(3.7)

for every nN.

We show that S τ ( n ) x τ ( n ) x τ ( n ) 0. Since S τ ( n ) is quasinonexpansive and wFFix( S τ ( n ) ), it follows from (3.6), (3.2), and Lemmas 2.3 and 3.2 that

0 x τ ( n ) w S τ ( n ) x τ ( n ) w x τ ( n ) + 1 w S τ ( n ) x τ ( n ) w α τ ( n ) f τ ( n ) ( x τ ( n ) ) w 0

as n. Since { x τ ( n ) } is bounded and { S n } is strongly quasinonexpansive type, Lemma 2.6 implies that S τ ( n ) x τ ( n ) x τ ( n ) 0.

Since { S n } satisfies the condition (Z), it follows from Lemma 2.7 that every weak cluster point of { x τ ( n ) } belongs to F. Thus Lemma 2.8 shows that

lim sup n S τ ( n ) x τ ( n ) w , f 1 ( w ) w 0.

Moreover, Lemmas 2.3 and 3.3 imply that α τ ( n ) 2 f τ ( n ) ( x τ ( n ) ) w 2 / β τ ( n ) 0 and 2 α τ ( n ) (1 α τ ( n ) )/ β τ ( n ) 1/(1θ). Therefore, we obtain

lim sup n γ τ ( n ) β τ ( n ) 0.
(3.8)

On the other hand, from (3.3) and (3.6), we know that

x τ ( n ) + 1 w 2 ( 1 β τ ( n ) ) x τ ( n ) w 2 + γ τ ( n ) ( 1 β τ ( n ) ) x τ ( n ) + 1 w 2 + γ τ ( n )

for every nN. Thus, by β τ ( n ) >0, this shows that

x τ ( n ) + 1 w 2 γ τ ( n ) β τ ( n )
(3.9)

for every nN.

Finally, we obtain a contradiction that x n w0. Using (3.7), (3.9), and (3.8), we conclude that

lim sup n x n w 2 lim sup n x τ ( n ) + 1 w 2 lim sup n γ τ ( n ) β τ ( n ) 0,

and hence x n w0, which is a contradiction. □

Proof of Theorem 3.1 We first show that S n x n x n 0. Since S n is quasinonexpansive, it follows from (3.2) that

0 x n w S n x n w x n w x n + 1 w+ α n f n ( x n ) w

for every nN, so that x n w S n x n w0 by Lemma 3.4, α n 0, and Lemma 3.2. Since { S n } is strongly quasinonexpansive type and { x n } is bounded, we conclude that S n x n x n 0.

Since { S n } satisfies the condition (Z), Lemma 2.8 implies that

lim sup n S n x n w , f 1 ( w ) w 0.

This shows that lim sup n γ n / β n 0 by using Lemmas 3.2 and 3.3. On the other hand, it follows from (3.3) that

x n + 1 w 2 (1 β n ) x n w 2 + β n γ n β n

for every nN. Therefore, noting that n = 1 β n = and using Lemma 2.9, we conclude that x n w0. □

A direct consequence of Theorem 3.1 is the following corollary, which is a slight generalization of [[8], Theorem 2.3].

Corollary 3.5 Let H be a real Hilbert space, C a nonempty closed convex subset of H, S:CC a strongly quasinonexpansive mapping, { α n } a sequence in (0,1] such that α n 0 and n = 1 α n =, and f:CC a θ-contraction with respect to F=Fix(S), where 0θ<1. Let { x n } be a sequence defined by x 1 C and

x n + 1 = α n f( x n )+(1 α n )S x n
(3.10)

for nN. Suppose that IS is demiclosed at 0. Then { x n } converges strongly to wF, where w is the unique fixed point of a contraction P F f.

Proof Set S n =S and f n =f for nN. Then it is clear that n = 1 Fix( S n )=Fix(S), { S n } is strongly quasinonexpansive type, { S n } satisfies the condition (Z), and { f n } is stable on C. Thus Theorem 3.1 implies the conclusion. □

4 Application to a variational inequality problem

In this section, applying Theorem 3.1, we study an approximation method for the following variational inequality problem.

Problem 4.1 Let κ and η be positive real numbers such that η 2 <2κ. Let F be a nonempty closed convex subset of H and A:HH a κ-strongly monotone and η-Lipschitz continuous mapping, that is, we assume that xy,AxAyκ x y 2 and AxAyηxy for all x,yH. Then find zF such that

yz,Az0for all yF.

The solution set of Problem 4.1 is denoted by VI(F,A). Under the assumptions of Problem 4.1, it is known that the following hold; see, for example, [22].

  • κη, 012κ+ η 2 <1 and IA is a θ-contraction, where θ= 1 2 κ + η 2 ;

  • Problem 4.1 has a unique solution and VI(F,A)=Fix( P F (IA)).

Remark 4.2 The assumption that η 2 <2κ in Problem 4.1 is not restrictive. Indeed, let F be a nonempty closed convex subset of H and A ˜ a κ ˜ -strongly monotone and η ˜ -Lipschitz continuous mapping, where κ ˜ >0 and η ˜ >0. Set A=μ A ˜ , κ=μ κ ˜ , and η=μ η ˜ , where μ is a positive constant such that μ η ˜ 2 <2 κ ˜ . Then it is easy to verify that A is κ-strongly monotone and η-Lipschitz continuous, η 2 <2κ, and moreover, VI(F,A)=VI(F, A ˜ ).

Using Theorem 3.1, we obtain the following convergence theorem for Problem 4.1.

Theorem 4.3 Let H, κ, η, and A be the same as in Problem  4.1. Let { S n } be a sequence of mappings of H into H such that F= n = 1 Fix( S n ) is nonempty, and { α n } the same as in Theorem  3.1. Let { x n } be a sequence defined by x 1 H and

x n + 1 = S n x n α n A S n x n
(4.1)

for nN. Suppose that { S n } is strongly quasinonexpansive type and { S n } satisfies the condition (Z). Then { x n } converges strongly to the unique solution of Problem  4.1.

Proof Set f n =(IA) S n for nN and θ= 1 2 κ + η 2 . Since IA is a θ-contraction and S n is quasinonexpansive, Lemma 2.2 implies that each f n is a θ-contraction with respect to F. It is obvious that { f n } is stable on F. Moreover, it follows from (4.1) that

x n + 1 = α n f n ( x n )+(1 α n ) S n x n

for nN. Thus Theorem 3.1 implies that { x n } converges strongly to w=( P F f 1 )(w)= P F (IA)w, which is the unique solution of Problem 4.1. □

Remark 4.4 The iteration (4.1) is called the hybrid steepest descent method; see [9, 10] for more details.

We finally construct an example of { S n } in Theorem 4.3 by using the notion of a subgradient projection.

Let g:HR be a continuous and convex function such that

C= { x H : g ( x ) 0 }

is nonempty and h:HH a mapping such that h(x)g(x) for all xH, where ∂g denotes the subdifferential mapping of g defined by

g(x)= { z H : g ( x ) + y x , z g ( y ) ( y H ) }

for all xH. Then the subgradient projection P g , h :HH with respect to g and h is defined by P g , h x= P L ( x ) x for all xH, where P L ( x ) denotes the metric projection of H onto the set L(x) defined by

L(x)= { y H : g ( x ) + y x , h ( x ) 0 }

for all xH. Note that C is a subset of L(x) for all xH and that L(x) is a closed half space for all xHC. According to [[23], Section 7], [[24], Proposition 2.3], and [[25], Proposition 1.1.11], we know the following:

(S1) Fix( P g , h )=C;

(S2) z P g , h x,x P g , h x0 for all zC and xH;

(S3) if g(V) is bounded for each bounded subset V of H, then I P g , h is demiclosed at 0.

It is known that the metric projection P D of H onto a nonempty closed convex subset D of H coincides with the subgradient projection P g , h with respect to g and h defined by g(x)= inf y D xy for all xH and

h(x)={ 0 ( x D ) ; ( x P D x ) / x P D x ( x H D ) .

The subgradient projection is not necessarily nonexpansive. In fact, if g:RR and h:RR are defined by g(x)=max{x,2x1} for all xR and h(x)=1 if x<1; h(x)=2 if x1, then P g , h is given by

P g , h (x)={ x ( x 0 ) ; 0 ( 0 < x < 1 ) ; 1 / 2 ( x 1 )

and is not nonexpansive.

Using (S1), (S2), and (S3), we show the following.

Example 4.5 Let g:HR be a continuous and convex function such that C={xH:g(x)0} is nonempty and g(V) is bounded for each bounded subset V of H, h:HH a mapping such that h(x)g(x) for all xH, and { S n } a sequence of mappings of H into H defined by

S n = β n I+(1 β n ) P g , h

for all nN, where { β n } is a sequence of real numbers such that 1< inf n β n and sup n β n <1. Then the following hold:

  1. (i)

    Fix( S n )=C for all nN;

  2. (ii)

    { S n } is strongly quasinonexpansive type;

  3. (iii)

    { S n } satisfies the condition (Z).

Proof Since β n 1 for all nN, the part (i) obviously follows from (S1).

We first show (ii). By (i), we know that n = 1 Fix( S n )=C is nonempty. Let nN, pC, and xH be given. Then we have

S n x p 2 + x S n x 2 x p 2 = 2 S n x x , S n x p = 2 ( 1 β n ) p S n x , x P g , h x .
(4.2)

It follows from (S2) that

p S n x,x P g , h x P g , h x S n x,x P g , h x.
(4.3)

On the other hand, we also know that

P g , h x S n x , x P g , h x = P g , h x x 2 + x S n x , x P g , h x ( P g , h x x 1 2 x S n x ) 2 + 1 4 x S n x 2 1 4 x S n x 2 .
(4.4)

By (4.2), (4.3), and (4.4), each S n satisfies

S n x p 2 + 1 2 (1+ β n ) x S n x 2 x p 2
(4.5)

for all pC and xH. Since (1+ β n )/2>0, we know that each S n is quasinonexpansive.

Let { x n } be a bounded sequence in H such that x n p S n x n p0 for some pC. Since { S n x n } is bounded, it follows from (4.5) that

1 2 (1+ β n ) x n S n x n 2 x n p 2 S n x n p 2 0

and hence S n x n x n 0 by inf n (1+ β n )>0. Thus { S n } is strongly quasinonexpansive type.

We finally show (iii). Let { y n } be a bounded sequence in H such that S n y n y n 0. By the definition of S n , we have

P g , h y n y n = 1 1 β n S n y n y n

for all nN. Since inf n (1 β n )>0, we obtain P g , h y n y n 0. Consequently, by (S1) and (S3), we know that { S n } satisfies the condition (Z). □

References

  1. 1.

    Moudafi A: Viscosity approximation methods for fixed-points problems. J. Math. Anal. Appl. 2000, 241: 46–55. 10.1006/jmaa.1999.6615

  2. 2.

    Aoyama K, Kimura Y, Kohsaka F: Strong convergence theorems for strongly relatively nonexpansive sequences and applications. J. Nonlinear Anal. Optim. 2012, 3: 67–77.

  3. 3.

    Xu H-K: Viscosity approximation methods for nonexpansive mappings. J. Math. Anal. Appl. 2004, 298: 279–291. 10.1016/j.jmaa.2004.04.059

  4. 4.

    Suzuki T: Moudafi’s viscosity approximations with Meir-Keeler contractions. J. Math. Anal. Appl. 2007, 325: 342–352. 10.1016/j.jmaa.2006.01.080

  5. 5.

    Aoyama, K, Kimura, Y: Viscosity approximation methods with a sequence of contractions. CUBO (to appear)

  6. 6.

    Halpern B: Fixed points of nonexpanding maps. Bull. Am. Math. Soc. 1967, 73: 957–961. 10.1090/S0002-9904-1967-11864-0

  7. 7.

    Maingé P-E: The viscosity approximation process for quasi-nonexpansive mappings in Hilbert spaces. Comput. Math. Appl. 2010, 59: 74–79. 10.1016/j.camwa.2009.09.003

  8. 8.

    Wongchan K, Saejung S: On the strong convergence of viscosity approximation process for quasinonexpansive mappings in Hilbert spaces. Abstr. Appl. Anal. 2011., 2011: Article ID 385843

  9. 9.

    Yamada I, Ogura N: Hybrid steepest descent method for variational inequality problem over the fixed point set of certain quasi-nonexpansive mappings. Numer. Funct. Anal. Optim. 2004, 25: 619–655.

  10. 10.

    Cegielski A, Zalas R: Methods for variational inequality problem over the intersection of fixed point sets of quasi-nonexpansive operators. Numer. Funct. Anal. Optim. 2013, 34: 255–283. 10.1080/01630563.2012.716807

  11. 11.

    Dotson WG Jr.: Fixed points of quasi-nonexpansive mappings. J. Aust. Math. Soc. 1972, 13: 167–170. 10.1017/S144678870001123X

  12. 12.

    Takahashi W: Introduction to Nonlinear and Convex Analysis. Yokohama Publishers, Yokohama; 2009.

  13. 13.

    Maingé P-E: Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization. Set-Valued Anal. 2008, 16: 899–912. 10.1007/s11228-008-0102-z

  14. 14.

    Aoyama K: Asymptotic fixed points of sequences of quasi-nonexpansive type mappings. In Banach and Function Spaces III. Yokohama Publishers, Yokohama; 2011:343–350.

  15. 15.

    Aoyama K, Kimura Y: Strong convergence theorems for strongly nonexpansive sequences. Appl. Math. Comput. 2011, 217: 7537–7545. 10.1016/j.amc.2011.01.092

  16. 16.

    Aoyama K, Kohsaka F, Takahashi W: Strong convergence theorems by shrinking and hybrid projection methods for relatively nonexpansive mappings in Banach spaces. In Nonlinear Analysis and Convex Analysis. Yokohama Publishers, Yokohama; 2009:7–26.

  17. 17.

    Aoyama K, Kohsaka F, Takahashi W: Strongly relatively nonexpansive sequences in Banach spaces and applications. J. Fixed Point Theory Appl. 2009, 5: 201–224. 10.1007/s11784-009-0108-7

  18. 18.

    Aoyama K, Kimura Y, Takahashi W, Toyoda M: On a strongly nonexpansive sequence in Hilbert spaces. J. Nonlinear Convex Anal. 2007, 8: 471–489.

  19. 19.

    Aoyama K, Kimura Y, Takahashi W, Toyoda M: Strongly nonexpansive sequences and their applications in Banach spaces. In Fixed Point Theory and Its Applications. Yokohama Publishers, Yokohama; 2008:1–18.

  20. 20.

    Xu H-K: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. (2) 2002, 66: 240–256. 10.1112/S0024610702003332

  21. 21.

    Aoyama K, Kimura Y, Takahashi W, Toyoda M: Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space. Nonlinear Anal. 2007, 67: 2350–2360. 10.1016/j.na.2006.08.032

  22. 22.

    arXiv: http://arxiv.org/abs/1101.0881

  23. 23.

    Bauschke HH, Borwein JM: On projection algorithms for solving convex feasibility problems. SIAM Rev. 1996, 38: 367–426. 10.1137/S0036144593251710

  24. 24.

    Bauschke HH, Combettes PL: A weak-to-strong convergence principle for Fejér-monotone methods in Hilbert spaces. Math. Oper. Res. 2001, 26: 248–264. 10.1287/moor.26.2.248.10558

  25. 25.

    Butnariu D, Iusem AN Applied Optimization 40. In Totally Convex Functions for Fixed Points Computation and Infinite Dimensional Optimization. Kluwer Academic, Dordrecht; 2000.

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Acknowledgements

This paper is dedicated to Professor Wataru Takahashi on the occasion of his 70th birthday.

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Correspondence to Fumiaki Kohsaka.

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The authors declare that they have no competing interests.

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All authors read and approved the final manuscript.

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Aoyama, K., Kohsaka, F. Viscosity approximation process for a sequence of quasinonexpansive mappings. Fixed Point Theory Appl 2014, 17 (2014) doi:10.1186/1687-1812-2014-17

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Keywords

  • viscosity approximation method
  • quasinonexpansive mapping
  • fixed point
  • hybrid steepest descent method