Open Access

Viscosity approximation process for a sequence of quasinonexpansive mappings

Fixed Point Theory and Applications20142014:17

https://doi.org/10.1186/1687-1812-2014-17

Received: 27 September 2013

Accepted: 18 December 2013

Published: 22 January 2014

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.

Keywords

viscosity approximation methodquasinonexpansive mappingfixed pointhybrid steepest descent method

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 n N , 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 x H is denoted by x n x and weak convergence by x n x .

Let T : C H 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 T x p x p for all x C and p Fix ( T ) ; T is said to be nonexpansive if T x T y x y for all x , y C ; 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 p T x n p 0 for some point p Fix ( T ) ; T is demiclosed at 0 if T p = 0 whenever { x n } is a sequence in C such that x n p and T x n 0 . We know that if T : C H is quasinonexpansive, then Fix ( T ) is closed and convex; see [[11], Theorem 1].

It is known that, for each x H , 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 : C C 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 ) θ x z for all x C and z F ; 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 : C C a θ-contraction with respect to F, where 0 θ < 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 : C C be a θ-contraction, where 0 θ < 1 and T : C C a quasinonexpansive mapping. Then f T 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 ) : n N } is a singleton for every z D . It is clear that if { f n } is stable on D, then f n ( z ) = f 1 ( z ) for all n N and z D .

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

Lemma 2.3 Let τ : N N 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 N N and an eventually increasing function τ : N N such that ξ τ ( n ) ξ τ ( n ) + 1 for all n N and ξ n ξ τ ( n ) + 1 for all n N .

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 p 0

for some point p F ;

  • { 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 : N N an eventually increasing function, and { z n } a bounded sequence in C such that z n p T τ ( n ) z n p 0 for some p F . 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 : N N 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 u H . Suppose that every weak cluster point of { z n } belongs to F. Then
lim sup n T n z n w , u w 0 ,

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 n N , 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 0 θ < 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 n N . Suppose that { S n } is strongly quasinonexpansive type and satisfies the condition (Z). Then { x n } converges strongly to w F , 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 n N .

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 n N .

Proof Since f n is a θ-contraction with respect to F, S n is quasinonexpansive, w F Fix ( 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 n N . Thus, by induction on n, we have
S n x n w x n w max { 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 n N . Therefore, (3.3) holds. □

Lemma 3.3 The following hold:

  • 0 < β n 1 for every n N ;

  • 2 α n ( 1 α n ) / β n 1 / ( 1 θ ) ;

  • α n 2 f n ( x n ) w 2 / β n 0 ;

  • n = 1 β n = .

Proof Since 0 < α n 1 and 1 < 1 2 θ 1 , 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 / ( 1 θ ) .

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 1 2 θ 0 . Then it is clear that β n α n for every n N . Thus, n = 1 β n n = 1 α n = . Next, we suppose that 1 2 θ < 0 . Then it is clear that β n > 2 ( 1 θ ) α n for every n N . Thus, n = 1 β n 2 ( 1 θ ) 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 N N and an eventually increasing function τ : N N such that
x τ ( n ) w x τ ( n ) + 1 w
(3.6)
for every n N and
x n w x τ ( n ) + 1 w
(3.7)

for every n N .

We show that S τ ( n ) x τ ( n ) x τ ( n ) 0 . Since S τ ( n ) is quasinonexpansive and w F Fix ( 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 / ( 1 θ ) . 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 n N . 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 n N .

Finally, we obtain a contradiction that x n w 0 . 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 w 0 , 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 n N , so that x n w S n x n w 0 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 n N . Therefore, noting that n = 1 β n = and using Lemma 2.9, we conclude that x n w 0 . □

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 : C C a strongly quasinonexpansive mapping, { α n } a sequence in ( 0 , 1 ] such that α n 0 and n = 1 α n = , and f : C C a θ-contraction with respect to F = Fix ( S ) , where 0 θ < 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 n N . Suppose that I S is demiclosed at 0. Then { x n } converges strongly to w F , where w is the unique fixed point of a contraction P F f .

Proof Set S n = S and f n = f for n N . 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 < 2 κ . Let F be a nonempty closed convex subset of H and A : H H a κ-strongly monotone and η-Lipschitz continuous mapping, that is, we assume that x y , A x A y κ x y 2 and A x A y η x y for all x , y H . Then find z F such that
y z , A z 0 for all  y F .

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].

  • κ η , 0 1 2 κ + η 2 < 1 and I A 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 ( I A ) ) .

Remark 4.2 The assumption that η 2 < 2 κ 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 < 2 κ , 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 n N . 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 = ( I A ) S n for n N and θ = 1 2 κ + η 2 . Since I A 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 n N . Thus Theorem 3.1 implies that { x n } converges strongly to w = ( P F f 1 ) ( w ) = P F ( I A ) 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 : H R be a continuous and convex function such that
C = { x H : g ( x ) 0 }
is nonempty and h : H H a mapping such that h ( x ) g ( x ) for all x H , 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 x H . Then the subgradient projection P g , h : H H with respect to g and h is defined by P g , h x = P L ( x ) x for all x H , 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 x H . Note that C is a subset of L ( x ) for all x H and that L ( x ) is a closed half space for all x H C . 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 x 0 for all z C and x H ;

(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 x y for all x H 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 : R R and h : R R are defined by g ( x ) = max { x , 2 x 1 } for all x R and h ( x ) = 1 if x < 1 ; h ( x ) = 2 if x 1 , 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 : H R be a continuous and convex function such that C = { x H : g ( x ) 0 } is nonempty and g ( V ) is bounded for each bounded subset V of H, h : H H a mapping such that h ( x ) g ( x ) for all x H , 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 n N , 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 n N ;

     
  2. (ii)

    { S n } is strongly quasinonexpansive type;

     
  3. (iii)

    { S n } satisfies the condition (Z).

     

Proof Since β n 1 for all n N , 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 n N , p C , and x H 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 p C and x H . 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 p 0 for some p C . 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 n N . 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). □

Declarations

Acknowledgements

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

Authors’ Affiliations

(1)
Department of Economics, Chiba University
(2)
Department of Computer Science and Intelligent Systems, Oita University

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© Aoyama and Kohsaka; licensee Springer. 2014

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