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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:

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 method
  • quasinonexpansive mapping
  • fixed point
  • hybrid 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, Yayoi-cho, Inage-ku, Chiba-shi, Chiba 263-8522, Japan
(2)
Department of Computer Science and Intelligent Systems, Oita University, Dannoharu, Oita-shi, Oita 870-1192, Japan

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