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A new explicit iterative algorithm for solving a class of variational inequalities over the common fixed points set of a finite family of nonexpansive mappings

Abstract

In this paper, we introduce a new explicit iterative algorithm for finding a solution for a class of variational inequalities over the common fixed points set of a finite family of nonexpansive mappings in Hilbert spaces. Under suitable assumptions, we prove that the sequence generated by the iterative algorithm converges strongly to the unique solution of the variational inequality. Our result improves and extends the corresponding results announced by many others. At the end of the paper, we extend our result to the more broad family of λ-strictly pseudo-contractive mappings.

1 Introduction

Let H be a real Hilbert space with inner product , and norm . Throughout this paper, we always assume that T is a nonexpansive operator on H. The fixed point set of T is denoted by Fix(T), i.e., Fix(T)={xH:Tx=x}. The typical problem is to minimize a quadratic function on a real Hilbert space H:

min x C 1 2 Ax,xx,u,
(1.1)

where C is a nonempty closed convex subset of H, u is a given point in H and A is a strongly positive bounded linear operator on H.

In 2003, Xu [1] introduced the following iterative scheme:

x n + 1 = α n u+(I α n A)T x n ,
(1.2)

where u is some point of H and { α n } is a sequence in (0,1). He proved that the sequence { x n } converges strongly to the unique solution of the minimization problem (1.1) with C=Fix(T).

In 2006, Marino and Xu [2] considered the viscosity method on the iterative scheme (1.2), and they gave the following general iterative method:

x n + 1 = α n γf( x n )+(I α n A)T x n ,
(1.3)

where f is a contraction on H. They proved the above sequence { x n } converges strongly to the unique solution of the variational inequality

( A γ f ) x , x x 0,xFix(T),

which is the optimality condition for the minimization problem

min x Fix ( T ) 1 2 Ax,xh(x),

where h is a potential function for γf (i.e., h (x)=γf(x) for xH).

In 2001, Yamada [3] considered the following hybrid iterative method:

x n + 1 =T x n μ λ n F(T x n ),
(1.4)

where F is L-Lipschitzian continuous and η-strongly monotone operator with L>0, η>0 and 0<μ<2η/ L 2 . Under some appropriate conditions, the sequence { x n } generated by (1.4) converges strongly to the unique solution of the variational inequality

F x , x x 0,xFix(T).

Combining (1.3) and (1.4), Tian [4] considered the following general viscosity type iterative method:

x n + 1 = α n γf( x n )+(Iμ α n F)T x n .
(1.5)

Improving and extending the corresponding results given by Marino et al., he proved that the sequence { x n } generated by (1.5) converges strongly to the unique solution x Fix(T) of the variational inequality

( γ f μ F ) x ˜ , x x ˜ 0,xFix(T).

In [5], Tian generalized the iterative method (1.5) replacing the contraction operator f with a Lipschitzian continuous operator V to solve the following variational inequality:

( γ V μ F ) x ˜ , x x ˜ 0,xFix(T).
(1.6)

On the other hand, let { T i } i = 1 N be a finite family of nonexpansive self-mappings of H. Assume i = 1 N Fix( T i ). In [1], Xu also defined the following sequence { x n }:

x n + 1 = α n u+(I α n A) T n + 1 x n ,n0,
(1.7)

where T n = T n mod N and the mod function takes values in {1,2,,N}. He found that the sequence { x n } generated by (1.7) converges strongly to the unique solution of the minimization problem (1.1) with C= i = 1 N Fix( T i ) under suitable conditions on { α n } and the following additional condition on { T n }:

F( T N T 2 T 1 )=F( T 1 T N T 3 T 2 )==F( T N 1 T 1 T N ).
(1.8)

In fact, there are many nonexpansive mappings which do not satisfy (1.8).

In 1999, Atsushiba and Takahashi [6] defined the W n -mappings generated by T 1 , T 2 ,, T N and { γ n , 1 },{ γ n , 2 },,{ γ n , N }[0,1] as follows:

U n , 0 = I , U n , 1 = γ n , 1 T 1 U n , 0 + ( 1 γ n , 1 ) I , U n , 2 = γ n , 2 T 2 U n , 1 + ( 1 γ n , 2 ) I , U n , N 1 = γ n , N 1 T N 1 U n , N 2 + ( 1 γ n , N 1 ) I , W n = U n , N = γ n , N T N U n , N 1 + ( 1 γ n , N ) I .

From [[6], Lemma 3.1], we know that F( W n )= i = 1 N F( T i ).

In 2006, Yao [7] introduced the following iterative method:

x n + 1 = α n γf( x n )+β x n + ( ( 1 β ) I α n A ) W n x n .
(1.9)

Without the condition (1.8), he proved that the sequence { x n } generated by (1.9) converges strongly to the unique solution of the following variational inequality:

( A γ f ) x , x x 0,x i = 1 N Fix( T i ),
(1.10)

which is the optimality condition for the minimization problem

min x C 1 2 Ax,xh(x),
(1.11)

where C= i = 1 N Fix( T i ) and h is a potential function for γf (i.e., h (x)=γf(x)).

Shang et al. [8] introduced the following scheme:

{ y n = β n x n + ( 1 β n ) W n x n , x n + 1 = α n γ f ( x n ) + ( I α n A ) y n .
(1.12)

Under certain appropriate conditions, without (1.8), they proved that { x n } defined by (1.12) converges strongly to the unique solution of (1.10) which is also the optimality condition for (1.11).

Recently, combining the Krasnoselskii-Mann type algorithm and the steepest-descent method, Buong and Duong [9] introduced a new explicit iterative algorithm:

x k + 1 = ( 1 β k 0 ) x k + β k 0 T 0 k T N k T 1 k x k ,
(1.13)

where T i k =(1 β k i )I+ β k i T i for i=1,2,,N, T 0 k =I λ k μF, and F is an L-Lipschitz continuous and η-strongly monotone mapping. Under some appropriate assumptions, they proved that the sequence { x k } converges strongly to the unique solution of the following variational inequality:

F ( x ) , x x 0,x i = 1 N Fix( T i ).
(1.14)

Very recently, Zhou and Wang [10] proposed a simpler iterative algorithm than the iterative algorithm (1.13) given by Buong and Duong:

x k + 1 =(I λ k μF) T N k T 1 k x k .
(1.15)

They proved that the sequence { x k } defined by (1.15) converges strongly to the unique solution of the variational inequality (1.14) in a faster rate of convergence.

Motivated and inspired by the results of Zhou et al., in this paper, we consider a new iterative algorithm to solve the class of variational inequalities (1.6). The iterative algorithm improves and extends the results of Yao et al., and the corresponding results announced by many others. At the end of this paper, we extend our iterative algorithm to the more broad family of λ-strictly pseudo-contractive mappings.

2 Preliminaries

Throughout this paper, we write x n x and x n x to indicate that { x n } converges weakly to x and converges strongly to x, respectively.

An operator T:HH is said to be nonexpansive if TxTyxy for all x,yH. It is well known that Fix(T) is closed and convex. A is called strongly positive if there exists a constant γ>0 such that Ax,xγ x 2 for all xH. The operator F is called η-strongly monotone if there exists a constant η>0 such that

xy,FxFyη x y 2

for all x,yH.

In order to prove our results, we collect some necessary conceptions and lemmas in this section.

Definition 2.1 A mapping T:HH is said to be an averaged mapping if there exists some number α(0,1) such that

T=(1α)I+αS,
(2.1)

where I:HH is the identity mapping and S:HH is nonexpansive. More precisely, when (2.1) holds, we say that T is α-averaged.

Lemma 2.1 ([11])

  1. (i)

    The composite of finitely many averaged mappings is averaged. That is, if each of the mappings { T i } i = 1 N is averaged, then so is the composite T 1 T N . In particular, if T 1 is α 1 -averaged and T 2 is α 2 -averaged, where α 1 , α 2 (0,1), then both T 1 T 2 and T 2 T 1 are α-averaged, where α= α 1 + α 2 α 1 α 2 .

  2. (ii)

    If the mappings { T i } i = 1 N are averaged and have a common fixed point, then

    i = 1 N Fix( T i )=Fix( T 1 T N ).

In particular, if N=2, we have Fix( T 1 )Fix( T 2 )=Fix( T 1 T 2 )=Fix( T 2 T 1 ).

Lemma 2.2 ([12])

Let C be a closed convex subset of a real Hilbert space H. Given xH and yC. Then y= P C x if and only if the following inequality holds:

xy,zy0

for every zC.

Lemma 2.3 ([5])

Assume V is a contraction on a Hilbert space H with coefficient α>0, and F:HH is an L-Lipschitzian continuous and η-strongly monotone operator with L>0, η>0. Then, for 0<γ< μ η α , μFγV is strongly monotone with coefficient μηγα.

Lemma 2.4 ([13])

Let H be a Hilbert space, C a closed convex subset of H, and T:CC a nonexpansive mapping with Fix(T). If { x n } is a sequence in C weakly converging to xC and {(IT) x n } converges strongly to yC, then (IT)x=y. In particular, if y=0, then xFix(T).

Lemma 2.5 ([14])

Let { x n } and { z n } be bounded sequences in a Banach space X and { β n } be a sequence in [0,1] which satisfies the following condition:

0< lim inf n β n lim sup n β n <1.

Suppose x n + 1 =(1 β n ) z n + β n x n for all integers n0 and

lim sup n ( z n + 1 z n x n + 1 x n ) 0.

Then lim n z n x n =0.

Lemma 2.6 ([1])

Assume { a n } is a sequence of nonnegative real numbers such that

a n + 1 (1 γ n ) a n + δ n ,n0,

where { γ n } is a sequence in (0,1) and { δ n } is a sequence such that

  1. (i)

    n = 1 γ n =,

  2. (ii)

    lim sup n δ n γ n 0 or n = 1 | δ n |<.

Then lim n a n =0.

Lemma 2.7 ([15])

Assume S is a λ-strictly pseudo-contractive mapping on a Hilbert space H. Define a mapping T by Tx=αx+(1α)Sx for all xH and α[λ,1). Then T is a nonexpansive mapping such that Fix(T)=Fix(S).

3 Main results

Now we state and prove our main results in this paper.

Theorem 3.1 Let { T i } i = 1 N be N nonexpansive mappings of a real Hilbert space H such that C= i = 1 N Fix( T i ), F be an L-Lipschitzian continuous and η-strongly monotone operator on H with L>0 and η>0, V be an α-Lipschitzian on H with α>0. Suppose x 1 H and 0<μ< 2 η L 2 . Define a sequence { x k } as follows:

x k + 1 = α k γV( x k )+(Iμ α k F) T N k T N 1 k T 1 k x k ,k0,
(3.1)

where 0<γ< τ α with τ=μ(η 1 2 μ L 2 ) and T i k =(1 β k i )I+ β k i T i for i=1,2,,N. Suppose α k (0,1) and β k i (ξ,ζ) for some ξ,ζ(0,1). If the following conditions are satisfied:

  1. (i)

    lim k α k =0;

  2. (ii)

    k = 1 α k =;

  3. (iii)

    lim k | β k + 1 i β k i |=0 for i=1,2,,N.

Then the sequence { x k } converges strongly to the unique solution x of the variational inequality:

( μ F γ V ) x , x x 0,x i = 1 N Fix( T i ).
(3.2)

Equivalently, we have P C (IμF+γV) x = x .

Proof Since our methods easily deduce the general case, we prove Theorem 3.1 for N=2.

First, we show { x k } is bounded. In fact, for some point pC, by (3.1) we have

x k + 1 p = α k γ V x k + ( I μ α k F ) T 2 k T 1 k x k p = ( I μ α k F ) T 2 k T 1 k x k ( I μ α k F ) p + α k ( γ V x k μ F p ) ( 1 α k τ ) T 2 k T 1 k x k T 2 k T 1 k p + α k ( γ V x k γ V p + γ V p μ F p ) ( 1 α k τ ) x k p + α k γ α x k p + α k γ V p μ F p = ( 1 α k ( τ γ α ) ) x k p + α k ( τ γ α ) γ V p μ F p τ γ α max { x k p , 1 τ γ α γ V p μ F p } max { x 0 p , 1 τ γ α γ V p μ F p } .

Therefore, { x k } is bounded. Hence we also see that { T 2 k T 1 k x k }, {F T 2 k T 1 k x k }, and {V x k } are all bounded. From (3.1), it follows that

lim k x k + 1 T 2 k T 1 k x k =0.
(3.3)

We next show that lim k x k + 1 x k =0. Noting that T 1 k and T 2 k are β k 1 -averaged and β k 2 -averaged, respectively, by Lemma 2.1, we find that T 2 k T 1 k is t k -averaged for every k, where t k = β k 1 + β k 2 β k 1 β k 2 . Set ξ =2ξ ξ 2 and ζ =2ζ ζ 2 . It is easy to deduce that 0< ξ t k ζ <1 for all k and

lim k t k + 1 t k =0.
(3.4)

Since for every k, T 2 k T 1 k is t k -averaged, we can find a family of nonexpansive mappings { S k } k 0 on H such that

T 2 k T 1 k =(1 t k )I+ t k S k ,k0.
(3.5)

Substituting (3.4) into (3.1) yields

x k + 1 = α k γ V x k + ( I μ α k F ) [ ( 1 t k ) x k + t k S k x k ] = ( 1 t k ) x k + t k [ S k x k + α k t k ( γ V x k μ F T 2 k T 1 k x k ) ] .

Define a sequence { z k } by z k = S k x k + α k t k (γV x k μF T 2 k T 1 k x k ), so

x k + 1 =(1 t k ) x k + t k z k .
(3.6)

Now, we claim that

lim sup k ( z k + 1 z k x k + 1 x k ) 0.

To this end, we observe that

z k + 1 z k S k + 1 x k + 1 S k x k + α k + 1 t k + 1 γ V x k + 1 μ F T 2 k + 1 T 1 k + 1 x k + 1 + α k t k γ V x k μ F T 2 k T 1 k x k S k + 1 x k + 1 S k + 1 x k + S k + 1 x k S k x k + α k + 1 t k + 1 ( γ V x k + 1 + μ F T 2 k + 1 T 1 k + 1 x k + 1 ) + α k t k ( γ V x k + μ F T 2 k T 1 k x k ) x k + 1 x k + S k + 1 x k S k x k + α k + 1 t k + 1 ( γ V x k + 1 + μ F T 2 k + 1 T 1 k + 1 x k + 1 ) + α k t k ( γ V x k + μ F T 2 k T 1 k x k )
(3.7)

and

S k + 1 x k S k x k = 1 t k + 1 T 2 k + 1 T 1 k + 1 x k 1 t k T 2 k T 1 k x k 1 t k + 1 t k + 1 x k + 1 t k t k x k | t k + 1 t k t k + 1 t k | ( T 2 k + 1 T 1 k + 1 x k + x k ) + 1 t k T 2 k + 1 T 1 k + 1 x k T 2 k T 1 k x k | t k + 1 t k t k + 1 t k | M + 1 t k ( T 2 k + 1 T 1 k + 1 x k T 2 k + 1 T 1 k x k + T 2 k + 1 T 1 k x k T 2 k T 1 k x k ) | t k + 1 t k t k + 1 t k | M + 1 ξ ( T 1 k + 1 x k T 1 k x k + T 2 k + 1 T 1 k x k T 2 k T 1 k x k ) ,
(3.8)

where M is a fixed constant satisfying

M sup k 0 { T 2 k + 1 T 1 k + 1 x k + x k } .

Note that

T 1 k + 1 x k T 1 k x k = ( 1 β k + 1 1 ) x k + β k + 1 1 T 1 x k ( 1 β k 1 ) x k β k 1 T 1 x k | β k + 1 1 β k 1 | ( x k + T 1 x k ) .

Since lim k | β k + 1 i β k i |=0 for i=1,2, and { x k } and { T 1 x k } are bounded, we easily obtain

lim k T 1 k + 1 x k T 1 k x k =0.
(3.9)

Similarly,

T 2 k + 1 T 1 k x k T 2 k T 1 k x k | β k + 1 2 β k 2 | ( T 1 k x k + T 2 T 1 k x k ) ,

from which it follows that

lim k T 2 k + 1 T 1 k x k T 2 k T 1 k x k =0.
(3.10)

Using (3.4), (3.9), and (3.10), from (3.8) we have

lim k S k + 1 x k S k x k =0.
(3.11)

Since lim k α k =0 and 0< ξ < t k < ζ <1, combining (3.7) and (3.11) we get

lim sup k ( z k + 1 z k x k + 1 x k ) 0.

By Lemma 2.5, we conclude that lim k z k x k =0, which implies that lim k x k + 1 x k =0 by (3.6). Thus from (3.3), it is true that

lim k x k T 2 k T 1 k x k =0.
(3.12)

From [[8], Theorem 3.2], we know that the solution of the variational inequality (3.2) is unique. We use x to denote the unique solution of (3.2). Since { x k } k 0 is bounded, there exists a subsequence { x k j } j 1 of { x k } k 0 such that x k j x ˆ as j and

lim sup k ( μ F γ V ) x , x x k = lim j ( μ F γ V ) x , x x k j .

Since { β k i } is bounded for i=1,2, we can assume that β k j i β i as j, where 0<ξ β i ζ<1 for i=1,2. Define T i =(1 β i )I+ β i T i (i=1,2). Then we have Fix( T i )=Fix( T i ) for i=1,2. Note that

T i k j x T i x | β k j i β i | ( x + T i x ) .

Hence, we deduce that

lim j sup x D T i k j x T i x =0,
(3.13)

where D is an arbitrary bounded subset of H.

Since Fix( T 1 )Fix( T 2 )=Fix( T 1 )Fix( T 2 )=C and T i is β i -averaged for i=1,2, by Lemma 2.1, we know that Fix( T 2 T 1 )=Fix( T 2 )Fix( T 1 )=C. Combining (3.12) and (3.13), we obtain

x k j T 2 T 1 x k j x k j T 2 k j T 1 k j x k j + T 2 k j T 1 k j x k j T 2 T 1 k j x k j + T 2 T 1 k j x k j T 2 T 1 x k j x k j T 2 k j T 1 k j x k j + T 2 k j T 1 k j x k j T 2 T 1 k j x k j + T 1 k j x k j T 1 x k j x k j T 2 k j T 1 k j x k j + sup x D T 2 k j x T 2 x + sup x D T 1 k j x T 1 x ,

where D is a bounded subset including { T 1 k j x k j } and D is a bounded subset including { x k j }. Hence lim j x k j T 2 T 1 x k j =0. From Lemma 2.4, we have x ˆ Fix( T 2 T 1 )=C. It follows that

lim sup k ( μ F γ V ) x , x T 2 k T 1 k x k = lim sup k ( μ F γ V ) x , x x k = lim j ( μ F γ V ) x , x x k j = ( μ F γ V ) x , x x ˆ 0 .
(3.14)

Finally, we show that x k x as k. From (3.1), we have

x k + 1 x 2 = α k γ V x k + ( I μ α k F ) T 2 k T 1 k x k x 2 = ( I μ α k F ) T 2 k T 1 k x k ( I μ α k F ) x + α k ( γ V x k μ F x ) 2 = ( I μ α k F ) T 2 k T 1 k x k ( I μ α k F ) x 2 + α k 2 γ V x k μ F x 2 + 2 α k ( I μ α k F ) T 2 k T 1 k x k ( I μ α k F ) x , γ V x k μ F x ( 1 α k τ ) 2 x k x 2 + α k 2 γ V x k μ F x 2 + 2 α k T 2 k T 1 k x k x , γ V x k μ F x 2 μ α k 2 F T 2 k T 1 k x k F x , γ V x k μ F x ( 1 α k τ ) 2 x k x 2 + α k 2 γ V x k μ F x 2 + 2 α k γ T 2 k T 1 k x k x , V x k V x + 2 α k T 2 k T 1 k x k x , γ V x μ F x 2 μ α k 2 F T 2 k T 1 k x k F x , γ V x k μ F x [ ( 1 α k τ ) 2 + 2 α α k γ ] x k x 2 + α k [ 2 T 2 k T 1 k x k x , ( γ V μ F ) x + α k γ V x k μ F x 2 + 2 α k L T 2 k T 1 k x k x γ V x k μ F x ] = [ 1 2 α k ( τ α γ ) ] x k x 2 + α k [ 2 T 2 k T 1 k x k x , ( γ V μ F ) x + α k ( γ V x k μ F x 2 + 2 L x k x γ V x k μ F x + τ 2 x k x 2 ) ] [ 1 2 α k ( τ α γ ) ] x k x 2 + α k [ 2 T 2 k T 1 k x k x , ( γ V μ F ) x + α k M ] ,

where M is a constant satisfying

M sup k 0 { γ V x k μ F x 2 + 2 L T 2 k T 1 k x k x γ V x k μ F x + τ 2 x k x 2 } .

Consequently, according to the conditions (i) and (ii), (3.14), and Lemma 2.6, we conclude that x k x as k. This completes the proof. □

4 An extension of our result

In this section, we extend our result to the more broad family of λ-strictly pseudo-contractive mappings. Now let us recall that a mapping S:HH is said to be λ-strictly pseudo-contractive if there exists a constant λ[0,1) such that

S x S y 2 x y 2 +λ ( I S ) x ( I S ) y 2 ,x,yH.

Let { S i } i = 1 N be a family of λ i -strictly pseudo-contractive self-mappings of H with 0 λ i <1. For i=1,2,,N, define

T ˆ i = ω i I+(1 ω i ) S i ,
(4.1)

where 0 λ i ω i <1. By virtue of Lemma 2.7, we know that { T ˆ i } i = 1 N is a family of nonexpansive mappings. Thus we extend Theorem 3.1 to the family of λ i -strictly pseudo-contractions.

Theorem 4.1 Let H be a real Hilbert space, F:HH be an L-Lipschitizian continuous and η-strongly monotone operator on H with L>0 and η>0, V be an α-Lipschitzian continuous on H with α>0. Let { S i } i = 1 N be N λ i -strictly pseudo-contractive mappings on H such that C= i = 1 N Fix( S i ). Suppose 0<μ< τ α , 0<γ< τ α with τ=μ(η 1 2 μ L 2 ), α k (0,1), β k i (ξ,ζ) for some ξ,ζ(0,1) and 0 λ i ω i <1 for i=1,2,,N. If the conditions (i)-(iii) of Theorem  3.1 are satisfied, the sequence { x k } k 0 defined by (3.1) with T i replaced by (4.1), converges strongly to the unique solution x of the following variational inequality:

( μ F γ V ) x , x x 0,x i = 1 N Fix( S i ).

References

  1. Xu HK: An iterative approach to quadratic optimization. J. Optim. Theory Appl. 2003, 116: 659–678. 10.1023/A:1023073621589

    Article  MathSciNet  Google Scholar 

  2. Marino G, Xu HK: A general iterative method for nonexpansive mappings in Hilbert spaces. J. Math. Anal. Appl. 2006, 318: 43–52. 10.1016/j.jmaa.2005.05.028

    Article  MathSciNet  Google Scholar 

  3. Yamada I: The hybrid steepest descent for the variational inequality problems over the intersection of fixed points sets of nonexpansive mapping. In Inherently Parallel Algorithms in Feasibility and Optimization and Their Application. Edited by: Butnariu D, Censor Y, Reich S. Elsevier, New York; 2001:473–504.

    Chapter  Google Scholar 

  4. Tian M: A general iterative algorithm for nonexpansive mappings in Hilbert spaces. Nonlinear Anal. 2010, 73: 689–694. 10.1016/j.na.2010.03.058

    Article  MathSciNet  Google Scholar 

  5. Tian M: A general iterative method based on the hybrid steepest descent scheme for nonexpansive mappings in Hilbert spaces. 2010 International Conference on Computational Intelligence and Software Engineering (CiSE) 2010.

    Google Scholar 

  6. Atsushiba S, Takahashi W: Strong convergence theorems for a finite family of nonexpansive mappings and applications. Indian J. Math. 2003, 116(3):659–678.

    MathSciNet  Google Scholar 

  7. Yao Y: A general iterative method for a finite family of nonexpansive mappings. Nonlinear Anal. 2007, 66: 2676–2687. 10.1016/j.na.2006.03.047

    Article  MathSciNet  Google Scholar 

  8. Shang M, Su Y, Qin X: Strong convergence theorems for a finite family of nonexpansive mappings. Fixed Point Theory Appl. 2007., 2007: Article ID 76971 10.1155/2007/76971

    Google Scholar 

  9. Buong N, Duong LT: An explicit iterative algorithm for a class of variational inequalities in Hilbert spaces. J. Optim. Theory Appl. 2011, 151: 513–524. 10.1007/s10957-011-9890-7

    Article  MathSciNet  Google Scholar 

  10. Zhou Y, Wang P: A simpler explicit iterative algorithm for a class of variational inequalities in Hilbert spaces. J. Optim. Theory Appl. 2013. (forthcoming) 10.1007/s10957-013-0470-x

    Google Scholar 

  11. López G, Martin V, Xu HK: Iterative algorithm for the multiple-sets split feasibility problem. Biomedical Mathematics: Promising Directions in Imaging, Therapy Planning and Inverse Problems 2009, 243–279.

    Google Scholar 

  12. Marino G, Xu HK: Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces. J. Math. Anal. Appl. 2007, 329: 336–346. 10.1016/j.jmaa.2006.06.055

    Article  MathSciNet  Google Scholar 

  13. Goebel K, Kirk WA: Topics in Metric Fixed-Point Theory. Cambridge University Press, Cambridge; 1990.

    Book  MATH  Google Scholar 

  14. Suzuki T: Strong convergence of Krasnoselskii and Mann’s type sequences for one parameter nonexpansive semigroups without Bochner integrals. J. Math. Anal. Appl. 2005, 35: 227–239.

    Article  Google Scholar 

  15. Zhou Y: Convergence theorems of fixed points for k -strict pseudo-contractions in Hilbert spaces. Nonlinear Anal. 2008, 69: 456–462. 10.1016/j.na.2007.05.032

    Article  MathSciNet  Google Scholar 

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Acknowledgements

This research is supported by the Fundamental Science Research Funds for the Central Universities (Program No. 3122013k004).

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Zhang, C., Yang, C. A new explicit iterative algorithm for solving a class of variational inequalities over the common fixed points set of a finite family of nonexpansive mappings. Fixed Point Theory Appl 2014, 60 (2014). https://doi.org/10.1186/1687-1812-2014-60

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