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Strong convergence theorems for common solutions of a family of nonexpansive mappings and an accretive operator

Abstract

In this paper, common solutions of a family of nonexpansive mappings and an accretive operator are investigated based on a viscosity iterative method. Strong convergence theorems for common solutions are established in a Banach space.

MSC:47H09, 47J05.

1 Introduction

Fixed point theory has emerged as an effective and powerful tool for studying a wide class of problems which arise in economics, finance, image reconstruction, ecology, transportation, network, elasticity and optimization; see [112] and the references therein. The computation of solutions is important in the study of many real world problems. The well-known convex feasibility problem which captures applications in various disciplines such as image restoration and radiation therapy treatment planning is to find a point in the intersection of common fixed point sets of a family of nonlinear mappings; see, for example, [1324] and the references therein.

The aim of this paper is to investigate a common solution problem of a family of nonexpansive mappings and an accretive operator based on a viscosity iterative method. The organization of this article is as follows. In Section 2, we provide some necessary preliminaries. In Section 3, a viscosity iterative method is discussed. Strong convergence theorems of common solutions are established in a reflexive and strictly convex Banach space E which enjoys weakly continuous duality mappings.

2 Preliminaries

Throughout this paper, we assume that E is a real Banach space. Let C be a nonempty, closed and convex subset of E, and let T:CC be a mapping. A point xC is a fixed point of T provided Tx=x. Denote by F(T) the set of fixed points of T; that is, F(T)={xC:Tx=x}.

Recall that T:CC is nonexpansive iff

TxTyxy,x,yC.

T:CC is a contraction iff there exists a constant α(0,1) such that

f ( x ) f ( y ) αxy,x,yC.

We use Π C to denote the collection of all contractions on C. That is, Π C :={f|f:CC a contraction}.

The Picard iterative algorithm is an efficient algorithm to study contractions. However, the Picard iterative algorithm fails to converge to fixed points of nonexpansive mappings even that their fixed point sets are not empty. One classical way to study nonexpansive mappings is to use contractions to approximate a nonexpansive mapping. More precisely, take t(0,1) and define a mapping T t :CC by

T t x=tf(u)+(1t)Tx,xC,

where uC is a fixed element and f is a contraction on C with the constant α. It is easy to see that T t is a contraction with the constant α. Indeed, we have the following:

T t x T t y t f ( x ) f ( y ) + ( 1 t ) T x T y t α x y + ( 1 t ) x y = [ 1 t ( 1 α ) ] x y .

Banach’s contraction mapping principle guarantees that T t has a unique fixed point. We denote the unique fixed point by x t . Reich [25] proved that if E is a uniformly smooth Banach space, then x t strongly converges to a fixed point of T, and the limit defines the (unique) sunny nonexpansive retraction from Π C onto F(T). Recently, Xu [26] further proved that the above results still hold in reflexive Banach spaces which have weakly continuous duality mappings.

Recall that the normal Mann iterative algorithm was introduced by Mann in 1953. Since then the construction of fixed points for nonexpansive mappings via the normal Mann iterative algorithm has been extensively investigated by many authors.

The normal Mann iterative algorithm generates a sequence { x n } in the following manner:

x 1 C, x n + 1 =(1 α n ) x n + α n T x n ,n1,

where the sequence { α n } is in the interval (0,1). If T is a nonexpansive mapping with a fixed point and the control sequence { α n } is chosen so that n = 0 α n (1 α n )=, then the sequence { x n } generated by the normal Mann iterative algorithm converges weakly to a fixed point of T (this is also valid in a uniformly convex Banach space with the Fréchet differentiable norm). Since the Mann iterative algorithm only has weak convergence in infinite dimension spaces, many authors tried to modify the normal Mann iteration algorithm to have strong convergence for nonexpansive mappings.

Kim and Xu [27] considered the following iterative algorithm.

{ x 0 C arbitrarily chosen , y n = β n x n + ( 1 β n ) T x n , x n + 1 = α n u + ( 1 α n ) y n , n 0 ,

where T is a nonexpansive mapping of C into itself, uC is a given point, { α n } and { β n } are two real number sequences in (0,1). They proved that the sequence { x n } generated by the above iterative algorithm strongly converges to a fixed point of the mapping T provided that the control sequences { α n } and { β n } satisfy appropriate conditions.

Recently, many authors have studied the following convex feasibility problem (CFP): x i = 1 N C i , where N1 is an integer, and each C i is assumed to be the fixed point set of a nonexpansive mapping T i , i=1,2,,N. There is a considerable investigation on CFP in the setting of Hilbert spaces which captures applications in various disciplines such as image restoration [28], computer tomography [29] and radiation therapy treatment planning [30].

In this paper, we consider the mapping W n defined by

U n , n + 1 = I , U n , n = γ n T n U n , n + 1 + ( 1 γ n ) I , U n , n 1 = γ n 1 T n 1 U n , n + ( 1 γ n 1 ) I , U n , k = γ k T k U n , k + 1 + ( 1 γ k ) I , U n , k 1 = γ k 1 T k 1 U n , k + ( 1 γ k 1 ) I , U n , 2 = γ 2 T 2 U n , 3 + ( 1 γ 2 ) I , W n = U n , 1 = γ 1 T 1 U n , 2 + ( 1 γ 1 ) I ,
(2.1)

where { γ 1 },{ γ 2 }, are real numbers such that 0 γ n 1 and T 1 , T 2 , are nonexpansive mappings of C into itself. Nonexpansivity of each T i ensures the nonexpansivity of  W n .

We have the following lemmas which are important to prove our main results.

Lemma 2.1 [31]

Let C be a nonempty closed convex subset of a strictly convex Banach space E. Let T 1 , T 2 , be nonexpansive mappings of C into itself such that n = 1 F( T n ) and let γ 1 , γ 2 , be real numbers such that 0< γ n b<1, where b is some real number, for any n1. Then, for every xC and kN, the limit lim n U n , k x exists.

Using Lemma 2.1, one can define the mapping W of C into itself as follows.

Wx= lim n W n x= lim n U n , 1 x,xC.
(2.2)

Such a mapping W is called the W-mapping generated by T 1 , T 2 , and γ 1 , γ 2 , . Throughout this paper, we will assume that 0< γ n b<1 for all n1.

Lemma 2.2 [31]

Let C be a nonempty closed convex subset of a strictly convex Banach space E. Let T 1 , T 2 , be nonexpansive mappings of C into itself such that n = 1 F( T n ) and let γ 1 , γ 2 , be real numbers such that 0< γ n b<1 for any n1. Then F(W)= n = 1 F( T n ).

Let I denote the identity operator on E. An operator AE×E with domain D(A)={zE:Az} and range R(A)={Az:zD(A)} is said to be accretive if for each x i D(A) and y i A x i , i=1,2, there exists j( x 1 x 2 )J( x 1 x 2 ) such that

y 1 y 2 , j ( x 1 x 2 ) 0.

An accretive operator A is said to be m-accretive if R(I+rA)=E for all r>0. In a real Hilbert space, an operator A is m-accretive if and only if A is maximal monotone. In this paper, we use A 1 (0) to denote the set of zero points of A. Interest in accretive operators, which is an important class of nonlinear operators, stems mainly from their firm connection with equations of evolution.

For an accretive operator A, we can define a nonexpansive single-valued mapping J r :R(I+rA)D(A) by J r = ( I + r A ) 1 for each r>0, which is called the resolvent of A. One of classical methods of studying the problem 0Ax, where AE×E is an accretive operator, is the following:

x 0 E, x n + 1 = J r n x n ,n0,

where J r n = ( I + r n A ) 1 and { r n } is a sequence of positive real numbers. Recently, different regularization iterative methods have been employed to treat zero points of accretive operators in the framework of Banach spaces; see [3236] and the references therein.

In this paper, we investigate common fixed point problems of a family of nonexpansive mappings generated in (2.1) and a zero point problem of an accretive operator based on a viscosity approximation method. Strong convergence theorems of common fixed points are established in a Banach space. In order to prove our main results, we need the following definitions and lemmas.

Recall that if C and D are nonempty subsets of a Banach space E such that C is nonempty closed convex and DC, then a map Q:CD is sunny provided that Q(x+t(xQ(x)))=Q(x) for all xC and t0 whenever x+t(xQ(x))C. A sunny nonexpansive retraction is a sunny retraction, which is also nonexpansive. Sunny nonexpansive retractions play an important role in our argument. They are characterized as follows: If E is a smooth Banach space, then Q:CD is a sunny nonexpansive retraction if and only if the following inequality holds:

x Q x , J ( y Q x ) 0for all xC and yD.

Chen and Zhu [34] showed that if E is a reflexive Banach space and has a weakly continuous duality, then there is a sunny nonexpansive retraction from Π C onto F(T) and it can be constructed as follows.

Lemma 2.3 [32]

Let E be a reflexive Banach space which has a weakly continuous duality mapping J φ (x). Let C be closed convex subset of E and let T:CC be a nonexpansive mapping. Let f:CC be a contractive mapping with F(f). For any t(0,1), let { x t } be defined by x t =tf( x t )+(1t)T x t , where T is a nonexpansive mapping. Then T has a fixed point if and only if { x t } remains bounded as t 0 + and, in this case, { x t } converges, as t 0 + , strongly to a fixed point of T.

Lemma 2.4 Under the condition of Lemma  2.3, we define the mapping Q: Π C F(T) by

Q(f):= lim t 0 x t ,f Π C .
(2.3)

Then the mapping Q is a sunny nonexpansive retraction from Π C onto F(T).

Proof From Theorem 3.1 of [34], for all t(0,1) and pF(T), we have

x t f ( x t ) , J φ ( x t p ) 0.

Letting t0, we have

( I f ) Q ( f ) , J φ ( Q ( f ) p ) 0.

Since J φ (x)= φ ( x ) x J(x) for any x0, we have

( I f ) Q ( f ) , J ( Q ( f ) p ) 0.

This completes the proof. □

Recall that a gauge is a continuous strictly increasing function φ:[0,)[0,) such that φ(0)=0 and φ(t) as t. The duality mapping J φ :X X associated to a gauge φ is defined by

J φ (x)= { x X : x , x = x φ ( x ) , x = φ ( x ) } ,xX.

Following Browder [37], we say that a Banach space E has a weakly continuous duality mapping if there exists a gauge φ for which the duality mapping J φ (x) is single-valued and weak-to-weak sequentially continuous (i.e., if { x n } is a sequence in E weakly convergent to a point x, then the sequence J φ ( x n ) converges weakly to J φ ). It is known that l p has a weakly continuous duality mapping with a gauge function φ(t)= t p 1 for all 1<p<. Set

Φ(t)= 0 t φ(τ) d τ ,t0.

Then

J φ (x)=Φ ( x ) ,xX,

where denotes the sub-differential in the sense of convex analysis.

The first part of the next lemma is an immediate consequence of the sub-differential inequality and the proof of the second part can be found in [38].

Lemma 2.5 Assume that a Banach space E has a weakly continuous duality mapping J φ with a gauge φ.

  1. (i)

    For all x,yE, the following inequality holds:

    Φ ( x + y ) Φ ( x ) + y , J φ ( x + y ) .

    In particular, for all x,yE

    x + y 2 x 2 +2 y , J ( x + y ) .
  2. (ii)

    Assume that a sequence { x n } in E converges weakly to a point xE.

Then the following identity holds:

lim sup n Φ ( x n y ) = lim sup n Φ ( x n x ) +Φ ( y x ) ,x,yE.

Lemma 2.6 [39]

Let { x n } and { y n } be bounded sequences in a Banach space X and let { β n } be a sequence in [0,1] with 0< lim inf n β n lim sup n β n <1. Suppose that x n + 1 =(1 β n ) y n + β n x n for all integers n0 and

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

Then lim n y n x n =0.

Lemma 2.7 [40]

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

α n + 1 (1 γ n ) α n + δ n ,

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 α n =0.

The following lemma can be obtained from Chang et al. [23]. For the sake of completeness, we still give the proof.

Lemma 2.8 Let C be a nonempty closed convex subset of a strictly convex Banach space E, let { T i :CC} be a family of infinitely nonexpansive mappings with i = 1 F( T i ), let { γ n } be a real sequence such that 0< γ n b<1 for each n1. If K is any bounded subset of C, then lim n sup x K Wx W n x=0.

Proof Let p i = 1 F( T i ). Since K is a bounded subset of C, there exists an M>0 such that sup x K xpM. It follows that

W n + 1 x W n x = γ 1 T 1 U n + 1 , 2 x + ( 1 γ 1 ) x γ 1 T 1 U n , 2 x ( 1 γ 1 ) x γ 1 U n + 1 , 2 x U n , 2 x = γ 1 γ 2 T 2 U n + 1 , 3 x + ( 1 γ 2 ) x γ 2 T 2 U n , 3 x ( 1 γ 2 ) x γ 1 γ 2 U n + 1 , 3 x U n , 3 x i = 1 n γ i U n + 1 , n + 1 x U n , n + 1 x i = 1 n + 1 γ i ( T n + 1 x p + p x ) 2 i = 1 n + 1 γ i M .

Since 0< γ n b<1, for any given ϵ>0, there exists a positive integer n 0 such that

b n 0 + 1 ϵ ( 1 b ) 2 M .

For any positive integers m>n> n 0 , we find that

W m x W n x j = n m 1 W j x W j x 2 M j = n m 1 i = 1 j + 1 γ i 2 M j = n m 1 b j + 1 2 M b n + 1 1 b ϵ , x K .

Letting m, we find that

Wx W n xϵ,n n 0 .

This implies that lim n sup x K Wx W n x=0. □

3 Main results

Theorem 3.1 Let E be a reflexive and strictly convex Banach space E which enjoys a weakly continuous duality map J φ (x) with gauge φ and let A be an m-accretive operator in E with the domain D(A). Assume that D ( A ) ¯ is convex. Let T i be a nonexpansive mapping from C=: D ( A ) ¯ into itself for i Z + . Let f Π C with the coefficient (0<α<1) and J r = ( I + r A ) 1 for some r>0. Assume that Ω:=F( J r W)=F( J r )F(W), where W is a mapping defined by (2.2). Let { x n } be a sequence generated in the following iterative algorithm:

{ x 0 C , y n = β n x n + ( 1 β n ) J r W n x n , x n + 1 = α n f ( y n ) + ( 1 α n ) y n , n 0 ,

where W n is generated in (2.1), { α n } and { β n } are real number sequences in (0,1) satisfying the following restrictions:

  1. (a)

    n = 0 α n =, lim n α n =0;

  2. (b)

    0< lim inf n β n lim sup n β n <1.

Then { x n } strongly converges to Q(f)Ω, where Q: Π C Ω is defined by (2.3).

Proof First we prove that sequences { x n } and { y n } are bounded. Fixing pΩ, we see that

y n p β n x n p + ( 1 β n ) J r W n x n p x n p .

It follows that

x n + 1 p = α n ( f ( y n ) p ) + ( 1 α n ) ( y n p ) α n f ( y n ) p + ( 1 α n ) y n p [ 1 α n ( 1 α ) ] x n p + α n f ( p ) p max { x n p , f ( p ) p 1 α } .

This in turn implies that

x n pmax { x 0 p , p f ( p ) 1 α } ,

which gives that the sequence { x n } is bounded, so is { y n }.

Next, we prove that x n + 1 x n 0 as n. Putting l n = x n + 1 β n x n 1 β n , we have

x n + 1 =(1 β n ) l n + β n x n .
(3.1)

In the light of

l n + 1 l n = α n + 1 f ( y n + 1 ) + ( 1 α n + 1 ) y n + 1 β n + 1 x n + 1 1 β n + 1 α n f ( y n ) + ( 1 α n ) y n β n x n 1 β n = α n + 1 ( f ( y n + 1 ) y n + 1 ) 1 β n + 1 α n ( f ( y n ) y n ) 1 β n + J r W n + 1 x n + 1 J r W n + 1 x n + J r W n + 1 x n J r W n x n ,
(3.2)

we obtain that

l n + 1 l n α n + 1 1 β n + 1 f ( y n + 1 ) y n + 1 + α n 1 β n y n f ( y n ) + x n + 1 x n + J r W n + 1 x n J r W n x n α n + 1 1 β n + 1 f ( y n + 1 ) y n + 1 + α n 1 β n y n f ( y n ) + x n + 1 x n + W n + 1 x n W n x n .
(3.3)

Since T i and U n , i are nonexpansive, we have

W n + 1 x n W n x n = γ 1 T 1 U n + 1 , 2 x n γ 1 T 1 U n , 2 x n γ 1 U n + 1 , 2 x n U n , 2 x n = γ 1 γ 2 T 2 U u + 1 , 3 x n γ 2 T 2 U n , 3 x n γ 1 γ 2 U u + 1 , 3 x n U n , 3 x n γ 1 γ 2 γ n U n + 1 , n + 1 x n U n , n + 1 x n M 1 i = 1 n γ i ,
(3.4)

where M 1 0 is an appropriate constant such that

U n + 1 , n + 1 x n U n , n + 1 x n M 1

for all n0. Substituting (3.4) into (3.3), we have

l n + 1 l n x n + 1 x n α n + 1 1 β n + 1 f ( y n + 1 ) y n + 1 + α n 1 β n y n f ( y n ) + M 1 i = 1 n γ i .

In view of conditions (a) and (b), we get that

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

We can obtain from Lemma 2.6 that lim n l n x n =0 easily. On the other hand, we see from (3.1) that

x n + 1 x n =(1 β n )( l n x n ).

This implies that

lim n x n + 1 x n =0.
(3.5)

Next, we prove that lim n J r W x n x n =0. In view of

x n + 1 y n = α n ( f ( y n ) y n ) ,

we obtain that

lim n x n + 1 y n =0.
(3.6)

On the other hand, we have

y n x n x n x n + 1 + x n + 1 y n .

In view of (3.5) and (3.6), we have

lim n y n x n =0.
(3.7)

Notice that

J r W n x n x n x n y n + y n J r W n x n x n y n + β n x n J r W n x n .

This implies that

(1 β n ) J r W n x n x n x n y n .

From condition (b) and (3.7), we obtain that

lim n J r W n x n x n =0.
(3.8)

On the other hand, we have

J r W x n x n J r W x n J r W n x n + J r W n x n x n W x n W n x n + J r W n x n x n .

In view of Lemma 2.8, we find that

lim n W x n W n x n =0.

This in turn implies that

lim n J r W x n x n =0.
(3.9)

Next, we show x n Q(f) as n. To show it, we first prove that

lim sup n ( I f ) Q ( f ) , J φ ( Q ( f ) x n ) 0.
(3.10)

In view of Lemma 2.4, we have the sunny nonexpansive retraction Q: Π C Ω. Take a subsequence { x n k } of { x n } such that

lim sup n ( I f ) Q ( f ) , J φ ( Q ( f ) x n ) = lim k ( I f ) Q ( f ) , J φ ( Q ( f ) x n k ) .
(3.11)

Since E is reflexive, we may further assume that x n k x ¯ for some x ¯ C. Since J φ is weakly continuous, we obtain from Lemma 2.5 that

lim sup n Φ ( x n k x ) = lim sup n Φ ( x n k x ¯ ) +Φ ( x x ¯ ) ,xE.

Put

g(x)= lim sup k Φ ( x n k x ) ,xE.

It follows that

g(x)=g( x ¯ )+Φ ( x x ¯ ) ,xE.

With the aid of (3.9), we arrive at

g ( J r W x ¯ ) = lim sup k Φ ( x n k J r W x ¯ ) = lim sup k Φ ( J r W x n k J r W x ¯ ) lim sup k Φ ( x n k x ¯ ) = g ( x ¯ ) .
(3.12)

Notice that

g( J r W x ¯ )=g( x ¯ )+Φ ( J r W x ¯ x ¯ ) .
(3.13)

From (3.12) and (3.13), we find that

Φ ( J r W x ¯ x ¯ ) 0.

This implies that J r W x ¯ = x ¯ . And hence x ¯ F( J r W). That is, x ¯ Ω. Since Q is the sunny nonexpansive retraction from Π C onto F, we have from (3.11)

lim sup n ( I f ) Q ( f ) , J φ ( Q ( f ) x n ) = ( I f ) Q ( f ) , J φ ( Q ( f ) x ¯ ) 0.

This shows that (3.10) holds. It follows from Lemma 2.5 that

Φ x n + 1 Q ( f ) = Φ ( α n ( f ( x n ) f ( Q ( f ) ) ) + α n ( f ( Q ( f ) ) Q ( f ) ) + ( 1 α n ) ( y n Q ( f ) ) ) Φ ( α n f ( x n ) f ( Q ( f ) ) + ( 1 α n ) y n Q ( f ) ) + α n f ( Q ( f ) ) Q ( f ) , J φ ( x n + 1 Q ( f ) ) Φ ( ( 1 α n ( 1 α ) ) x n Q ( f ) ) + α n f ( Q ( f ) ) Q ( f ) , J φ ( x n + 1 Q ( f ) ) ( 1 α n ( 1 α ) ) Φ ( x n Q ( f ) ) + α n f ( Q ( f ) ) Q ( f ) , J φ ( x n + 1 Q ( f ) ) .

We find that x n Q(f)0 as n from Lemma 2.7. That is, x n Q(f). This completes the proof. □

Remark 3.2 Taking T i =I, the identity mapping, i1, we see that W n =I. Then the strict convexity of E in Theorem 3.1 may not be needed.

Corollary 3.3 Let E be a reflexive Banach space E which enjoys a weakly continuous duality map J φ (x) with gauge φ and A be an m-accretive operator in E with the domain D(A). Assume that D ( A ) ¯ is convex. Let f Π D ( A ) ¯ with the coefficient (0<α<1) and J r = ( I + r A ) 1 for some r>0. Assume that A 1 (0). Let { x n } be a sequence generated in the following iterative algorithm:

{ x 0 A 1 ( 0 ) , y n = β n x n + ( 1 β n ) J r x n , x n + 1 = α n f ( y n ) + ( 1 α n ) y n , n 0 ,

where { α n } and { β n } are real number sequences in (0,1) satisfying the following restrictions:

  1. (a)

    n = 0 α n =, lim n α n =0;

  2. (b)

    0< lim inf n β n lim sup n β n <1.

Then { x n } strongly converges to Q(f) A 1 (0), where Q: Π D ( A ) ¯ A 1 (0) is defined by (2.3).

If f(x)=u, where u is a fixed element in D ( A ) ¯ , then Theorem 3.1 is reduced to the following.

Corollary 3.4 Let E be a reflexive and strictly convex Banach space E which enjoys a weakly continuous duality map J φ (x) with gauge φ and A be an m-accretive operator in E with the domain D(A). Assume that D ( A ) ¯ is convex. Let T i be a nonexpansive mapping from C=: D ( A ) ¯ into itself for i Z + . Let J r = ( I + r A ) 1 for some r>0. Assume that Ω:=F( J r W)=F( J r )F(W), where W is a mapping defined by (2.2). Let { x n } be a sequence generated in the following iterative algorithm:

{ x 0 C , y n = β n x n + ( 1 β n ) J r W n x n , x n + 1 = α n u + ( 1 α n ) y n , n 0 ,

where W n is generated in (2.1), { α n } and { β n } are real number sequences in (0,1) satisfying the following restrictions:

  1. (a)

    n = 0 α n =, lim n α n =0;

  2. (b)

    0< lim inf n β n lim sup n β n <1.

Then { x n } strongly converges to Q(u)Ω, where Q: Π C Ω is defined by (2.3).

If A=I, then Theorem 3.1 is reduced to the following.

Corollary 3.5 Let E be a reflexive and strictly convex Banach space E which enjoys a weakly continuous duality map J φ (x) with gauge φ and let C be a closed and convex subset of E. Let T i be a nonexpansive mapping from C into itself for i Z + . Let f Π C with the coefficient (0<α<1). Assume that Ω:= i = 1 F( T i ). Let { x n } be a sequence generated in the following iterative algorithm:

{ x 0 C , y n = β n x n + ( 1 β n ) W n x n , x n + 1 = α n f ( y n ) + ( 1 α n ) y n , n 0 ,

where W n is generated in (2.1), { α n } and { β n } are real number sequences in (0,1) satisfying the following restrictions:

  1. (a)

    n = 0 α n =, lim n α n =0;

  2. (b)

    0< lim inf n β n lim sup n β n <1.

Then { x n } strongly converges to Q(f)Ω, where Q: Π C Ω is defined by (2.3).

References

  1. Dhage BC, Jadhav NS: Differential inequalities and comparison theorems for first order hybrid integro-differential equations. Adv. Inequal. Appl. 2013, 2: 61–80.

    Google Scholar 

  2. Ansari QH, Schaible S, Yao JC: The system of generalized vector equilibrium problems with applications. J. Glob. Optim. 2002, 22: 3–16. 10.1023/A:1013857924393

    Article  MathSciNet  Google Scholar 

  3. Noor MA, Noor KI, Waseem M: Decomposition method for solving system of linear equations. Eng. Math. Lett. 2013, 2: 34–41.

    Google Scholar 

  4. Qin X, Chang SS, Cho YJ: Iterative methods for generalized equilibrium problems and fixed point problems with applications. Nonlinear Anal. 2010, 11: 2963–2972. 10.1016/j.nonrwa.2009.10.017

    Article  MathSciNet  Google Scholar 

  5. Abdel-Salam HS, Al-Khaled K: Variational iteration method for solving optimization problems. J. Math. Comput. Sci. 2012, 2: 1475–1497.

    MathSciNet  Google Scholar 

  6. Park S: On generalizations of the Ekeland-type variational principles. Nonlinear Anal. 2000, 39: 881–889. 10.1016/S0362-546X(98)00253-3

    Article  MathSciNet  Google Scholar 

  7. Al-Bayati AY, Al-Kawaz RZ: A new hybrid WC-FR conjugate gradient-algorithm with modified secant condition for unconstrained optimization. J. Math. Comput. Sci. 2012, 2: 937–966.

    MathSciNet  Google Scholar 

  8. Tanaka Y: Constructive proof of the existence of Nash equilibrium in a strategic game with sequentially locally non-constant payoff functions. Adv. Fixed Point Theory 2012, 2: 398–416.

    Google Scholar 

  9. Khanh PQ, Luu LM: On the existence of solutions to vector quasivariational inequalities and quasicomplementarity problems with applications to traffic network equilibria. J. Optim. Theory Appl. 2004, 123: 533–548. 10.1007/s10957-004-5722-3

    Article  MathSciNet  Google Scholar 

  10. Husain S, Gupta S: A resolvent operator technique for solving generalized system of nonlinear relaxed cocoercive mixed variational inequalities. Adv. Fixed Point Theory 2011, 1: 18–28.

    Google Scholar 

  11. Iiduka H: Fixed point optimization algorithm and its application to network bandwidth allocation. J. Comput. Appl. Math. 2012, 236: 1733–1742. 10.1016/j.cam.2011.10.004

    Article  MathSciNet  Google Scholar 

  12. Shen J, Pang LP: An approximate bundle method for solving variational inequalities. Comm. Optim. Theory 2012, 1: 1–18.

    Google Scholar 

  13. Shimoji K, Takahashi W: Strong convergence to common fixed points of infinite nonexpansive mappings and applications. Taiwan. J. Math. 2001, 5: 387–404.

    MathSciNet  Google Scholar 

  14. Cho SY, Kang SM: Approximation of common solutions of variational inequalities via strict pseudocontractions. Acta Math. Sci. 2012, 32: 1607–1618.

    Article  Google Scholar 

  15. Cho SY, Kang SM: Approximation of fixed points of pseudocontraction semigroups based on a viscosity iterative process. Appl. Math. Lett. 2011, 24: 224–228. 10.1016/j.aml.2010.09.008

    Article  MathSciNet  Google Scholar 

  16. Kim JK: Strong convergence theorems by hybrid projection methods for equilibrium problems and fixed point problems of the asymptotically quasi- ϕ -nonexpansive mappings. Fixed Point Theory Appl. 2011., 2011: Article ID 10

    Google Scholar 

  17. Qin X, Cho YJ, Kang SM: Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces. J. Comput. Appl. Math. 2009, 225: 20–30. 10.1016/j.cam.2008.06.011

    Article  MathSciNet  Google Scholar 

  18. Qin X, Cho SY, Kang SM: On hybrid projection methods for asymptotically quasi- ϕ -nonexpansive mappings. Appl. Math. Comput. 2010, 215: 3874–3883. 10.1016/j.amc.2009.11.031

    Article  MathSciNet  Google Scholar 

  19. Zegeye H, Shahzad N: Strong convergence theorem for a common point of solution of variational inequality and fixed point problem. Adv. Fixed Point Theory 2012, 2: 374–397.

    Google Scholar 

  20. Qin X, Cho SY, Kang SM: Iterative algorithms for variational inequality and equilibrium problems with applications. J. Glob. Optim. 2010, 48: 423–445. 10.1007/s10898-009-9498-8

    Article  MathSciNet  Google Scholar 

  21. Saini RK, Kumar M: Common fixed point of expansion type maps in cone metric space using implicit relations. Adv. Fixed Point Theory 2012, 2: 326–339.

    Google Scholar 

  22. Qin X, Cho SY, Kang SM: Strong convergence of shrinking projection methods for quasi- ϕ -nonexpansive mappings and equilibrium problems. J. Comput. Appl. Math. 2010, 234: 750–760. 10.1016/j.cam.2010.01.015

    Article  MathSciNet  Google Scholar 

  23. Luo H, Wang Y: Iterative approximation for the common solutions of a infinite variational inequality system for inverse-strongly accretive mappings. J. Math. Comput. Sci. 2012, 2: 1660–1670.

    MathSciNet  Google Scholar 

  24. Qing Y, Cho SY, Shang M: Strong convergence of an iterative process for a family of strictly pseudocontractive mappings. Fixed Point Theory Appl. 2013., 2013: Article ID 117

    Google Scholar 

  25. Reich S: Strong convergence theorems for resolvents of accretive operators in Banach spaces. J. Math. Anal. Appl. 1980, 75: 287–292. 10.1016/0022-247X(80)90323-6

    Article  MathSciNet  Google Scholar 

  26. Xu HK: Strong convergence of an iterative method for nonexpansive and accretive operators. J. Math. Anal. Appl. 2006, 314: 631–643. 10.1016/j.jmaa.2005.04.082

    Article  MathSciNet  Google Scholar 

  27. Kim TH, Xu HK: Strong convergence of modified Mann iterations. Nonlinear Anal. 2005, 61: 51–60. 10.1016/j.na.2004.11.011

    Article  MathSciNet  Google Scholar 

  28. Kotzer T, Cohen N, Shamir J: Image restoration by a novel method of parallel projection onto constraint sets. Opt. Lett. 1995, 20: 1172–1174. 10.1364/OL.20.001172

    Article  Google Scholar 

  29. Sezan MI, Stark H: Application of convex projection theory to image recovery in tomograph and related areas. In Image Recovery: Theory and Application. Edited by: Stark H. Academic Press, Orlando; 1987:155–270.

    Google Scholar 

  30. Censor Y, Zenios SA: Parallel Optimization. Theory, Algorithms, and Applications, Numerical Mathematics and Scientific Computation. Oxford University Press, New York; 1997.

    Google Scholar 

  31. Takahashi W, Shimoji K: Convergence theorems for nonexpansive mappings and feasibility problems. Math. Comput. Model. 2000, 32: 1463–1471. 10.1016/S0895-7177(00)00218-1

    Article  MathSciNet  Google Scholar 

  32. Qin X, Su Y: Approximation of a zero point of accretive operator in Banach spaces. J. Math. Anal. Appl. 2007, 329: 415–424. 10.1016/j.jmaa.2006.06.067

    Article  MathSciNet  Google Scholar 

  33. Chang SS, Lee HWJ, Chan CK: A new method for solving equilibrium problem fixed point problem and variational inequality problem with application to optimization. Nonlinear Anal. 2009, 70: 3307–3319. 10.1016/j.na.2008.04.035

    Article  MathSciNet  Google Scholar 

  34. Chen R, Zhu Z: Viscosity approximation fixed points for nonexpansive and m -accretive operators. Fixed Point Theory Appl. 2006., 2006: Article ID 81325

    Google Scholar 

  35. Ceng LC, Petrusel A, Wong MM: Hybrid viscosity iterative approximation of zeros of m -accretive operators in Banach spaces. Taiwan. J. Math. 2011, 15: 2459–2481.

    MathSciNet  Google Scholar 

  36. Qin X, Kang SM, Cho YJ: Approximating zeros of monotone operators by proximal point algorithms. J. Glob. Optim. 2010, 46: 75–87. 10.1007/s10898-009-9410-6

    Article  MathSciNet  Google Scholar 

  37. Browder FE: Fixed point theorems for noncompact mappings in Hilbert spaces. Proc. Natl. Acad. Sci. USA 1965, 53: 1272–1276. 10.1073/pnas.53.6.1272

    Article  MathSciNet  Google Scholar 

  38. Lim TC, Xu HK: Fixed point theorems for asymptotically nonexpansive mappings. Nonlinear Anal. 1994, 22: 1345–1355. 10.1016/0362-546X(94)90116-3

    Article  MathSciNet  Google Scholar 

  39. Suzuki T: Strong convergence of Krasnoselskii and Mann’s type sequences for one-parameter nonexpansive semigroups without Bochner integrals. J. Math. Anal. Appl. 2005, 305: 227–239. 10.1016/j.jmaa.2004.11.017

    Article  MathSciNet  Google Scholar 

  40. Liu LS: Ishikawa and Mann iteration process with errors for nonlinear strongly accretive mappings in Banach spaces. J. Math. Anal. Appl. 1995, 194: 114–125. 10.1006/jmaa.1995.1289

    Article  MathSciNet  Google Scholar 

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Cheng, P., Wu, H. Strong convergence theorems for common solutions of a family of nonexpansive mappings and an accretive operator. Fixed Point Theory Appl 2013, 172 (2013). https://doi.org/10.1186/1687-1812-2013-172

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