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Wiener-Hopf equation technique for solving equilibrium problems and variational inequalities and fixed points of a nonexpansive mapping

Abstract

In this paper, we introduce some new iterative schemes based on the Wiener-Hopf equation technique and auxiliary principle for finding common elements of the set of solutions of equilibrium problems, the set of fixed points of a nonexpansive mapping and the set of solutions of a variational inequality. Several strong convergence results for the sequences generated by these iterative schemes are established in Hilbert spaces. As the generation, we also consider two generalized variational inequalities, and obtain some iterative schemes and the proposed strong convergence theorems for solving these generalized variational inequalities, equilibrium problems, and a nonexpansive mapping. Our results and proof are new, and they extend the corresponding results of Verma (Appl. Math. Lett. 10:107-109, 1997), Wu and Li (4th International Congress on Image and Signal Processing, pp. 2802-2805, 2011), and Noor and Huang (Appl. Math. Comput. 191:504-510, 2007).

1 Introduction

Let H be a real Hilbert space, whose inner product and norm are denoted by , and , respectively. Let K be a nonempty closed convex subset of H. Let T,S:KK be nonlinear mappings. Let M:H 2 H be a multi-valued operator and let f:K×KR be a bifunction, where R is the set of real numbers. Let P K be the projection of H onto the closed convex set K and Q K =I P K , where the I is the identity operator.

In 1994, Blum and Oettli (see [1]) introduced the equilibrium problem (EP) which is to find x ¯ K, such that

f( x ¯ ,y)0,yK.
(1.1)

The set of solutions of (1.1) is denoted by EP(f). Let f(x,y)=Tx,yx for all x,yK, then the EP reduces to the variational inequality problem (VIP) which is to find x ¯ K such that

T x ¯ ,yx0,yK.
(1.2)

This problem was introduced by Stampacchia (see [2]) in 1964. Related to EP and VIP, fixed point problems (FP) of a nonexpansive mapping are also considered by many authors. Recall that a mapping S is nonexpansive if SxSyxy for all x,yK. Let us denote the set of fixed points of S and set of solutions of problem (1.2) by F(S), VI(K,T), respectively.

Recently, for solving the EP, VIP, and FP, many authors have introduced and extended lots of iterative schemes.

For solving variational inequality problems:

In 1991, Shi (see [3]) demonstrated the equivalence between the VIP: Tuf,vu0, vK and Wiener-Hopf equation: (T P K + Q K )v=f, where fH. Noor (see [4]) established the equivalence between the generalized VIP: Tu,g(v)g(u)0, g(v)K and the generalized Wiener-Hopf equation: T g 1 P K z+ ρ 1 Q K z=0 and introduced an iterative scheme based on this equivalence,

{ g ( u n ) = P K z n , z n + 1 = g ( u n ) ρ T u n ,
(1.3)

where g 1 exists.

Afterwards, by using different generalized Wiener-Hopf equation techniques, Verma and Al-Shemas et al. introduced several algorithms for solving generalized VIPs, respectively (see, for example, [57]). It has been shown that the Wiener-Hopf equation techniques are more flexible and general than projection methods.

On the other hand, for getting the unified approach to solve EP, VIP and FP, many authors also suggest and analyze lots of iterative schemes for common elements of F(S), EP(f), VI(K,T).

For solving EP(f)F(S):

Takahashi and Takahashi (see [8]) introduced the following iterative schemes based on the viscosity approximation method:

{ F ( u n , y ) + 1 r n y u n , u n x n 0 , y K , x n + 1 = α n f ( x n ) + ( 1 α n ) S u n .

For solving EP(f)VI(K,T):

Li and Su (see [9]) introduced the following iterative schemes:

{ y n = Π C J 1 ( J x n r n A x n ) , u n C : f ( u n , y ) + 1 r n y u n , J u n J y n 0 , y K , C n + 1 = { z C n : φ ( z , u n ) φ ( z , x n ) } , x n + 1 = Π C n + 1 x 0 .

For solving EP(f)VIP(K,T)F(S):

Plubtieng and Punpaeng (see [10]) introduced the following iterative schemes:

{ f ( u n , y ) + 1 r n y u n , u n x n 0 , y K , y n = P C ( u n λ n A u n ) , x n + 1 = α n x 1 + β n x n + γ n S P C ( y n λ n A y n ) .

For more algorithms, please see, for instance, [133] and the references therein.

Summarizing the above algorithms, we know that these papers have mainly used the auxiliary principle, the projection technique and iterative schemes of fixed points. However, the Wiener-Hopf equation technique which is more flexible and general than projection methods has not been used for solving EP(f)F(S), EP(f)VI(K,T), and EP(f)VI(K,T)F(S).

Remark 1.1 It is worth mentioning that although there are some papers which have applied the Wiener-Hopf equation technique to solve VIP and VI(K,T)F(S), their research does not include EP(f). However, our idea is just to apply the Wiener-Hopf equation technique to study the common element problem which is related to EP(f).

In this paper, motivated and inspired by the above analysis and ongoing research in this field, we combine the Wiener-Hopf equation technique and auxiliary principle to introduce some iterative schemes for solving the common element problem which is related to EP(f). This paper is organized as follows: In Section 2, some preliminaries are presented. Section 3 is devoted to solving the EP(f)VI(K,T). In Section 4, we consider a nonexpansive mapping S and obtain some iterative schemes and strong convergent results for solving EP(f)VI(K,T)F(S). In Section 5, we extend the VIPs in Section 3 and Section 4, and we get some iterative schemes and strong convergent theorems for solving GVI(K,T)EP(f) and GVI(K,T)EP(f)F(S), respectively. Our results extend the corresponding results of Verma (see [6]), Wu and Li (see [29]), and Noor and Huang (see [33]).

2 Preliminaries

In the rest of this paper, let H be a real Hilbert space, whose inner product and norm are denoted by , and , respectively. Let K be a nonempty closed convex subset of H. Let T,S:KK be nonlinear mapping. Let M:H 2 H be a multi-valued operator and let f:K×KR be a bifunction, where R is the set of real numbers. Let P K be the projection of H onto the closed convex set K and Q K =I P K , where I is the identity operator.

Definition 2.1 The operator T:KK is said to be:

  1. (i)

    μ-Lipschitz continuous, if there exists a constant μ>0 such that TxTyμxy for all x,yK;

  2. (ii)

    r-strongly monotone, if there exists a constant r>0 such that TxTy,xyr x y 2 for all x,yK;

  3. (iii)

    γ-co-coercive, if there exists a constant γ>0 such that TxTy,xyγ T x T y 2 for all x,yK;

  4. (iv)

    relaxed γ-co-coercive, if there exists a constant γ>0 such that TxTy,xyγ T x T y 2 for all x,yK;

  5. (v)

    relaxed (γ,r)-co-coercive, if there exist two constants γ,r>0 such that TxTy,xyγ T x T y 2 +r x y 2 for all x,yK.

Definition 2.2 The multi-valued operator M:H 2 H is said to be:

  1. (i)

    a relaxed monotone operator, if there exists a constant k>0 such that w 1 w 2 ,uvk u v 2 , w 1 Mu, w 2 Mv;

  2. (ii)

    Lipschitz continuous if there exists a constant λ>0 such that w 1 w 2 λuv, w 1 Mu, w 2 Mv.

Lemma 2.1 (see [10])

Let the bifunction f:K×KR satisfy the following conditions:

  1. (i)

    f(x,x)=0 for all xK;

  2. (ii)

    f is monotone, i.e. f(x,y)+f(y,x)0 for all x,yK;

  3. (iii)

    for each x,y,zK, lim t 0 f(tz+(1t)x,y)f(x,y);

  4. (iv)

    for each xK, f(x,) is convex and lower semicontinuous.

Then EP(f).

Lemma 2.2 (see [10])

Let r>0, xH, and f satisfy the conditions (i)-(iv) in Lemma  2.1. Then there exists zK such that f(z,y)+ 1 r yz,zx0, yK.

Lemma 2.3 (see [10])

Let r>0, xH, and f satisfy the conditions (i)-(iv) in Lemma  2.1. Define a mapping T r :HK as T r (x)={zK:f(z,y)+ 1 r yz,zx0,yK}.

Then the following hold:

  1. (a)

    T r is single-valued;

  2. (b)

    T r is firmly nonexpansive, i.e. T r x T r y T r x T r y,xy for all x,yH;

  3. (c)

    EP(f)=F( T r ), where F( T r ) denotes the sets of fixed point of T r ;

  4. (d)

    EP(f) is closed and convex.

In [4], Noor introduced the following generalized Wiener-Hopf equation:

T g 1 P K z+ ρ 1 Q K z=0.
(2.1)

Here he assumed g 1 exists, and note that if g=I, the identity operator, then (2.1) reduces to

T P K z+ ρ 1 Q K z=0,
(2.2)

which was introduced by Shi (see [3]). Denote the sets of solutions of (2.1) and (2.2) by WHE(T,g) and WHE(T), respectively.

Lemma 2.4 (see [3])

The variational inequality (1.2) has a solution x ¯ H if and only if the Wiener-Hopf equation (2.2) has a solution z ¯ H, where x ¯ = P K z ¯ , z ¯ = x ¯ ρT x ¯ .

In 1988, Noor (see [19]) introduced the generalized variational inequality (GVIP) which is to find x ¯ H such that g( x ¯ )H and

T x ¯ , g ( y ) g ( x ¯ ) 0,g(y)K.
(2.3)

Denote the set of solutions of (2.3) by GVI(K,T). Clearly, if g=I, the identity operator, the GVIP (2.3) reduces to VIP (1.2).

Lemma 2.5 (see [4])

The variational inequality (2.3) has a solution x ˜ H if and only if the Wiener-Hopf equation (2.1) has a solution z ˜ H, where g( x ¯ )= P K z ˜ , z ˜ =g( x ¯ )ρT x ¯ , and ρ>0 is a constant.

For finding the common element of the set of fixed points of a nonexpansive mapping and the set of solution of the variational inequality, Noor and Huang [33] introduced the Wiener-Hopf equation which included a nonexpansive mapping:

TS P K z+ ρ 1 Q K z=0.
(2.4)

Furthermore the equivalence was established between the Wiener-Hopf equation (2.4) and the variational inequality (1.2) as follows.

Lemma 2.6 (see [33])

The variational inequality (1.2) has a solution x ˜ if and only if the Wiener-Hopf equation (2.4) has a solution z ˜ , where z ˜ = x ˜ ρT x ˜ , x ˜ =S P K z ˜ .

Wu and Li (see [29]) introduced the Wiener-Hopf equation which includes a nonexpansive mapping S:

TS P K z+w+ ρ 1 Q K z=0,wMS P K z
(2.5)

and established the equivalence between the Wiener-Hopf equation (2.5) and the generalized variational inequality which is to find uK such that

Tu+w,vu0,vC,wMu.
(2.6)

Next, denote the sets of solutions of (2.5) by WHE(T,S).

Lemma 2.7 (see [29])

The variational inequality (2.6) has a solution c ˜ H if and only if the Wiener-Hopf equation (2.5) has a solution z ˜ H, where z ˜ = c ˜ ρ(T c ˜ +w), c ˜ =S P K z ˜ .

Lemma 2.8 (see [30])

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

a n + 1 (1 λ n ) a n + b n ,n n 0 ,

where n 0 is some nonnegative integer, and { λ n } is a sequence in [0,1] such that n = 1 λ n =, b n =o( λ n ). Then lim n a n =0.

Lemma 2.9 For given x,zK, if

yz,zx0,
(2.7)

for any yK. Then x=z.

Proof Assume xz. Put y=x, then (2.7) reduces to 0xz,zx= x z 2 <0, which is a contradiction. □

3 Results for solving EP(f)VI(K,T)

In this section, we firstly use Lemma 2.4 to introduce some iterative schemes and the convergence theorems for solving EP(f)VI(K,T).

Algorithm 3.1 For a given z 0 , compute the approximate solution z n + 1 by the iterative schemes:

{ u n = P K z n , f ( v n , y ) + 1 r y v n , v n u n 0 , y K , z n + 1 = v n ρ T v n .

By an appropriate rearrangement, Algorithm 3.1 can be written in the following form.

Algorithm 3.2 For a given z 0 , compute the approximate solution z n + 1 by the iterative schemes:

{ f ( v n , y ) + 1 r y v n , v n P K z n 0 , y K , z n + 1 = v n ρ T v n .

If f(x,y)=0 for all x,yK, Algorithm 3.1 collapses to the following iterative method for solving variational inequalities (1.2), which is mainly due to Shi [3].

Algorithm 3.3 For a given z 0 , compute the approximate solution z n + 1 by the iterative schemes:

{ u n = P K z n , z n + 1 = u n ρ T u n .

Theorem 3.1 Let K be the nonempty closed convex subset of H. The bifunction f:K×KR satisfies the conditions (i)-(iv) of Lemma  2.1. Let T:KK be a α-strongly monotone and β-Lipschitz continuous operator such that EP(f)VI(K,T). Let { z n }, { u n }, { v n } be the sequences generated by Algorithm 3.1, where ρ>0 is a constant and

0<12αρ+ β 2 ρ 2 <1.

Then { u n }, { v n } generated by Algorithm 3.1 converge to sEP(f)VI(K,T), and { z n } generated by Algorithm 3.1 converges to z ˜ WHE(T).

Proof Let z ˜ WHE(T) and sEP(f)VI(K,T).

Step 1. Estimate z n + 1 z ˜ .

From Lemma 2.4, we have

z ˜ = s ρ T s , s = P K z ˜ .

Hence

z n + 1 z ˜ = v n ρT v n s+ρTs
(3.1)
= v n s ( ρ T v n ρ T s ) 2
(3.2)
v n s 2 2 ρ T v n T s , v n s + T v n T s 2
(3.3)
1 2 α ρ + β 2 ρ 2 v n s.
(3.4)

Here, we have used the α-strong monotonicity and μ-Lipschitz continuity of T in (3.3) and (3.4).

Step 2. Estimate v n s.

Since sEP(f)VI(K,T), we have

f(s,y)0,yK.
(3.5)

Putting y= v n in (3.5) and y=s in Algorithm 3.1, we have

f(s, v n )0andf( v n ,s)+ 1 r s v n , v n u n 0.
(3.6)

From the monotonicity of f, we get

f(s, v n )0f( v n ,s)0.
(3.7)

Combining (3.6) and (3.7), we obtain

s v n , v n u n 0.

It follows that

s v n , v n s + s u n 0 s v n , v n s + s v n , s u n 0 s v n 2 s v n , s u n s v n s u n s v n s u n v n s u n s .

Step 3. Estimate u n s and prove the strong convergence of sequences generated by Algorithm 3.1.

Due to the nonexpansivity of P K , we find

u n s= P K z n P K z ˜ z n z ˜ .

By the above three steps, we have

z n + 1 z ˜ ( 1 2 α ρ + β 2 ρ 2 ) v n s ( 1 2 α ρ + β 2 ρ 2 ) u n s ( 1 2 α ρ + β 2 ρ 2 ) z n z ˜ ,

which implies that

z n + 1 z ˜ ( 1 2 α ρ + β 2 ρ 2 ) z n z ˜ ( ( 1 2 α ρ + β 2 ρ 2 ) ) n + 1 z 0 z ˜ .

Since

( 1 2 α ρ + β 2 ρ 2 ) <1,

we conclude

z n + 1 z ˜ 0(n).

Therefore, from

v n s u n s= P K z n P K z z n z,

we have

v n s 0 ( n ) , u n s 0 ( n ) .

 □

4 Results for solving EP(f)VI(K,T)F(S)

In this section, applying Lemma 2.6, we consider a nonexpansive mapping and analyze several iterative schemes and convergence theorems for solving EP(f)VI(K,T)F(S).

Algorithm 4.1 For a given z 0 H, compute the approximate solution z n + 1 by the iterative schemes

{ u n = α n P C z n + ( 1 α n ) S P C z n , f ( v n , y ) + 1 r y v n , v n u n 0 , y K , z n + 1 = ( 1 α n ) z n + α n ( v n ρ T v n ) .

For S=I, the identity operator, Algorithm 4.1 collapses to the following iterative method.

Algorithm 4.2 For a given z 0 H, compute the approximate solution z n + 1 by the iterative schemes

{ u n = P C z n , f ( v n , y ) + 1 r y v n , v n u n 0 , y K , z n + 1 = ( 1 α n ) z n + α n ( v n ρ T v n ) .

For α n =1, S=I, the identity operator, Algorithm 4.1 collapses to the following iterative method.

Algorithm 4.3 For a given z 0 H, compute the approximate solution z n + 1 by the iterative schemes

{ u n = P C z n , f ( v n , y ) + 1 r y v n , v n u n 0 , y K , z n + 1 = v n ρ T v n .

For f=0 and via Lemma 2.9, Algorithm 4.1 reduces to the following iterative method for solving VI(K,T)F(S).

Algorithm 4.4 For a given z 0 H, compute the approximate solution z n + 1 by the iterative schemes

{ u n = α n P C z n + ( 1 α n ) S P C z n , z n + 1 = ( 1 α n ) z n + α n ( u n ρ T u n ) .

For f=0, S=I, α n =1, and via Lemma 2.9, Algorithm 4.1 reduces to the following iterative method for solving VIP.

Algorithm 4.5 For a given z 0 H, compute the approximate solution z n + 1 by the iterative schemes

{ u n = P C z n , z n + 1 = u n ρ T u n .

Theorem 4.1 Let K be the nonempty closed convex subset of H. The bifunction f satisfies the conditions (i)-(iv) of Lemma  2.1. Let T:KK be relaxed (γ,r)-co-coercive and μ-Lipschitz continuous, and let S:KK be a k-strictly pseudocontractive mapping such that EP(f)VIP(K,T)F(S), where

n = 0 α n =,α[k,1),0<ρ< 2 ( r γ μ k ) ( μ + m ) 2 ,r>γμ+k.

Then the sequences { u n }, { v n } generated by Algorithm 4.1 converge to c ˜ EP(f)VI(K,T)F(S) and the sequence { z n } generated by Algorithm 4.1 converges to z ˜ WHE(T).

Proof Let z ˜ WHE(T) and c ˜ EP(f)VI(K,T)F(S).

Step 1. Estimate z n + 1 z ˜ .

From Lemma 2.6, we have

z ˜ = c ˜ ρ T c ˜ , c ˜ = S P K z ˜ .

This implies that

c ˜ = α n P C z ˜ + ( 1 α n ) S P C z ˜ , z ˜ = ( 1 α n ) z ˜ + α n ( c ˜ ρ T c ˜ ) .

Hence

z n + 1 z ˜ = ( 1 α n ) z n + α n ( v n ρ T v n ) z ˜
(4.1)
= ( 1 α n ) z n + α n ( v n ρ T v n ) ( 1 α n ) z ˜ α n ( c ˜ ρ T c ˜ )
(4.2)
= ( 1 α n ) z n ( 1 α n ) z ˜ + α n ( v n ρ T v n ) α n ( c ˜ ρ T c ˜ )
(4.3)
(1 α n ) z n z ˜ + α n v n c ˜ ρ ( T v n T c ˜ )
(4.4)
=(1 α n ) z n z ˜ + α n v n c ˜ ρ ( T v n T c ˜ ) 2
(4.5)
=(1 α n ) z n z ˜ + α n v n c ˜ 2 2 ρ T v n ρ T c ˜ , v n c ˜ + ρ 2 T v n T c ˜ 2
(4.6)
(1 α n ) z n z ˜
(4.7)
+ α n v n c ˜ 2 2 ρ [ γ T v n ρ T c ˜ 2 + r v n c ˜ 2 ] + ρ 2 T v n ρ T c ˜ 2
(4.8)
(1 α n ) z n z ˜
(4.9)
+ α n v n c ˜ 2 2 ρ [ γ μ 2 v n c ˜ 2 + r v n c ˜ 2 ] + ρ 2 μ 2 v n c ˜ 2
(4.10)
=(1 α n ) z n z ˜ + α n ( 1 + 2 ρ γ μ 2 2 ρ r + ρ 2 μ 2 ) v n c ˜ .
(4.11)

In (4.8), (4.10), (4.11) of the above induction, we have used the (γ,r)-co-coercivity and μ-Lipschitz continuity of the operator T.

Step 2. Estimate v n c ˜ .

Since c ˜ EP(f)VI(K,T)F(S), we get

f( c ˜ ,y)0,yK.
(4.12)

Putting y= v n in (4.12) and y= c ˜ in Algorithm 4.1, respectively, we have

f( c ˜ , v n )0andf( v n , c ˜ )+ 1 r c ˜ v n , v n u n 0.
(4.13)

From the monotonicity of f, we obtain

f( c ˜ , v n )0f( v n , c ˜ )0.
(4.14)

Combining (4.13) and (4.14), we know

c ˜ v n , v n u n 0.

It follows that

c ˜ v n , v n c ˜ + c ˜ u n c ˜ v n , v n c ˜ + c ˜ v n , c ˜ u n c ˜ v n 2 c ˜ v n , c ˜ u n c ˜ v n c ˜ u n c ˜ v n c ˜ u n v n c ˜ u n c ˜ .

Step 3. Estimate u n c ˜ and prove the strong convergence of sequences generated by Algorithm 4.1.

Since

u n = α n P C z n + ( 1 α n ) S P C z n , c ˜ = α n P C z ˜ + ( 1 α n ) S P C z ˜ ,

we have

u n c ˜ = α n P C z n + ( 1 α n ) S P C z n α n P C z ˜ ( 1 α n ) S P C z ˜ = α n P C z n α n P C z ˜ + ( 1 α n ) S P C z n ( 1 α n ) S P C z ˜ α n P C z n P C z ˜ + ( 1 α n ) S P C z n S P C z ˜ z n z ˜ .

By the above three steps, we get

z n + 1 z ˜ ( 1 α n ) z n z ˜ + α n ( 1 + 2 ρ γ μ 2 2 ρ r + ρ 2 μ 2 ) v n c ˜ ( 1 α n ) z n z ˜ + α n ( 1 + 2 ρ γ μ 2 2 ρ r + ρ 2 μ 2 ) u n c ˜ ( 1 α n ) z n z ˜ + α n ( 1 + 2 ρ γ μ 2 2 ρ r + ρ 2 μ 2 ) z n c ˜ = [ 1 α n ( 1 θ ) ] z n c ˜ ,

where θ= ( 1 + 2 ρ γ μ 2 2 ρ r + ρ 2 μ 2 ) .

From Lemma 2.8, it is easy to see that

z n z ˜ 0(n),

and from

v n s u n s= P K z n P K z z n z

we have

u n s 0 ( n ) , v n s 0 ( n ) .

 □

5 Generation

Ever since the classical variational inequality was introduced, it has been extended to many forms in different directions, such as the nonconvex variational inequality, the multi-valued variational inequality, the extended general quasi-variational inequalities and so on. Related to the variational inequality, the Wiener-Hopf equation has also been extended, and the equivalence between generalized variational inequalities and generalized Wiener-Hopf equations has been studied.

In this section, we consider the two generalized variational inequalities (2.3) and (2.6), respectively. Furthermore, applying Lemma 2.5, we firstly suggest and give some iterative schemes for solving the variational inequality (2.3) and the equilibrium problem; secondly, via Lemma 2.7, we also analyze and introduce several iterative schemes for solving the variational inequality (2.6), the equilibrium problem, and a nonexpansive mapping.

5.1 Results for solving variational inequality (2.3) and equilibrium problem

In this subsection, we use a similar technique to Section 3, and the proposed results in this subsection can be considered as the generation of Section 3.

Algorithm 5.1 For a given z 0 H, compute the approximate solution z n + 1 by the iterative schemes

{ g ( u n ) = P K z n , f ( v n , y ) + 1 r y v n , v n u n 0 , y K , z n + 1 = g ( v n ) ρ T v n .

Theorem 5.1 Let K be the nonempty closed convex subset of H and the bifunction f:K×KR satisfy the conditions (i)-(iv) of Lemma  2.1. Let T:KK be a α-strongly monotone and β-Lipschitz continuous single-valued operator and let g:KK be a σ-strongly monotone and δ-Lipschitz continuous single-valued operator such that g 1 exists and EP(f)GVI(K,T), where

k= 1 2 σ + δ 2 1,t= 1 2 ρ α + β 2 ρ 2 ,0< k + t 1 k <1,ρ>0.
(5.1)

Then the sequences { u n }, { v n } generated by Algorithm 5.1 converge strongly to sEP(f)GVI(K,T) and { z n } generated by Algorithm 5.1 converges strongly to z ˜ WHE(T,g).

Proof Let sEP(f)GVI(K,T), z ˜ WHE(T,g).

Step 1. Estimate z n + 1 z ˜ .

From the Lemma 2.5, we have

g(s)= P K z ˜ ,
(5.2)
z ˜ =g(s)ρTs.
(5.3)

This implies that

z n + 1 z ˜ = g ( v n ) g ( s ) ρ ( T v n T s )
(5.4)
v n s ( g ( v n ) g ( s ) ) + v n s ρ ( T v n T s )
(5.5)
= v n s 2 2 g ( v n ) g ( s ) , v n s + g ( v n ) g ( s ) 2
(5.6)
+ v n s 2 2 ρ T v n T s , v n s + ρ 2 T v n T s 2
(5.7)
1 2 σ + δ 2 v n s+ 1 2 ρ α + β 2 ρ 2 v n s
(5.8)
(k+t) v n s 2 .
(5.9)

Note that we use the strong monotonicity and Lipschitz continuity of T, g in (5.8), where k= 1 2 σ + δ 2 1, t= 1 2 ρ α + β 2 ρ 2 .

Step 2. Estimate v n s.

Employing the technique that we use in the proof of Theorem 3.1, we get

v n s u n s.

Step 3. Estimate u n s and prove the strong convergence of sequences generated by Algorithm 5.1.

From (5.3), we obtain

u n s u n s ( g ( u n ) g ( s ) ) + P K z n P K z k u n s + z n z .

It follows that

u n s ( 1 1 k ) z n z(k1).

Combining the above three steps, we have

z n + 1 z ( k + t ) v n s ( k + t ) u n s ( k + t 1 k ) z n z .

Since

0< k + t 1 k <1,

we get

z n z ˜ 0 ( n ) , u n s 0 ( n ) , v n s 0 ( n ) .

 □

Remark 5.1 Clearly, if g is the identity mapping, Theorem 5.1 reduces to Theorem 3.1.

Remark 5.2 Via Lemma 2.9, it is easy to see that if the bifunction f(x,y)=0, x,yK, Algorithm 5.1 reduces to

{ g ( u n ) = P K z n , z n + 1 = g ( u n ) ρ T u n .

This algorithm was introduced by Noor in [4], which implies that Algorithm 5.1 in this paper extends the results in [4].

Remark 5.3 It is easy to show the condition (5.1) can be satisfied, for instance

α= 7 4 ,β= 2 ,δ= 2 10 ,σ= 49 100 ,ρ= 1 4 .

5.2 Results for solving variational inequality (2.6), an equilibrium problem, and a nonexpansive mapping

In this subsection, we use a similar technique to Section 4, and the proposed results in this subsection can be considered as the generation of Section 4.

Algorithm 5.2 For a given z 0 H, compute the approximate solution z n + 1 by the iterative schemes

{ u n = α P C z n + ( 1 α ) S P C z n , f ( v n , y ) + 1 r y v n , v n u n 0 , y K , z n + 1 = ( 1 α n ) z n + α n [ v n ρ ( T v n + w n ) ] .

If S=I, Algorithm 5.2 reduces to the following.

Algorithm 5.3 For a given z 0 H, compute the approximate solution z n + 1 by the iterative schemes

{ u n = P C z n , f ( v n , y ) + 1 r y v n , v n u n 0 , y K , z n + 1 = ( 1 α n ) z n + α n [ v n ρ ( T v n + w n ) ] .

If S=I and α n =1, Algorithm 5.2 reduces to the following.

Algorithm 5.4 For a given z 0 H, compute the approximate solution z n + 1 by the iterative schemes

{ u n = P C z n , f ( v n , y ) + 1 r y v n , v n u n 0 , y K , z n + 1 = v n ρ ( T v n + w n ) .

If f=0, Algorithm 5.2 reduces to the following.

Algorithm 5.5 For a given z 0 H, compute the approximate solution z n + 1 by the iterative schemes

{ u n = α P C z n + ( 1 α ) S P C z n , z n + 1 = ( 1 α n ) z n + α n [ v n ρ ( T v n + w n ) ] ,

which was introduced in [8].

If f=0, S=I, Algorithm 5.2 reduces to the following.

Algorithm 5.6 For a given z 0 H, compute the approximate solution z n + 1 by the iterative schemes

{ u n = S P C z n , z n + 1 = ( 1 α n ) z n + α n [ v n ρ ( T v n + w n ) ] ,

which was introduced in [8].

If f=0, S=I, and α n =1, Algorithm 5.2 reduces to the following.

Algorithm 5.7 For a given z 0 H, compute the approximate solution z n + 1 by the iterative schemes

{ u n = S P C z n , z n + 1 = u n ρ ( T u n + w n ) ,

which was introduced in [6].

Theorem 5.2 Let K be the nonempty closed convex subset of H and the bifunction f satisfy the conditions (i)-(iv) of Lemma  2.1. Let T:KK be a relaxed (γ,r)-co-coercive and μ-Lipschitz continuous operator, and S:KK be k-strictly such that EP(f)VI(K,T)F(S). Let M:H 2 H be a multi-valued relaxed monotone and Lipschitz continuous operator with the corresponding constant k>0, m>0, where

n = 0 α n =,α[k,1),0<ρ< 2 ( r γ μ k ) ( μ + m ) 2 ,r>γμ+k.

Then the sequences { u n }, { v n } generated by Algorithm 5.2 converge strongly to c ˜ EP(f)VI(K,T)F(S), and { z n } generated by Algorithm 5.2 converges strongly to z ˜ WHE(T,S).

Proof Let c ˜ EP(f)VI(K,T)F(S) and z ˜ WHE(T,S).

Step 1. Estimate z n + 1 z ˜ and u n c ˜ .

By the same technique in [29], we have

z n + 1 z ˜ ( 1 α n ) z n z ˜ + α n θ v n c ˜ , u n c ˜ z n z ˜ ,

where θ= 1 + 2 ρ ( γ μ r + k ) + ρ 2 ( μ + m ) 2 .

Step 2. Estimate v n c ˜ .

Applying the technique in Theorem 3.2 of this paper, we get

v n c ˜ u n c ˜ .

Combining the above two steps, we can obtain

z n + 1 z ˜ ( 1 α n ) z n z ˜ + α θ u n c ˜ ( 1 α n ) z n z ˜ + α θ z n c ˜ = [ 1 α n ( 1 θ ) ] z n z ˜ .

From Lemma 2.8, it follows that

z n z ˜ 0(n).

Note that

v n s u n s z n z.

We conclude that

u n s 0 ( n ) , v n s 0 ( n ) .

 □

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Acknowledgements

The authors would like to thank the referees for the valuable suggestions, which helped to improve this manuscript. This project is supported by the National Natural Science Foundation of China (11071109).

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Wang, Y., Zhang, C. Wiener-Hopf equation technique for solving equilibrium problems and variational inequalities and fixed points of a nonexpansive mapping. J Inequal Appl 2014, 286 (2014). https://doi.org/10.1186/1029-242X-2014-286

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Keywords

  • equilibrium problems
  • algorithms
  • Wiener-Hopf equation technique
  • auxiliary principle