Open Access

A new iterative algorithm for solving common solutions of generalized mixed equilibrium problems, fixed point problems and variational inclusion problems with minimization problems

Fixed Point Theory and Applications20122012:111

https://doi.org/10.1186/1687-1812-2012-111

Received: 25 January 2012

Accepted: 21 June 2012

Published: 12 July 2012

Abstract

In this article, we introduce a new general iterative method for solving a common element of the set of solutions of fixed point for nonexpansive mappings, the set of solutions of generalized mixed equilibrium problems and the set of solutions of the variational inclusion for a β-inverse-strongly monotone mapping in a real Hilbert space. We prove that the sequence converges strongly to a common element of the above three sets under some mild conditions. Our results improve and extend the corresponding results of Marino and Xu (J. Math. Anal. Appl. 318:43-52, 2006), Su et al. (Nonlinear Anal. 69:2709-2719, 2008), Tan and Chang (Fixed Point Theory Appl. 2011:915629, 2011) and some authors.

MSC:46C05, 47H09, 47H10.

Keywords

nonexpansive mappinginverse-strongly monotone mappinggeneralized mixed equilibrium problemvariational inclusion

1 Introduction

Let C be a nonempty closed convex subset of a real Hilbert space H with the inner product , and the norm , respectively. A mapping S : C C is said to be nonexpansive if S x S y x y x , y C . If C is bounded closed convex and S is a nonexpansive mapping of C into itself, then F ( S ) : = { x C : S x = x } is nonempty [1]. A mapping S : C C is said to be a k-strictly pseudo-contraction[2] if there exists 0 k < 1 such that S x S y 2 x y 2 + k ( I S ) x ( I S ) y 2 x , y C , where I denotes the identity operator on C. We denote weak convergence and strong convergence by notations and →, respectively. A mapping A of C into H is called monotone if A x A y , x y 0 x , y C . A mapping A is called α-inverse-strongly monotone if there exists a positive real number α such that A x A y , x y α A x A y 2 x , y C . A mapping A is called α-strongly monotone if there exists a positive real number α such that A x A y , x y α x y 2 x , y C . It is obvious that any α-inverse-strongly monotone mappings A is a monotone and 1 α -Lipschitz continuous mapping. A linear bounded operator A is called strongly positive if there exists a constant γ ¯ > 0 with the property A x , x γ ¯ x 2 x H . A self mapping f : C C is called contraction on C if there exists a constant α ( 0 , 1 ) such that f ( x ) f ( y ) α x y x , y C .

Let B : H H be a single-valued nonlinear mapping and M : H 2 H be a set-valued mapping. The variational inclusion problem is to find x H such that
θ B ( x ) + M ( x ) ,
(1.1)

where θ is the zero vector in H. The set of solutions of (1.1) is denoted by I ( B , M ) . The variational inclusion has been extensively studied in the literature. See, e.g.[310] and the reference therein.

A set-valued mapping M : H 2 H is called monotone if x , y H , f M ( x ) and g M ( y ) imply x y , f g 0 . A monotone mapping M is maximal if its graph G ( M ) : = { ( f , x ) H × H : f M ( x ) } of M is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping M is maximal if and only if for ( x , f ) H × H , x y , f g 0 for all ( y , g ) G ( M ) imply f M ( x ) .

Let B be an inverse-strongly monotone mapping of C into H and let N C v be normal cone to C at v C i.e. N C v = { w H : v u , w 0 , u C } , and define
M v = { B v + N C v , if v C , , if v C .

Then M is a maximal monotone and θ M v if and only if v VI ( C , B ) (see [11]).

Let M : H 2 H be a set-valued maximal monotone mapping, then the single-valued mapping J M , λ : H H defined by
J M , λ ( x ) = ( I + λ M ) 1 ( x ) , x H
(1.2)

is called the resolvent operator associated with M, where λ is any positive number and I is the identity mapping. In the worth mentioning that the resolvent operator is nonexpansive, 1-inverse-strongly monotone and that a solution of problem (1.1) is a fixed point of the operator J M , λ ( I λ B ) for all λ > 0 (see [12]).

Let F be a bifunction of C × C into R , where R is the set of real numbers, Ψ : C H be a mapping and ψ : C R be a real-valued function. The generalized mixed equilibrium problem for finding x C such that
F ( x , y ) + Ψ x , y x + ψ ( y ) ψ ( x ) 0 , y C .
(1.3)
The set of solutions of (1.3) is denoted by GMEP ( F , ψ , Ψ ) , that is
GMEP ( F , ψ , Ψ ) = { x C : F ( x , y ) + Ψ x , y x + ψ ( y ) ψ ( x ) 0 , y C } .
If Ψ 0 and ψ 0 , the problem (1.3) is reduced into the equilibrium problem (see also [13]) for finding x C such that
F ( x , y ) 0 , y C .
(1.4)
The set of solutions of (1.4) is denoted by EP ( F ) , that is
EP ( F ) = { x C : F ( x , y ) 0 , y C } .

This problem contains fixed point problems, includes as special cases numerous problems in physics, optimization and economics. Some methods have been proposed to solve the equilibrium problem, please consult [1416].

If F 0 and ψ 0 , the problem (1.3) is reduced into the Hartmann-Stampacchia variational inequality[17] for finding x C such that
Ψ x , y x 0 , y C .
(1.5)

The set of solutions of (1.5) is denoted by VI ( C , Ψ ) . The variational inequality has been extensively studied in the literature [18].

If F 0 and Ψ 0 , the problem (1.3) is reduced into the minimize problem for finding x C such that
ψ ( y ) ψ ( x ) 0 , y C .
(1.6)
The set of solutions of (1.6) is denoted by Argmin ( ψ ) . Iterative methods for nonexpansive mappings have recently been applied to solve convex minimization problems. Convex minimization problems have a great impact and influence in the development of almost all branches of pure and applied sciences. A typical problem is to minimize a quadratic function over the set of the fixed points of a nonexpansive mapping on a real Hilbert space H:
θ ( x ) = 1 2 A x , x x , y , x F ( S ) ,
(1.7)

where A is a linear bounded operator, F ( S ) is the fixed point set of a nonexpansive mapping S and y is a given point in H[19].

In 2000, Moudafi [20] introduced the viscosity approximation method for nonexpansive mapping and prove that if H is a real Hilbert space, the sequence { x n } defined by the iterative method below, with the initial guess x 0 C is chosen arbitrarily,
x n + 1 = α n f ( x n ) + ( 1 α n ) S x n , n 0 ,
(1.8)
where { α n } ( 0 , 1 ) satisfies certain conditions, converge strongly to a fixed point of S (say x ¯ C ) which is the unique solution of the following variational inequality:
( I f ) x ¯ , x x ¯ 0 , x F ( S ) .
(1.9)
In 2005, Iiduka and Takahashi [21] introduced following iterative process x 0 C
x n + 1 = α n u + ( 1 α n ) S P C ( x n λ n A x n ) , n 0 ,
(1.10)
where u C { α n } ( 0 , 1 ) and { λ n } [ a , b ] for some a b with 0 < a < b < 2 β . They proved that under certain appropriate conditions imposed on { α n } and { λ n } , the sequence { x n } generated by (1.10) converges strongly to a common element of the set of fixed points of nonexpansive mapping and the set of solutions of the variational inequality for an inverse-strongly monotone mapping (say x ¯ C ) which solve some variational inequality
x ¯ u , x x ¯ 0 , x F ( S ) .
(1.11)
In 2006, Marino and Xu [19] introduced a general iterative method for nonexpansive mapping. They defined the sequence { x n } generated by the algorithm x 0 C
x n + 1 = α n γ f ( x n ) + ( I α n A ) S x n , n 0 ,
(1.12)
where { α n } ( 0 , 1 ) and A is a strongly positive linear bounded operator. They proved that if C = H and the sequence { α n } satisfies appropriate conditions, then the sequence { x n } generated by (1.12) converge strongly to a fixed point of S (say x ¯ H ) which is the unique solution of the following variational inequality:
( A γ f ) x ¯ , x x ¯ 0 , x F ( S ) ,
(1.13)
which is the optimality condition for the minimization problem
min x F ( S ) 1 2 A x , x h ( x ) ,
(1.14)

where h is a potential function for γf (i.e. h ( x ) = γ f ( x ) for x H ).

In 2008, Su et al.[22] introduced the following iterative scheme by the viscosity approximation method in a real Hilbert space: x 1 , u n H
{ F ( u n , y ) + 1 r n y u n , u n x n 0 , y C , x n + 1 = α n f ( x n ) + ( 1 α n ) S P C ( u n λ n A u n ) ,
(1.15)

for all n N , where { α n } [ 0 , 1 ) and { r n } ( 0 , ) satisfy some appropriate conditions. Furthermore, they proved { x n } and { u n } converge strongly to the same point z F ( S ) VI ( C , A ) EP ( F ) where z = P F ( S ) VI ( C , A ) EP ( F ) f ( z ) .

In 2011, Tan and Chang [10] introduced following iterative process for { T n : C C } is a sequence of nonexpansive mappings. Let { x n } be the sequence defined by
x n + 1 = α n x n + ( 1 α n ) ( S P C ( ( 1 t n ) J M , λ ( I λ A ) T n ( I μ B ) ) x n ) , n 0 ,
(1.16)

where { α n } ( 0 , 1 ) λ ( 0 , 2 α ] and μ ( 0 , 2 β ] . The sequence { x n } defined by (1.16) converges strongly to a common element of the set of fixed points of nonexpansive mappings, the set of solutions of the variational inequality and the generalized equilibrium problem.

In this article, we mixed and modified the iterative methods (1.12), (1.15) and (1.16) by purposing the following new general viscosity iterative method: x 0 , u n C and
{ u n = T r n ( F 1 , ψ 1 ) ( x n r n B 1 x n ) , v n = T s n ( F 2 , ψ 2 ) ( x n s n B 2 x n ) , x n + 1 = ξ n P C [ α n γ f ( x n ) + ( I α n A ) S J M , λ ( I λ B ) u n ] + ( 1 ξ n ) v n , n 0 ,

where { α n } , { ξ n } ( 0 , 1 ) , λ ( 0 , 2 β ) such that 0 < a λ b < 2 β , { r n } ( 0 , 2 η ) with 0 < c d 1 η and { s n } ( 0 , 2 ρ ) with 0 < e f 1 ρ satisfy some appropriate conditions. The purpose of this article, we show that under some control conditions the sequence { x n } converges strongly to a common element of the set of fixed points of nonexpansive mappings, the common solutions of the generalized mixed equilibrium problem and the set of solutions of the variational inclusion in a real Hilbert space.

2 Preliminaries

Let H be a real Hilbert space with the inner product , and the norm , respectively. Let C be a nonempty closed convex subset of H. Recall that the metric (nearest point) projection P C from H onto C assigns to each x H , the unique point in P C x C satisfying the property
x P C x = min y C x y .

The following characterizes the projection P C . We recall some lemmas which will be needed in the rest of this article.

Lemma 2.1 The function u C is a solution of the variational inequality (1.5) if and only if u C satisfies the relation u = P C ( u λ Ψ u ) for all λ > 0 .

Lemma 2.2 For a given z H , u C , u = P C z u z , v u 0 , v C .

It is well known that P C is a firmly nonexpansive mapping of H onto C and satisfies
P C x P C y 2 P C x P C y , x y , x , y H .
(2.1)
Moreover, P C x is characterized by the following properties: P C x C and for all x H , y C ,
x P C x , y P C x 0 .
(2.2)

Lemma 2.3 ([23])

Let M : H 2 H be a maximal monotone mapping and let B : H H be a monotone and Lipshitz continuous mapping. Then the mapping L = M + B : H 2 H is a maximal monotone mapping.

Lemma 2.4 ([24])

Each Hilbert space H satisfies Opial’s condition, that is, for any sequence { x n } H with x n x , the inequality lim inf n x n x < lim inf n x n y , hold for each y H with y x .

Lemma 2.5 ([25])

Assume { a n } is a sequence of nonnegative real numbers such that
a n + 1 ( 1 γ n ) a n + δ n , n 0 ,
where { γ n } ( 0 , 1 ) and { δ n } is a sequence in R 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 2.6 ([26])

Let C be a closed convex subset of a real Hilbert space H and let T : C C be a nonexpansive mapping. Then I T is demiclosed at zero, that is,
x n x , x n T x n 0

implies x = T x .

For solving the generalized mixed equilibrium problem, let us assume that the bifunction F : C × C R , the nonlinear mapping Ψ : C H is continuous monotone and ψ : C R satisfies the following conditions:

(A1) F ( x , x ) = 0 for all x C ;

(A2) F is monotone, i.e., F ( x , y ) + F ( y , x ) 0 for any x , y C ;

(A3) for each fixed y C , x F ( x , y ) is weakly upper semicontinuous;

(A4) for each fixed x C , y F ( x , y ) is convex and lower semicontinuous;

(B1) for each x C and r > 0 , there exist a bounded subset D x C and y x C such that for any z C D x ,
F ( z , y x ) + ψ ( y x ) ψ ( z ) + 1 r y x z , z x < 0 ,
(2.3)

(B2) C is a bounded set.

Lemma 2.7 ([27])

Let C be a nonempty closed convex subset of a real Hilbert space H. Let F : C × C R be a bifunction mapping satisfies (A 1)-(A 4) and let ψ : C R is convex and lower semicontinuous such that C dom ψ . Assume that either (B 1) or (B 2) holds. For r > 0 and x H , then there exists u C such that
F ( u , y ) + ψ ( y ) ψ ( u ) + 1 r y u , u x 0 .
Define a mapping T r ( F , ψ ) : H C as follows:
T r ( F , ψ ) ( x ) = { u C : F ( u , y ) + ψ ( y ) ψ ( u ) + 1 r y u , u x 0 , y C }
(2.4)
for all x H . Then, the following hold:
  1. (i)

    T r ( F , ψ ) is single-valued;

     
  2. (ii)
    T r ( F , ψ ) is firmly nonexpansive, i.e., for any x , y H ,
    T r ( F , ψ ) x T r ( F , ψ ) y 2 T r ( F , ψ ) x T r ( F , ψ ) y , x y ;
     
  3. (iii)

    F ( T r ( F , ψ ) ) = MEP ( F , ψ ) ;

     
  4. (iv)

    MEP ( F , ψ ) is closed and convex.

     

Lemma 2.8 ([19])

Assume A is a strongly positive linear bounded operator on a Hilbert space H with coefficient γ ¯ > 0 and 0 < ρ A 1 , then I ρ A 1 ρ γ ¯ .

Lemma 2.9 ([28])

Let H be a real Hilbert space and A : H H a mapping.
  1. (i)

    If A is δ-strongly monotone and μ-strictly pseudo-contraction with δ + μ > 1 , then I A is contraction with constant ( 1 δ ) / μ .

     
  2. (ii)

    If A is δ-strongly monotone and μ-strictly pseudo-contraction with δ + μ > 1 , then for any fixed number τ ( 0 , 1 ) , I τ A is contraction with constant 1 τ ( 1 ( 1 δ ) / μ ) .

     

3 Strong convergence theorems

In this section, we show a strong convergence theorem which solves the problem of finding a common element of F ( S ) , GMEP ( F 1 , ψ 1 , B 1 ) , GMEP ( F 2 , ψ 2 , B 2 ) and I ( B , M ) of an inverse-strongly monotone mapping in a real Hilbert space.

Theorem 3.1 Let H be a real Hilbert space, C be a closed convex subset of H. Let F 1 , F 2 be two bifunctions of C × C into R satisfying (A 1)-(A 4) and B , B 1 , B 2 : C H be β , η , ρ -inverse-strongly monotone mappings, ψ 1 , ψ 2 : C R be convex and lower semicontinuous function, f : C C be a contraction with coefficient α ( 0 < α < 1 ), M : H 2 H be a maximal monotone mapping and A be a δ-strongly monotone and μ-strictly pseudo-contraction mapping with δ + μ > 1 , γ is a positive real number such that γ < 1 α ( 1 1 δ μ ) . Assume that either (B 1) or (B 2) holds. Let S be a nonexpansive mapping of H into itself such that
Θ : = F ( S ) GMEP ( F 1 , ψ 1 , B 1 ) GMEP ( F 2 , ψ 2 , B 2 ) I ( B , M ) .
Suppose { x n } is a sequence generated by the following algorithm x 0 C arbitrarily:
{ u n = T r n ( F 1 , ψ 1 ) ( x n r n B 1 x n ) , v n = T s n ( F 2 , ψ 2 ) ( x n s n B 2 x n ) , x n + 1 = ξ n P C [ α n γ f ( x n ) + ( I α n A ) S J M , λ ( I λ B ) u n ] + ( 1 ξ n ) v n , n 0 ,
(3.1)

where { α n } , { ξ n } ( 0 , 1 ) , λ ( 0 , 2 β ) such that 0 < a λ b < 2 β , { r n } ( 0 , 2 η ) with 0 < c d 1 η and { s n } ( 0 , 2 ρ ) with 0 < e f 1 ρ satisfy the following conditions:

(C1): lim n α n = 0 , Σ n = 0 α n = , Σ n = 1 | α n + 1 α n | < ,

(C2): 0 < lim inf n ξ n < lim sup n ξ n < 1 , Σ n = 1 | ξ n + 1 ξ n | < ,

(C3): lim inf n r n > 0 and lim n | r n + 1 r n | = 0 ,

(C4): lim inf n s n > 0 and lim n | s n + 1 s n | = 0 .

Then { x n } converges strongly to q Θ , where q = P Θ ( γ f + I A ) ( q ) which solves the following variational inequality:
( γ f A ) q , p q 0 , p Θ
which is the optimality condition for the minimization problem
min q Θ 1 2 A q , q h ( q ) ,
(3.2)

where h is a potential function for γf (i.e., h ( q ) = γ f ( q ) for q H ).

Proof Since B is β-inverse-strongly monotone mappings, we have
( I λ B ) x ( I λ B ) y 2 = ( x y ) λ ( B x B y ) 2 = x y 2 2 λ x y , B x B y + λ 2 B x B y 2 x y 2 + λ ( λ 2 β ) B x B y 2 x y 2 .
(3.3)
And B 1 , B 2 are η , ρ -inverse-strongly monotone mappings, we have
( I r n B 1 ) x ( I r n B 1 ) y 2 = ( x y ) r n ( B 1 x B 1 y ) 2 = x y 2 2 r n x y , B 1 x B 1 y + r n 2 B 1 x B 1 y 2 x y 2 + r n ( r n 2 η ) B 1 x B 1 y 2 x y 2 .
(3.4)
In similar way, we can obtain
( I s n B 2 ) x ( I s n B 2 ) y 2 x y 2 .
(3.5)

It is clear that if 0 < λ < 2 β , 0 < r n < 2 η , 0 < s n 2 ρ then I λ B , I r n B 1 , I s n B 2 are all nonexpansive. We will divide the proof into six steps.

Step 1. We will show { x n } is bounded. Put y n = J M , λ ( u n λ B u n ) , n 0 . It follows that
y n q = J M , λ ( u n λ B u n ) J M , λ ( q λ B q ) u n q .
(3.6)
By Lemma 2.7, we have u n = T r n ( F 1 , ψ 1 ) ( x n r n B 1 x n ) for all n 0 . Then, we note that
u n q 2 = T r n ( F 1 , ψ 1 ) ( x n r n B 1 x n ) T r n ( F 1 , ψ 1 ) ( q r n B 1 q ) 2 ( x n r n B 1 x n ) ( q r n B 1 q ) 2 x n q 2 + r n ( r n 2 η ) B 1 x n B 1 q 2 x n q 2 .
(3.7)
In similar way, we can obtain
v n q 2 = T s n ( F 2 , ψ 2 ) ( x n s n B 2 x n ) T s n ( F 2 , ψ 2 ) ( q s n B 2 q ) 2 ( x n s n B 2 x n ) ( q s n B 2 q ) 2 x n q 2 + s n ( s n 2 ρ ) B 2 x n B 2 q 2 x n q 2 .
(3.8)
Put z n = P C [ α n γ f ( x n ) + ( I α n A ) S y n ] for all n 0 . From (3.1) and Lemma 2.9(ii), we deduce that
(3.9)
It follows from induction that
x n q max { x 0 q , γ f ( q ) A q 1 1 δ μ γ α } , n 0 .

Therefore { x n } is bounded, so are { v n } , { y n } , { z n } , { S y n } , { f ( x n ) } and { A S y n } .

Step 2. We claim that lim n x n + 2 x n + 1 = 0 . From (3.1), we have
x n + 2 x n + 1 = ξ n + 1 z n + 1 + ( 1 ξ n + 1 ) v n + 1 ξ n z n ( 1 ξ n ) v n = ξ n + 1 ( z n + 1 z n ) + ( ξ n + 1 ξ n ) z n + ( 1 ξ n + 1 ) ( v n + 1 v n ) + ( ξ n + 1 ξ n ) v n ξ n + 1 z n + 1 z n + ( 1 ξ n + 1 ) v n + 1 v n + | ξ n + 1 ξ n | ( z n + v n ) .
(3.10)
We will estimate v n + 1 v n . On the other hand, from v n 1 = T s n 1 ( F 2 , ψ 2 ) ( x n 1 s n 1 B 2 x n 1 ) and v n = T s n ( F 2 , ψ 2 ) ( x n s n B 2 x n ) , it follows that
(3.11)
and
F 2 ( v n , y ) + B 2 x n , y v n + ψ 2 ( y ) ψ 2 ( v n ) + 1 s n y v n , v n x n 0 , y C .
(3.12)
Substituting y = v n in (3.11) and y = v n 1 in (3.12), we get
F 2 ( v n 1 , v n ) + B 2 x n 1 , v n v n 1 + ψ 2 ( v n ) ψ 2 ( v n 1 ) + 1 s n 1 v n v n 1 , v n 1 x n 1 0
and
F 2 ( v n , v n 1 ) + B 2 x n , v n 1 v n + ψ 2 ( v n 1 ) ψ 2 ( v n ) + 1 s n v n 1 v n , v n x n 0 .
From (A2), we obtain
v n v n 1 , B 2 x n 1 B 2 x n + v n 1 x n 1 s n 1 v n x n s n 0 ,
and then
v n v n 1 , s n 1 ( B 2 x n 1 B 2 x n ) + v n 1 x n 1 s n 1 s n ( v n x n ) 0 ,
so
v n v n 1 , s n 1 B 2 x n 1 s n 1 B 2 x n + v n 1 v n + v n x n 1 s n 1 s n ( v n x n ) 0 .
It follows that
Without loss of generality, let us assume that there exists a real number e such that s n 1 > e > 0 , for all n N . Then, we have
v n v n 1 2 v n v n 1 , x n x n 1 + ( 1 s n 1 s n ) ( v n x n ) v n v n 1 { x n x n 1 + | 1 s n 1 s n | v n x n }
and hence
v n v n 1 x n x n 1 + 1 s n | s n s n 1 | v n x n x n x n 1 + M 1 e | s n s n 1 | ,
(3.13)
where M 1 = sup { v n x n : n N } . Substituting (3.13) into (3.10) that
x n + 2 x n + 1 ξ n + 1 z n + 1 z n + ( 1 ξ n + 1 ) { x n + 1 x n + M 1 e | s n s n 1 | } + | ξ n + 1 ξ n | ( z n + v n ) .
(3.14)
We note that
z n + 1 z n = P C [ α n + 1 γ f ( x n + 1 ) + ( I α n + 1 A ) S y n + 1 ] P C [ α n γ f ( x n ) ( I α n A ) S y n ] α n + 1 γ f ( x n + 1 ) + ( I α n + 1 A ) S y n + 1 ( α n γ f ( x n ) ( I α n A ) S y n ) α n + 1 γ ( f ( x n + 1 ) f ( x n ) ) + ( α n + 1 α n ) γ f ( x n ) + ( I α n + 1 A ) ( S y n + 1 S y n ) + ( α n α n + 1 ) A S y n α n + 1 γ α x n + 1 x n + | α n + 1 α n | γ f ( x n ) + ( 1 α n + 1 ( 1 1 δ μ ) ) y n + 1 y n + | α n + 1 α n | A S y n α n + 1 γ α x n + 1 x n + | α n + 1 α n | ( γ f ( x n ) + A S y n ) + ( 1 α n + 1 ( 1 1 δ μ ) ) y n + 1 y n .
(3.15)
Since I λ B be nonexpansive, we have
y n + 1 y n = J M , λ ( u n + 1 λ B u n + 1 ) J M , λ ( u n λ B u n ) ( u n + 1 λ B u n + 1 ) ( u n λ B u n ) u n + 1 u n .
(3.16)
On the other hand, from u n 1 = T r n 1 ( F 1 , ψ 1 ) ( x n 1 r n 1 B 1 x n 1 ) and u n = T r n ( F 1 , ψ 1 ) ( x n r n B 1 x n ) , it follows that
(3.17)
and
F 1 ( u n , y ) + B 1 x n , y u n + ψ 1 ( y ) ψ 1 ( u n ) + 1 r n y u n , u n x n 0 , y C .
(3.18)
Substituting y = u n in (3.17) and y = u n 1 in (3.18), we get
F 1 ( u n 1 , u n ) + B 1 x n 1 , u n u n 1 + ψ 1 ( u n ) ψ 1 ( u n 1 ) + 1 r n 1 u n u n 1 , u n 1 x n 1 0
and
F 1 ( u n , u n 1 ) + B 1 x n , u n 1 u n + ψ 1 ( u n 1 ) ψ 1 ( u n ) + 1 r n u n 1 u n , u n x n 0 .
From (A2), we obtain
u n u n 1 , B 1 x n 1 B 1 x n + u n 1 x n 1 r n 1 u n x n r n 0 ,
and then
u n u n 1 , r n 1 ( B 1 x n 1 B 1 x n ) + u n 1 x n 1 r n 1 r n ( u n x n ) 0 ,
so
u n u n 1 , r n 1 B 1 x n 1 r n 1 B 1 x n + u n 1 u n + u n x n 1 r n 1 r n ( u n x n ) 0 .
It follows that
Without loss of generality, let us assume that there exists a real number c such that r n 1 > c > 0 , for all n N . Then, we have
u n u n 1 2 u n u n 1 , x n x n 1 + ( 1 r n 1 r n ) ( u n x n ) u n u n 1 { x n x n 1 + | 1 r n 1 r n | u n x n }
and hence
u n u n 1 x n x n 1 + 1 r n | r n r n 1 | u n x n x n x n 1 + M 2 c | r n r n 1 | ,
(3.19)
where M 2 = sup { u n x n : n N } . Substituting (3.19) into (3.16), we have
y n y n 1 x n x n 1 + M 2 c | r n r n 1 | .
(3.20)
Substituting (3.20) into (3.15), we obtain that
z n + 1 z n α n + 1 γ α x n + 1 x n + | α n + 1 α n | ( γ f ( x n ) + A S y n ) + ( 1 α n + 1 ( 1 1 δ μ ) ) { x n x n 1 + M 2 c | r n r n 1 | } .
(3.21)
And substituting (3.13), (3.21) into (3.10), we get
x n + 2 x n + 1 ξ n + 1 { α n + 1 γ α x n + 1 x n + | α n + 1 α n | ( γ f ( x n ) + A S y n ) + ( 1 α n + 1 ( 1 1 δ μ ) ) x n x n 1 + M 2 c | r n r n 1 | } + ( 1 ξ n + 1 ) { x n x n 1 + M 1 e | s n s n 1 | } + | ξ n + 1 ξ n | ( z n + v n ) ( 1 ( ( 1 1 δ μ ) γ α ) ξ n + 1 α n + 1 ) x n + 1 x n + ( | α n + 1 α n | + | ξ n + 1 ξ n | ) M 3 + M 1 e | s n s n 1 | + M 2 c | r n r n 1 | ,
(3.22)
where M 3 > 0 is a constant satisfying
sup n { γ f ( x n ) + A S y n , z n + v n } M 3 .
This together with (C1)-(C4) and Lemma 2.5, imply that
lim n x n + 2 x n + 1 = 0 .
(3.23)

From (3.20), we also have y n + 1 y n 0 as n .

Step 3. We show the followings:
  1. (i)

    lim n B u n B q = 0 ;

     
  2. (ii)

    lim n B 1 x n B 1 q = 0 ;

     
  3. (iii)

    lim n B 2 x n B 2 q = 0 .

     
For q Θ and q = J M , λ ( q λ B q ) , then we get
y n q 2 = J M , λ ( u n λ B u n ) J M , λ ( q λ B q ) 2 ( u n λ B u n ) ( q λ B q ) 2 u n q 2 + λ ( λ 2 β ) B u n B q 2 x n q 2 + λ ( λ 2 β ) B u n B q 2 .
(3.24)
It follows that
z n q 2 = P C ( α n γ f ( x n ) + ( I α n A ) S y n ) P C ( q ) 2 α n ( γ f ( x n ) A q ) + ( I α n A ) ( S y n q ) 2 α n γ f ( x n ) A q 2 + ( 1 α n ( 1 1 δ μ ) ) y n q 2 + 2 α n ( 1 α n ( 1 1 δ μ ) ) γ f ( x n ) A q y n q α n γ f ( x n ) A q 2 + 2 α n ( 1 α n ( 1 1 δ μ ) ) γ f ( x n ) A q y n q + ( 1 α n ( 1 1 δ μ ) ) { x n q 2 + λ ( λ 2 β ) B u n B q 2 } α n γ f ( x n ) A q 2 + 2 α n ( 1 α n ( 1 1 δ μ ) ) γ f ( x n ) A q y n q + x n q 2 + ( 1 α n ( 1 1 δ μ ) ) λ ( λ 2 β ) B u n B q 2 .
(3.25)
By the convexity of the norm , we have
x n + 1 q 2 = ξ n z n + ( 1 ξ n ) v n q 2 ξ n ( z n q ) + ( 1 ξ n ) ( v n q ) 2 ξ n z n q 2 + ( 1 ξ n ) v n q 2 .
(3.26)
Substituting (3.8), (3.25) into (3.26), we obtain
So, we obtain
where ϵ n = 2 ξ n α n ( 1 α n ( 1 1 δ μ ) ) γ f ( x n ) A q y n q . Since conditions (C1)-(C3) and lim n x n + 1 x n = 0 , then we obtain that B u n B q 0 as n . We consider this inequality in (3.25) that
z n q 2 α n γ f ( x n ) A q 2 + ( 1 α n ( 1 1 δ μ ) ) y n q 2 + 2 α n ( 1 α n ( 1 1 δ μ ) ) γ f ( x n ) A q y n q .
(3.27)
Substituting (3.6) and (3.8) into (3.27), we have
z n q 2 α n γ f ( x n ) A q 2 + ( 1 α n ( 1 1 δ μ ) ) × { x n q 2 + r n ( r n 2 η ) B 1 x n B 1 q 2 } + 2 α n ( 1 α n ( 1 1 δ μ ) ) γ f ( x n ) A q y n q = α n γ f ( x n ) A q 2 + ( 1 α n ( 1 1 δ μ ) ) x n q 2 + ( 1 α n ( 1 1 δ μ ) ) r n ( r n 2 η ) B 1 x n B 1 q 2 + 2 α n ( 1 α n ( 1 1 δ μ ) ) γ f ( x n ) A q y n q α n γ f ( x n ) A q 2 + x n q 2 + ( 1 α n ( 1 1 δ μ ) ) r n ( r n 2 η ) B 1 x n B 1 q 2 + 2 α n ( 1 α n ( 1 1 δ μ ) ) γ f ( x n ) A q y n q .
(3.28)
Substituting (3.7) and (3.28) into (3.26), we obtain
x n + 1 q 2 ξ n { α n γ f ( x n ) A q 2 + x n q 2 + ( 1 α n ( 1 1 δ μ ) ) r n ( r n 2 η ) B 1 x n B 1 q 2 + 2 α n ( 1 α n ( 1 1 δ μ ) ) γ f ( x n ) A q y n q } + ( 1 ξ n ) x n q 2 = ξ n α n γ f ( x n ) A q 2 + ξ n x n q 2 + ξ n ( 1 α n ( 1 1 δ μ ) ) r n ( r n 2 η ) B 1 x n B 1 q 2 + 2 ξ n α n ( 1 α n ( 1 1 δ μ ) ) γ f ( x n ) A q y n q + ( 1 ξ n ) x n q 2 .
(3.29)
So, we also have
where ϵ n = 2 ξ n α n ( 1 α n ( 1 1 δ μ ) ) γ f ( x n ) A q y n q . Since conditions (C1)-(C3), lim n x n + 1 x n = 0 , then we obtain that B 1 x n B 1 q 0 as n . Substituting (3.24) into (3.27), we have
(3.30)
Substituting (3.8) and (3.30) into (3.26), we obtain
(3.31)
So, we also have

where ϵ n = 2 ξ n α n ( 1 α n ( 1 1 δ μ ) ) γ f ( x n ) A q y n q . Since conditions (C1), (C2), (C4), lim n x n + 1 x n = 0 and lim n B u n B q = 0 , then we obtain that B 2 x n B 2 q 0 as n .

Step 4. We show the followings:
  1. (i)

    lim n x n u n = 0 ;

     
  2. (ii)

    lim n u n y n = 0 ;

     
  3. (iii)

    lim n y n S y n = 0 .

     
Since T r n ( F 1 , ψ 1 ) is firmly nonexpansive, we observe that
u n q 2 = T r n ( F 1 , ψ 1 ) ( x n r n B 1 x n ) T r n ( F 1 , ψ 1 ) ( q r n B 1 q ) 2 ( x n r n B 1 x n ) ( q r n B 1 q ) , u n q = 1 2 ( ( x n r n B 1 x n ) ( q r n B 1 q ) 2 + u n q 2 ( x n r n B 1 x n ) ( q r n B 1 q ) ( u n q ) 2 ) 1 2 ( x n q 2 + u n q 2 ( x n u n ) r n ( B 1 x n B 1 q ) 2 ) = 1 2 ( x n q 2 + u n q 2 x n u n 2 + 2 r n B 1 x n B 1 q , x n u n r n 2 B 1 x n B 1 q 2 ) .
Hence, we have
u n q 2 x n q 2 x n u n 2 + 2 r n B 1 x n B 1 q x n u n .
(3.32)
Since J M , λ is 1-inverse-strongly monotone, we compute
y n q 2 = J M , λ ( u n λ B u n ) J M , λ ( q λ B q ) 2 ( u n λ B u n ) ( q λ B q ) , y n q = 1 2 ( ( u n λ B u n ) ( q λ B q ) 2 + y n q 2 ( u n λ B u n ) ( q λ B q ) ( y n q ) 2 ) = 1 2 ( u n q 2 + y n q 2 ( u n y n ) λ ( B u n B q ) 2 ) 1 2 ( u n q 2 + y n q 2 u n y n 2 + 2 λ u n y n , B u n B q λ 2 B u n B q 2 ) ,
(3.33)
which implies that
y n q 2 u n q 2 u n y n 2 + 2 λ u n y n B u n B q .
(3.34)
Substitute (3.32) into (3.34), we have
y n q 2 { x n q 2 x n u n 2 + 2 r n B 1 x n B 1 q x n u n } u n y n 2 + 2 λ u n y n B u n B q .
(3.35)
Substitute (3.35) into (3.27), we have
z n q 2 α n γ f ( x n ) A q 2 + ( 1 α n ( 1 1 δ μ ) ) { x n q 2 x n u n 2 + 2 r n B 1 x n B 1 q x n u n u n y n 2 + 2 λ u n y n B u n B q } + 2 α n ( 1 α n ( 1 1 δ μ ) ) γ f ( x n ) A q y n q α n γ f ( x n ) A q 2 + x n q 2 x n u n 2 + 2 ( 1 α n ( 1 1 δ μ ) ) r n B 1 x n B 1 q x n u n u n y n 2 + 2 ( 1 α n ( 1 1 δ μ ) ) λ u n y n B u n B q + 2 α n ( 1 α n ( 1 1 δ μ ) ) γ f ( x n ) A q y n q .
(3.36)
Since T s n ( F 2 , ψ 2 ) is firmly nonexpansive, we observe that
v n q 2 = T s n ( F 2 , ψ 2 ) ( x n s n B 2 x n ) T s n ( F 2 , ψ 2 ) ( q s n B 2 q ) 2 ( x n s n B 2 x n ) ( q s n B 2 q ) , v n q = 1 2 ( ( x n s n B 2 x n ) ( q s n B 2 q ) 2 + v n q 2 ( x n s n B 2 x n ) ( q s n B 2 q ) ( v n q ) 2 ) 1 2 ( x n q 2 + v n q 2 ( x n v n ) s n ( B 2 x n B 2 q ) 2 ) = 1 2 ( x n q 2 + v n q 2 x n v n 2 + 2 s n B 2 x n B 2 q , x n v n s n 2 B 2 x n B 2 q 2 ) .
Hence, we have
v n q 2 x n q 2 x n v n 2 + 2 s n B 2 x n B 2 q x n v n .
(3.37)
Substitute (3.36) and (3.37) into (3.26), we obtain
(3.38)
Then, we derive
(3.39)
By conditions (C1)-(C4), lim n x n + 1 x n = 0 , lim n B u n B q = 0 , lim n B 1 x n B 1 q = 0 and lim n B 2 x n B 2 q = 0 . So, we have x n u n 0 , u n y n 0 , x n v n 0 as n . We note that x n + 1 x n = ξ n ( z n x n ) + ( 1 ξ n ) ( v n x n ) . From lim n x n v n = 0 , lim n x n + 1 x n = 0 , and hence
lim n z n x n = 0 .
(3.40)
It follows that
x n y n x n u n + u n y n 0 , as n .
(3.41)
Since
z n y n z n x n + x n y n .
So, by (3.40) and lim n x n y n = 0 , we obtain
lim n z n y n = 0 .
(3.42)
Therefore, we observe that
S y n z n = P C S y n P C ( α n γ f ( x n ) + ( I α n A ) S y n ) S y n α n γ f ( x n ) ( I α n A ) S y n = α n γ f ( x n ) A S y n .
(3.43)
By condition (C1), we have S y n z n 0 as n . Next, we observe that
S y n y n S y n z n + z n y n .

By (3.42) and (3.43), we have S y n y n 0 as n .

Step 5. We show that q Θ : = F ( S ) GMEP ( F 1 , ψ 1 , B 1 ) GMEP ( F 2 , ψ 2 , B 2 ) I ( B , M ) and lim sup n ( γ f A ) q , S y n q 0 . It is easy to see that P Θ ( γ f + ( I A ) ) is a contraction of H into itself. In fact, from Lemma 2.9, we have
P Θ ( γ f + ( I A ) ) x P Θ ( γ f + ( I A ) ) y ( γ f + ( I A ) ) x ( γ f + ( I A ) ) y γ f ( x ) f ( y ) + ( I A ) x y γ α x y + ( 1 ( 1 1 δ μ ) ) x y = ( 1 δ μ + γ α ) x y .

Hence H is complete, there exists a unique fixed point q H such that q = P Θ ( γ f + ( I A ) ) ( q ) . By Lemma 2.2 we obtain that ( γ f A ) q , w q 0 for all w Θ .

Next, we show that lim sup n ( γ f A ) q , S y n q 0 , where q = P Θ ( γ f + I A ) ( q ) is the unique solution of the variational inequality ( γ f A ) q , p q r 0 , p Θ . We can choose a subsequence { y n i } of { y n } such that
lim sup n ( γ f A ) q , S y n q = lim i ( γ f A ) q , S y n i q .

As { y n i } is bounded, there exists a subsequence { y n i j } of { y n i } which converges weakly to w. We may assume without loss of generality that y n i w . We claim that w Θ . Since y n S y n 0 and by Lemma 2.6, we have w F ( S ) .

Next, we show that w GMEP ( F 1 , ψ 1 , B 1 ) . Since u n = T r n ( F 1 , ψ 1 ) ( x n r n B 1 x n ) , we know that
F 1 ( u n , y ) + ψ 1 ( y ) ψ 1 ( u n ) + B 1 x n , y u n + 1 r n y u n , u n x n 0 , y C .
It follows by (A2) that
ψ 1 ( y ) ψ 1 ( u n ) + B 1 x n , y u n + 1 r n y u n , u n x n F 1 ( y , u n ) , y C .
Hence,
ψ 1 ( y ) ψ 1 ( u n i ) + B 1 x n i , y u n i + 1 r n i y u n i , u n i x n i F 1 ( y , u n i ) , y C .
(3.44)
For t ( 0 , 1 ] and y H , let y t = t y + ( 1 t ) w . From (3.44), we have
y t u n i , B 1 y t y t u n i , B 1 y t ψ 1 ( y t ) + ψ 1 ( u n i ) B 1 x n i , y t u n i 1 r n i y t u n i , u n i x n i + F 1 ( y t , u n i ) = y t u n i , B 1 y t B 1 u n i + y t u n i , B 1 u n i B 1 x n i ψ 1 ( y t ) + ψ 1 ( u n i ) 1 r n i y t u n i , u n i x n i + F 1 ( y t , u n i ) .
From u n i x n i 0 , we have B 1 u n i B 1 x n i 0 . Further, from (A4) and the weakly lower semicontinuity of ψ 1 , u n i x n i r n i 0 and u n i w , we have
y t w , B 1 y t ψ 1 ( y t ) + ψ 1 ( w ) + F 1 ( y t , w ) .
(3.45)
From (A1), (A4) and (3.45), we have
0 = F 1 ( y t , y t ) ψ 1 ( y t ) + ψ 1 ( y t ) t F 1 ( y t , y ) + ( 1 t ) F 1 ( y t , w ) + t ψ 1 ( y ) + ( 1 t ) ψ 1 ( w ) ψ 1 ( y t ) = t [ F 1 ( y t , y ) + ψ 1 ( y ) ψ 1 ( y t ) ] + ( 1 t ) [ F 1 ( y t , w ) + ψ 1 ( w ) ψ 1 ( y t ) ] t [ F 1 ( y t , y ) + ψ 1 ( y ) ψ 1 ( y t ) ] + ( 1 t ) y t w , B 1 y t = t [ F 1 ( y t , y ) + ψ 1 ( y ) ψ 1 ( y t ) ] + ( 1 t ) t y w , B 1 y t ,
and hence
0 F 1 ( y t , y ) + ψ 1 ( y ) ψ 1 ( y t ) + ( 1 t ) y w , B 1 y t .
Letting t 0 , we have, for each y C ,
F 1 ( w , y ) + ψ 1 ( y ) ψ 1 ( w ) + y w , B 1 w 0 .

This implies that w GMEP ( F 1 , ψ 1 , B 1 ) . By following the same arguments, we can show that w GMEP ( F 2 , ψ 2 , B 2 ) .

Lastly, we show that w I ( B , M ) . In fact, since B is a β-inverse-strongly monotone, B is monotone and Lipschitz continuous mapping. It follows from Lemma 2.3 that M + B is a maximal monotone. Let ( v , g ) G ( M + B ) , since g B v M ( v ) . Again since y n i = J M , λ ( u n i λ B u n i ) , we have u n i λ B u n i ( I + λ M ) ( y n i ) , that is, 1 λ ( u n i y n i λ B u n i ) M ( y n i ) . By virtue of the maximal monotonicity of M + B , we have
v y n i , g B v 1 λ ( u n i y n i λ B u n i ) 0 ,
and hence
v y n i , g v y n i , B v + 1 λ ( u n i y n i λ B u n i ) = v y n i , B v B y n i + v y n i , B y n i B u n i + v y n i , 1 λ ( u n i y n i ) .
It follows from lim n u n y n = 0 , we have lim n B u n B y n = 0 and y n i w that
lim sup n v y n , g = v w , g 0 .
It follows from the maximal monotonicity of B + M that θ ( M + B ) ( w ) , that is, w I ( B , M ) . Therefore, w Θ . It follows that
lim sup n ( γ f A ) q , S y n q = lim i ( γ f A ) q , S y n i q = ( γ f A ) q , w q 0 .
Step 6. We prove x n q . By using (3.1) and together with Schwarz inequality, we have
x n + 1 q 2 = ξ n P C [ ( α n γ f ( x n ) + ( I α n A ) S y n ) q ] + ( 1 ξ n ) ( v n q ) 2 ξ n P C [ ( α n γ f ( x n ) + ( I α n A ) S y n ) P C ( q ) ] 2 + ( 1 ξ n ) v n q 2 ξ n α n ( γ f ( x n ) A q ) + ( I α n A ) ( S y n q ) 2 + ( 1 ξ n ) x n q 2 ξ n ( I α n A ) 2 S y n q 2 + ξ n α n 2 γ f ( x n ) A q 2 + 2 ξ n α n ( I α n A ) ( S y n q ) , γ f ( x n ) A q + ( 1 ξ n ) x n q 2 ξ n ( 1 α n ( 1 1 δ μ ) ) 2 y n q 2 + ξ n α n 2 γ f ( x n ) A q 2 + 2 ξ n α n S y n q , γ f ( x n ) A q 2 ξ n α n 2 A ( S y n q ) , γ f ( x n ) A q + ( 1 ξ n ) x n q 2 ξ n ( 1 α n ( 1 1 δ μ ) ) 2 x n q 2 + ξ n α n 2 γ f ( x n ) A q 2 + 2 ξ n α n S y n q , γ f ( x n ) γ f ( q ) + 2 ξ n α n S y n q , γ f ( q ) A q 2 ξ n α n 2 A ( S y n q ) , γ f ( x n ) A q + ( 1 ξ n ) x n q 2 ξ n ( 1 α n ( 1 1 δ μ ) ) 2 x n q 2 + ξ n α n 2 γ f ( x n ) A q 2 + 2 ξ n α n S y n q γ f ( x n ) γ f ( q ) + 2 ξ n α n S y n q , γ f ( q ) A q 2 ξ n α n 2 A ( S y n q ) , γ f ( x n ) A q + ( 1 ξ n ) x n q 2 ξ n ( 1 α n ( 1 1 δ μ ) ) 2 x n q 2 + ξ n α n 2 γ f ( x n ) A q 2 + 2 ξ n γ α α n y n q x n q + 2 ξ n α n S y n q , γ f ( q ) A q 2 ξ n α n 2 A ( S y n q ) , γ f ( x n ) A q + ( 1 ξ n ) x n q 2 ( ξ n 2 ξ n α n ( 1 1 δ μ ) + ξ n α n 2 ( 1 1 δ μ ) 2 ) x n q 2 + ξ n α n 2 γ f ( x n ) A q 2 + 2 ξ n γ α α n x n q 2 + 2 ξ n α n S y n q , γ f ( q ) A q 2 ξ n α n 2 A ( S y n q ) , γ f ( x n ) A q + ( 1 ξ n ) x n q 2 ( 1 2 ξ n α n ( 1 1 δ μ ) + 2 ξ n γ α α n ) x n q 2 + α n { ξ n α n γ f ( x n ) A q 2 + 2 ξ n S y n q , γ f ( q ) A q 2 ξ n α n A ( S y n q ) γ f ( x n ) A q + ξ n α n ( 1 1 δ μ ) 2 x n q 2 } = ( 1 2 ( ( 1 1 δ μ ) γ α ) ξ n α n ) x n q 2 + α n { ξ n α n γ f ( x n ) A q 2 + 2 ξ n S y n q , γ f ( q ) A q 2 ξ n α n A ( S y n q ) γ f ( x n ) A q + ξ n α n ( 1 1 δ μ ) 2 x n q 2 } .
Since { x n } is bounded, where η ξ n γ f ( x n ) A q 2 2 ξ n A ( S y n q ) γ f ( x n ) A q + ξ n ( 1 1 δ μ ) 2 x n q 2 for all n 0 . It follows that
x n + 1 q 2 ( 1 2 ( ( 1 1 δ μ ) γ α ) ξ n α n ) x n q 2 + α n ς n ,
(3.46)

where ς n = 2 ξ n S y n q , γ f ( q ) A q + η α n . By lim sup n ( γ f A ) q , S y n q 0 , we get lim sup n ς n 0 . Applying Lemma 2.5, we can conclude that x n q . This completes the proof. □

Corollary 3.2 Let H be a real Hilbert space, C be a closed convex subset of H. Let F 1 , F 2 be two bifunctions of C × C into R satisfying (A 1)-(A 4) and B , B 1 , B 2 : C H be β , η , ρ -inverse-strongly monotone mappings, ψ 1 , ψ 2 : C R be convex and lower semicontinuous function, f : C C be a contraction with coefficient α ( 0 < α < 1 ), M : H 2 H be a maximal monotone mapping. Assume that either (B 1) or (B 2) holds. Let S be a nonexpansive mapping of H into itself such that
Θ : = F ( S ) GMEP ( F 1 , ψ 1 , B 1 ) GMEP ( F 2 , ψ 2 , B 2 ) I ( B , M ) .
Suppose { x n } is a sequence generated by the following algorithm x 0 C arbitrarily:
{ u n = T r n ( F 1 , ψ 1 ) ( x n r n B 1 x n ) , v n = T s n ( F 2 , ψ 2 ) ( x n s n B 2 x n ) , x n + 1 = ξ n P C [ α n f ( x n ) + ( I α n ) S J M , λ ( I λ B ) u n ] + ( 1 ξ n ) v n ,

where { α n } , { ξ n } ( 0 , 1 ) , λ ( 0 , 2 β ) such that 0 < a λ b < 2 β , { r n } ( 0 , 2 η ) with 0 < c d 1 η and { s n } ( 0 , 2 ρ ) with 0 < e f 1 ρ satisfy the conditions (C 1)-(C 4).

Then { x n } converges strongly to q Θ , where q = P Θ ( f + I ) ( q ) which solves the following variational inequality:
( f I ) q , p q 0 , p Θ .

Proof Putting A I and γ 1 in Theorem 3.1, we can obtain desired conclusion immediately. □

Corollary 3.3 Let H be a real Hilbert space, C be a closed convex subset of H. Let F 1 , F 2 be two bifunctions of C × C into R satisfying (A 1)-(A 4) and B , B 1 , B 2 : C H be β , η , ρ -inverse-strongly monotone mappings, ψ 1 , ψ 2 : C R be convex and lower semicontinuous function. Assume that either (B 1) or (B 2) holds. Let S be a nonexpansive mapping of H into itself such that
Θ : = F ( S ) GMEP ( F 1 , ψ 1 , B 1 ) GMEP ( F 2 , ψ 2 , B 2 ) I ( B , M ) .
Suppose { x n } is a sequence generated by the following algorithm x 0 C arbitrarily:
{ u n = T r n ( F 1 , ψ 1 ) ( x n r n B 1 x n ) , v n = T s n ( F 2 , ψ 2 ) ( x n s n B 2 x n ) , x n + 1 = ξ n P C [ α n u + ( I α n ) S J M , λ ( I λ B ) u n ] + ( 1 ξ n ) v n ,

where { α n } , { ξ n } ( 0 , 1 ) , λ ( 0 , 2 β ) such that 0 < a λ b < 2 β , { r n } ( 0 , 2 η ) with 0 < c d 1 η and { s n } ( 0 , 2 ρ ) with 0 < e f 1 ρ satisfy the conditions (C 1)-(C 4).

Then { x n } converges strongly to q Θ , where q = P Θ ( q ) which solves the following variational inequality:
u q , p q 0 , p Θ .

Proof Putting f u C is a constant in Corollary 3.2, we can obtain desired conclusion immediately. □

Corollary 3.4 Let H be a real Hilbert space, C be a closed convex subset of H. Let F 1 , F 2 be two bifunctions of C × C into R satisfying (A 1)-(A 4) and B , B 1 , B 2 : C H be β , η , ρ -inverse-strongly monotone mappings, ψ 1 , ψ 2 : C R be convex and lower semicontinuous function, f : C C be a contraction with coefficient α ( 0 < α < 1 ) and A is δ-strongly monotone and μ-strictly pseudo-contraction with δ + μ > 1 , γ is a positive real number such that γ < 1 α ( 1 1 δ μ ) . Assume that either (B 1) or (B 2) holds. Let S be a nonexpansive mapping of C into itself such that
Θ : = F ( S ) GMEP ( F 1 , ψ 1 , B 1 ) GMEP ( F 2 , ψ 2 , B 2 ) VI ( C , B ) .
Suppose { x n } is a sequence generated by the following algorithm x 0 C arbitrarily:
{ u n = T r n ( F 1 , ψ 1 ) ( x n r n B 1 x n ) , v n = T s n ( F 2 , ψ 2 ) ( x n s n B 2 x n ) , x n + 1 = ξ n P C [ α n γ f ( x n ) + ( I α n A ) S P C ( I λ B ) u n ] + ( 1 ξ n ) v n ,

where { α n } , { ξ n } ( 0 , 1 ) , λ ( 0 , 2 β ) such that 0 < a λ b < 2 β , { r n } ( 0 , 2 η ) with 0 < c d 1 η and { s n } ( 0 , 2 ρ ) with 0 < e f 1 ρ satisfy the conditions (C 1)-(C 4).

Then { x n } converges strongly to q Θ , where q = P Θ ( γ f + I A ) ( q ) which solves the following variational inequality:
( γ f A ) q , p q 0 , p Θ .

Proof Taking J M , λ = P C in Theorem 3.1, we can obtain desired conclusion immediately. □

Corollary 3.5 Let H be a real Hilbert space, C be a closed convex subset of H. Let f : C C be a contraction with coefficient α ( 0 < α < 1 ), A is δ-strongly monotone and μ-strictly pseudo-contraction with δ + μ > 1 , γ is a positive real number such that γ < 1 α ( 1 1 δ μ ) . Let S be a nonexpansive mapping of C into itself such that
Θ : = F ( S ) .
Suppose { x n } is a sequence generated by the following algorithm x 0 C arbitrarily:
x n + 1 = α n γ f ( x n ) + ( I α n A ) S x n ,
where { α n } ( 0 , 1 ) and satisfy the condition lim n α n = 0 . Then { x n } converges strongly to q Θ , where q = P Θ ( γ f + I A ) ( q ) which solves the following variational inequality:
( γ f A ) q , p q 0 , p Θ .

Proof Taking ξ n 1 , P C I and B , B 1 , B 2 0 in Corollary 3.4, we can obtain desired conclusion immediately. □

Remark 3.6 Corollary 3.5 generalizes and improves the result of Marino and Xu [19].

Corollary 3.7 Let H be a real Hilbert space, C be a closed convex subset of H. Let F 1 , F 2 be two bifunctions of C × C into R satisfying (A 1)-(A 4) and B 1 , B 2 : C H be η , ρ -inverse-strongly monotone mappings, ψ 1 , ψ 2 : C R be convex and lower semicontinuous function, f : C C be a contraction with coefficient α ( 0 < α < 1 ). Assume that either (B 1) or (B 2) holds. Let S be a nonexpansive mapping of C into itself such that
Θ : = F ( S ) GMEP ( F 1 , ψ 1 , B 1 ) GMEP ( F 2 , ψ 2 , B 2 ) .
Suppose { x n } is a sequence generated by the following algorithm x 0 C arbitrarily:
{ u n = T r n ( F 1 , ψ 1 ) ( x n r n B 1 x n ) , v n = T s n ( F 2 , ψ 2 ) ( x n s n B 2 x n ) , x n + 1 = ξ n P C [ α n f ( x n ) + ( I α n ) S u n ] + ( 1 ξ n ) v n ,

where { α n } , { ξ n } ( 0 , 1 ) , { r n } ( 0 , 2 η ) with 0 < c d 1 η and { s n } ( 0 , 2 ρ ) with 0 < e f 1 ρ satisfy the conditions (C 1)-(C 4).

Then { x n } converges strongly to q Θ , where q = P Θ ( f + I ) ( q ) which solves the following variational inequality:
( f I ) q , p q 0 , p Θ .

Proof Taking γ 1 , A I , J M , λ I and B 0 in Theorem 3.1, we can obtain desired conclusion immediately. □

Remark 3.8 Corollary 3.7 generalizes and improves the result of Yao and Liou [29].

Declarations

Acknowledgements

This work was supported by the Higher Education Research Promotion and the National Research University Project of Thailand, Office of the Higher Education Commission (under NRU-CSEC Project No. 55000613). Furthermore, the authors are grateful for the reviewers for the careful reading of the article and for the suggestions which improved the quality of this study.

Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT)
(2)
Computational Science and Engineering Research Cluster (CSEC), King Mongkut’s University of Technology Thonburi (KMUTT)

References

  1. Kirk WA: Fixed point theorem for mappings which do not increase distance. Am. Math. Mon. 1965, 72: 1004–1006. 10.2307/2313345MathSciNetView ArticleGoogle Scholar
  2. Browder FE, Petryshym WV: Construction of fixed points of nonlinear mappings in Hilbert spaces. J. Math. Anal. Appl. 1963, 20: 197–228.View ArticleGoogle Scholar
  3. Chamnarnpan T, Kumam P: Iterative algorithms for solving the system of mixed equilibrium problems, fixed-point problems, and variational inclusions with application to minimization problem. J. Appl. Math. 2012., 2012:Google Scholar
  4. Hao Y: Some results of variational inclusion problems and fixed point problems with applications. Appl. Math. Mech. 2009, 30(12):1589–1596. 10.1007/s10483-009-1210-xMathSciNetView ArticleGoogle Scholar
  5. Jitpeera T, Kumam P: Hybrid algorithms for minimization problems over the solutions of generalized mixed equilibrium and variational inclusion problems. Math. Probl. Eng. 2011., 2011:Google Scholar
  6. Jitpeera T, Kumam P: Approximating common solution of variational inclusions and generalized mixed equilibrium problems with applications to optimization problems. Dyn. Contin. Discrete Impuls. Syst. 2011, 18(6):813–837.MathSciNetGoogle Scholar
  7. Liu M, Chang SS, Zuo P: An algorithm for finding a common solution for a system of mixed equilibrium problem, quasivariational inclusion problems of nonexpansive semigroup. J. Inequal. Appl. 2010., 2010:Google Scholar
  8. Petrot N, Wangkeeree R, Kumam P: A viscosity approximation method of common solutions for quasi variational inclusion and fixed point problems. Fixed Point Theory 2011, 12(1):165–178.MathSciNetGoogle Scholar
  9. Zhang SS, Lee JHW, Chan CK: Algorithms of common solutions to quasi variational inclusion and fixed point problems. Appl. Math. Mech. 2008, 29(5):571–581. 10.1007/s10483-008-0502-yMathSciNetView ArticleGoogle Scholar
  10. Tan JF, Chang SS: Iterative algorithms for finding common solutions to variational inclusion equilibrium and fixed point problems. Fixed Point Theory Appl. 2011., 2011:Google Scholar
  11. Rockafellar RT: On the maximality of sums of nonlinear monotone operators. Trans. Am. Math. Soc. 2000, 149: 46–55.Google Scholar
  12. Takahashi W: Nonlinear Functional Analysis. Yokohama Publishers, Yokohama; 2000.Google Scholar
  13. Blum E, Oettli W: From optimization and variational inequalities to equilibrium problems. Math. Stud. 1994, 63: 123–145.MathSciNetGoogle Scholar
  14. Combettes PL, Hirstoaga SA: Equilibrium programming in Hilbert spaces. J. Nonlinear Convex Anal. 2005, 6: 117–136.MathSciNetGoogle Scholar
  15. Flam SD, Antipin AS: Equilibrium programming using proximal-link algorithms. Math. Program. 1997, 78: 29–41.MathSciNetView ArticleGoogle Scholar
  16. Takahashi S, Takahashi W: Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces. J. Math. Anal. Appl. 2007, 331: 506–515. 10.1016/j.jmaa.2006.08.036MathSciNetView ArticleGoogle Scholar
  17. Hartman P, Stampacchia G: On some nonlinear elliptic differential functional equations. Acta Math. 1966, 115: 271–310. 10.1007/BF02392210MathSciNetView ArticleGoogle Scholar
  18. Yao JC, Chadli O: Pseudomonotone complementarity problems and variational inequalities. In Handbook of Generalized Convexity and Monotonicity Edited by: Crouzeix JP, Haddjissas N, Schaible S. 2005, 501–558.View ArticleGoogle Scholar
  19. Marino G, Xu HK: A general iterative method for nonexpansive mapping in Hilbert space. J. Math. Anal. Appl. 2006, 318: 43–52. 10.1016/j.jmaa.2005.05.028MathSciNetView ArticleGoogle Scholar
  20. Moudafi A: Viscosity approximation methods for fixed points problems. J. Math. Anal. Appl. 2000, 241: 46–55. 10.1006/jmaa.1999.6615MathSciNetView ArticleGoogle Scholar
  21. Iiduka H, Takahashi W: Strong convergence theorems for nonexpansive mapping and inverse-strong monotone mappings. Nonlinear Anal. 2005, 61: 341–350. 10.1016/j.na.2003.07.023MathSciNetView ArticleGoogle Scholar
  22. Su Y, Shang M, Qin X: An iterative method of solution for equilibrium and optimization problems. Nonlinear Anal. 2008, 69: 2709–2719. 10.1016/j.na.2007.08.045MathSciNetView ArticleGoogle Scholar
  23. Brézis H: Opérateur Maximaux Monotones. North-Holland, Amsterdam; 1973.Google Scholar
  24. Opial Z: Weak convergence of successive approximations for nonexpansive mappings. Bull. Am. Math. Soc. 1967, 73: 591–597. 10.1090/S0002-9904-1967-11761-0MathSciNetView ArticleGoogle Scholar
  25. Xu HK: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. 2002, 66: 240–256. 10.1112/S0024610702003332View ArticleGoogle Scholar
  26. Browder FE: Nonlinear operators and nonlinear equations of evolution in Banach spaces. Proc. Symp. Pure Math. 1976, 18: 78–81.Google Scholar
  27. Peng J-W, Liou Y-C, Yao J-C: An iterative algorithm combining viscosity method with parallel method for a generalized equilibrium problem and strict pseudocontractions. Fixed Point Theory Appl. 2009., 2009:Google Scholar
  28. Wangkeeree R, Bantaojai T: A general composite algorithms for