Open Access

Convergence of a regularization algorithm for nonexpansive and monotone operators in Hilbert spaces

Fixed Point Theory and Applications20142014:180

https://doi.org/10.1186/1687-1812-2014-180

Received: 28 May 2014

Accepted: 15 August 2014

Published: 2 September 2014

Abstract

Variational inequality, fixed point and generalized equilibrium problems are investigated via a regularization algorithm. It is proved that the sequence generated in the regularization algorithm converges strongly to a common solution of the three problems in the framework of Hilbert spaces. The results presented in this paper improve and extend the corresponding ones announced by many authors.

Keywords

equilibrium problem fixed point variational inequality zero point

1 Introduction and preliminaries

In this paper, we always assume that H is a real Hilbert space with inner product x , y and induced norm x = x , x for x , y H . Let C be a nonempty, closed, and convex subset of H.

Let A : C H be a mapping. Recall that A is said to be monotone iff
A x A y , x y 0 , x , y C .
Recall that A is said to be strongly monotone iff there exists a constant α > 0 such that
A x A y , x y α x y 2 , x , y C .
For such a case, A is also said to be α-strongly monotone. Recall that A is said to be inverse-strongly monotone iff there exists a constant α > 0 such that
A x A y , x y α A x A y 2 , x , y C .

For such a case, A is also said to be α-inverse-strongly monotone.

Recall that the classical variational inequality is to find an x C such that
A x , y x 0 , y C .
(1.1)

In this paper, we always use V I ( C , A ) to denote the solution set of (1.1) and use P C denote the metric projection from H onto C. It is well known that x C is a solution of (1.1) iff x is a fixed point of the mapping P C ( I r A ) , where r > 0 is a constant, I stands for the identity mapping. If A is strongly monotone and Lipschitz continuous, the existence and uniqueness of solutions of equilibrium (1.1) is guaranteed by the Banach contraction principle.

Recall that a set-valued mapping M : H H is said to be monotone iff, for all x , y H , f M x , and g M y imply x y , f g > 0 . M is maximal iff the graph Graph ( M ) of R is not properly contained in the graph of any other monotone mapping. It is well known that a monotone mapping M is maximal if and only if, for any ( x , f ) H × H , x y , f g 0 , for all ( y , g ) Graph ( M ) implies f R x .

Let S : C C be a mapping. F ( S ) stands for the fixed point set of S; that is, F ( S ) : = { x C : x = S x } .

Recall that S is said to be contractive iff there exists a constant α ( 0 , 1 ) such that
S x S y α x y , x , y C .

For such a case, S is also said to be α-contractive. We know that the mapping enjoys a unique fixed point and Picard’s algorithm can be employed to approximate its unique fixed point.

Recall that S is said to be nonexpansive iff
S x S y x y , x , y C .

If C is a closed, bounded and convex subset of H, then F ( S ) is not empty; see [1].

Let { S i : C C } be a family of infinitely nonexpansive mappings and { γ i } be a nonnegative real sequence with 0 γ i < 1 , i 1 . For n 1 , define a mapping W n : C C as follows:
U n , n + 1 = I , U n , n = γ n S n U n , n + 1 + ( 1 γ n ) I , U n , n 1 = γ n 1 S n 1 U n , n + ( 1 γ n 1 ) I , U n , k = γ k S k U n , k + 1 + ( 1 γ k ) I , U n , k 1 = γ k 1 S k 1 U n , k + ( 1 γ k 1 ) I , U n , 2 = γ 2 S 2 U n , 3 + ( 1 γ 2 ) I , W n = U n , 1 = γ 1 S 1 U n , 2 + ( 1 γ 1 ) I .
(1.2)

Such a mapping W n is nonexpansive from C to C and it is called a W-mapping generated by S n , S n 1 , , S 1 and γ n , γ n 1 , , γ 1 ; see [2] and the references therein.

Let T : C H be a monotone mapping and let F be a bifunction of C × C into , where denotes the set of real numbers. We consider the following generalized equilibrium problem:
Find  x C  such that  F ( x , y ) + T x , y x 0 , y C .
(1.3)
In this paper, the set of such x C is denoted by E P ( F , T ) , i.e.,
GEP ( F , A ) = { x C : F ( x , y ) + T x , y x 0 , y C } .
If T 0 , the zero mapping, then the problem (1.3) is reduced to the following equilibrium problem [3]:
Find  x C  such that  F ( x , y ) 0 , y C .
(1.4)

In this paper, the set of such an x C is denoted by E P ( F ) .

If F 0 , then the problem (1.3) is reduced to the classical variational inequality (1.1).

To study equilibrium problems (1.3) and (1.4), we may assume that F 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 all x , y C ;

(A3) for each x , y , z C ,
lim sup t 0 F ( t z + ( 1 t ) x , y ) F ( x , y ) ;

(A4) for each x C , y F ( x , y ) is convex and weakly lower semi-continuous.

Many important problems have reformulations which require finding solutions of equilibriums (1.3) and (1.4), for instance, image recovery, inverse problems, network allocation, transportation problems and optimization problems; see [311] and the references therein. For solving solutions of equilibriums (1.3) and (1.4), regularization methods recently have been extensively studied; see [1128] and the references therein.

In this paper, motivated and inspired by the research going on in this direction, we study the variational inequality (1.1), and the fixed point and equilibrium problem (1.3) based on a regularization algorithm. It is proved that the sequence generated in the regularization algorithm converges strongly to a common solutions of the three problems in the framework of Hilbert spaces. The results presented in this paper improve and extend the corresponding results in Chang et al. [11], Takahashi and Takahashi [13] and Hao [29].

The following lemmas play an important role in our paper.

Lemma 1.1 [3]

Let F : C × C R be a bifunction satisfying (A1)-(A4). Then, for any r > 0 and x H , there exists z C such that
F ( z , y ) + 1 r y z , z x 0 , y C .
Define a mapping T r : H C as follows:
T r x = { z C : F ( z , y ) + 1 r y z , z x 0 , y C } , x H ,
then the following conclusions hold:
  1. (1)

    T r is single-valued;

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

    F ( T r ) = E P ( F ) ;

     
  4. (4)

    E P ( F ) is closed and convex.

     

Lemma 1.2 [30]

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. (1)

    n = 1 γ n = ;

     
  2. (2)

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

     

Then lim n α n = 0 .

Lemma 1.3 [2]

Let { S i : C C } be a family of infinitely nonexpansive mappings with a nonempty common fixed point set and let { γ i } be a real sequence such that 0 < γ i l < 1 , where l is some real number, i 1 . Then
  1. (1)

    W n is nonexpansive and F ( W n ) = i = 1 F ( S i ) , for each n 1 ;

     
  2. (2)

    for each x C and for each positive integer k, the limit lim n U n , k exists.

     
  3. (3)
    the mapping W : C C defined by
    W x : = lim n W n x = lim n U n , 1 x , x C ,
    (1.5)
     

is a nonexpansive mapping satisfying F ( W ) = i = 1 F ( S i ) and it is called the W-mapping generated by S 1 , S 2 , and γ 1 , γ 2 ,  .

Lemma 1.4 [31]

Let { x n } and { y n } be bounded sequences in H 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 n 0 and
lim sup n ( y n + 1 y n x n + 1 x n ) 0 .

Then lim n y n x n = 0 .

Lemma 1.5 [11]

Let { S i : C C } be a family of infinitely nonexpansive mappings with a nonempty common fixed point set and let { γ i } be a real sequence such that 0 < γ i l < 1 , i 1 . If K is any bounded subset of C, then
lim n sup x K W x W n x = 0 .

Throughout this paper, we always assume that 0 < γ i l < 1 , i 1 .

Lemma 1.6 [10]

Let A : C H a Lipschitz monotone mapping and let N C x be the normal cone to C at x C ; that is, N C x = { y H : x u , y , u C } . Define
D x = { A x + N C x , x C , x C .

Then D is maximal monotone and 0 D x if and only if x V I ( C , A ) .

2 Main results

Theorem 2.1 Let C be a nonempty closed convex subset of H. Let F be a bifunction from C × C to which satisfies (A1)-(A4) and let f : C C be a κ-contraction. Let A : C H be an α-inverse-strongly monotone mapping and let B : C H be a β-inverse-strongly monotone mapping. Let T : C H be a τ-inverse-strongly monotone mapping. Let { S i : C C } be a family of infinitely nonexpansive mappings. Assume that Σ = i = 1 F ( S i ) GEP ( F , T ) V I ( C , A ) V I ( C , B ) is not empty. Let { α n } , { β n } , and { γ n } be sequences in ( 0 , 1 ) such that α n + β n + γ n = 1 . Let { r n } , { s n } , and { λ n } be positive number sequences. Let x 1 C and let { x n } be a sequence generated by
{ y n = P C ( u n s n B u n ) , x n + 1 = α n f ( y n ) + β n W n P C ( y n r n A y n ) + γ n x n , n 1 ,
(2.1)
where { u n } is such that F ( u n , y ) + T x n , y u n + 1 λ n y u n , u n x n 0 , y C , and { W n } is the sequence generated in (1.5). Assume that the following restrictions hold:
  1. (a)

    0 < a λ n b < 2 τ and lim n | λ n + 1 λ n | = 0 ,

     
  2. (b)

    0 < a r n b < 2 α and lim n | r n + 1 r n | = 0 ,

     
  3. (c)

    0 < a s n b < 2 β and lim n | s n + 1 s n | = 0 ,

     
  4. (d)

    lim n α n = 0 , n = 1 α n = ,

     
  5. (e)

    0 < lim inf n γ n lim sup n γ n < 1 ,

     
where a, a , a , b, b , and b are real constants. Then { x n } converges strongly to x ¯ Σ , which solves uniquely the following variational inequality:
x ¯ f ( x ¯ ) , x ¯ x 0 , x Σ .
Proof Since A is inverse-strongly monotone, we see from restriction (b) that
( I r n A ) x ( I r n A ) y 2 = x y 2 2 r n x y , A x A y + r n 2 A x A y 2 x y 2 2 r n α A x A y 2 + r n 2 A x A y 2 = x y 2 + r n ( r n 2 α ) A x A y 2 x y 2 , x , y C .
This shows that I r n A is nonexpansive. In the same way, we find that I s n B and I λ n T are nonexpansive. Note that u n can be re-written as u n = T λ n ( I λ n T ) x n . Let x Σ . It follows that
u n x ( I λ n T ) x n ( I λ n T ) x x n x .
Putting z n = P C ( y n r n ) A y n , we see that z n x y n x x n x .
x n + 1 x α n f ( y n ) x + β n W n z n x + γ n x n x α n κ y n x + α n f ( x ) x + β n z n x + γ n x n x ( 1 α n ( 1 κ ) ) x n x + α n ( 1 κ ) f ( x ) x 1 κ .
This implies that
x n x max { x 1 x , f ( x ) x 1 κ } < .
This yields the result that the sequence { x n } is bounded, and so are { y n } , { z n } , and { u n } . Without loss of generality, we can assume that there exists a bounded set K C such that x n , y n , z n , u n K . Since u n = T λ n ( I λ n ) x n , we find that
F ( u n + 1 , y ) + 1 λ n + 1 y u n + 1 , u n + 1 ( I r n + 1 T ) x n + 1 0 , y C ,
(2.2)
and
F ( u n , y ) + 1 λ n y u n , u n ( I λ n T ) x n 0 , y C .
(2.3)
Let y = u n in (2.2) and y = u n + 1 in (2.3). By adding up these two inequalities, we obtain
u n + 1 u n , u n u n + 1 + u n + 1 ( I λ n A 3 ) x n λ n λ n + 1 ( u n + 1 ( I λ n + 1 A 3 ) x n + 1 ) 0 .
This implies that
u n + 1 u n 2 u n + 1 u n , ( I λ n + 1 T ) x n + 1 ( I λ n T ) x n + ( 1 λ n λ n + 1 ) ( u n + 1 ( I λ n + 1 T ) x n + 1 ) u n + 1 u n ( ( I λ n + 1 T ) x n + 1 ( I λ n T ) x n + | 1 λ n λ n + 1 | u n + 1 ( I λ n + 1 T ) x n + 1 ) .
It follows that
u n + 1 u n ( I λ n + 1 T ) x n + 1 ( I λ n T ) x n + | λ n + 1 λ n | λ n + 1 u n + 1 ( I λ n + 1 T ) x n + 1 x n + 1 x n + | λ n + 1 λ n | M 1 ,
(2.4)
where M 1 is an appropriate constant such that
M 1 = sup n 1 { T x n + u n + 1 ( I λ n + 1 T ) x n + 1 a } .
It follows from (2.4) that
y n + 1 y n P C ( u n + 1 s n + 1 B u n + 1 ) P C ( u n s n + 1 B u n ) + P C ( u n s n + 1 B u n ) P C ( u n s n B u n ) u n + 1 u n + | s n + 1 s n | B u n x n + 1 x n + | λ n + 1 λ n | M 1 + | s n + 1 s n | B u n .
Hence, we have
z n + 1 z n P C ( y n + 1 r n + 1 A y n + 1 ) P C ( y n r n + 1 A y n ) + P C ( y n r n + 1 A y n ) P C ( y n r n A y n ) y n + 1 y n + | r n + 1 r n | A y n x n + 1 x n + M 2 ( | λ n + 1 λ n | + | s n + 1 s n | + | r n + 1 r n | ) ,
(2.5)
where M 2 = max { M 1 , sup n 1 { A y n } , sup n 1 { B u n } } . This implies from (2.5) that
W n + 1 z n + 1 W n z n W n + 1 z n + 1 W z n + 1 + W z n + 1 W z n + W z n W n z n sup x K { W n + 1 x W x + W x W n x } + x n + 1 x n + M 2 ( | λ n + 1 λ n | + | s n + 1 s n | + | r n + 1 r n | ) ,
(2.6)
where K is the bounded subset of C defined above. Let x n + 1 = ( 1 γ n ) q n + γ n x n . It follows that
q n + 1 q n = α n + 1 f ( y n + 1 ) + β n + 1 W n + 1 z n + 1 1 γ n + 1 α n f ( y n ) + β n W n z n 1 γ n = α n + 1 1 γ n + 1 f ( y n + 1 ) + 1 α n + 1 γ n + 1 1 γ n + 1 W n + 1 z n + 1 ( α n 1 γ n f ( y n ) + 1 α n γ n 1 γ n W n z n ) = α n + 1 1 γ n + 1 ( f ( y n + 1 ) W n + 1 z n + 1 ) α n 1 γ n ( f ( y n ) W n z n ) + W n + 1 z n + 1 W n z n .
By use of (2.6), we find that
q n + 1 q n α n + 1 1 γ n + 1 f ( y n + 1 ) W n + 1 z n + 1 + α n 1 γ n f ( y n ) W n z n + W n + 1 z n + 1 W n z n α n + 1 1 γ n + 1 f ( y n + 1 ) W n + 1 z n + 1 + α n 1 γ n f ( y n ) W n z n + sup x K { W n + 1 x W x + W x W n x } + x n + 1 x n + M 2 ( | λ n + 1 λ n | + | s n + 1 s n | + | r n + 1 r n | ) .
This implies that
q n + 1 q n x n + 1 x n α n + 1 1 γ n + 1 f ( y n + 1 ) W n + 1 z n + 1 + α n 1 γ n f ( y n ) W n z n + sup x K { W n + 1 x W x + W x W n x } + M 2 ( | λ n + 1 λ n | + | s n + 1 s n | + | r n + 1 r n | ) .
It follows from restrictions (a)-(e) that
lim sup n ( q n + 1 q n x n + 1 x n ) 0 .
This implies from Lemma 1.4 that lim n q n x n = 0 . It follows that
lim n x n + 1 x n = 0 .
(2.7)
Since A is inverse-strongly monotone, we find that
z n x 2 ( I r n A ) y n ( I r n A ) x 2 y n x 2 2 r n α A y n A x 2 + r n 2 A y n A x 2 x n x 2 + r n ( r n 2 α ) A y n A x 2 .
It follows that
x n + 1 x 2 α n f ( y n ) x 2 + β n W n z n x 2 + γ n x n x 2 α n f ( y n ) x 2 + β n z n x 2 + γ n x n x 2 α n f ( y n ) x 2 + r n ( r n 2 α ) β n A y n A x 2 + x n x 2 .
Hence, we have
r n ( 2 α r n ) β n A y n A x 2 α n f ( y n ) x 2 + ( x n x + x n + 1 x ) x n x n + 1 .
By use of the restrictions (a), (d), and (e), we obtain from (2.7)
lim n A y n A x = 0 .
(2.8)
Since the metric projection is firmly nonexpansive, we find that
z n x 2 ( I r n A ) y n ( I r n A ) x , z n x = 1 2 { ( I r n A ) y n ( I r n A ) x 2 + z n x 2 ( I r n A ) y n ( I r n A ) x ( z n x ) 2 } 1 2 { y n x 2 + z n x 2 y n z n r n ( A y n A x ) 2 } 1 2 { x n x 2 + z n x 2 y n z n 2 + 2 r n y n z n A y n A x } .
Hence, we have
z n x 2 x n x 2 y n z n 2 + 2 r n y n z n A y n A x .
This further implies that
x n + 1 x 2 α n f ( y n ) x 2 + β n W n z n x 2 + γ n x n x 2 α n f ( y n ) x 2 + β n z n x 2 + γ n x n x 2 α n f ( y n ) x 2 β n y n z n 2 + 2 r n y n z n A y n A x + x n x 2 ,
which yields
β n y n z n 2 α n f ( y n ) x 2 + 2 r n y n z n A y n A x + x n x 2 x n + 1 x 2 α n f ( y n ) x 2 + 2 r n y n z n A y n A x + ( x n x + x n + 1 x ) x n + 1 x n .
By use of restrictions (b), (d), and (e), we find from (2.7) that
lim n y n z n = 0 .
(2.9)
Since B is inverse-strongly monotone, we find that
y n x 2 ( I s n B ) u n ( I s n B ) x 2 u n x 2 2 s n β B u n B x 2 + s n 2 B u n B x 2 x n x 2 + s n ( s n 2 β ) B u n B x 2 .
It follows that
x n + 1 x 2 α n f ( y n ) x 2 + β n W n z n x 2 + γ n x n x 2 α n f ( y n ) x 2 + β n y n x 2 + γ n x n x 2 α n f ( y n ) x 2 + s n ( s n 2 β ) β n B u n B x 2 + x n x 2 .
Hence, we have
s n ( 2 β s n ) β n B u n B x 2 α n f ( y n ) x 2 + ( x n x + x n + 1 x ) x n x n + 1 .
By use of the restrictions (c), (d), and (e), we obtain from (2.7)
lim n B u n B x = 0 .
(2.10)
Since the metric projection is firmly nonexpansive, we find that
y n x 2 ( I s n B ) u n ( I s n B ) x , y n x = 1 2 { ( I s n B ) u n ( I s n B ) x 2 + y n x 2 ( I s n B ) y n ( I s n B ) x ( y n x ) 2 } 1 2 { u n x 2 + y n x 2 u n y n s n ( B u n B x ) 2 } 1 2 { x n x 2 + y n x 2 u n y n 2 + 2 s n u n y n B u n B x } .
Hence, we have
y n x 2 x n x 2 u n y n 2 + 2 s n u n y n B u n B x .
This further implies that
x n + 1 x 2 α n f ( y n ) x 2 + β n W n z n x 2 + γ n x n x 2 α n f ( y n ) x 2 + β n y n x 2 + γ n x n x 2 α n f ( y n ) x 2 β n u n y n 2 + 2 s n u n y n B u n B x + x n x 2 ,
which yields
β n u n y n 2 α n f ( y n ) x 2 + 2 s n u n y n B u n B x + x n x 2 x n + 1 x 2 α n f ( y n ) x 2 + 2 s n u n y n B u n B x + ( x n x + x n + 1 x ) x n + 1 x n .
By use of restrictions (c), (d), and (e), we find from (2.7) that
lim n y n u n = 0 .
(2.11)
Since T is inverse-strongly monotone, we find that
x n + 1 x 2 α n f ( y n ) x 2 + β n u n x 2 + γ n x n x 2 α n f ( y n ) x 2 + β n x n x λ n ( T x n T x ) 2 + γ n x n x 2 α n f ( y n ) x 2 + x n x 2 λ n β n ( 2 τ λ n ) T x n T x 2 .
This implies that
λ n β n ( 2 τ λ n ) T x n T x 2 α n f ( y n ) x 2 + ( x n x + x n + 1 x ) x n x n + 1 .
In view of the restrictions (a), (d), and (e), we see from (2.7) that
lim n T x n T x = 0 .
(2.12)
Since T λ n is firmly nonexpansive, we find that
u n x 2 = T λ n ( I λ n T ) x n T λ n ( I λ n T ) x 2 ( I λ n T ) x n ( I λ n T ) x , u n x 1 2 ( x n x 2 + u n x 2 x n u n 2 + 2 λ n T x n T x x n u n ) .
This in turn implies that
u n x 2 x n x 2 x n u n 2 + 2 λ n T x n T x x n u n .
It follows that
x n + 1 x 2 α n f ( y n ) x 2 + β n u n x 2 + γ n x n x 2 α n f ( y n ) x 2 β n x n u n 2 + 2 λ n T x n T x x n u n + x n x 2 .
By use of restrictions (a), (d), and (e), we see from (2.7) and (2.12) that
lim n x n u n = 0 .
(2.13)
Next, we prove that
lim sup n f ( x ¯ ) x ¯ , x n x ¯ 0 ,
where x ¯ = P Σ f ( x ¯ ) . To see this, we choose a subsequence { x n i } of { x n } such that
lim sup n ( f I ) x ¯ , x n x ¯ = lim i ( f I ) x ¯ , x n i x ¯ .
Since { x n i } is bounded, there exists a subsequence { x n i j } of { x n i } which converges weakly to w. Without loss of generality, we may assume that x n i w . Since
β n W n z n x n x n x n + 1 + α n f ( y n ) x n .
In view of the restrictions (d) and (e), we obtain from (2.7)
lim n W n z n x n = 0 .
(2.14)
Note that
W n z n z n W n z n x n + x n u n + u n y n + y n z n .
In view of (2.8), (2.11), (2.13), and (2.14), we find that
lim n W n z n z n = 0 .
(2.15)
Suppose the contrary, w i = 1 F ( S i ) , i.e., W w w . Since y n i w , we find from Opial’s condition [32] that
lim inf i z n i w < lim inf i z n i W w lim inf i { z n i W z n i + W z n i W w } lim inf i { z n i W z n i + z n i w } .
On the other hand, we have
W z n z n W z n W n z n + W n z n y n sup x K W x W n x + W n z n z n .

In view of Lemma 1.5, we obtain from (2.15) lim n W z n z n = 0 . It follows that lim inf i z n i w < lim inf i z n i w . Thus one derives a contradiction. Thus, we have w i = 1 F ( S i ) .

Next, we show that x ¯ V I ( C , A ) . Let T be the maximal monotone mapping defined by
D x = { B x + N C x , x C , , x C .
For any given ( x , y ) Graph ( D ) , we have y B x N C x . Since y n C , by the definition of N C , we have x y n , y B x 0 . Since y n = P C ( u n s n B u n ) , we see that x y n , y n u n s n + B u n 0 . It follows that
x y n i , y x y n i , B x x y n i , B x x y n i , y n i u n i s n i + B u n i = x y n i , B x B y n i + x y n i , B y n i B u n i x y n i , y n i u n i s n i x y n i , B y n i B u n i x y n i , y n i u n i s n i .

Since B is Lipschitz continuous, we see that x w , y 0 . Notice that D is maximal monotone and hence 0 T w . This shows that w V I ( C , A ) . In the same way, we find that w V I ( C , B ) .

Next, we show that w GEP ( F , T ) . Since u n = T λ n ( I λ n T ) x n , for any y C , we find from (A2) that
T x n i , y u n i + y u n i , u n i x n i λ n i F ( y , u n i ) , y C .
(2.16)
Putting y t = t y + ( 1 t ) w for any t ( 0 , 1 ] and y C , we see that y t C . It follows from (2.16) that
y t u n i , T y t y t u n i , T y t T x n i , y t u n i y t u n i , u n i x n i λ n i + F ( y t , u n i ) = y t u n i , T y t T u n i + y t u n i , T u n i T x n i y t u n i , u n i x n i λ n i + F ( y t , u n i ) .
In view of the monotonicity of T, and the restriction (a), we obtain from (A4)
y t w , T y t F ( y t , q ) .
(2.17)
From (A1) and (A4), we see that
0 = F ( y t , y t ) t F ( y t , y ) + ( 1 t ) F ( y t , w ) t F ( y t , y ) + ( 1 t ) y t w , T y t = t F ( y t , y ) + ( 1 t ) t y w , T y t .

It follows from (A3) that w GEP ( F , T ) . This proves that lim sup n f ( x ¯ ) x ¯ , x n x ¯ 0 .

Finally, we show that x n x ¯ , as n . Note that
x n + 1 x ¯ 2 = α n f ( y n ) x ¯ , x n + 1 x ¯ + β n W n z n x ¯ , x n + 1 x ¯ + γ n x n x ¯ , x n + 1 x ¯ α n κ y n x ¯ x n + 1 x ¯ + α n f ( x ¯ ) x ¯ , x n + 1 x ¯ + β n z n x ¯ x n + 1 x ¯ + γ n x n x ¯ x n + 1 x ¯ ( 1 α n ( 1 κ ) ) x n x ¯ x n + 1 x ¯ + α n f ( x ¯ ) x ¯ , x n + 1 x ¯ .
This implies that
x n + 1 x ¯ 2 ( 1 α n ( 1 κ ) ) x n x ¯ 2 + 2 α n f ( x ¯ ) x ¯ , x n + 1 x ¯ .

From the restriction (d), we obtain from Lemma 1.2 lim n x n x ¯ = 0 . This completes the proof. □

Corollary 2.2 Let C be a nonempty closed convex subset of H. Let F be a bifunction from C × C to which satisfies (A1)-(A4) and let f : C C be a κ-contraction. Let T : C H be a τ-inverse-strongly monotone mapping. Let { S i : C C } be a family of infinitely nonexpansive mappings. Assume that Σ = i = 1 F ( S i ) GEP ( F , T ) is not empty. Let { α n } , { β n } , and { γ n } be sequences in ( 0 , 1 ) such that α n + β n + γ n = 1 . Let { λ n } be a positive number sequence. Let x 1 C and let { x n } be a sequence generated by
x n + 1 = α n f ( y n ) + β n W n u n + γ n x n , n 1 ,
where { u n } is such that F ( u n , y ) + T x n , y u n + 1 λ n y u n , u n x n 0 , y C , and { W n } is the sequence generated in (1.5). Assume that the following restrictions hold:
  1. (a)

    0 < a λ n b < 2 τ and lim n | λ n + 1 λ n | = 0 ,

     
  2. (b)

    lim n α n = 0 , n = 1 α n = ,

     
  3. (c)

    0 < lim inf n γ n lim sup n γ n < 1 ,

     
where a and b are real constants. Then { x n } converges strongly to x ¯ Σ , which solves uniquely the following variational inequality:
x ¯ f ( x ¯ ) , x ¯ x 0 , x Σ .
Corollary 2.3 Let C be a nonempty closed convex subset of H. Let F be a bifunction from C × C to which satisfies (A1)-(A4) and let f : C C be a κ-contraction. Let B : C H be a β-inverse-strongly monotone mapping. Let T : C H be a τ-inverse-strongly monotone mapping. Let { S i : C C } be a family of infinitely nonexpansive mappings. Assume that Σ = i = 1 F ( S i ) GEP ( F , T ) V I ( C , B ) is not empty. Let { α n } , { β n } , and { γ n } be sequences in ( 0 , 1 ) such that α n + β n + γ n = 1 . Let { s n } and { λ n } be positive number sequences. Let x 1 C and let { x n } be a sequence generated by
{ y n = P C ( u n s n B u n ) , x n + 1 = α n f ( y n ) + β n W n y n + γ n x n , n 1 ,
where { u n } is such that F ( u n , y ) + T x n , y u n + 1 λ n y u n , u n x n 0 , y C , and { W n } is the sequence generated in (1.5). Assume that the following restrictions hold:
  1. (a)

    0 < a λ n b < 2 τ and lim n | λ n + 1 λ n | = 0 ,

     
  2. (b)

    0 < a s n b < 2 β and lim n | s n + 1 s n | = 0 ,

     
  3. (c)

    lim n α n = 0 , n = 1 α n = ,

     
  4. (d)

    0 < lim inf n γ n lim sup n γ n < 1 ,

     
where a, a , b, and b are real constants. Then { x n } converges strongly to x ¯ Σ , which solves uniquely the following variational inequality:
x ¯ f ( x ¯ ) , x ¯ x 0 , x Σ .
Corollary 2.4 Let C be a nonempty closed convex subset of H. Let F be a bifunction from C × C to which satisfies (A1)-(A4) and let f : C C be a κ-contraction. Let A : C H be an α-inverse-strongly monotone mapping and let B : C H be a β-inverse-strongly monotone mapping. Let { S i : C C } be a family of infinitely nonexpansive mappings. Assume that Σ = i = 1 F ( S i ) E P ( F ) V I ( C , A ) V I ( C , B ) is not empty. Let { α n } , { β n } , and { γ n } be sequences in ( 0 , 1 ) such that α n + β n + γ n = 1 . Let { r n } , { s n } , and { λ n } be positive number sequences. Let x 1 C and let { x n } be a sequence generated by
{ y n = P C ( u n s n B u n ) , x n + 1 = α n f ( y n ) + β n W n P C ( y n r n A y n ) + γ n x n , n 1 ,
where { u n } is such that F ( u n , y ) + 1 λ n y u n , u n x n 0 , y C , and { W n } is the sequence generated in (1.5). Assume that the following restrictions hold:
  1. (a)

    0 < a λ n b < 2 τ and lim n | λ n + 1 λ n | = 0 ,

     
  2. (b)

    0 < a r n b < 2 α and lim n | r n + 1 r n | = 0 ,

     
  3. (c)

    0 < a s n b < 2 β and lim n | s n + 1 s n | = 0 ,

     
  4. (d)

    lim n α n = 0 , n = 1 α n = ,

     
  5. (e)

    0 < lim inf n γ n lim sup n γ n < 1 ,

     
where a, a , a , b, b , and b are real constants. Then { x n } converges strongly to x ¯ Σ , which solves uniquely the following variational inequality:
x ¯ f ( x ¯ ) , x ¯ x 0 , x Σ .
Proof In Theorem 2.1, put T = 0 . Then, for all τ ( 0 , ) , we have
x , y , T x T y τ T x T y 2 , x , y C .

Taking a , b ( 0 , ) with 0 < a < b < and choosing a sequence { λ n } of real numbers with a λ n b , we obtain the desired result by Theorem 2.1. □

Corollary 2.5 Let C be a nonempty closed convex subset of H. Let f : C C be a κ-contraction and let T : C H be a τ-inverse-strongly monotone mapping. Let A : C H be an α-inverse-strongly monotone mapping and let B : C H be a β-inverse-strongly monotone mapping. Let { S i : C C } be a family of infinitely nonexpansive mappings. Assume that Σ = i = 1 F ( S i ) V I ( C , T ) V I ( C , A ) V I ( C , B ) is not empty. Let { α n } , { β n } , and { γ n } be sequences in ( 0 , 1 ) such that α n + β n + γ n = 1 . Let { r n } , { s n } , and { λ n } be positive number sequences. Let x 1 C and let { x n } be a sequence generated by
{ u n = P C ( x n λ n T x n ) , y n = P C ( u n s n B u n ) , x n + 1 = α n f ( y n ) + β n W n P C ( y n r n A y n ) + γ n x n , n 1 ,
where { W n } is the sequence generated in (1.5). Assume that the following restrictions hold:
  1. (a)

    0 < a λ n b < 2 τ and lim n | λ n + 1 λ n | = 0 ,

     
  2. (b)

    0 < a r n b < 2 α and lim n | r n + 1 r n | = 0 ,

     
  3. (c)

    0 < a s n b < 2 β and lim n | s n + 1 s n | = 0 ,

     
  4. (d)

    lim n α n = 0 , n = 1 α n = ,

     
  5. (e)

    0 < lim inf n γ n lim sup n γ n < 1 ,

     
where a, a , a , b, b , and b are real constants. Then { x n } converges strongly to x ¯ Σ , which solves uniquely the following variational inequality:
x ¯ f ( x ¯ ) , x ¯ x 0 , x Σ .
Proof Putting F = 0 , we find that
T x n , y u n + 1 λ n y u n , u n x n 0 , y C ,
is equivalent to
y u n , x n λ n T x n u n 0 , y C ,

that is, u n = P C ( x n λ n T x n ) . This completes the proof. □

Declarations

Acknowledgements

The authors are grateful to the referees for useful suggestions which improved the contents of the paper.

Authors’ Affiliations

(1)
College of Science, Hangzhou Normal University
(2)
Department of Mathematics, Jinming Institute of Education

References

  1. Browder FE: Nonlinear operators and nonlinear equations of evolution in Banach spaces. Proc. Symp. Pure Math. 1976, 18: 78–81.Google Scholar
  2. Shimoji K, Takahashi W: Strong convergence to common fixed points of infinite nonexpansive mappings and applications. Taiwan. J. Math. 2001, 5: 387–404.MathSciNetGoogle Scholar
  3. Blum E, Oettli W: From optimization and variational inequalities to equilibrium problems. Math. Stud. 1994, 63: 123–145.MathSciNetGoogle Scholar
  4. He RH: Coincidence theorem and existence theorems of solutions for a system of Ky Fan type minimax inequalities in FC-spaces. Adv. Fixed Point Theory 2012, 2: 47–57.Google Scholar
  5. Kim KS, Kim JK, Lim WH: Convergence theorems for common solutions of various problems with nonlinear mapping. J. Inequal. Appl. 2014., 2014: Article ID 2Google Scholar
  6. Park S: A review of the KKM theory on ϕ A -spaces or GFC-spaces. Adv. Fixed Point Theory 2013, 3: 355–382.Google Scholar
  7. Cho SY, Kang SM: Approximation of common solutions of variational inequalities via strict pseudocontractions. Acta Math. Sci. 2012, 32: 1607–1618. 10.1016/S0252-9602(12)60127-1View ArticleGoogle Scholar
  8. 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
  9. 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.MathSciNetGoogle Scholar
  10. Rockafellar RT: On the maximality of sums of nonlinear monotone operators. Trans. Am. Math. Soc. 1970, 149: 75–88. 10.1090/S0002-9947-1970-0282272-5View ArticleMathSciNetGoogle Scholar
  11. 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.2008.04.035View ArticleMathSciNetGoogle Scholar
  12. 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.2009.10.017View ArticleMathSciNetGoogle Scholar
  13. 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.2006.08.036View ArticleMathSciNetGoogle Scholar
  14. Zhang L, Tong H: An iterative method for nonexpansive semigroups, variational inclusions and generalized equilibrium problems. Adv. Fixed Point Theory 2014, 4: 325–343.Google Scholar
  15. 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/s10898-009-9498-8View ArticleMathSciNetGoogle Scholar
  16. Lv S: Strong convergence of a general iterative algorithm in Hilbert spaces. J. Inequal. Appl. 2013., 2013: Article ID 19Google Scholar
  17. Li DF, Zhao J: On variational inequality, fixed point and generalized mixed equilibrium problems. J. Inequal. Appl. 2014., 2014: Article ID 203Google Scholar
  18. Chen JH: Iterations for equilibrium and fixed point problems. J. Nonlinear Funct. Anal. 2013., 2013: Article ID 4Google Scholar
  19. Mahato NK, Nahak C: Equilibrium problem under various types of convexities in Banach space. J. Math. Comput. Sci. 2011, 1: 77–88.MathSciNetGoogle Scholar
  20. Qin X, Cho SY, Kang SM: Some results on variational inequalities and generalized equilibrium problems with applications. Comput. Appl. Math. 2010, 29: 393–421.MathSciNetGoogle Scholar
  21. Wu C, Liu A: Strong convergence of a hybrid projection iterative algorithm for common solutions of operator equations and of inclusion problems. Fixed Point Theory Appl. 2012., 2012: Article ID 90Google Scholar
  22. Wang ZM, Zhang X: Shrinking projection methods for systems of mixed variational inequalities of Browder type, systems of mixed equilibrium problems and fixed point problems. J. Nonlinear Funct. Anal. 2014., 2014: Article ID 15Google Scholar
  23. Qin X, Shang M, Su Y: Strong convergence of a general iterative algorithm for equilibrium problems and variational inequality problems. Math. Comput. Model. 2008, 48: 1033–1046. 10.1016/j.mcm.2007.12.008View ArticleMathSciNetGoogle Scholar
  24. Jeong JU: Strong convergence theorems for a generalized mixed equilibrium problem and variational inequality problems. Fixed Point Theory Appl. 2013., 2013: Article ID 65Google Scholar
  25. Cho SY: Iterative processes for common fixed points of two different families of mappings with applications. J. Glob. Optim. 2013, 57: 1429–1446. 10.1007/s10898-012-0017-yView ArticleGoogle Scholar
  26. Rodjanadid B, Sompong S: A new iterative method for solving a system of generalized equilibrium problems, generalized mixed equilibrium problems and common fixed point problems in Hilbert spaces. Adv. Fixed Point Theory 2013, 3: 675–705.Google Scholar
  27. Wen DJ, Chen YA: General iterative methods for generalized equilibrium problems and fixed point problems of k-strict pseudo-contractions. Fixed Point Theory Appl. 2012., 2012: Article ID 125Google Scholar
  28. Cho SY, Qin X: On the strong convergence of an iterative process for asymptotically strict pseudocontractions and equilibrium problems. Appl. Math. Comput. 2014, 235: 430–438.View ArticleMathSciNetGoogle Scholar
  29. Hao Y: On variational inclusion and common fixed point problems in Hilbert spaces with applications. Appl. Math. Comput. 2010, 217: 3000–3010. 10.1016/j.amc.2010.08.033View ArticleMathSciNetGoogle Scholar
  30. Liu LS: Ishikawa and Mann iterative process with errors for nonlinear strongly accretive mappings in Banach spaces. J. Math. Anal. Appl. 1995, 194: 114–125. 10.1006/jmaa.1995.1289View ArticleMathSciNetGoogle Scholar
  31. 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.2004.11.017View ArticleMathSciNetGoogle Scholar
  32. Opial Z: Weak convergence of the sequence of successive approximation for nonexpansive mappings. Bull. Am. Math. Soc. 1967, 73: 591–597. 10.1090/S0002-9904-1967-11761-0View ArticleMathSciNetGoogle Scholar

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© Yuan and Zhang; licensee Springer. 2014

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