Open Access

Strong convergence of a Halpern-type algorithm for common solutions of fixed point and equilibrium problems

Journal of Inequalities and Applications20142014:313

https://doi.org/10.1186/1029-242X-2014-313

Received: 5 May 2014

Accepted: 16 July 2014

Published: 21 August 2014

Abstract

In this article, fixed points of nonexpansive mappings and equilibrium problems based on a Halpern-type algorithm are investigated. Strong convergence theorems for common solutions of the two problems are obtained in the framework of real Hilbert spaces.

Keywords

nonexpansive mapping equilibrium problem fixed point variational inequality

1 Introduction

The study of equilibrium problems is an important branch of optimization theory and nonlinear functional analysis. Numerous problems in physics, optimization, transportation, signal processing, and economics are reduced to find a solution to equilibrium problems, which cover fixed point problems, variational inequalities, saddle problems, inclusion problems, and so on. A closely related subject of current interest is the problem of finding common elements in the fixed point set of nonlinear operators and in the solution set of monotone variational inequalities; see [115] and the references therein. The motivation for this subject is mainly due to its possible applications to mathematical modeling of concrete complex problems. The aim of this paper is to investigate a common element problem based on a Halpern-type algorithm. Strong convergence of the algorithm is obtained in the framework of real Hilbert spaces. The organization of this paper is as follows. In Section 2, we provide some necessary preliminaries. In Section 3, a Halpern-type algorithm is proposed and analyzed. Strong convergence theorems for common solutions of two problems are established in the framework of Hilbert spaces. In Section 4, applications of the main results are provided.

2 Preliminaries

Let H be a real Hilbert space with inner product , and norm . Let C be a nonempty, closed, and convex subset of H and let Proj C be the metric projection from H onto C.

Let T : C C be a mapping. In this paper, we use F ( T ) to denote the fixed point set of T. Recall that T is said to be contractive iff there exists a constant α ( 0 , 1 ) such that
T x T y α x y , x , y C .
For such a case, T is also said to be α-contractive. Recall that T is said to be nonexpansive iff
T x T y x y , x , y C .

It is well known that the fixed point set of nonexpansive mappings is nonempty provided that the subset C is bounded, convex, and closed.

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 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 .
(2.1)

In this paper, we use VI ( C , A ) to denote the solution set of (2.1). It is well known that x C is a solution of the variational inequality (2.1) iff x is a solution of the fixed point equation P C ( I r A ) x = x , where r > 0 is a constant.

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 .

For a maximal monotone operator M on H, and r > 0 , we may define the single-valued resolvent J r : H D ( M ) , where D ( M ) denotes the domain of M. It is known that J r is firmly nonexpansive, and M 1 ( 0 ) = F ( J r ) , where F ( J r ) : = { x D ( M ) : x = J r x } , and M 1 ( 0 ) : = { x H : 0 M x } .

Let A : 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 ) + A x , y x 0 , y C .
(2.2)

In this paper, we use EP ( F , A ) to denote the solution set of the generalized equilibrium problem (2.2).

Next, we give some special cases of the generalized equilibrium problem (2.2).
  1. (I)

    If F 0 , then problem (2.2) is reduced to the classical variational inequality (2.1).

     
  2. (II)
    If A 0 , the zero mapping, then problem (2.2) is reduced to the following equilibrium problem:
    Find  x C  such that  F ( x , y ) 0 , y C .
    (2.3)
     

In this paper, we use EP ( F ) to denote the solution set of the equilibrium problem (2.3).

To study the equilibrium problems, we may assume that F satisfies the following conditions:
  1. (A1)

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

     
  2. (A2)

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

     
  3. (A3)
    for each x , y , z C ,
    lim sup t 0 F ( t z + ( 1 t ) x , y ) F ( x , y ) ;
     
  4. (A4)

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

     

Recently, many authors have studied fixed point problems of nonexpansive mappings and solution problems of the equilibrium problems (2.2) and (2.3); for more details, see [1625] and the references therein. In this paper, motivated and inspired by the research going on in this direction, we consider common element problems based on a mean iterative process. Strong convergence of the iterative process is obtained in the framework of real Hilbert spaces. The results presented in this paper improve and extend the corresponding results in Hao [1], Qin et al. [24], Chang et al. [25].

In order to prove our main results, we need the following lemmas.

Lemma 2.1 [26]

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 2.2 [27]

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 ) = EP ( F ) ;

     
  4. (4)

    EP ( F ) is closed and convex.

     
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 .
(2.4)

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 [28] and the references therein.

Lemma 2.3 [28]

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 ,
    (2.5)
     

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 2.4 [29]

Let B : C H be a mapping and let M : H H be a maximal monotone operator. Then F ( J r ( I r B ) ) = ( B + M ) 1 ( 0 ) , where r is some positive real number.

Lemma 2.5 [25]

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 2.6 [30]

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
W x = { A x + N C x , x C , x C .

Then W is maximal monotone and 0 W x if and only if x VI ( C , A ) .

Lemma 2.7 [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 .

3 Main results

Theorem 3.1 Let C be a nonempty closed convex subset of a Hilbert space H and let F be a bifunction from C × C to which satisfies (A1)-(A4). Let A : C H be an α-inverse-strongly monotone mapping and let B : C H be a β-inverse-strongly monotone mapping. Let M : H H be a maximal monotone operator. Let { S i : C C } be a family of infinitely nonexpansive mappings. Assume that F : = i = 1 F ( S i ) EP ( F , B ) ( A + M ) 1 ( 0 ) . Let f : C C be a κ-contraction. Let { x n } be a sequence generated by the process: x 1 C and
{ F ( u n , y ) + B x n , y u n + 1 s n y u n , u n x n 0 , y C , x n + 1 = α n f ( x n ) + β n x n + γ n W n Proj C J r n ( u n r n A u n ) , n 1 ,
where { W n : C C } is the sequence generated in (2.4), { α n } , { β n } , and { γ n } are sequences in ( 0 , 1 ) such that α n + β n + γ n = 1 and { r n } and { s n } are positive number sequences. Assume that the above control sequences satisfy the following restrictions:
  1. (a)

    0 < r r n r < 2 α , 0 < r s n r < 2 β ;

     
  2. (b)

    lim n α n = 0 and n = 1 α n = ;

     
  3. (c)

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

     
  4. (d)

    lim n | s n s n + 1 | = lim n | r n r n + 1 | = 0 .

     

Then the sequence { x n } converges strongly to x ¯ = Proj F f ( x ¯ ) .

Proof First, we show that I r n A is nonexpansive. For x , y C , we have
( 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 .
Using restriction (a), we have I r n A is nonexpansive, so is I s n B . Fix x F . It follows that u n x ( I s n B ) x n ( I s n B ) x x n x . Putting y n = J r n ( u n r n A u n ) , one finds that y n x u n x x n x . It follows that
x n + 1 x α n f ( x n ) x + β n x n x + γ n W n Proj C y n x α n f ( x n ) f ( x ) + α n f ( x ) x + β n x n x + γ n y n x ( 1 α n ( 1 κ ) ) x n x + α n f ( x ) x .
Hence, we have x n x max { x 1 x , f ( x ) x 1 α } . This yields the result that { x n } is bounded. Therefore, both { y n } and { u n } are also bounded. Next, without loss of generality, we assume that there exists a bounded set K C such that x n , y n , u n K . Notice that F ( u n + 1 , y ) + 1 s n + 1 y u n + 1 , u n + 1 ( I s n + 1 B ) x n + 1 0 , y C , and F ( u n , y ) + 1 s n y u n , u n ( I s n B ) x n 0 , y C . It follows that
u n + 1 u n , u n ( I s n B ) x n s n u n + 1 ( I s n + 1 B ) x n + 1 s n + 1 0 .
Hence, we have
u n + 1 u n 2 u n + 1 u n , ( I s n + 1 B ) x n + 1 ( I s n B ) x n + ( 1 s n s n + 1 ) ( u n + 1 ( I s n + 1 B ) x n + 1 ) u n + 1 u n ( ( I s n + 1 B ) x n + 1 ( I s n B ) x n + | 1 s n s n + 1 | u n + 1 ( I s n + 1 B ) x n + 1 ) .
This yields the result that
u n + 1 u n ( I s n + 1 B ) x n + 1 ( I s n B ) x n + | s n + 1 s n | s n + 1 u n + 1 ( I s n + 1 B ) x n + 1 = ( I s n + 1 B ) x n + 1 ( I s n + 1 B ) x n + ( I s n + 1 B ) x n ( I s n B ) x n + | s n + 1 s n | s n + 1 u n + 1 ( I s n + 1 B ) x n + 1 x n + 1 x n + | s n + 1 s n | M 1 ,
(3.1)
where M 1 is an appropriate constant such that
M 1 = sup n 1 { B x n + u n + 1 ( I s n + 1 B ) x n + 1 a ¯ } .
Since J r n is firmly nonexpansive, one sees that
y n + 1 y n = J r n ( u n + 1 r n + 1 A u n + 1 ) J r n ( u n r n A u n ) u n + 1 r n + 1 A u n + 1 ( u n r n A u n ) = ( I r n + 1 A ) u n + 1 ( I r n + 1 A ) u n + ( r n r n + 1 ) A u n u n + 1 u n + | r n r n + 1 | A u n .
(3.2)
Substituting (3.1) into (3.2), one finds that
y n + 1 y n x n + 1 x n + ( | s n + 1 s n | + | r n r n + 1 | ) M 2 ,
(3.3)
where M 2 is an appropriate constant such that M 2 = max { sup n 1 { A u n } , M 1 } . On the other hand, one has
W n + 1 Proj C y n + 1 W n Proj C y n W n + 1 y n + 1 W n y n W n + 1 y n + 1 W y n + 1 + W y n + 1 W y n + W y n W n y n sup x K { W n + 1 x W x + W x W n x } + y n + 1 y n ,
(3.4)
where K is the bounded subset of C defined above. Combining (3.3) with (3.4), one finds
W n + 1 Proj C y n + 1 W n Proj C y n sup x K { W n + 1 x W x + W x W n x } + x n + 1 x n + ( | r n + 1 r n | + | s n s n + 1 | ) M 2 .
(3.5)
Letting x n + 1 = ( 1 β n ) z n + β n x n we see that
z n + 1 z n α n + 1 1 β n + 1 f ( x n + 1 ) W n + 1 y n + 1 + α n 1 β n f ( x n ) W n y n + W n + 1 Proj C y n + 1 W n Proj C y n .
(3.6)
Substituting (3.5) into (3.6), we see that
z n + 1 z n x n + 1 x n α n + 1 1 β n + 1 f ( x n + 1 ) W n + 1 y n + 1 + α n 1 β n f ( x n ) W n y n + sup x K { W n + 1 x W x + W x W n x } + ( | r n + 1 r n | + | s n s n + 1 | ) M 2 .
It follows from restrictions (a), (c), and (d) that lim sup n ( z n + 1 z n x n + 1 x n ) 0 . Using Lemma 2.7, we find that lim n z n x n = 0 . It follows that
lim n x n + 1 x n = 0 .
(3.7)
For any x F , we see that
x n + 1 x 2 α n f ( x n ) x 2 + β n x n x 2 + γ n W n Proj C y n x 2 α n f ( x n ) x 2 + β n x n x 2 + γ n y n x 2 .
(3.8)
Since
y n x 2 = J r n ( u n r n A u n ) x 2 ( I r n A ) u n ( I r n A ) x 2 = u n x 2 2 r n u n x , A u n A x + r n 2 A u n A x 2 x n x 2 + r n ( r n 2 α ) A u n A x 2 ,
we find from (3.8) that
lim n A u n A x = 0 .
(3.9)
It also follows from (3.8) that
x n + 1 x 2 α n f ( x n ) x 2 + β n x n x 2 + γ n x n x s n ( B x n B x ) 2 α n f ( x n ) x 2 + β n x n x 2 + γ n ( x n x 2 + s n 2 B x n B x 2 2 s n B x n B x , x n x ) α n f ( x n ) x 2 + β n x n x 2 + γ n x n x 2 s n γ n ( 2 β s n ) B x n B x 2 .
Using (3.7), one arrives at
lim n B x n B x = 0 .
(3.10)
Since T s n is firmly nonexpansive, we find that
u n x 2 ( I s n B ) x n ( I s n B ) x , u n x 1 2 ( x n x 2 + u n x 2 x n u n 2 s n 2 B x n B x 2 + 2 s n B x n B x , x n u n ) ,
which implies that u n x 2 x n x 2 x n u n 2 + 2 s n B x n B x x n u n . Hence
γ n x n u n 2 α n f ( x n ) x 2 + ( x n x + x n + 1 x ) x n x n + 1 + 2 s n B x n B x x n u n .
Using (3.7) and (3.10), one has
lim n x n u n = 0 .
(3.11)
Similarly, one also has
lim n y n u n = 0 .
(3.12)
Since
W n y n y n y n u n + u n x n + x n W n y n ,
we find from (3.11) and (3.12) that
lim n W n y n y n = 0 .
(3.13)
Now, we are in a position to show lim sup n f ( x ¯ ) x ¯ , x n z 0 , where x ¯ = Proj F f ( x ¯ ) . To prove this, we choose a subsequence { x n i } of { x n } such that
lim sup n f ( x ¯ ) x ¯ , x n x ¯ = lim i f ( x ¯ ) x ¯ , x n i x ¯ .
(3.14)
Since { x n i } is bounded, without loss of generality, we may assume that x n i q . Using (3.11) and (3.12), we have lim n x n y n = 0 . Therefore, we see that y n i q . Now, we are in a position to prove q ( A + M ) 1 ( 0 ) . Notice that u n y n r n A u n M y n . Let μ M ν . Since M is monotone, we find that u n y n r n A u n μ , y n ν 0 . This implies that A q μ , q ν 0 . This implies that A q M q , that is, q ( A + M ) 1 ( 0 ) . Next, we show that q EP ( F , B ) . Since u n = T s n ( I s n B ) x n , we find from (A2) that
B x n i , y u n i + y u n i , u n i x n i s n i F ( y , u n i ) , y C .
(3.15)
Putting y t = t y + ( 1 t ) q for any t ( 0 , 1 ] and y C , we see that y t C . Using (3.15), we find that
y t u n i , B y t y t u n i , B y t B x n i , y t u n i y t u n i , u n i x n i s n i + F ( y t , u n i ) = y t u n i , B y t B u n i + y t u n i , B u n i B x n i y t u n i , u n i x n i s n i + F ( y t , u n i ) .
Since B is monotone, we obtain from (A4) that y t w , B y t F ( y t , w ) . Using (A1) and (A4), we find 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 , B y t = t F ( y t , y ) + ( 1 t ) t y w , B y t .
Hence, 0 F ( y t , y ) + ( 1 t ) y w , B y t , y C . It follows from (A3) that w EP ( F , B ) . Next, we prove that q i = 1 F ( S i ) . Suppose to the contrary, q i = 1 F ( S i ) , i.e., W q q . Since y n i q and the space satisfies Opial’s condition, one has
lim inf i y n i q < lim inf i y n i W q lim inf i { y n i W y n i + W y n i W q } lim inf i { y n i W y n i + y n i q } .
(3.16)
Since W y n y n sup x K W x W n x + W n y n y n , we find from Lemma 2.5 that lim n W y n y n = 0 . It follows that lim inf i y n i q < lim inf i y n i q . This leads to a contradiction. Thus, we have q i = 1 F ( S i ) . This proves that q F . Therefore, one has
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 ( x n ) f ( x ¯ ) , x n + 1 x ¯ + α n f ( x ¯ ) x ¯ , x n + 1 x ¯ + β n x n x ¯ x n + 1 x ¯ + γ n y n x ¯ x n + 1 x ¯ κ 2 α n ( x n x ¯ 2 + x n + 1 x ¯ 2 ) + α n f ( x ¯ ) x ¯ , x n + 1 x ¯ + ( 1 α n ) x n x ¯ x n + 1 x ¯ 1 α n ( 1 κ ) 2 x n x ¯ 2 + 1 2 x n + 1 x ¯ 2 + α n f ( x ¯ ) x ¯ , x n + 1 x ¯ ,
which 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 ¯ .

Using Lemma 2.1, we find that lim n x n x ¯ = 0 . This completes the proof. □

4 Applications

Recall that a mapping T : C C is said to be a k-strict pseudo-contraction if there exists a constant k [ 0 , 1 ) such that
T x T y 2 x y 2 + k ( I T ) x ( I T ) y 2

for all x , y C . Note that the class of k-strict pseudo-contractions strictly includes the class of nonexpansive mappings. Put A = I T , where T : C C is a k-strict pseudo-contraction. Then A is 1 k 2 -inverse-strongly monotone. Now, we are in a position to state a results on fixed points of strict pseudo-contractions.

Theorem 4.1 Let C be a nonempty closed convex subset of a Hilbert space H and let F be a bifunction from C × C to which satisfies (A1)-(A4). Let T : C H be a k-strict pseudo-contraction, B : C H be a β-inverse-strongly monotone mapping, and { S i : C C } be a family of infinitely nonexpansive mappings. Assume that F : = i = 1 F ( S i ) EP ( F , B ) F ( T ) . Let f : C C be a κ-contraction. Let { x n } be a sequence generated by x 1 C and
{ F ( u n , y ) + B x n , y u n + 1 s n y u n , u n x n 0 , y C , y n = ( 1 r n ) u n + r n T u n , x n + 1 = α n f ( x n ) + β n x n + γ n W n y n , n 1 ,
where { W n : C C } is the sequence generated in (2.4), { α n } , { β n } , and { γ n } are sequences in ( 0 , 1 ) such that α n + β n + γ n = 1 and { r n } , and { s n } are positive number sequences. Assume that the above control sequences satisfy the following restrictions:
  1. (a)

    0 < r s n r < 2 β , 0 < r r n r < 1 k ;

     
  2. (b)

    lim n α n = 0 and n = 1 α n = ;

     
  3. (c)

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

     
  4. (d)

    lim n | s n s n + 1 | = lim n | r n r n + 1 | = 0 .

     

Then the sequence { x n } converges strongly to x ¯ = Proj F f ( x ¯ ) .

Proof Taking A = I T , wee see that A : C H is a α-strict pseudo-contraction with α = 1 k 2 and F ( T ) = VI ( C , A ) . Using Theorem 3.1, we find the desired conclusion immediately. □

Let g : H ( , + ] be a proper convex lower semi-continuous function. Then the subdifferential ∂g of g is defined as follows:
f g ( x ) = { y H : g ( z ) g ( x ) + z x , y , z H } , x H .

From Rockafellar [30], we know that ∂g is maximal monotone. It is not hard to verify that 0 g ( x ) if and only if g ( x ) = min y H g ( y ) .

Let I C be the indicator function of C, i.e.,
I C ( x ) = { 0 , x C , + , x C .

Since I C is a proper lower semi-continuous convex function on H, we see that the subdifferential I C of I C is a maximal monotone operator. It is clear that J r x = P C x , x H . Notice that ( A + I C ) 1 ( 0 ) = VI ( C , A 1 ) . Now, we are in a position to state the result on variational inequalities.

Theorem 4.2 Let C be a nonempty closed convex subset of a Hilbert space H and let F be a bifunction from C × C to which satisfies (A1)-(A4). Let A : C H be an α-inverse-strongly monotone mapping, B : C H be a β-inverse-strongly monotone mapping, and { S i : C C } be a family of infinitely nonexpansive mappings. Assume that F : = i = 1 F ( S i ) EP ( F , B ) VI ( C , A ) . Let f : C C be a κ-contraction. Let { x n } be a sequence generated by x 1 C and
{ F ( u n , y ) + B x n , y u n + 1 s n y u n , u n x n 0 , y C , x n + 1 = α n f ( x n ) + β n x n + γ n W n P C ( u n s n A u n ) , n 1 ,
where { W n : C C } is the sequence generated in (2.4), { α n } , { β n } , and { γ n } are sequences in ( 0 , 1 ) such that α n + β n + γ n = 1 and { r n } , and { s n } are positive number sequences. Assume that the above control sequences satisfy the following restrictions:
  1. (a)

    0 < r s n r < 2 β , 0 < r r n r < 2 α ;

     
  2. (b)

    lim n α n = 0 and n = 1 α n = ;

     
  3. (c)

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

     
  4. (d)

    lim n | s n s n + 1 | = lim n | r n r n + 1 | = 0 .

     

Then the sequence { x n } converges strongly to x ¯ = Proj F f ( x ¯ ) .

Declarations

Acknowledgements

The author is grateful to Prof. Tian Cheng and the reviewers’ suggestions which improved the contents of the article.

Authors’ Affiliations

(1)
School of Mathematics and Information Science, North China University of Water Resources and Electric Power

References

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© Zhang; licensee Springer 2014

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