Open Access

A regularization method for treating zero points of the sum of two monotone operators

Fixed Point Theory and Applications20142014:75

https://doi.org/10.1186/1687-1812-2014-75

Received: 5 December 2013

Accepted: 11 March 2014

Published: 25 March 2014

Abstract

In this paper, a regularization method for treating zero points of the sum of two monotone operators is investigated. Strong convergence theorems are established in the framework of Hilbert spaces.

Keywords

maximal monotone operator fixed point nonexpansive mapping proximal point algorithm zero point

1 Introduction

In the real world, many important problems have reformulations which require finding zero points of some nonlinear operator, for instance, evolution equations, complementarity problems, mini-max problems, variational inequalities and optimization problems; see [113] and the references therein. It is well known that minimizing a convex function f can be reduced to finding zero points of the subdifferential mapping A = f . Splitting methods have recently received much attention due to the fact that many nonlinear problems arising in applied areas such as image recovery, signal processing, and machine learning are mathematically modeled as a nonlinear operator equation and this operator is decomposed as the sum of two nonlinear operators. The central problem is to iteratively find a zero point of the sum of two monotone operators; that is, 0 ( A + B ) ( x ) . Many problems can be formulated as a problem of the above form. For instance, a stationary solution to the initial value problem of the evolution equation 0 F u + u t , u 0 = u ( 0 ) , can be recast as the above inclusion problem when the governing maximal monotone F is of the form F = A + B ; for more details; see [14] and the references therein.

In this paper, we study a regularization method for treating zero points of the sum of an inverse-strongly monotone and a maximal monotone operator. The main contribution of the paper is establish a strong convergence theorem for viscosity zero points under mild restrictions imposed on the control sequences. The main results include the corresponding results in Xu [15] as a special case. The organization of this paper is as follows. In Section 2, we provide some necessary preliminaries. In Section 3, a regularization method is investigated. A strong convergence theorem for zero points of the sum operator is established. In Section 4, applications of the main results are discussed.

2 Preliminaries

In what follows, we always assume that H is a real Hilbert space with inner product , and norm . Let C be a nonempty, closed and convex subset of H. 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 κ ( 0 , 1 ) such that
S x S y κ x y , x , y C .
It is well known that every contractive mapping has a unique fixed point in metric spaces. S is said to be nonexpansive iff
S x S y x y , x , y C .

If C is a bounded, closed, and convex subset of H, then F ( S ) is not empty, closed, and convex; see [16] and the references therein.

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. It is not hard to see that every inverse-strongly monotone mapping is monotone and continuous.

Recall that a set-valued mapping B : H H is said to be monotone iff, for all x , y H , f B x and g B y imply x y , f g > 0 . In this paper, we use B 1 ( 0 ) to stand for the zero point of B. A monotone mapping B : H H is maximal iff the graph Graph ( B ) of B is not properly contained in the graph of any other monotone mapping. It is well known that a monotone mapping B is maximal if and only if, for any ( x , f ) H × H , x y , f g 0 , for all ( y , g ) Graph ( B ) implies f B x . For a maximal monotone operator B on H, and r > 0 , we may define the single-valued resolvent J r : H Dom ( B ) , where Dom ( B ) denote the domain of B. It is well known that J r is firmly nonexpansive, and B 1 ( 0 ) = F ( J r ) .

Recently, many authors studied zero points of monotone operators based on different regularization methods; see [1729] and the references therein. The main motivation is from Xu [15]. We propose a regularization method for treating zero points of the sum of two monotone operators. Strong convergence theorems are established in the framework of Hilbert spaces.

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

Lemma 2.1 [30]

Let A : C H be a mapping, and B : H H a maximal monotone operator. Then F ( J r ( I r A ) ) = ( A + B ) 1 ( 0 ) .

Lemma 2.2 [31]

Let { a n } be a sequence of nonnegative numbers satisfying the condition a n + 1 ( 1 t n ) a n + t n b n + c n , n 0 , where { t n } is a number sequence in ( 0 , 1 ) such that lim n t n = 0 and n = 0 t n = , { b n } is a number sequence such that lim sup n b n 0 , and { c n } is a positive number sequence such that n = 0 c n < . Then lim n a n = 0 .

Lemma 2.3 [32]

Let H be a Hilbert space, and A a maximal monotone operator. For λ > 0 , μ > 0 , and x E , we have J λ x = J μ ( μ λ x + ( 1 μ λ ) J λ x ) , where J λ = ( I + λ A ) 1 and J μ = ( I + μ A ) 1 .

3 Main results

Theorem 3.1 Let A : C H be an α-inverse-strongly monotone mapping and let B be a maximal monotone operator on H. Assume that Dom ( B ) C and ( A + B ) 1 ( 0 ) is not empty. Let f : C C be a fixed κ-contraction and let J r n = ( I + r n B ) 1 . Let { x n } be a sequence in C in the following process: x 0 C and
{ y n = α n f ( x n ) + ( 1 α n ) x n , x n + 1 = J r n ( y n r n A y n + e n ) , n 0 ,
where { α n } is a real number sequence in ( 0 , 1 ) , { e n } is sequence in H and { r n } is a positive real number sequence in ( 0 , 2 α ) . If the control sequences satisfy the following restrictions:
  1. (a)

    lim n α n = 0 , n = 0 α n = and n = 1 | α n α n 1 | < ;

     
  2. (b)

    0 < a r n b < 2 α and n = 1 | r n r n 1 | < ;

     
  3. (c)

    n = 0 e n < ,

     

then { x n } converges strongly to a point x ¯ ( A + B ) 1 ( 0 ) , where x ¯ = Proj ( A + B ) 1 ( 0 ) f ( x ¯ ) .

Proof First, we show that { x n } is bounded. Notice that I r n A is nonexpansive. Indeed, 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 r n ( 2 α r n ) A x A y 2 .
In view of the restriction (b), we find that I r n A is nonexpansive. Fixing p ( A + B ) 1 ( 0 ) , we find that
y n p α n f ( x n ) p + ( 1 α n ) x n p ( 1 α n ( 1 κ ) ) x n p + α n f ( p ) p .
It follows that
x n + 1 p ( y n r n A y n + e n ) ( I r n A ) p ( I r n A ) y n ( I r n A ) p + e n ( 1 α n ( 1 κ ) ) x n p + α n f ( p ) p + e n max { x n p , f ( p ) p 1 κ } + e n max { x n 1 p , f ( p ) p 1 κ } + e n 1 + e n max { x 0 p , f ( p ) p 1 κ } + i = 0 n e i max { x 0 p , f ( p ) p 1 κ } + i = 0 e i < .
This proves that the sequence { x n } is bounded, and so is { y n } . Notice that
y n y n 1 ( 1 α n ( 1 κ ) ) x n x n 1 + | α n α n 1 | f ( x n 1 ) x n 1 .
Putting z n = y n r n A y n + e n , we find that
z n z n 1 y n y n 1 + r n r n 1 A y n 1 + e n + e n 1 ( 1 α n ( 1 κ ) ) x n x n 1 + | α n α n 1 | f ( x n 1 ) x n 1 + | r n r n 1 | A y n 1 + e n + e n 1 .
It follows from Lemma 2.3 that
x n + 1 x n = J r n z n J r n 1 z n 1 = J r n 1 ( r n 1 r n z n + ( 1 r n 1 r n ) J r n z n ) J r n 1 z n 1 r n 1 r n ( z n z n 1 ) + ( 1 r n 1 r n ) ( J r n z n z n 1 ) ( z n z n 1 ) + ( 1 r n 1 r n ) ( J r n z n z n ) z n z n 1 + | r n r n 1 | a J r n z n z n ( 1 α n ( 1 κ ) ) x n x n 1 + f n ,
where
f n = | α n α n 1 | f ( x n 1 ) x n 1 + | r n r n 1 | ( A y n 1 + J r n z n z n a ) + e n + e n 1 .
It follows from the restrictions (a), (b), and (c) that n = 1 f n < . In view of Lemma 2.2, we find that lim n x n + 1 x n = 0 . In view of y n x n α n f ( x n ) x n , we find from the above that
lim n y n x n + 1 = lim n y n x n = 0 .
(3.1)
Next, we show that
lim sup n f ( x ¯ ) x ¯ , y n x ¯ 0 ,
(3.2)
where x ¯ is the unique fixed point of the mapping Proj ( A + B ) 1 ( 0 ) f . To show this inequality, we choose a subsequence { y n i } of { y n } such that
lim sup n f ( x ¯ ) x ¯ , y n x ¯ = lim i f ( x ¯ ) x ¯ , y n i x ¯ 0 .

Since { y n i } is bounded, we find that there exists a subsequence { y n i j } of { y n i } which converges weakly to x ˆ . Without loss of generality, we can assume that y n i x ˆ .

Now, we show that x ˆ ( A + B ) 1 ( 0 ) . Notice that y n r n A y n + e n x n + 1 + r n B x n + 1 ; that is,
y n r n A y n + e n x n + 1 r n B x n + 1 .
Let μ B ν . Since B is monotone, we find that
y n + e n x n + 1 r n A y n μ , x n + 1 ν 0 .
In view of the restriction (b), we see from (3.1) that A x ˆ μ , x ˆ ν 0 . This implies that A x ˆ B x ˆ , that is, x ˆ ( A + B ) 1 ( 0 ) . This proves that (3.2) holds. Notice that
y n x ¯ 2 α n κ x n x ¯ y n x ¯ + α n f ( x ¯ ) x ¯ , y n x ¯ + ( 1 α n ) x n x ¯ y n x ¯ .
It follows that y n x ¯ 2 ( 1 α n ( 1 κ ) ) x n x ¯ 2 + 2 α n f ( x ¯ ) x ¯ , y n x ¯ . On the other hand, we have
x n + 1 x ¯ 2 J r n ( y n r n A y n + e n ) x ¯ 2 ( y n r n A y n ) ( I r n A ) x ¯ 2 + e n ( e n + 2 ( y n r n A y n ) ( I r n A ) x ¯ ) y n x ¯ 2 + e n ( e n + 2 ( y n r n A y n ) ( I r n A ) x ¯ ) ( 1 α n ( 1 κ ) ) x n x ¯ 2 + 2 α n f ( x ¯ ) x ¯ , y n x ¯ + e n ( e n + 2 ( y n r n A y n ) ( I r n A ) x ¯ ) .

An application of Lemma 2.2 to the above inequality yields lim n x n x ¯ = 0 . This completes the proof. □

4 Applications

First, we consider the problem of finding a minimizer of a proper convex lower semicontinuous function.

For a proper lower semicontinuous convex function g : H ( , ] , the subdifferential mapping ∂g of g is defined by
g ( x ) = { x H : g ( x ) + y x , x g ( y ) , y H } , x H .

Rockafellar [33] proved that ∂g is a maximal monotone operator. It is easy to verify that 0 g ( v ) if and only if g ( v ) = min x H g ( x ) .

Theorem 4.1 Let g : H ( , + ] be a proper convex lower semicontinuous function such that ( g ) 1 ( 0 ) is not empty. Let f : H H be a κ-contraction and let { x n } be a sequence in H in the following process: x 0 H and
{ y n = α n f ( x n ) + ( 1 α n ) x n , x n + 1 = arg min z H { g ( z ) + z y n e n 2 2 r n } , n 0 ,
where { α n } is a real number sequence in ( 0 , 1 ) , { e n } is sequence in H and { r n } is a positive real number sequence. If the control sequences satisfy the following restrictions:
  1. (a)

    lim n α n = 0 , n = 0 α n = and n = 1 | α n α n 1 | < ;

     
  2. (b)

    0 < a r n ;

     
  3. (c)

    n = 0 e n < ,

     

then { x n } converges strongly to a point x ¯ ( g ) 1 ( 0 ) , where x ¯ = Proj ( g ) 1 ( 0 ) f ( x ¯ ) .

Proof Since g : H ( , ] is a proper convex and lower semicontinuous function, we see that subdifferential ∂g of g is maximal monotone. Noting that
x n + 1 = arg min z H { g ( z ) + z y n e n 2 2 r n }
is equivalent to
0 g ( x n + 1 ) + 1 r n ( x n + 1 y n e n ) .
It follows that
y n + e n x n + 1 + r n g ( x n + 1 ) .

Putting A = 0 , we immediately derive from Theorem 3.1 the desired conclusion. □

Next, we consider the problem of finding a solution of a classical variational inequality.

Let C be a nonempty closed and convex subset of a Hilbert space H. Let i C be the indicator function of C, that is,
i C ( x ) = { 0 , x C , , x C .
Since i C is a proper lower and semicontinuous convex function on H, the subdifferential i C of i C is maximal monotone. So, we can define the resolvent J r of i C for r > 0 , i.e., J r : = ( I + r i C ) 1 . Letting x = J r y , we find that
y x + r i C x y x + r N C x y x , v x 0 , v C x = Proj C y ,

where Proj C is the metric projection from H onto C and N C x : = { e H : e , v x , v C } .

Theorem 4.2 Let A : C H be an α-inverse-strongly monotone mapping. Assume that V I ( C , A ) is not empty. Let f : C C be a fixed κ-contraction. Let { x n } be a sequence in C in the following process: x 0 C and
{ y n = α n f ( x n ) + ( 1 α n ) x n , x n + 1 = Proj C ( y n r n A y n + e n ) , n 0 ,
where { α n } is a real number sequence in ( 0 , 1 ) , { e n } is sequence in H and { r n } is a positive real number sequence in ( 0 , 2 α ) . If the control sequences satisfy the following restrictions:
  1. (a)

    lim n α n = 0 , n = 0 α n = and n = 1 | α n α n 1 | < ;

     
  2. (b)

    0 < a r n b < 2 α and n = 1 | r n r n 1 | < ;

     
  3. (c)

    n = 0 e n < ,

     

then { x n } converges strongly to a point x ¯ V I ( C , A ) , where x ¯ = Proj V I ( C , A ) f ( x ¯ ) .

Proof Putting B = i C in Theorem 3.1, we find that J r n = Proj C . We can draw the desired conclusion from Theorem 3.1.

Next, we consider the problem of finding a solution of a Ky Fan inequality, which is known as an equilibrium problem in the terminology of Blum and Oettli; see [34] and [35] and the references therein.

Let F be a bifunction of C × C into , where denotes the set of real numbers. Recall the following equilibrium problem:
Find  x C  such that  F ( x , y ) 0 , y C .
(4.1)

To study the equilibrium problem (4.1), we may assume that F satisfies the following restrictions:

(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 lower semicontinuous.

 □

The following lemma can be found in [35].

Lemma 4.3 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 . Further, define
T r x = { z C : F ( z , y ) + 1 r y z , z x 0 , y C }
(4.2)

for all r > 0 and x H . Then (1) T r is single-valued and firmly nonexpansive; (2) F ( T r ) = E P ( F ) is closed and convex.

Lemma 4.4 [36]

Let F be a bifunction from C × C to which satisfies (A1)-(A4), and let A F be a multivalued mapping of H into itself defined by
A F x = { { z H : F ( x , y ) y x , z , y C } , x C , , x C .
(4.3)

Then A F is a maximal monotone operator with the domain D ( A F ) C , E P ( F ) = A F 1 ( 0 ) , where F P ( F ) stands for the solution set of (4.1), and T r x = ( I + r A F ) 1 x , x H , r > 0 , where T r is defined as in (4.2).

Theorem 4.5 Let F : C × C R be a bifunction satisfying (A1)-(A4). Assume that E P ( F ) is not empty. Let f : C C be a fixed κ-contraction and let T r n = ( I + r n A F ) 1 . Let { x n } be a sequence in C in the following process: x 0 C and
x n + 1 = T r n ( α n f ( x n ) + ( 1 α n ) x n + e n ) , n 0 ,
where { α n } is a real number sequence in ( 0 , 1 ) , { e n } is sequence in H and { r n } is a positive real number sequence. If the control sequences satisfy the following restrictions:
  1. (a)

    lim n α n = 0 , n = 0 α n = and n = 1 | α n α n 1 | < ;

     
  2. (b)

    0 < a r n b < and n = 1 | r n r n 1 | < ;

     
  3. (c)

    n = 0 e n < ,

     

then { x n } converges strongly to a point x ¯ E P ( F ) , where x ¯ = Proj E P ( F ) f ( x ¯ ) .

Proof Putting A = 0 in Theorem 3.1, we find that J r n = T r n . From Theorem 3.1, we can draw the desired conclusion immediately.

Recall that a mapping T : C T is said to be α-strictly pseudocontractive if there exists a constant α [ 0 , 1 ) such that
T x T y 2 x y 2 + α ( I T ) x ( I T ) y 2 , x , y C .

The class of strictly pseudocontractive mappings was first introduced by Browder and Petryshyn [37]. It is well known that if T is α-strictly-pseudocontractive, then I T is 1 α 2 -inverse-strongly monotone. □

Finally, we consider fixed point problem of α-strictly pseudocontractive mappings.

Theorem 4.6 Let T : C C be an α-strictly pseudocontractive mapping with a nonempty fixed point set and let f : C C be a fixed κ-contraction. Let { x n } be a sequence generated in the following manner: x 0 C and
{ y n = α n f ( x n ) + ( 1 α n ) x n , x n + 1 = ( 1 r n ) y n + r n T y n , n 0 ,
where { α n } is a real number sequence in ( 0 , 1 ) and { r n } is a positive real number sequence in ( 0 , 1 α ) . If the control sequences satisfy the following restrictions:
  1. (a)

    lim n α n = 0 , n = 0 α n = and n = 1 | α n α n 1 | < ;

     
  2. (b)

    0 < a r n b < 1 α and n = 1 | r n r n 1 | < ;

     

then { x n } converges strongly to a point x ¯ F ( T ) , where x ¯ = Proj F ( T ) f ( x ¯ ) .

Proof Putting A = I T , we find A is 1 α 2 -inverse-strongly monotone. We also have F ( T ) = V I ( C , A ) and Proj C ( y n r n A y n ) = ( 1 r n ) y n + r n T y n . In view of Theorem 3.1, we obtain the desired result. □

Declarations

Acknowledgements

The authors are grateful to the three anonymous referees for useful suggestions, which improved the contents of the article.

Authors’ Affiliations

(1)
Department of Mathematics, Hangzhou Normal University
(2)
Department of Mathematics, Gyeongsang National University
(3)
College of Statistics and Mathematics, Yunnan University of Finance and Economics

References

  1. Kinderlehrer D, Stampacchia G: An Introduction to Variational Inequalities and Their Applications. Academic Press, New York; 1980.MATHGoogle Scholar
  2. Facchinei F, Pang JS: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, New York; 2003.Google Scholar
  3. Kamimura S, Takahashi W: Weak and strong convergence of solutions to accretive operator inclusions and applications. Set-Valued Anal. 2000, 8: 361–374. 10.1023/A:1026592623460View ArticleMathSciNetMATHGoogle Scholar
  4. Cho SY, Qin X, Kang SM: 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 ArticleMathSciNetMATHGoogle Scholar
  5. Cho SY, Li W, Kang SM: Convergence analysis of an iterative algorithm for monotone operators. J. Inequal. Appl. 2013., 2013: Article ID 199Google Scholar
  6. 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 ArticleMATHGoogle Scholar
  7. Wang Z, Lou W: A new iterative algorithm of common solutions to quasi-variational inclusion and fixed point problems. J. Math. Comput. Sci. 2013, 3: 57–72.Google Scholar
  8. Wu C: Mann iteration for zero theorems of accretive operators. J. Fixed Point Theory 2013., 2013: Article ID 3Google Scholar
  9. Qin X, Su Y: Approximation of a zero point of accretive operator in Banach spaces. J. Math. Anal. Appl. 2007, 329: 415–424. 10.1016/j.jmaa.2006.06.067View ArticleMathSciNetMATHGoogle Scholar
  10. 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
  11. Kim JK: Strong convergence theorems by hybrid projection methods for equilibrium problems and fixed point problems of the asymptotically quasi- ϕ -nonexpansive mappings. Fixed Point Theory Appl. 2011., 2011: Article ID 10Google Scholar
  12. Yang S: Zero theorems of accretive operators in reflexive Banach spaces. J. Nonlinear Funct. Anal. 2013., 2013: Article ID 2Google Scholar
  13. Qin X, Cho YJ, Kang SM: Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces. J. Comput. Appl. Math. 2009, 225: 20–30. 10.1016/j.cam.2008.06.011View ArticleMathSciNetMATHGoogle Scholar
  14. Lions PL, Mercier B: Splitting algorithms for the sum of two nonlinear operators. SIAM J. Numer. Anal. 1979, 16: 964–979. 10.1137/0716071View ArticleMathSciNetMATHGoogle Scholar
  15. Xu HK: A regularization method for the proximal point algorithm. J. Glob. Optim. 2006, 36: 115–125. 10.1007/s10898-006-9002-7View ArticleMATHGoogle Scholar
  16. Browder FE: Nonlinear operators and nonlinear equations of evolution in Banach spaces. Proc. Symp. Pure Math. 1976, 18: 78–81.MATHGoogle Scholar
  17. Qin X, Cho SY, Wang L: Iterative algorithms with errors for zero points of m -accretive operators. Fixed Point Theory Appl. 2013., 2013: Article ID 148Google Scholar
  18. Hao Y: Zero theorems of accretive operators. Bull. Malays. Math. Soc. 2011, 34: 103–112.MATHGoogle Scholar
  19. Bruck RE Jr.: A strongly convergent iterative method for the solution of 0 U x for a maximal monotone operator U in Hilbert space. J. Math. Anal. Appl. 1974, 48: 114–126. 10.1016/0022-247X(74)90219-4View ArticleMathSciNetMATHGoogle Scholar
  20. Yuan Q, Cho SY: Proximal point algorithms for zero points of nonlinear operators. Fixed Point Theory Appl. 2014., 2014: Article ID 42Google Scholar
  21. Qin X, Su Y: Strong convergence theorems for relatively nonexpansive mappings in a Banach space. Nonlinear Anal. 2007, 67: 1958–1965. 10.1016/j.na.2006.08.021View ArticleMathSciNetMATHGoogle Scholar
  22. 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
  23. Wu C, Lv S: Bregman projection methods for zeros of monotone operators. J. Fixed Point Theory 2013., 2013: Article ID 7Google Scholar
  24. Cho SY, Qin X, Kang SM: Hybrid projection algorithms for treating common fixed points of a family of demicontinuous pseudocontractions. Appl. Math. Lett. 2012, 25: 854–857. 10.1016/j.aml.2011.10.031View ArticleMathSciNetMATHGoogle Scholar
  25. Cho SY, Kang SM: Zero point theorems for m -accretive operators in a Banach space. Fixed Point Theory 2012, 13: 49–58.MathSciNetMATHGoogle Scholar
  26. Chen JW, Wan Z, Zou Y: Strong convergence theorems for firmly nonexpansive-type mappings and equilibrium problems in Banach spaces. Optimization 2013, 62: 483–497. 10.1080/02331934.2011.626779View ArticleMathSciNetMATHGoogle Scholar
  27. Cho SY, Kang SM: Approximation of fixed points of pseudocontraction semigroups based on a viscosity iterative process. Appl. Math. Lett. 2011, 24: 224–228. 10.1016/j.aml.2010.09.008View ArticleMathSciNetMATHGoogle Scholar
  28. Kamimura S, Takahashi W: Approximating solutions of maximal monotone operators in Hilbert spaces. J. Approx. Theory 2000, 106: 226–240. 10.1006/jath.2000.3493View ArticleMathSciNetMATHGoogle Scholar
  29. Kang SM, Cho SY, Liu Z: Convergence of iterative sequences for generalized equilibrium problems involving inverse-strongly monotone mappings. J. Inequal. Appl. 2010., 2010: Article ID 827082Google Scholar
  30. Cho SY: Strong convergence of an iterative algorithm for sums of two monotone operators. J. Fixed Point Theory 2013., 2013: Article ID 6Google Scholar
  31. Liu L: 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 ArticleMathSciNetMATHGoogle Scholar
  32. Barbu V: Nonlinear Semigroups and Differential Equations in Banach Space. Noordhoff, Groningen; 1976.View ArticleGoogle Scholar
  33. Rockafellar RT: Characterization of the subdifferentials of convex functions. Pac. J. Math. 1966, 17: 497–510. 10.2140/pjm.1966.17.497View ArticleMathSciNetMATHGoogle Scholar
  34. Fan K: A minimax inequality and applications. In Inequality. III. Edited by: Shisha O. Academic Press, New York; 1972:103–113.Google Scholar
  35. Blum E, Oettli W: From optimization and variational inequalities to equilibrium problems. Math. Stud. 1994, 63: 123–145.MathSciNetMATHGoogle Scholar
  36. Takahashi S, Takahashi W, Toyoda M: Strong convergence theorems for maximal monotone operators with nonlinear mappings in Hilbert spaces. J. Optim. Theory Appl. 2010, 147: 27–41. 10.1007/s10957-010-9713-2View ArticleMathSciNetMATHGoogle Scholar
  37. Browder FE, Petryshyn WV: Construction of fixed points of nonlinear mappings in Hilbert space. J. Math. Anal. Appl. 1967, 20: 197–228. 10.1016/0022-247X(67)90085-6View ArticleMathSciNetMATHGoogle Scholar

Copyright

© Qin et al.; licensee Springer. 2014

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.