Open Access

Convergence of a proximal point algorithm for maximal monotone operators in Hilbert spaces

Journal of Inequalities and Applications20122012:137

https://doi.org/10.1186/1029-242X-2012-137

Received: 9 February 2012

Accepted: 13 June 2012

Published: 13 June 2012

Abstract

In this article, we consider the proximal point algorithm for the problem of approximating zeros of maximal monotone mappings. Strong convergence theorems for zero points of maximal monotone mappings are established in the framework of Hilbert spaces.

2000 AMS Subject Classification: 47H05; 47H09; 47J25.

Keywords

fixed pointnonexpansive mappingmaximal monotone mappingzero

1. Introduction

The theory of maximal monotone operators has emerged as an effective and powerful tool for studying many real world problems arising in various branches of social, physical, engineering, pure and applied sciences in unified and general framework. Recently, much attention has been payed to develop efficient and implementable numerical methods including the projection method and its variant forms, auxiliary problem principle, proximal-point algorithm and descent framework for solving variational inequalities and related optimization problems (see [132] and the references therein). The proximal point algorithm, can be traced back to Martinet [33] in the context of convex minimization and Rockafellar [34] in the general setting of maximal monotone operators, has been extended and generalized in different directions by using novel and innovative techniques and ideas.

In this article, we investigate the problem of approximating a zero of the maximal monotone mapping based on a proximal point algorithm in the framework of Hilbert spaces. Strong convergence of the iterative algorithm is obtained.

2. Preliminaries

Throughout this article, we assume that H is a real Hilbert space, whose inner product and norm are denoted by 〈·,·〉 and ǀǀ · ǀǀ, respectively. Let T be a set-valued mapping.
  1. (a)
    The set D(T) defined by
    D ( T ) = { u H : T ( u ) }
     
is called the effective domain of T.
  1. (b)
    The set R(T) defined by
    R ( T ) = u H T ( u )
     
is called the range of T.
  1. (c)
    The set G(T) defined by
    G ( T ) = { ( u , v ) H × H : u D ( T ) , v R ( T ) }
     

is said to be the graph of T.

Recall the following definitions.
  1. (c)
    T is said to be monotone if
    u - v , x - y 0 , ( u , x ) , ( v , y ) G ( T ) .
     
  2. (d)

    T is said to be maximal monotone if it is not properly contained in any other monotone operator.

     
For a maximal monotone T : D(T) → 2 H , we can defined the resolvent of T by
J t = ( I + t T ) - 1 , t > 0 .
(2.1)
It is well known that J t : HD(T) is nonexpansive, and F(J t ) = T-1(0), where F(J t ) denotes the set of fixed points of J t . The Yosida approximation T t is defined by
T t = 1 t ( I - J t ) , t > 0 .
It is well known that T t x T J t x, x H and ǁT t x ǁ ≤ ǀTx ǀ, where
T x = inf { y : y T x } ,

for all x D(T).

Let C be a nonempty, closed and convex subset of H. Next, we always assume that T: C → 2 H is a maximal monotone mapping with T - 1 ( 0 ) , where T-1(0) denotes the set of zeros of T.

The class of monotone mappings is one of the most important classes of mappings among nonlinear mappings. Within the past several decades, many authors have been devoting to the studies on the existence and convergence of zero points for maximal monotone mappings. A classical method to solve the following set-valued equation
0 T x ,
(2.2)
is the proximal point algorithm. To be more precise, start with any point x0 H, and update x n +1 iteratively conforming to the following recursion
x n x n + 1 + β n T x n + 1 , n 0 ,
(2.3)

where {β n } [β, ∞), (β > 0) is a sequence of real numbers. However, as pointed in [15], the ideal form of the method is often impractical since, in many cases, to solve the problem (2.3) exactly is either impossible or the same difficult as the original problem (2.2). Therefore, one of the most interesting and important problems in the theory of maximal monotone operators is to find an efficient iterative algorithm to compute approximately zeroes of T.

In 1976, Rockafellar [35] gave an inexact variant of the method
x 0 H , x n + e n + 1 x n + 1 + λ n T x n + 1 , n 0 ,
(2.4)
where {e n } is regarded as an error sequence. This is an inexact proximal point algorithm. It was shown that, if
n = 0 e n < ,
(A)

then the sequence {x n } defined by (1.4) converges weakly to a zero of T provided that T - 1 ( 0 ) . In [16], Güller obtained an example to show that Rockafellar's proximal point algorithm (1.4) does not converge strongly, in general.

Recently, many authors studied the problems of modifying Rockafellar's proximal point algorithm so that strong convergence is guaranteed. Cho et al. [13] proved the following result.

Theorem CKZ. Let H be a real Hilbert space, Ω a nonempty closed convex subset of H, and T: Ω → 2 H a maximal monotone operator with T - 1 ( 0 ) . Let PΩbe the metric projection of H onto Ω. Suppose that, for any given x n H, β n > 0 and e n H, there exists x ̄ n Ω conforming to the SVME (2.4), where {β n } (0, + ∞) with β n → ∞ as n → ∞ and
n = 1 e n 2 < .
(B)
Let {α n } be a real sequence in [0, 1] such that
  1. (i)

    α n → 0 as n → ∞,

     
  2. (ii)

    n = 0 α n = .

     
for any fixed u Ω, define the sequence {x n } iteratively as follows:
x n + 1 = α n u + ( 1 - α n ) P Ω ( x ̄ n - e n ) , n 0 .

Then {x n } converges strongly to a fixed point z of T, where z = limt→ ∞J t u.

In this article, motivated by Theorem CKZ, we continue to consider the problem of approximating a zero of the maximal monotone mapping T. Strong convergence theorems are established under mild restrictions imposed on the error sequence {e n } comparing with the restriction (B). The results which include Cho et al. [13] as a special case also improve the corresponding results announced by many others.

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

Lemma 2.1. (Bruck [[35], Lemma 1]). Let H be a Hilbert space and C a nonempty, closed and convex subset H. For all u C, limt→ ∞J t u exists and it is the point of T-1(0) nearest u.

Lemma 2.2 (Eckstein [[15], Lemma 2]). For any given x n C, λ n > 0, and e n H, there exists x ̄ n C conforming to the following set-valued mapping equation (in short, SVME):
x n + e n x ̄ n + λ n T x ̄ n .
(2.5)
Furthermore, for any p T-1(0), we have
x n - x ̄ n , x n - x ̄ n + e n x n - p , x n - x ̄ n + e n
and
x ̄ n - e n - p 2 x n - p 2 - x n - x ̄ n 2 + e n 2 .
Lemma 2.3 (Liu [36]). Assume that {α n } is a sequence of nonnegative real numbers such that
α n + 1 ( 1 - γ n ) α n + δ n ,
where n } is a sequence in (0,1) and {δ n } is a sequence such that
  1. (i)

    n = 1 γ n = ;

     
  2. (ii)

    lim supn→∞ δ n /γ n ≤ 0 or n = 1 δ n < .

     

Then limn→∞α n = 0.

3. Main results

Theorem 3.1. Let H be a real Hilbert space, C a nonempty, closed and convex subset of H and T: C → 2 H a maximal monotone operator with T - 1 ( 0 ) . Let P C be a metric projection from H onto C. For any x n H and λ n > 0, find x ̄ n C and e n H conforming to the SVME (2.5), where {λ n } (0, ∞) with λ n → ∞ as n → ∞ and
e n η n x n - x ̄ n
(C)
with sup n ≥0η n = η < 1. Let {α n } and {β n } be real sequences in [0, 1] satisfying α n + β n < 1 and the following control conditions:
lim n α n = lim n β n = 0 a n d n = 0 α n = .
Let {x n } be a sequence generated by the following manner:
x 0 H , x n + 1 = α n u + β n x n + ( 1 - α n - β n ) P C ( x ̄ n - e n ) . n 0 ,
(3.1)

where u C is a fixed element. Then the sequence {x n } generated by (3.1) strongly converges to a zero point z of T, where z = limt→∞J t u, if and only if e n → 0 as n → ∞.

Proof. First, show that the necessity. Assume that x n z as n → ∞, where z T-1(0). It follows from (2.5) that
x ̄ n - z x n - z + e n x n - z + η n x n - x ̄ n ( 1 + η n ) x n - z + η n x ̄ n - z .
This implies that
x ̄ n - z 1 + η n 1 - η n x n - z .
It follows that x ̄ n z as n → ∞. Note that
e n η n x n - x ̄ n η n ( x n - z + z - x ̄ n ) .

This shows that e n → 0 as n → ∞.

Next, we show the sufficiency. The proof is divided into three steps.

Step 1. Show that {x n } is bounded.

From the assumption (C), we see that
e n x n - x ̄ n .
For any p T-1 (0), it follows from Lemma 2.2 that
P C ( x ̄ n - e n ) - p 2 x ̄ n - e n - p 2 x n - p 2 - x n - x ̄ n 2 + e n 2 x n - p 2 .
That is,
P C ( x ̄ n - e n ) - p x n - p .
(3.2)
It follows from (3.2) that
x n + 1 - p = α n ( u - p ) + ( 1 - α n ) [ P C ( x ̄ n - e n ) - p ] α n u - p + ( 1 - α n ) P C ( x ̄ n - e n ) - p α n u - p + ( 1 - α n ) x n - p .
(3.3)
Putting
M = max { x 0 - p , u - p } ,
we show that ǀǀx n ǀǀ ≤ M for all n ≥ 0. It is easy to see that the result holds for n = 0. Assume that the result holds for some n ≥ 0. That is, ǀǀx n - p ǀǀ ≤ M. Next, we prove that ǀǀx n +1 - p ǀǀ ≤ M. Indeed, we see from (3.3) that
x n + 1 - p M .

This shows that the sequence {x n } is bounded.

Step 2. Show that lim supn→∞u - z, x n +1 -z〉 ≤ 0, where z = limt→∞J t u.

From Lemma 2.1, we see that limt→∞J t u exists, which is the point of T-1(0) nearest to u. Since T is maximal monotone, T t u TJ t u and T λn x n TJ λn x n , we see
u - J t u , J λ n x n - J t u = - t T t u , J t u - J λ n x n = - t T t u - T λ n x n , J t u - J λ n x n - t T λ n x n , J t u - J λ n x n = - t λ n x n - J λ n x n , J t u - J λ n x n .
Since λ n → ∞ as n → ∞, for any t > 0, we have
lim sup n u - J t u , J λ n x n - J t u 0 .
(3.4)
On the other hand, by the nonexpansivity of J λn , we obtain
J λ n ( x n + e n ) - J λ n x n ( x n + e n ) - x n = e n .
From the assumption e n → 0 as n → ∞ and (3.4), we arrive at
lim sup n u - J t u , J λ n ( x n + e n ) - J t u 0 .
(3.5)
From (2.5), we see that
P C ( x ̄ n - e n ) - J λ n ( x n + e n ) ( x ̄ n - e n ) - J λ n ( x n + e n ) e n .
That is,
lim n P C ( x ̄ n - e n ) - J λ n ( x n + e n ) = 0 .
(2.6)
Combining (3.5) with (3.6), we arrive at
lim sup n u - J t u , P C ( x ̄ n - e n ) - J t u 0 .
(3.7)
On the other hand, we see from the algorithm (3.1) that
x n + 1 - P C ( x ̄ n - e n ) = α n [ u - P C ( x ̄ n - e n ) ] + β n [ x n - P C ( x ̄ n - e n ) ] .
It follows from the condition limn→∞α n = limn→∞β n = 0 that
x n + 1 - P C ( x ̄ n - e n ) 0 as n ,
which combines with (3.7) yields that
lim sup n u - J t u , x n + 1 - J t u 0 , t 0 .
(3.8)
From z = limt→∞J t u and (3.8), we arrive at
lim sup n u - z , x n + 1 - z 0 .
(3.9)

Step 3. Show that x n z as n → ∞.

It follows from (3.2) that
x n + 1 - z 2 = α n u + β n x n + ( 1 - α n - β n ) P C ( x ̄ n - e n ) - z , x n + 1 - z α n u - z , x n + 1 - z + β n x n - z , x n + 1 - z + ( 1 - α n - β n ) P C ( x ̄ n - e n ) - z , x n + 1 - z α n u - z , x n + 1 - z + β n x n - z x n + 1 - z + ( 1 - α n - β n ) P C ( x ̄ n - e n ) - z x n + 1 - z α n u - z , x n + 1 - z + β n x n - z x n + 1 - z + ( 1 - α n - β n ) x n - z x n + 1 - z = α n u - z , x n + 1 - z + ( 1 - α n ) x n - z x n + 1 - z α n u - z , x n + 1 - z + 1 - α n 2 ( x n - z 2 + x n + 1 - z 2 ) .
This implies that
x n + 1 - z 2 ( 1 - α n ) x n - z 2 + α n u - z , x n + 1 - z .
(3.10)

Applying Lemma 2.3 to (3.10), we obtain that x n z as n → ∞. This completes the proof.

As a corollary of Theorem 3.1, we have the following.

Corollary 3.2. Let H be a real Hilbert space, C a nonempty, closed and convex subset of H and T: C → 2 H a maximal monotone operator with T - 1 ( 0 ) . Let P C be a metric projection from H onto C. For any x n H and λ n > 0, find x ̄ n C and e n H conforming to the SVME (2.5), where {λ n } (0, ∞) with λ n → ∞ as n → ∞ and
e n η n x n - x ̄ n
with sup n ≥0η n = η < 1. Let {α n } be a real sequence in (0,1) satisfying the following control conditions:
lim n α n = 0 a n d n = 0 α n = .
Let {x n } be a sequence generated by the following manner:
x 0 H , x n + 1 = α n u + ( 1 - α n ) P C ( x ̄ n - e n ) . n 0 ,

where u C is a fixed element. Then the sequence {x n } strongly converges to a zero point z of T, where z = limt→ ∞, J t u, if and only if e n → 0 as n → ∞.

Remark 3.3. Corollary 3.2 improves Theorem CKZ by relaxing the restriction imposed on the sequence {e n }. In [34], Rockafellar obtained a weak convergence by assuming that n = 0 e n < , see [34] for more details.

Next, as applications of Theorem 3.1, we consider the problem of finding a minimizer of a convex function.

Let H be a Hilbert space, and f: H → (-∞, +∞] be a proper convex lower semi-continuous function. Then the subdifferential ∂f of f is defined as follows:
f ( x ) = { y H : f ( z ) f ( x ) + z - x , y , z H } , x H .
Theorem 3.4. Let H be a real Hilbert space and f: H → (-∞, +∞] a proper convex lower semi-continuous function. Let {λ n } be a sequence in (0, +∞) with λ n → ∞ as n → ∞ and {e n } a sequence in H with e n → ∞ as n → ∞. Assume that
e n η n x n - x ̄ n
with sup n ≥0η n = η < 1. Let x ̄ n be the solution of SVME (2.5) with T replacing byf. That is,
x n + e n x ̄ n + λ n f ( x ̄ n ) , n 0 .
Let {α n } and {β n } be real sequences in [0, 1] satisfying α n + β n < 1 and the following control conditions:
lim n α n = lim n β n = 0 a n d n = 0 α n = .
Let {x n } be a sequence generated by the following manner:
x 0 H , x ̄ n = a r g m i n x H { f ( x ) + 1 2 λ n x - x n - e n 2 } , x n + 1 = α n u + β n x n + ( 1 - α n - β n ) ( x ̄ n - e n ) . n 0 ,

where u H is a fixed element. If f ( 0 ) , the sequence {x n } converges strongly to a minimizer of f nearest to u.

Proof. Since f: H → (-∞, +∞] is a proper convex lower semi-continuous function, we have that the subdifferential ∂ f of f is maximal monotone by Theorem 1 of [34]. Notice that
x ̄ n = arg  min x H f ( x ) + 1 2 β n x - x n - e n 2
is equivalent to the following
0 f ( x ̄ n ) + 1 λ n ( x ̄ n - x n - e n ) .
It follows that
x n + e n x ̄ n + λ n f ( x ̄ n ) , n 0 .

By Theorem 3.1, we can obtain the desired conclusion immediately.

As a corollary of Theorem 3.4, we have the following.

Corollary 3.5. Let H be a real Hilbert space and f: H → (-∞, +∞] a proper convex lower semi-continuous function. Let {λ n } be a sequence in (0, +∞) with λ n → ∞ as n → ∞ and {e n } a sequence in H with e n → ∞ as n → ∞. Assume that
e n η n x n - x ̄ n
with sup n ≥0η n = η < 1. Let x ̄ n be the solution of SVME (2.5) with T replacing byf. That is,
x n + e n x ̄ n + λ n f ( x ̄ n ) , n 0 .
Let {α n } be a real sequence in [0, 1] ssatisfying the following control conditions:
lim n α n = 0 a n d n = 0 α n = .
Let {x n } be a sequence generated by the following manner:
x 0 H , x ̄ n = arg min x H { f ( x ) + 1 2 λ n x - x n - e n 2 } , x n + 1 = α n u + ( 1 - α n ) ( x ̄ n - e n ) . n 0 ,

where u H is a fixed element. If f ( 0 ) , the sequence {x n } converges strongly to a minimizer of f nearest to u.

Declarations

Acknowledgements

The authors are grateful to the referees for their valuable comments and suggestions which improve the contents of the article.

Authors’ Affiliations

(1)
School of Mathematics and Information Science, North China University of Water Resources and Electric Power, Zhengzhou, China
(2)
College of Science, Hebei University of Engineering, Handan, China

References

  1. Kang SM, Cho SY, Liu Z: Convergence of iterative sequences for generalized equilibrium problems involving inverse-strongly monotone mappings. J Inequal Appl 2010, 2010: 827082.MathSciNetGoogle Scholar
  2. Kim JK, Cho SY, Qin X: Hybrid projection algorithms for generalized equilibrium problems and strictly pseudocontractive mappings. J Inequal Appl 2010, 2010: 18. (Article ID 312602)MathSciNetGoogle Scholar
  3. Lin LJ, Huang YJ: Generalized vector quasi-equilibrium problems with applications to common fixed point theorems and optimization problems. Nonlinear Anal TMA 2007, 66: 1275–1289. 10.1016/j.na.2006.01.025MathSciNetView ArticleGoogle Scholar
  4. Husain S, Gupta S: A resolvent operator technique for solving generalized system of nonlinear relaxed cocoercive mixed variational inequalities. Adv Fixed Point Theory 2012, 2: 18–28.Google Scholar
  5. 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.2008.06.011MathSciNetView ArticleGoogle Scholar
  6. 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:Google Scholar
  7. Cho SY, Kang SM: Approximation of common solutions of variational inequalities via strict pseudocontraction. Acta Mathematica Scientia 2012, 32: 1607–1618.View ArticleGoogle Scholar
  8. 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.2008.04.035MathSciNetView ArticleGoogle Scholar
  9. Mahato NK, Nahak C: Equilibrium problem under various types of convexities in Banach space. J Math Comput Sci 2011, 1: 77–88.MathSciNetGoogle Scholar
  10. Qin X, Cho SY, Kang SM: Strong convergence of shrinking projection methods for quasi- ϕ -nonexpansive mappings and equilibrium problems. J Comput Appl Math 2010, 234: 750–760. 10.1016/j.cam.2010.01.015MathSciNetView ArticleGoogle Scholar
  11. Qin X, Cho SY, Kang SM: Convergence of an iterative algorithm for systems of variational inequalities and nonexpansive mappings. J Comput Appl Math 2009, 233: 231–240. 10.1016/j.cam.2009.07.018MathSciNetView ArticleGoogle Scholar
  12. Mathato NK, Nahak C: Equilibrium problem under various types of convexities in Banach space. J Math Comput Sci 2011, 1: 77–88.MathSciNetGoogle Scholar
  13. Cho YJ, Kang SM, Zhou H: Approximate proximal point algorithms for finding zeroes of maximal monotone operators in Hilbert spaces. J Inequal Appl 2008, 2008: 10. (Article ID 598191)MathSciNetGoogle Scholar
  14. Qin X, Kang SM, Cho YJ: Approximation zeros of monotone operators by proximal point algorithms. J Global Optim 2010, 46: 75–87. 10.1007/s10898-009-9410-6MathSciNetView ArticleGoogle Scholar
  15. Eckstein J: Approximate iterations in Bregman-function-based proximal algorithms. Math Program 1998, 83: 113–123.MathSciNetGoogle Scholar
  16. Güller O: On the convergence of the proximal point algorithmfor convex minimization. SIAM J Control Optim 1991, 29: 403–419. 10.1137/0329022MathSciNetView ArticleGoogle Scholar
  17. Tossings P: The perturbed proximal point algorithm and some of its applications. Appl Math Optim 1994, 29: 125–159. 10.1007/BF01204180MathSciNetView ArticleGoogle Scholar
  18. Tikhonov AN: Solution of incorrectly formulated problems and a regularization method. Soviet Math Doklady 1963, 4: 1035–1038.Google Scholar
  19. 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.2006.06.067MathSciNetView ArticleGoogle Scholar
  20. Minty GJ: On the maximal domain of a monotone function. Michigan Math J 1961, 8: 135–137.MathSciNetView ArticleGoogle Scholar
  21. Sahu DR, Yao JC: The prox-Tikhonov regularization method for the proximal point algorithm in Banach spaces. J Optim Theory Appl 2011, 151: 641–655.MathSciNetGoogle Scholar
  22. Verma RU: Rockafellar's celebrated theorem based on A-maximal monotonicity design. Appl Math Lett 2008, 21: 355–360. 10.1016/j.aml.2007.05.004MathSciNetView ArticleGoogle Scholar
  23. Agarwa RP, Verma RU: Inexact a-proximal point algorithm and applications to nonlinear varia-tional inclusion problems. J Optim Theory Appl 2010, 144: 431–444. 10.1007/s10957-009-9615-3MathSciNetView ArticleGoogle Scholar
  24. 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.2006.08.021MathSciNetView ArticleGoogle Scholar
  25. Burachik BS, Scheimberg S, Svaiter BF: Robustness of the hybrid extragradient proximal-point algorithm. J Optim Theory Appl 2001, 111: 117–136. 10.1023/A:1017523331361MathSciNetView ArticleGoogle Scholar
  26. Noor MA: Splitting algorithms for general pseudomonotone mixed variational inequalities. J Global Optim 2000, 18: 75–89. 10.1023/A:1008322118873MathSciNetView ArticleGoogle Scholar
  27. Anh PN, Muu LD, Nguyen VH, Srodiot JJ: Using the Banach contraction principe to implement the proximal point method for multivalued monotone variational inequalities. J Optim Theory Appl 2005, 124: 285–306. 10.1007/s10957-004-0926-0MathSciNetView ArticleGoogle Scholar
  28. 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.2010.09.008MathSciNetView ArticleGoogle Scholar
  29. Shehu Y: A new iterative scheme for a countable family of relatively nonexpansive mappings and an equilibrium problem in Banach spaces. J Global Optim 2011.Google Scholar
  30. Kamimura S, Kohsaka F, Takahashi W: Weak and strong convergence theorems for maximal monotone operators in a Banach space. Set-valed Anal 2004, 12: 417–429. 10.1007/s11228-004-8196-4MathSciNetView ArticleGoogle Scholar
  31. Moudafi A: On the regularization of the sum of two maximal monotone operators. Nonlinear Anal 2000, 42: 1203–1208. 10.1016/S0362-546X(99)00136-4MathSciNetView ArticleGoogle Scholar
  32. Qin X, Cho YJ, Kang SM: Viscosity approximation methods for generalized equilibrium problems and fixed point problems with applications. Nonlinear Anal 2010, 72: 99–112. 10.1016/j.na.2009.06.042MathSciNetView ArticleGoogle Scholar
  33. Martinet B: Algorithmes pour la Resolution des Problemes d'Optimisation et de Minmax. These d'etat Universite de Grenoble, France; 1972.Google Scholar
  34. Rockafellar RT: Monotone operators and the proximal point algorithm. SIAM J Control Optim 1976, 14: 877–898. 10.1137/0314056MathSciNetView ArticleGoogle Scholar
  35. Bruck RE: A strongly convergent iterative method for the solution of 0 Ux for a maximal monotone operator U in Hilbert space. J Math Appl Anal 1974, 48: 114–126. 10.1016/0022-247X(74)90219-4MathSciNetView ArticleGoogle Scholar
  36. Liu L: Ishikawa-type and Mann-type iterative processes with errors for constructing solutions of nonlinear equations involving m -accretive operators in Banach spaces. Nonlinear Anal 1998, 34: 307–317. 10.1016/S0362-546X(97)00579-8MathSciNetView ArticleGoogle Scholar

Copyright

© Wei and Shi; licensee Springer. 2012

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 cited.