Open Access

Weak and strong convergence theorems for common zeros of accretive operators

Journal of Inequalities and Applications20142014:282

https://doi.org/10.1186/1029-242X-2014-282

Received: 23 March 2014

Accepted: 16 July 2014

Published: 15 August 2014

Abstract

In this paper, we propose two proximal point algorithms for investigating common zeros of a family of accretive operators. Weak and strong convergence of the two algorithms are obtained in a Banach space.

MSC:47H06, 47H09, 47J25.

Keywords

accretive operator convergence resolvent proximal point algorithm zero point

1 Introduction and preliminaries

Let E be a real Banach space, and let E be the dual space of E. Let R + be a positive real number set. Let φ : [ 0 , ] : = R + R + be a continuous strictly increasing function such that φ ( 0 ) = 0 and φ ( t ) as t . This function φ is called a gauge function. The duality mapping J φ : E E associated with a gauge function φ is defined by
J φ ( x ) = { f E : x , f = x φ ( x ) , f = φ ( x ) } , x E ,

where , denotes the generalized duality pairing. In the case that φ ( t ) = t , we write J for J φ and call J the normalized duality mapping.

Following Browder [1], we say that a Banach space E has a weakly continuous duality mapping if there exists a gauge φ for which the duality mapping J φ ( x ) is single-valued and weak-to-weak sequentially continuous (i.e., if { x n } is a sequence in E weakly convergent to a point x, then the sequence J φ ( x n ) converges weakly to J φ ). It is known that l p has a weakly continuous duality mapping with a gauge function φ ( t ) = t p 1 for all 1 < p < .

Let U E = { x E : x = 1 } . E is said to be smooth or is said to be have a Gâteaux differentiable norm if the limit lim t 0 x + t y x t exists for each x , y U E . E is said to have a uniformly Gâteaux differentiable norm if for each y U E , the limit is attained uniformly for all x U E . E is said to be uniformly smooth or is said to have a uniformly Fréchet differentiable norm if the limit is attained uniformly for x , y U E .

It is well known that Fréchet differentiability of the norm of E implies Gâteaux differentiability of the norm of E. It is known that if the norm of E is uniformly Gâteaux differentiable, then the duality mapping J is single-valued and uniformly norm-to-weak continuous on each bounded subset of E.

Let D be a nonempty subset of a set C. Let Proj D : C D . Q is said to be
  1. (1)

    a contraction if Proj D 2 = Proj D ;

     
  2. (2)

    sunny if for each x C and t ( 0 , 1 ) , we have Proj D ( t x + ( 1 t ) Proj D x ) = Proj D x ;

     
  3. (3)

    a sunny nonexpansive retraction if Proj D is sunny, nonexpansive and a contraction.

     

D is said to be a nonexpansive retract of C if there exists a nonexpansive retraction from C onto D. The following result, which was established in [24], describes a characterization of sunny nonexpansive retractions on a smooth Banach space.

Let E be a smooth Banach space, and let C be a nonempty subset of E. Let Proj C : E C be a retraction and J φ be the duality mapping on E. Then the following are equivalent:
  1. (1)

    Proj C is sunny and nonexpansive;

     
  2. (2)

    Proj C x Proj C y 2 x y , J φ ( Proj C x Proj C y ) , x , y E ;

     
  3. (3)

    x Proj C x , J φ ( y Proj C x ) 0 , x E , y C .

     

It is well known that if E is a Hilbert space, then a sunny nonexpansive retraction Proj C is coincident with the metric projection from E onto C. Let C be a nonempty closed convex subset of a smooth Banach space E, let x E and let x 0 C . Then we have from the above that x 0 = Proj C x if and only if x x 0 , J φ ( y x 0 ) 0 for all y C , where Proj C is a sunny nonexpansive retraction from E onto C.

A Banach space E is said to be strictly convex if and only if
x = y = ( 1 λ ) x + λ y

for x , y E and 0 < λ < 1 implies that x = y .

E is said to be uniformly convex if for any ϵ ( 0 , 2 ] there exists δ > 0 such that for any x , y U E ,
x y ϵ implies x + y 2 1 δ .

It is known that a uniformly convex Banach space is reflexive and strictly convex.

Let C be a nonempty closed convex subset of E. Let S : C C be a mapping. In this paper, we use F ( S ) to denote the set of fixed points of S. Recall that S is said to be nonexpansive iff S x S y x y , x , y C . For the existence of fixed points of a nonexpansive mapping, we refer the readers to [5]. Let x be a fixed element in C, and let S be a nonexpansive mapping with a nonempty fixed point set. For each t ( 0 , 1 ) , let x t be the unique solution of the equation y = t x + ( 1 t ) S y . In the framework of uniformly smooth Banach spaces, Reich [6] proved that { x t } converges strongly to a fixed point Proj F ( S ) x , where Proj F ( S ) is the unique sunny nonexpansive retraction from C onto F ( S ) , of S as t 0 . Xu [7] further extended the results to the framework of reflexive Banach spaces; for more details, see [7] and [8] and the references therein.

Let I denote the identity operator on E. An operator A E × E with domain D ( A ) = { z E : A z } and range R ( A ) = { A z : z D ( A ) } is said to be accretive if for each x i D ( A ) and y i A x i , i = 1 , 2 , there exists j ( x 1 x 2 ) J ( x 1 x 2 ) such that y 1 y 2 , j ( x 1 x 2 ) 0 . An accretive operator A is said to be M-accretive if R ( I + r A ) = E for all r > 0 . In this paper, we use A 1 ( 0 ) to denote the set of zero points of A. For an accretive operator A, we can define a nonexpansive single-valued mapping J r : R ( I + r A ) D ( A ) by J r = ( I + r A ) 1 for each r > 0 , which is called the resolvent of A. Set
Φ ( t ) = 0 t φ ( τ ) d τ , t 0 ,
then
J φ ( x ) = Φ ( x ) , x E ,

where denotes the sub-differential in the sense of convex analysis.

Zero problems of accretive operators recently have been extensively studied (see [621] and the references therein) because of their important applications in real world. Proximal point algorithm, which was proposed by Martinet [22, 23] and generalized by Rockafellar [24, 25], is a classical method for investigating zeros of monotone operators. In this paper, we propose two proximal point algorithms for investigating common zeros of a family of m-accretive operators. Weak and strong convergence theorems are established in Banach spaces.

The first part of the next lemma is an immediate consequence of the subdifferential inequality and the proof of the second part can be found in [26].

Lemma 1.1 Assume that a Banach space E has a weakly continuous duality mapping J φ with gauge φ.
  1. (i)
    For all x , y E , the following inequality holds:
    Φ ( x + y ) Φ ( x ) + y , J φ ( x + y ) .
     
In particular, for all x , y E ,
x + y 2 x 2 + 2 y , J ( x + y ) .
  1. (ii)

    Assume that a sequence { x n } in E converges weakly to a point x E .

     
Then the following identity holds:
lim sup n Φ ( x n y ) = lim sup n Φ ( x n x ) + Φ ( y x ) , x , y E .

Lemma 1.2 [7]

Let E be a reflexive Banach space and have a weakly continuous duality map J φ ( x ) with gauge φ. Let C be a closed convex subset of E, and let S : C C be a nonexpansive mapping. Fix x C and t ( 0 , 1 ) . Let x t C be the unique fixed point of the mapping t x + ( 1 t ) S . Then S has a fixed point if and only if { x t } remains bounded as t 0 + , and in this case, { x t } converges as t 0 + strongly to a fixed point of S. Define a mapping Proj F ( S ) : C F ( S ) by Proj F ( S ) x : = lim t 0 x t . Then Proj F ( S ) is the sunny nonexpansive retraction from C onto F ( S ) .

Lemma 1.3 [27]

Let C be a closed convex subset of a strictly convex Banach space E. Let N 1 be some positive integer, and let S m : C C be a nonexpansive mapping. Suppose that m = 1 N F ( S m ) is nonempty. Then the mapping m = 1 N β m S m , where { β m } is a real number sequence in ( 0 , 1 ) such that m = 1 N β m = 1 , is nonexpansive with F ( m = 1 N β m S m ) = m = 1 N F ( S m ) .

Lemma 1.4 [28]

Let { a n } , { b n } and { c n } be three nonnegative real sequences satisfying b n + 1 ( 1 a n ) b n + a n c n , n n 0 , where n 0 is some positive integer, { a n } is a number sequence in ( 0 , 1 ) such that n = n 0 a n = , { c n } is a number sequence such that lim sup n c n 0 . Then lim n a n = 0 .

Lemma 1.5 [29]

Let E be a uniformly convex Banach space, s > 0 be a positive number, and B s ( 0 ) be a closed ball of E. There exits a continuous, strictly increasing and convex function g : [ 0 , ) [ 0 , ) with g ( 0 ) = 0 such that
m = 1 N β m x m 2 m = 1 N β m x m 2 β 1 β 2 g ( x 1 x 2 )

for all x 1 , x 2 , , x N B s ( 0 ) = { x E : x < s } and β 1 , β 2 , , β N ( 0 , 1 ) such that m = 1 N β m = 1 .

Lemma 1.6 [30]

Let E be a uniformly convex Banach space. Let C be a nonempty closed convex subset of E, and let S : C C be a nonexpansive mapping. Then I S is demiclosed at zero.

2 Main results

Theorem 2.1 Let E be a uniformly convex Banach space which has the Opial condition. Let N 1 be some positive integer, and let A m be an M-accretive operator in E for each 1 m N . Assume that m = 1 N D ( A i ) ¯ is convex. Let { α n } and { β n , m } be real number sequences in ( 0 , 1 ) , and let { r m } be a positive real number sequence. Assume that m = 1 N A m 1 ( 0 ) is not empty. Let { x n } be a sequence generated in the following manner: x 1 m = 1 N D ( A i ) ¯ and
x n + 1 = α n x n + ( 1 α n ) m = 1 N β n , m J r m x n , n 1 ,
where J r m = ( I + r m A m ) 1 . Assume that the following restrictions are satisfied:
  1. (a)

    0 < a α n b < 1 ;

     
  2. (b)

    m = 1 N β n , m = 1 and 0 < c β n , m < 1 ,

     

where a, b and c are real numbers. Then the sequence { x n } converges weakly to x m = 1 N A m 1 ( 0 ) .

Proof We start the proof with the boundedness of the sequence { x n } . Fixing p m = 1 N A m 1 ( 0 ) , we find that
x n + 1 p α n x n p + ( 1 α n ) m = 1 N β n , m J r m x n p α n x n p + ( 1 α n ) m = 1 N β n , m J r m x n p x n p .
This shows that the limit lim n x n p exists. This implies that { x n } is bounded. Using Lemma 1.5, we find that
x n + 1 p 2 α n x n p 2 + ( 1 α n ) m = 1 N β n , m J r m x n p 2 α n ( 1 α n ) g ( x n m = 1 N β n , m J r m x n ) x n p 2 α n ( 1 α n ) g ( m = 1 N β n , m x n J r m x n ) .
This implies that
α n ( 1 α n ) g ( m = 1 N β n , m x n J r m x n ) x n p 2 x n + 1 p 2 .

In view of restriction (a), we find that lim n g ( m = 1 N β n , m x n J r m x n ) = 0 . It follows that lim n m = 1 N β n , m x n J r m x n = 0 . Using restriction (b), we arrive at lim n x n J r m x n = 0 for each m { 1 , 2 , , N } . Since { x n } is bounded, we see that there exists a subsequence { x n i } of { x n } converging weakly to x . Using Lemma 1.6, we obtain that x F ( J r m ) . This proves that x m = 1 N A m 1 ( 0 ) .

Next we show that { x n } converges weakly to x . Supposing the contrary, we see that there exists some subsequence { x n j } of { x n } such that { x n j } converges weakly to x ˆ C , where x ˆ x . Similarly, we can show x ˆ m = 1 N A m 1 ( 0 ) . Note that we have proved that lim n x n p exists for every p m = 1 N A m 1 ( 0 ) . Assume that lim n x n x = d , where d is a nonnegative number. Since the space has the Opial condition [31], we see that
d = lim inf i x n i x < lim inf i x n i x ˆ = lim inf j x n j x ˆ < lim inf j x n j x = d .

This is a contradiction. Hence x = x ˆ . This completes the proof. □

Theorem 2.2 Let E be a strictly convex and reflexive Banach space which has a weakly continuous duality map J φ . Let N 1 be some positive integer, and let A m be an M-accretive operator in E for each 1 m N . Assume that m = 1 N D ( A i ) ¯ is convex. Let { α n } and { β n , m } be real number sequences in ( 0 , 1 ) , and let { r m } be a positive real number sequence for each 1 m N . Assume that m = 1 N A m 1 ( 0 ) is not empty. Let { x n } be a sequence generated in the following manner: x 1 m = 1 N D ( A i ) ¯ and
x n + 1 = α n x + ( 1 α n ) m = 1 N β n , m J r m x n , n 1 ,
where x is a fixed element in m = 1 N D ( A m ) ¯ and J r m = ( I + r m A m ) 1 . Assume that the following restrictions are satisfied:
  1. (a)

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

     
  2. (b)

    m = 1 N β n , m = 1 , lim n β n , m = β m and n = 1 | β n + 1 , m β n , m | < .

     

Then the sequence { x n } converges strongly to x = Proj m = 1 N A m 1 ( 0 ) x , where Proj m = 1 N A m 1 ( 0 ) is the unique sunny nonexpansive retract from m = 1 N D ( A m ) ¯ onto m = 1 N A m 1 ( 0 ) .

Proof We start the proof with the boundedness of the sequence { x n } . Fixing p m = 1 N A m 1 ( 0 ) , we find that
x n + 1 p α n x p + ( 1 α n ) m = 1 N β n , m J r m x n p α n x p + ( 1 α n ) m = 1 N β n , m J r m x n p α n x p + ( 1 α n ) x n p .
This implies that x n + 1 p max { x p , x 1 p } . This shows that { x n } is bounded. Put y n = m = 1 N β n , m J r m x n . It follows that
y n y n 1 m = 1 N β n , m J r m x n m = 1 N β n , m J r m x n 1 + m = 1 N β n , m J r m x n 1 m = 1 N β n 1 , m J r m x n 1 x n x n 1 + m = 1 N | β n , m β n 1 , m | J r m x n 1 .
This implies that
x n + 1 x n ( 1 α n ) y n y n 1 + | α n α n 1 | x y n 1 ( 1 α n ) x n x n 1 + m = 1 N | β n , m β n 1 , m | J r m x n 1 + | α n α n 1 | x y n 1 .
In light of restrictions (a) and (b), we find that
lim n x n + 1 x n = 0 .
(2.1)
Set S = m = 1 N β m J r m . It follows from Lemma 1.3 that S is nonexpansive with F ( S ) = m = 1 N F ( J r m ) = m = 1 N A m 1 ( 0 ) . Note that
S x n x n x n x n + 1 + x n + 1 S x n x n x n + 1 + α n x S x n + β n m = 1 N β n , m J r m x n S x n x n x n + 1 + α n x S x n + m = 1 N | β n , m β m | J r m x n .
In view of (2.1), we find from the restrictions (a) and (b) that
lim n S x n x n = 0 .
(2.2)
Now, we are in a position to prove
lim sup n x Proj m = 1 N A m 1 ( 0 ) x , J φ ( x n Proj m = 1 N A m 1 ( 0 ) x ) 0 .
(2.3)
By Lemma 1.2, we have the sunny nonexpansive retraction Proj m = 1 N A m 1 ( 0 ) : m = 1 N D ( A m ) ¯ m = 1 N A m 1 ( 0 ) . Take a subsequence { x n k } of { x n } such that
lim sup n x Proj m = 1 N A m 1 ( 0 ) x , J φ ( x n Proj m = 1 N A m 1 ( 0 ) ) x = lim k x Proj m = 1 N A m 1 ( 0 ) x , J φ ( x n k Proj m = 1 N A m 1 ( 0 ) ) x .
(2.4)
Since E is reflexive, we may further assume that x n k x ¯ for some x ¯ m = 1 N D ( A m ) ¯ . Since j φ is weakly continuous, we have from Lemma 1.1
lim sup k Φ ( x n k y ) = lim sup k Φ ( x n k x ¯ ) + Φ ( y x ¯ ) , y E .
Put f ( y ) = lim sup k Φ ( x n k y ) , y E . It follows that
f ( y ) = f ( x ¯ ) + Φ ( y x ¯ ) , y E .
(2.5)
From (2.2), we have
f ( S x ¯ ) = lim sup k Φ ( x n k S x ¯ ) = lim sup k Φ ( S x n k S x ¯ ) lim sup k Φ ( x n k x ¯ ) = f ( x ¯ ) .
(2.6)
Using (2.5), we have
f ( S x ¯ ) = f ( x ¯ ) + Φ ( S x ¯ x ¯ ) .
(2.7)
Combining (2.6) with (2.7), we obtain that
Φ ( S x ¯ x ¯ ) 0 .
Hence S x ¯ = x ¯ ; that is, x ¯ F ( S ) = m = 1 N A m 1 ( 0 ) . It follows from (2.4) that
lim sup n x Proj m = 1 N A m 1 ( 0 ) x , j φ ( x n Proj m = 1 N A m 1 ( 0 ) x ) 0 .
Finally, we prove that x n Proj m = 1 N A m 1 ( 0 ) x as n . Using Lemma 1.1, we find that
Φ ( x n + 1 Proj m = 1 N A m 1 ( 0 ) x ) = Φ ( α n ( x Proj m = 1 N A m 1 ( 0 ) x ) + ( 1 α n ) ( m = 1 N β n , m J r m x n Proj m = 1 N A m 1 ( 0 ) x ) ) ( 1 α n ) Φ ( x n Proj m = 1 N A m 1 ( 0 ) x ) + α n x Proj m = 1 N A m 1 ( 0 ) x , J φ ( x n + 1 Proj m = 1 N A m 1 ( 0 ) x ) .
Using Lemma 1.4, we see that Φ ( x n Proj m = 1 N A m 1 ( 0 ) x ) 0 . This implies that
lim n x n Proj m = 1 N A m 1 ( 0 ) x = 0 .

This completes the proof. □

3 Applications

In this section, we give an application of Theorem 2.1 in the framework of Hilbert spaces.

Let C be a nonempty closed and convex subset of a Hilbert space H. 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 .
(3.1)

To study the equilibrium problem (3.1), 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 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 semi-continuous.

Lemma 3.1 [32]

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 .
Then A F is a maximal monotone operator with the domain D ( A F ) C , EP ( F ) = A F 1 ( 0 ) , where EP ( F ) stands for the solution set of (3.1), and
T r x = ( I + r A F ) 1 x , x H , r > 0 ,
where T s is defined by
T r x = { z C : F ( z , y ) + 1 r y z , z x 0 , y C } , x H .
Corollary 3.2 Let C be a nonempty closed and convex subset of a Hilbert space H. Let N 1 be some positive integer, and let F m : C × C R be a bifunction satisfying (A1)-(A4). Let { α n } and { β n , m } be real number sequences in ( 0 , 1 ) , and let { r m } be a positive real number sequence. Assume that m = 1 N EP ( F m ) is not empty. Let { x n } be a sequence generated in the following manner: x 1 C and
x n + 1 = α n x n + ( 1 α n ) m = 1 N β n , m T r m x n , n 1 ,
where T r m = ( I + r m A F m ) 1 . Assume that the following restrictions are satisfied:
  1. (a)

    0 < a α n b < 1 ;

     
  2. (b)

    m = 1 N β n , m = 1 and 0 < c β n , m < 1 ,

     

where a, b and c are real numbers. Then the sequence { x n } converges weakly to x m = 1 N EP ( F m ) .

Corollary 3.3 Let C be a nonempty closed and convex subset of a Hilbert space H. Let N 1 be some positive integer, and let F m : C × C R be a bifunction satisfying (A1)-(A4). Let { α n } and { β n , m } be real number sequences in ( 0 , 1 ) , and let { r m } be a positive real number sequence. Assume that m = 1 N EP ( F m ) is not empty. Let { x n } be a sequence generated in the following manner: x 1 C and
x n + 1 = α n x + ( 1 α n ) m = 1 N β n , m T r m x n , n 1 ,
where x is a fixed element in C and T r m = ( I + r m A F m ) 1 . Assume that the following restrictions are satisfied:
  1. (a)

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

     
  2. (b)

    m = 1 N β n , m = 1 , lim n β n , m = β m and n = 1 | β n + 1 , m β n , m | < .

     

Then the sequence { x n } converges strongly to x = Proj m = 1 N EP ( F m ) x , where Proj m = 1 N EP ( F m ) is the metric projection from C onto m = 1 N EP ( F m ) .

Declarations

Acknowledgements

This first author thanks the Fundamental Research Funds for the Central Universities (2014ZD44). The authors are grateful to the reviewers’ suggestions which improved the contents of the article.

Authors’ Affiliations

(1)
Department of Mathematics and Physics, North China Electric Power University

References

  1. Browder FE: Convergence theorems for sequences of nonlinear operators in Banach spaces. Math. Z. 1967, 100: 201–225. 10.1007/BF01109805MathSciNetView ArticleMATHGoogle Scholar
  2. Bruck RE: Nonexpansive projections on subsets of Banach spaces. Pac. J. Math. 1973, 47: 341–355. 10.2140/pjm.1973.47.341MathSciNetView ArticleMATHGoogle Scholar
  3. Goebel K, Reich S: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. Dekker, New York; 1984.MATHGoogle Scholar
  4. Reich S: Asymptotic behavior of contractions in Banach spaces. J. Math. Anal. Appl. 1973, 44: 57–70. 10.1016/0022-247X(73)90024-3MathSciNetView ArticleMATHGoogle Scholar
  5. Browder FE: Fixed point theorems for noncompact mappings in Hilbert spaces. Proc. Natl. Acad. Sci. USA 1965, 53: 1272–1276. 10.1073/pnas.53.6.1272MathSciNetView ArticleMATHGoogle Scholar
  6. Reich S: Strong convergence theorems for resolvents of accretive operators in Banach spaces. J. Math. Anal. Appl. 1980, 75: 287–292. 10.1016/0022-247X(80)90323-6MathSciNetView ArticleMATHGoogle Scholar
  7. Xu HK: Strong convergence of an iterative method for nonexpansive and accretive operators. J. Math. Anal. Appl. 2006, 314: 631–643. 10.1016/j.jmaa.2005.04.082MathSciNetView ArticleMATHGoogle Scholar
  8. 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
  9. Kim JK: Convergence of Ishikawa iterative sequences for accretive Lipschitzian mappings in Banach spaces. Taiwan. J. Math. 2006, 10: 553–561.MathSciNetMATHGoogle Scholar
  10. Zegeye H, Shahzad N: Strong convergence theorems for a common zero of a finite family of m -accretive mappings. Nonlinear Anal. 2007, 66: 1161–1169. 10.1016/j.na.2006.01.012MathSciNetView ArticleMATHGoogle Scholar
  11. 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 ArticleMATHGoogle Scholar
  12. Yang S: Zero theorems of accretive operators in reflexive Banach spaces. J. Nonlinear Funct. Anal. 2013., 2013: Article ID 2Google Scholar
  13. Song J, Chen M: A modified Mann iteration for zero points of accretive operators. Fixed Point Theory Appl. 2013., 2013: Article ID 347Google Scholar
  14. Cho SY, Kang SM: Zero point theorems for m -accretive operators in a Banach space. Fixed Point Theory 2012, 13: 49–58.MathSciNetMATHGoogle Scholar
  15. Hao Y: Zero theorems of accretive operators. Bull. Malays. Math. Soc. 2011, 34: 103–112.MathSciNetMATHGoogle Scholar
  16. Yuan Q, Cho SY: A regularization algorithm for zero points of accretive operators. Fixed Point Theory Appl. 2013., 2013: Article ID 341Google Scholar
  17. Kim JK, Buong N: Regularization inertial proximal point algorithm for monotone hemicontinuous mapping and inverse strongly monotone mappings in Hilbert spaces. J. Inequal. Appl. 2010., 2010: Article ID 451916Google Scholar
  18. Qing Y, Cho SY, Qin X: Convergence of iterative sequences for common zero points of a family of m -accretive mappings in Banach spaces. Fixed Point Theory Appl. 2011., 2011: Article ID 216173Google Scholar
  19. 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.MathSciNetMATHGoogle Scholar
  20. Wu C, Lv S, Zhang Y: Some results on zero points of m -accretive operators in reflexive Banach spaces. Fixed Point Theory Appl. 2014., 2014: Article ID 118Google Scholar
  21. Cho SY, Li W, Kang SM: Convergence analysis of an iterative algorithm for monotone operators. J. Inequal. Appl. 2013., 2013: Article ID 199Google Scholar
  22. Martinet B: Régularisation d’inéquations variationelles par approximations successives. Rev. Fr. Autom. Inform. Rech. Opér. 1970, 4: 154–158.MathSciNetMATHGoogle Scholar
  23. Martinet B: Détermination approchée d’un point fixe d’une application pseudo-contractante. C. R. Acad. Sci. Paris, Ser. A-B 1972, 274: 163–165.MathSciNetMATHGoogle Scholar
  24. Rockafellar RT: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 1976, 14: 877–898. 10.1137/0314056MathSciNetView ArticleMATHGoogle Scholar
  25. Rockafellar RT: Augmented Lagrangians and applications of the proximal point algorithm in convex programming. Math. Oper. Res. 1976, 1: 97–116. 10.1287/moor.1.2.97MathSciNetView ArticleMATHGoogle Scholar
  26. Lim TC, Xu HK: Fixed point theorems for asymptotically nonexpansive mappings. Nonlinear Anal. 1994, 22: 1345–1355. 10.1016/0362-546X(94)90116-3MathSciNetView ArticleMATHGoogle Scholar
  27. Bruck RE: Properties of fixed-point sets of nonexpansive mappings in Banach spaces. Trans. Am. Math. Soc. 1973, 179: 251–262.MathSciNetView ArticleMATHGoogle Scholar
  28. 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 ArticleMATHGoogle Scholar
  29. Hao Y: Some weak convergence theorems for a family of asymptotically nonexpansive nonself mappings. Fixed Point Theory Appl. 2010., 2010: Article ID 218573Google Scholar
  30. Gornicki J: Weak convergence theorems for asymptotically nonexpansive mappings in uniformly convex Banach spaces. Comment. Math. Univ. Carol. 1989, 30: 249–252.MathSciNetMATHGoogle Scholar
  31. Opial Z: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Am. Math. Soc. 1967, 73: 591–597. 10.1090/S0002-9904-1967-11761-0MathSciNetView ArticleMATHGoogle Scholar
  32. 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/s10957-010-9713-2MathSciNetView ArticleMATHGoogle Scholar

Copyright

© Wang and Li; 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/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.