Open Access

Hybrid projection methods for a bifunction and relatively asymptotically nonexpansive mappings

Fixed Point Theory and Applications20132013:294

https://doi.org/10.1186/1687-1812-2013-294

Received: 8 July 2013

Accepted: 4 September 2013

Published: 9 November 2013

Abstract

The purpose of this paper is to investigate a bifunction equilibrium problem and a fixed point problem of relatively asymptotically nonexpansive mappings based on a generalized projection method. A weak convergence theorem for common solutions is established in a uniformly smooth and uniformly convex Banach space.

Keywords

bifunction equilibrium problem fixed point generalized projection relatively asymptotically nonexpansive mapping

1 Introduction and preliminaries

Let E be a real Banach space, E be the dual space of E, and C be a nonempty subset of E. Let F be a bifunction from C × C to , where denotes the set of real numbers. Recall the following equilibrium problem: Find x ¯ C such that
F ( x ¯ , y ) 0 , y C .
(1.1)
From now on, we use EP ( F ) to denote the solution set of equilibrium problem (1.1) and assume that F satisfies the following conditions:
  1. (A1)

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

     
  2. (A2)

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

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

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

     
Let U E = { x E : x = 1 } be the unit sphere of E. Then the Banach space E is said to be smooth iff
lim t 0 x + t y x t

exists for each x , y U E . It is also said to be uniformly smooth iff the above limit is attained uniformly for x , y U E . It is well known that if E is uniformly smooth, then J is uniformly norm-to-norm continuous on each bounded subset of E. Recall that E is said to be uniformly convex iff lim n x n y n = 0 for any two sequences { x n } and { y n } in E such that x n = y n = 1 and lim n x n + y n 2 = 1 . It is well known that E is uniformly smooth if and only if E is uniformly convex.

Recall that a Banach space E enjoys the Kadec-Klee property if for any sequence { x n } E , and x E with x n x , and x n x , then x n x 0 as n . For more details on the Kadec-Klee property, the readers can refer to [1] and the references therein. It is well known that if E is a uniformly convex Banach space, then E enjoys the Kadec-Klee property.

Let T : C C be a mapping. From now on, we use F ( T ) to denote the fixed point set of T. Recall that T is said to be closed if for any sequence { x n } C such that lim n x n = x 0 and lim n T x n = y 0 , then T x 0 = y 0 . In this paper, we use → and to denote the strong convergence and the weak convergence, respectively.

Recall that the normalized duality mapping J from E to 2 E is defined by
J x = { f E : x , f = x 2 = f 2 } ,
where , denotes the generalized duality pairing. Next, we assume that E is a smooth Banach space. Consider the functional defined by
ϕ ( x , y ) = x 2 2 x , J y + y 2 , x , y E .
Observe that in a Hilbert space H the equality is reduced to ϕ ( x , y ) = x y 2 , x , y H . As we all know, if C is a nonempty closed convex subset of a Hilbert space H and P C : H C is the metric projection of H onto C, then P C is nonexpansive. This fact actually characterizes Hilbert spaces and, consequently, it is not available in more general Banach spaces. In this connection, Alber [2] recently introduced a generalized projection operator Π C in a Banach space E which is an analogue of the metric projection P C in Hilbert spaces. Recall that the generalized projection Π C : E C is a map that assigns to an arbitrary point x E the minimum point of the functional ϕ ( x , y ) , that is, Π C x = x ¯ , where x ¯ is the solution to the minimization problem
ϕ ( x ¯ , x ) = min y C ϕ ( y , x ) .
Existence and uniqueness of the operator Π C follow from the properties of the functional ϕ ( x , y ) and strict monotonicity of the mapping J. In Hilbert spaces, Π C = P C . It is obvious from the definition of a function ϕ that
( x y ) 2 ϕ ( x , y ) ( y + x ) 2 , x , y E .
(1.2)

Remark 1.1 If E is a reflexive, strictly convex, and smooth Banach space, then ϕ ( x , y ) = 0 if and only if x = y ; for more details, see [2] and the references therein.

Recall that a point p in C is said to be an asymptotic fixed point of a mapping T iff C contains a sequence { x n } which converges weakly to p so that lim n x n T n x n = 0 . The set of asymptotic fixed points of T will be denoted by F ˜ ( T ) .

Recall that a mapping T is said to be relatively nonexpansive iff
F ˜ ( T ) = F ( T ) , ϕ ( p , T x ) ϕ ( p , x ) , x C , p F ( T ) .
Recall that a mapping T is said to be relatively asymptotically nonexpansive iff
F ˜ ( T ) = F ( T ) , ϕ ( p , T n x ) ( 1 + μ n ) ϕ ( p , x ) , x C , p F ( T ) , n 1 ,

where { μ n } [ 0 , ) is a sequence such that μ n 0 as n .

Remark 1.2 The class of relatively nonexpansive mappings was first considered in Butnariu et al. [3]. The class of relatively asymptotically nonexpansive mappings was first considered in Agarwal et al. [4] and the references therein.

Recently, many authors investigated fixed point problems of a (relatively) nonexpansive mapping based on hybrid projection methods; for more details, see [537] and the references therein. However, most of the results are on strong convergence. In this article, we investigate a bifunction equilibrium problem and a fixed point problem of relatively asymptotically nonexpansive mappings based on a generalized projection method. A weak convergence theorem for common solutions is established in a uniformly smooth and uniformly convex Banach space.

The following lemmas play an important role in this paper.

Lemma 1.3 [37, 38]

Let C be a closed convex subset of a uniformly smooth and uniformly convex Banach space E. Let F be a bifunction from C × C to satisfying (A1)-(A4). Let r > 0 and x E . Then there exists z C such that F ( z , y ) + 1 r y z , J z J x 0 , y C . Define a mapping S r : E C by S r x = { z C : F ( z , y ) + 1 r y z , J z J x , y C } . Then the following conclusions hold:
  1. (a)

    S r is single-valued;

     
  2. (b)
    S r is a firmly nonexpansive-type mapping, i.e., for all x , y E ,
    S r x S r y , J S r x J S r y S r x S r y , J x J y ;
     
  3. (c)

    F ( S r ) = EP ( F ) is closed and convex;

     
  4. (d)

    S r is relatively nonexpansive;

     
  5. (e)

    ϕ ( q , S r x ) + ϕ ( S r x , x ) ϕ ( q , x ) , q F ( S r ) .

     

Lemma 1.4 [4]

Let E be a uniformly smooth and uniformly convex Banach space. Let C be a nonempty closed and convex subset of E. Let T : C C be a relatively asymptotically nonexpansive mapping. Then F ( T ) is a closed convex subset of C.

Lemma 1.5 [2]

Let E be a reflexive, strictly convex, and smooth Banach space, let C be a nonempty, closed, and convex subset of E, and let x E . Then
ϕ ( y , Π C x ) + ϕ ( Π C x , x ) ϕ ( y , x ) , y C .

Lemma 1.6 [2]

Let C be a nonempty, closed, and convex subset of a smooth Banach space E, and let x E . Then x 0 = Π C x if and only if
x 0 y , J x J x 0 0 , y C .

Lemma 1.7 [39]

Let E be a smooth and uniformly convex Banach space, and let r > 0 . Then there exists a strictly increasing, continuous, and convex function g : [ 0 , 2 r ] R such that g ( 0 ) = 0 and
t x + ( 1 t ) y 2 t x 2 + ( 1 t ) y 2 t ( 1 t ) g ( x y )

for all x , y B r = { x E : x r } and t [ 0 , 1 ] .

Lemma 1.8 [40]

Let { a n } , { b n } , and { c n } be three nonnegative sequences satisfying the following condition:
a n + 1 ( 1 + b n ) a n + c n , n n 0 ,

where n 0 is some nonnegative integer. If n = 1 b n < and n = 1 c n < , then the limit of the sequence { a n } exists. If, in addition, there exists a subsequence { α n i } { α n } such that α n i 0 , then α n 0 as n .

Lemma 1.9 [41]

Let E be a smooth and uniformly convex Banach space, and let r > 0 . Then there exists a strictly increasing, continuous, and convex function g : [ 0 , 2 r ] R such that g ( 0 ) = 0 and g ( x y ) ϕ ( x , y ) for all x , y B r .

2 Main results

Theorem 2.1 Let E be a uniformly smooth and uniformly convex Banach space, and let C be a nonempty closed and convex subset of E. Let F be a bifunction from C × C to satisfying (A1)-(A4). Let T : C C be a relatively asymptotically nonexpansive mapping with the sequence { μ n , 1 } , and let S : C C be a relatively asymptotically nonexpansive mapping with the sequence { μ n , 2 } . Assume that Φ : = F ( T ) F ( S ) EP ( F ) is nonempty. Let { x n } be a sequence generated in the following manner:
{ y 0 E chosen arbitrarily , x n C  such that  F ( x n , x ) + 1 r n x x n , J x n J y n 0 , x C , y n + 1 = J 1 ( α n J x n + β n J T n x n + γ n J S n x n ) , n 0 ,
where { α n } , { β n } , { γ n } are real sequences in [ 0 , 1 ] and { r n } is a real number sequence in [ r , ) , where r > 0 is some real number. Assume that J is weakly sequentially continuous and the following restrictions hold:
  1. (i)

    α n + β n + γ n = 1 ;

     
  2. (ii)

    n = 1 μ n < ;

     
  3. (iii)

    lim inf n α n β n > 0 , lim inf n α n γ n > 0 .

     

Then the sequence { x n } converges weakly to x ¯ Φ , where x ¯ = lim n Π Φ x n .

Proof Set μ n = max { μ n , 1 , μ n , 2 } . Fixing p Φ , we find that
ϕ ( p , x n + 1 ) = ϕ ( p , S r n + 1 y n + 1 ) ϕ ( p , y n + 1 ) = p 2 2 p , α n J x n + β n J T n x n + γ n J S n x n + α n J x n + β n J T n x n + γ n J S n x n 2
(2.1)
p 2 2 α n p , J x n 2 β n p , J T n x n 2 γ n p , J S n x n + α n x n 2 + β n T n x n 2 + γ n S n x n 2 = α n ϕ ( p , x n ) + β n ϕ ( p , T n x n ) + γ n ϕ ( p , S n x n ) ϕ ( p , x n ) + β n μ n ϕ ( p , x n ) + γ n μ n ϕ ( p , x n ) ( 1 + μ n ) ϕ ( p , x n ) .
(2.2)
In view of Lemma 1.8, we obtain that lim n ϕ ( p , x n ) exits. This implies that the sequence { x n } is bounded. In the light of Lemma 1.7, we find that
ϕ ( p , x n + 1 ) = ϕ ( p , S r n + 1 y n + 1 ) p 2 2 p , α n J x n + β n J T n x n + γ n J S n x n + α n J x n + β n J T n x n + γ n J S n x n 2 p 2 2 α n p , J x n 2 β n p , J T n x n 2 γ n p , J S n x n + α n x n 2 + β n T n x n 2 + γ n S n x n 2 α n β n g ( J T n x n J x n ) ϕ ( p , x n ) + β n μ n ϕ ( p , x n ) + γ n μ n ϕ ( p , x n ) α n β n g ( J T n x n J x n ) ( 1 + μ n ) ϕ ( p , x n ) α n β n g ( J T n x n J x n ) .
It follows that
α n β n g ( J T n x n J x n ) ( 1 + μ n ) ϕ ( p , x n ) ϕ ( p , x n + 1 ) .
This finds from the restrictions (ii) and (iii) that
lim n g ( J T n x n J x n ) = 0 .
This implies that
lim n J T n x n J x n = 0 .
Since J 1 is uniformly norm-to-norm continuous on bounded sets, we find that
lim n T n x n x n = 0 .
In the same way, we find that
lim n S n x n x n = 0 .
Since { x n } is bounded, we see that there exists a subsequence { x n i } of { x n } such that { x n i } converges weakly to p C . It follows that p F ( T ) F ( S ) . Next, we prove that p EP ( F ) . Let r = sup n 1 { x n , y n } . In view of Lemma 1.9, we find that there exists a continuous, strictly increasing and convex function h with h ( 0 ) = 0 such that
h ( x , y ) ϕ ( x , y ) , x , y B r .
It follows from (2.1) that
h ( x n y n ) ϕ ( x n , y n ) ϕ ( p , y n ) ϕ ( p , x n ) ϕ ( p , x n 1 ) ϕ ( p , x n ) + μ n 1 ϕ ( p , x n 1 ) .
This implies that
lim n h ( x n y n ) = 0 .
It follows from the property of h that
lim n x n y n = 0 .
Since J is uniformly norm-to-norm continuous on bounded sets, one has
lim n J x n J y n = 0 .
Since { r n } is a real number sequence in [ r , ) , where r > 0 is some real number, one finds that
lim n J x n J y n r n = 0 .
Notice that x n = S r n y n , one sees that
F ( x n , x ) + 1 r n x x n , J x n J y n 0 , x C .
By replacing n by n i , one finds from (A2) that
x x n i J x n i J y n i r n i 1 r n i x x n i , J x n i J y n i F ( x , x n i ) .
Letting i in the above inequality, one obtains from (A4) that
F ( x , p ) 0 , x C .
For 0 < t < 1 and y C , define x t = t x + ( 1 t ) p . It follows that x t C , which yields that F ( x t , p ) 0 . It follows from (A1) and (A4) that
0 = F ( x t , x t ) t F ( x t , x ) + ( 1 t ) F ( x x , p ) t F ( x t , x ) .
That is,
F ( x t , x ) 0 .
Letting t 0 , we obtain from (A3) that F ( p , x ) 0 , x C . This implies that p EP ( F ) . This completes the proof that p F ( T ) F ( S ) EP ( F ) . Define z n = Π F ( T ) F ( S ) EP ( F ) x n . It follows from (2.1) that
ϕ ( z n , x n + 1 ) ( 1 + μ n ) ϕ ( z n , x n ) .
(2.3)
This in turn implies from Lemma 1.5 that
ϕ ( z n + 1 , x n + 1 ) = ϕ ( Π F ( T ) F ( S ) EP ( F ) x n + 1 , x n + 1 ) ϕ ( z n , x n + 1 ) ϕ ( z n , Π F ( T ) F ( S ) EP ( F ) x n + 1 ) ϕ ( z n , x n + 1 ) ϕ ( z n , z n + 1 ) ϕ ( z n , x n + 1 ) .
It follows from (2.3) that
ϕ ( z n + 1 , x n + 1 ) ( 1 + μ n ) ϕ ( z n , x n ) .
This finds from Lemma 1.8 that the sequence { ϕ ( z n , x n ) } is a convergence sequence. It follows from (2.1) that
ϕ ( p , x n + m ) ϕ ( p , x n ) + L ( i = 1 m μ n + m i ) ,
(2.4)
where L = sup n 1 ϕ ( p , x n ) . Since z n F ( T ) F ( S ) EP ( F ) , we find that
ϕ ( z n , x n + m ) ϕ ( z n , x n ) + M ( i = 1 m μ n + m i ) ,
where M = sup n 1 ϕ ( z n , x n ) . Since z n + m = Π F ( T ) F ( S ) EP ( F ) x n + m , we find from Lemma 1.5 that
ϕ ( z n , z n + m ) + ϕ ( z n + m , x n + m ) ϕ ( z n , x n + m ) ϕ ( z n , x n ) + M ( i = 1 m μ n + m i ) .
It follows that
ϕ ( z n , z n + m ) ϕ ( z n , x n ) ϕ ( z n + m , x n + m ) + M ( i = 1 m μ n + m i ) .
In view of Lemma 1.9, we find that there exists a continuous, strictly increasing, and convex function g with
g ( z n z m ) ϕ ( z n , z m ) ϕ ( z n , x n ) ϕ ( z n + m , x n + m ) + M ( i = 1 m μ n + m i ) .
This shows that { z n } is a Cauchy sequence. Since F ( T ) F ( S ) EP ( F ) is closed, one sees that { z n } converges strongly to z F ( T ) F ( S ) EP ( F ) . Since p F ( T ) F ( S ) EP ( F ) , we find from Lemma 1.6 that
z n k p , J x n k J z n k 0 .

Notice that J is weakly sequentially continuous. Letting k , we find that z p , J p J z 0 . It follows from the monotonicity of J that z p , J p J z 0 . Since the space is uniformly convex, we find that z = p . This completes the proof. □

Remark 2.2 Theorem 2.1 improves Theorem 2.5 in Qin et al. [36] on the mappings from the class of relatively nonexpansive mappings to the class of relatively asymptotically nonexpansive mappings.

If T = S , then Theorem 2.1 is reduced to the following.

Corollary 2.3 Let E be a uniformly smooth and uniformly convex Banach space, and let C be a nonempty closed and convex subset of E. Let F be a bifunction from C × C to satisfying (A1)-(A4). Let T : C C be a relatively asymptotically nonexpansive mapping with the sequence { μ n } . Assume that Φ : = F ( T ) EP ( F ) is nonempty. Let { x n } be a sequence generated in the following manner:
{ y 0 E chosen arbitrarily , x n C  such that  F ( x n , x ) + 1 r n x x n , J x n J y n 0 , x C , y n + 1 = J 1 ( α n J x n + ( 1 α n ) J T n x n ) , n 0 ,
where { α n } is a real sequence in [ 0 , 1 ] and { r n } is a real number sequence in [ r , ) , where r > 0 is some real number. Assume that J is weakly sequentially continuous and the following restrictions hold:
  1. (i)

    n = 1 μ n < ;

     
  2. (ii)

    lim inf n α n ( 1 α n ) > 0 .

     

Then the sequence { x n } converges weakly to x ¯ Φ , where x ¯ = lim n Π Φ x n .

Remark 2.4 Corollary 2.3 is an improvement of Theorem 4.1 in Zembayashi and Takahashi [37] on the nonlinear mapping.

Declarations

Acknowledgements

This work is supported by the Fundamental Research Funds for the Central Universities (Grant No.: 9161013002). The authors are grateful to the editor and the anonymous reviewers’ suggestions which improved the contents of the article.

Authors’ Affiliations

(1)
Department of Applied Mathematics and Physics, North China Electric Power University
(2)
Department of Mathematics and Sciences, Shijiazhuang University of Economics

References

  1. Cioranescu I: Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems. Kluwer Academic, Dordrecht; 1990.View ArticleMATHGoogle Scholar
  2. Alber YI: Metric and generalized projection operators in Banach spaces: properties and applications. In Theory and Applications of Nonlinear Operators of Accretive and Monotone Type. Edited by: Kartsatos AG. Dekker, New York; 1996.Google Scholar
  3. Butnariu D, Reich S, Zaslavski AJ: Weak convergence of orbits of nonlinear operators in reflexive Banach spaces. Numer. Funct. Anal. Optim. 2003, 24: 489–508. 10.1081/NFA-120023869MathSciNetView ArticleGoogle Scholar
  4. Agarwal RP, Cho YJ, Qin X: Generalized projection algorithms for nonlinear operators. Numer. Funct. Anal. Optim. 2007, 28: 1197–1215. 10.1080/01630560701766627MathSciNetView ArticleGoogle Scholar
  5. Qin X, Agarwal RP: Shrinking projection methods for a pair of asymptotically quasi- ϕ -nonexpansive mappings. Numer. Funct. Anal. Optim. 2010, 31: 1072–1089. 10.1080/01630563.2010.501643MathSciNetView ArticleGoogle Scholar
  6. Ye J, Huang J: Strong convergence theorems for fixed point problems and generalized equilibrium problems of three relatively quasi-nonexpansive mappings in Banach spaces. J. Math. Comput. Sci. 2011, 1: 1–18.MathSciNetView ArticleGoogle Scholar
  7. 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
  8. Shen J, Pang LP: An approximate bundle method for solving variational inequalities. Commun. Optim. Theory 2012, 1: 1–18.Google Scholar
  9. 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
  10. Qin X, Chang SS, Cho YJ: Iterative methods for generalized equilibrium problems and fixed point problems with applications. Nonlinear Anal. 2010, 11: 2963–2972. 10.1016/j.nonrwa.2009.10.017MathSciNetView ArticleGoogle Scholar
  11. Noor MA, Noor KI, Waseem M: Decompsition method for solving system of linear equations. Eng. Math. Lett. 2012, 2: 34–41.Google Scholar
  12. Luo H, Wang Y: Iterative approximation for the common solutions of a infinite variational inequality system for inverse-strongly accretive mappings. J. Math. Comput. Sci. 2012, 2: 1660–1670.MathSciNetGoogle Scholar
  13. Qin X, Cho SY, Kang SM: On hybrid projection methods for asymptotically quasi- ϕ -nonexpansive mappings. Appl. Math. Comput. 2010, 215: 3874–3883. 10.1016/j.amc.2009.11.031MathSciNetView ArticleGoogle Scholar
  14. Chang SS, Wang L, Tang YK, Zao YH, Wang B: Strong convergence theorems of quasi- ϕ -asymptotically nonexpansive semi-groups in Banach spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 15Google Scholar
  15. Chang SS, Kim JK, Wang L: Total quasi- ϕ -asymptotically nonexpansive semigroups and strong convergence theorems in Banach spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 153Google Scholar
  16. Cho SY, Li W, Kang SM: Convergence analysis of an iterative algorithm for monotone operators. J. Inequal. Appl. 2013., 2013: Article ID 199Google Scholar
  17. Zuo P, Chang SS, Liu M: On a hybrid algorithm for a family of total quasi- ϕ -asymptotically nonexpansive mappings in Banach spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 70Google Scholar
  18. 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
  19. Kim JK: Hybrid projection algorithms for generalized equilibrium problems and strictly pseudocontractive mappings. J. Inequal. Appl. 2010., 2010: Article ID 312602Google Scholar
  20. 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
  21. Ali B, Minjibir M: Convergence of a hybrid iterative method for finite families of generalized quasi- ϕ -asymptotically nonexpansive mappings. Fixed Point Theory Appl. 2012., 2012: Article ID 121Google Scholar
  22. 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
  23. Li Y, Liu H: Strong convergence theorems for modifying Halpern-Mann iterations for a quasi- ϕ -asymptotically nonexpansive multi-valued mapping in Banach spaces. Fixed Point Theory Appl. 2013., 2013: Article ID 132Google Scholar
  24. Zhang J, Su Y, Chen Q: Simple projection algorithm for a countable family of weak relatively nonexpansive mappings and applications. Fixed Point Theory Appl. 2012., 2012: Article ID 205Google Scholar
  25. Qin X, Agarwal RP, Cho SY, Kang SM: Convergence of algorithms for fixed points of generalized asymptotically quasi- ϕ -nonexpansive mappings with applications. Fixed Point Theory Appl. 2012., 2012: Article ID 58Google Scholar
  26. Zhou H, Gao X: A strong convergence theorem for a family of quasi- ϕ -nonexpansive mappings in a Banach space. Fixed Point Theory Appl. 2009., 2009: Article ID 351265Google Scholar
  27. Qin X, Cho SY, Kang SM: An extragradient-type method for generalized equilibrium problems involving strictly pseudocontractive mappings. J. Glob. Optim. 2011, 49: 679–693. 10.1007/s10898-010-9556-2MathSciNetView ArticleGoogle Scholar
  28. Kohsaka F, Takahashi W: Block iterative methods for a finite family of relatively nonexpansive mappings in Banach spaces. Fixed Point Theory Appl. 2007., 2007: Article ID 021972Google Scholar
  29. Hao Y: Some results on a modified Mann iterative scheme in a reflexive Banach space. Fixed Point Theory Appl. 2013., 2013: Article ID 227Google Scholar
  30. Qin X, Cho YJ, Kang SM, Zhou H: Convergence of a modified Halpern-type iteration algorithm for quasi- ϕ -nonexpansive mappings. Appl. Math. Lett. 2009, 22: 1051–1055. 10.1016/j.aml.2009.01.015MathSciNetView ArticleGoogle Scholar
  31. Wang X, Hu C, Guan J: Strong convergence theorems for fixed point problems of infinite family of asymptotically quasi-phi-nonexpansive mappings and a system of equilibrium problems. Fixed Point Theory Appl. 2013., 2013: Article ID 221Google Scholar
  32. Kim JK: Some results on generalized equilibrium problems involving strictly pseudocontractive mappings. Acta Math. Sci. 2011, 31: 2041–2057.View ArticleGoogle Scholar
  33. Su Y, Qin X: Monotone CQ iteration processes for nonexpansive semigroups and maximal monotone operators. Nonlinear Anal. 2008, 68: 3657–3664. 10.1016/j.na.2007.04.008MathSciNetView ArticleGoogle Scholar
  34. Cho SY, Kang SM: Approximation of common solutions of variational inequalities via strict pseudocontractions. Acta Math. Sci. 2012, 32: 1607–1618.View ArticleGoogle Scholar
  35. Wangkeeree R: A new hybrid approximation algorithm based on the shrinking projection method for two asymptotically quasi- ϕ -nonexpansive mappings. J. Comput. Anal. Appl. 2012, 14: 298–313.MathSciNetGoogle Scholar
  36. 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
  37. Zembayashi WK, Takahashi W: Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces. Nonlinear Anal. 2009, 70: 45–57. 10.1016/j.na.2007.11.031MathSciNetView ArticleGoogle Scholar
  38. Blum E, Oettli W: From optimization and variational inequalities to equilibrium problems. Math. Stud. 1994, 63: 123–145.MathSciNetGoogle Scholar
  39. Zǎlinescu C: On uniformly convex functions. J. Math. Anal. Appl. 1983, 95: 344–374. 10.1016/0022-247X(83)90112-9MathSciNetView ArticleGoogle Scholar
  40. Chang SS, Cho YJ, Zhou H: Iterative Methods for Nonlinear Operator Equations in Banach Spaces. Nova Science Publishers, Huntington; 2002.MATHGoogle Scholar
  41. Kamimura S, Takahashi W: Strong convergence of a proximal-type algorithm in a Banach space. SIAM J. Optim. 2002, 13: 938–945. 10.1137/S105262340139611XMathSciNetView ArticleGoogle Scholar

Copyright

© Wang and Song; licensee Springer. 2013

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.