Skip to main content

Advertisement

You are viewing the new article page. Let us know what you think. Return to old version

Research | Open | Published:

Approximation of fixed points for nonexpansive semigroups in Hilbert spaces

Abstract

In this paper, we propose two new algorithms for finding a common fixed point of a nonexpansive semigroup in Hilbert spaces and prove some strong convergence theorems for nonexpansive semigroups. Our results improve and generalize the corresponding results given by Shimizu and Takahashi (J. Math. Anal. Appl. 211:71-83, 1997), Shioji and Takahashi (Nonlinear Anal. TMA 34:87-99, 1998), Lau et al. (Nonlinear Anal. TMA 67:1211-1225, 2007) and many others.

MSC:47H05, 47H10, 47H17.

1 Introduction

Let H be a real Hilbert space with the inner product , and the norm . Let C be a nonempty closed convex subset of H. A mapping T:CC is said to be nonexpansive if

TxTyxy,x,yC.

Recall that a family S:= { T ( s ) } s 0 of mappings of C into itself is called a nonexpansive semigroup if it satisfies the following conditions:

  1. (S1)

    T(0)x=x for all xC;

  2. (S2)

    T(s+t)=T(s)T(t) for all s,t0;

  3. (S3)

    T(s)xT(s)yxy for all x,yC and s0;

  4. (S4)

    for each xH, sT(s)x is continuous.

We denote by Fix(T(s)) the set of fixed points of T(s) and by Fix(S) the set of all common fixed points of S, i.e., Fix(S)= s 0 Fix(T(s)). It is known that Fix(S) is closed and convex [[1], Lemma 1].

Approximation of fixed points of nonexpansive mappings by a sequence of finite means has been considered by many authors; see, for instance, [127]. This work was originated with the beautiful work of Baillon [5] in 1975 (see also [6] and [7] for a generalization): If C is a closed convex subset of a Hilbert space and T is a nonexpansive mapping from C into itself such that the set Fix(T) of fixed points of T is nonempty, then for each xC, the Cesàro mean

1 n k = 1 n T k x

converges weakly to x Fix(T). In this case, if we put x = P Fix ( T ) x for each xC, then P Fix ( T ) is a nonexpansive retraction from C onto F(T). In [18], Takahashi proved the existence of such a retraction for an amenable semigroup of nonexpansive mappings on a Hilbert space. In [19], Rodé also found a sequence of means on a semigroup generalizing the Cesàro means and extended Baillon’s theorem. In [28], Lau, Shioji and Takahashi extended Takahashi’s result and Rode’s result to a closed convex subset of a uniformly convex Banach space.

In the literature, a nonlinear ergodic theorem for nonexpansive semigroups has been considered by many authors (see [2946]). Especially, Shioji and Takahashi [17] introduced an implicit iteration { x n } in a Hilbert space defined by

x n = α n x+(1 α n ) 1 λ n 0 λ n T(s) x n ds,n0,
(1.1)

where { α n } is a sequence in (0,1) and { λ n } is a sequence of positive real numbers divergent to ∞. Under certain restrictions on the sequence { α n }, Shioji and Takahashi [17] proved strong convergence of { x n } generated by (1.1) to a member of Fix(T(s)). In [16], Shimizu and Takahashi studied the strong convergence of the iterative sequence { x n } defined by

x n + 1 = α n x+(1 α n ) 1 λ n 0 λ n T(s) x n ds,n0.
(1.2)

The corresponding viscosity approximations of (1.1) and (1.2) have been extended in [29]. Lau et al. [37] studied the iterative schemes of Browder and Halpern types for a nonexpansive semigroup { T ( s ) } s 0 on a compact convex subset C of a smooth (and strictly convex) Banach space with respect to a sequence { μ n } of strongly asymptotically invariant means defined on an appropriate invariant subspace of l (S), the space of bounded real-valued functions on a semigroup S.

Motivated and inspired by the works in the literature, in this paper, we introduce two new algorithms for finding a common fixed point of a nonexpansive semigroup { T ( s ) } s 0 in Hilbert spaces and prove that both approaches converge strongly to a common fixed point of { T ( s ) } s 0 .

2 Preliminaries

Let C be a nonempty closed convex subset of a real Hilbert space H. The metric (or nearest point) projection from H onto C is the mapping P C :HC which assigns to each point xC the unique point P C xC satisfying the property

x P C x= inf y C xy=:d(x,C).

It is well known that P C is a nonexpansive mapping and satisfies

xy, P C x P C y P C x P C y 2 ,x,yH.

Moreover, P C is characterized by the following properties:

x P C x,y P C x0
(2.1)

and

x y 2 x P C x 2 + y P C x 2 ,xH,yC.

We need the following lemmas for proving our main results.

Lemma 2.1 [16]

Let C be a nonempty bounded closed convex subset of a Hilbert space H and { T ( s ) } s 0 be a nonexpansive semigroup on C. Then, for any h0,

lim t sup x C 1 t 0 t T ( s ) x d s T ( h ) 1 t 0 t T ( s ) x d s =0.

Lemma 2.2 [8]

Let C be a closed convex subset of a real Hilbert space H and S:CC be a nonexpansive mapping. Then the mapping IS is demiclosed. That is, if { x n } is a sequence in C such that x n x weakly and (IS) x n y strongly, then (IS) x =y.

Lemma 2.3 [13]

Let { x n } and { y n } be bounded sequences in a Banach space X and { γ n } be a sequence in [0,1] with 0< lim inf n β n lim sup n β n <1. Suppose that

x n + 1 =(1 γ n ) x n + γ n y n ,n0,

and

lim sup n ( y n y n 1 x n x n 1 ) 0.

Then lim n y n x n =0.

Lemma 2.4 [12]

Assume that { a n } is a sequence of nonnegative real numbers such that

a n + 1 (1 γ n ) a n + δ n γ n ,n1,

where { γ n } is a sequence in (0,1) and { δ n } is a sequence such that

  1. (a)

    n = 1 γ n =;

  2. (b)

    lim sup n δ n 0 or n = 1 | δ n γ n |<.

Then lim n a n =0.

3 Main results

In this section, we show our main results.

Theorem 3.1 Let C be a nonempty closed convex subset of a real Hilbert space H. Let S= { T ( s ) } s 0 :CC be a nonexpansive semigroup with Fix(S). Let { γ t } 0 < t < 1 and { λ t } 0 < t < 1 be two continuous nets of positive real numbers such that γ t (0,1), lim t 0 γ t =1 and lim t 0 λ t =+. Let { x t } be the net defined in the following implicit manner:

x t = P C [ t ( γ t x t ) + ( 1 t ) 1 λ t 0 λ t T ( s ) x t d s ] ,t(0,1).
(3.1)

Then, as t0+, the net { x t } strongly converges to x Fix(S).

Proof First, we note that the net { x t } defined by (3.1) is well defined. We define the mapping

Wx:= P C [ t ( γ t x ) + ( 1 t ) 1 λ t 0 λ t T ( s ) x d s ] ,t(0,1).

It follows that

W x W y t γ t ( x y ) + ( 1 t ) 1 λ t 0 λ t ( T ( s ) x T ( s ) y ) d s t γ t x y + ( 1 t ) 1 λ t 0 λ t ( T ( s ) x T ( s ) y ) d s t γ t x y + ( 1 t ) x y = [ 1 ( 1 γ t ) t ] x y .

This implies that the mapping W is a contraction and so it has a unique fixed point. Therefore, the net { x t } defined by (3.1) is well defined.

Take pFix(S). By (3.1), we have

x t p = P C [ t ( γ t x t ) + ( 1 t ) 1 λ t 0 λ t T ( s ) x t d s ] p t γ t ( x t p ) t ( 1 γ t ) p + ( 1 t ) ( 1 λ t 0 λ t T ( s ) x t d s p ) t γ t x t p + t ( 1 γ t ) p + ( 1 t ) 1 λ t 0 λ t T ( s ) x t T ( s ) p d s t γ t x t p + t ( 1 γ t ) p + ( 1 t ) x t p = [ 1 ( 1 γ t ) t ] x t p + t ( 1 γ t ) p .

It follows that

x t pp,

which implies that the net { x t } is bounded. Set R:=p. It is clear that { x t }B(p,R). Notice that

1 λ t 0 λ t T ( s ) x t d s p x t pR.

Moreover, we observe that if xB(p,R), then

T ( s ) x p T ( s ) x T ( s ) p xpR,

i.e., B(p,R) is T(s)-invariant for all s. Set y t =t( γ t x t )+(1t) 1 λ t 0 λ t T(s) x t ds. Then x t = P C [ y t ]. It follows that

By Lemma 2.1, we deduce that for all 0τ<,

lim t 0 T ( τ ) x t x t =0.
(3.2)

Note that x t = P C [ y t ]. By using the property of the metric projection (2.1), we have

x t p 2 = x t y t , x t p + y t p , x t p y t p , x t p = t γ t x t p , x t p t ( 1 γ t ) p , x t p + ( 1 t ) 1 λ t 0 λ t T ( s ) x t d s p , x t p [ 1 ( 1 γ t ) t ] x t p 2 t ( 1 γ t ) p , x t p .

Therefore, we have

x t p 2 p,p x t ,pFix(S).
(3.3)

From this inequality, immediately it follows that ω w ( x t )= ω s ( x t ), where ω w ( x t ) and ω s ( x t ) denote the sets of weak and strong cluster points of { x t }, respectively.

Let { t n }(0,1) be a sequence such that t n 0 as n. Put x n := x t n , y n := y t n and λ n := λ t n . Since { x n } is bounded, without loss of generality, we may assume that the sequence { x n } converges weakly to a point x C. Also, y n x weakly. Noticing (3.2), we can use Lemma 2.2 to get x Fix(S). From (3.3), we have

x n p 2 p,p x n ,pFix(S).
(3.4)

In particular, if we substitute x for p in (3.4), then we have

x n x 2 x , x x n .
(3.5)

However, x n x . This together with (3.5) guarantees that x n x and so the net { x t } is relatively compact, as t 0 + , in the norm topology.

Now, in (3.4), taking n, we get

x p 2 p , p x ,pFix(S).

This is equivalent to the following:

0 x , p x ,pFix(S).

Therefore, x = P Fix ( T ) (0), which is obviously unique. This is sufficient to conclude that the entire net { x t } converges in norm to x . This completes the proof. □

Remark 3.2

It is known that the algorithm

x t = P C [ t x t + ( 1 t ) 1 λ t 0 λ t T ( s ) x t d s ] ,t(0,1),

has only weak convergence. However, our similar algorithm (3.1) (with γ t 1) has strong convergence.

Next, we introduce an explicit algorithm for the nonexpansive semigroup S= { T ( s ) } s 0 :CC and prove the strong convergence theorems of this algorithm.

Theorem 3.3 Let C be a nonempty closed convex subset of a real Hilbert space H. Let S= { T ( s ) } s 0 :CC be a nonexpansive semigroup with Fix(S). Let { x n } be the sequence generated iteratively by the following explicit algorithm:

x n + 1 =(1 β n ) x n + β n P C [ α n ( γ n x n ) + ( 1 α n ) 1 λ n 0 λ n T ( s ) x n d s ] ,n0,
(3.6)

where { α n }, { β n } and { γ n } are sequences of real numbers in [0,1] and { λ n } is a sequence of positive real numbers. Suppose that the following conditions are satisfied:

  1. (i)

    lim n α n =0, n = 0 α n = and lim n γ n =1;

  2. (ii)

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

  3. (iii)

    lim n λ n = and lim n λ n 1 λ n =1.

Then the sequence { x n } generated by (3.6) strongly converges to a point x Fix(S).

Proof Take pFix(S). From (3.6), we have

It follows that, by induction,

x n pmax { x 0 p , p } .

Set y n = P C [ α n ( γ n x n )+(1 α n ) z n ] for all n0, where z n = 1 λ n 0 λ n T(s) x n ds. We have

and

Therefore, we have

where M>0 is a constant such that

sup n 1 { x n 1 , z n 1 , 2 x n 1 p } M.

Hence we get

lim sup n ( y n y n 1 x n x n 1 ) 0.

This together with Lemma 2.3 implies that

lim n y n x n =0.

Therefore, it follows that

lim n x n + 1 x n = lim n β n y n x n =0.

Note that

T ( τ ) x n x n T ( τ ) x n T ( τ ) 1 λ n 0 λ n T ( s ) x n d s + T ( τ ) 1 λ n 0 λ n T ( s ) x n d s 1 λ n 0 λ n T ( s ) x n d s + 1 λ n 0 λ n T ( s ) x n d s x n T ( τ ) 1 λ n 0 λ n T ( s ) x n d s 1 λ n 0 λ n T ( s ) x n d s + 2 x n 1 λ n 0 λ n T ( s ) x n d s .
(3.7)

From (3.6), we have

It follows that

(3.8)

From (3.7), (3.8) and Lemma 2.1, we have

lim n T ( τ ) x n x n =0,τ0.
(3.9)

Notice that { x n } is a bounded sequence and x ˜ is a weak limit of { x n }. Putting x = P Fix ( S ) (0). Then there exists a positive number R such that B( x ,R) contains { x n }. Moreover, B( x ,R) is T(s)-invariant for all s0 and so, without loss of generality, we can assume that { T ( s ) } s 0 is a nonexpansive semigroup on B( x ,R). By the demiclosedness principle (Lemma 2.2) and (3.9), we have x ˜ Fix(S) and hence

lim sup n x , x n + 1 x = lim n x , x ˜ x 0.

Finally, we prove that x n x . Set u n = α n ( γ n x n )+(1 α n ) 1 λ n 0 λ n T(s) x n ds. It follows that y n = P C [ u n ] for all n0. By using the property of the metric projection (2.1), we have

y n u n , y n x 0

and so

y n x 2 = y n x , y n x = y n u n , y n x + u n x , y n x u n x , y n x = α n γ n x n x , y n x α n ( 1 γ n ) x , y n x + ( 1 α n ) z n x , y n x α n γ n x n x y n x α n ( 1 γ n ) x , y n x + ( 1 α n ) z n x y n x [ 1 ( 1 γ n ) α n ] x n x y n x α n ( 1 γ n ) x , y n x 1 ( 1 γ n ) α n 2 x n x 2 + 1 2 y n x α n ( 1 γ n ) x , y n x ,

that is,

y n x 2 [ 1 ( 1 γ n ) α n ] x n x 2 2 α n (1 γ n ) x , y n x .

By the convexity of the norm, we have

x n + 1 x 2 ( 1 β n ) x n x 2 + β n y n x 2 [ 1 ( 1 γ n ) α n β n ] x n x 2 2 ( 1 γ n ) α n β n x , y n x .

Hence all the conditions of Lemma 2.4 are satisfied. Therefore, we immediately deduce that x n x . This completes the proof. □

In Theorem 3.3, if we put β n =1 for each n1, we have the following corollary.

Corollary 3.4 Let C be a nonempty closed convex subset of a real Hilbert space H. Let S= { T ( s ) } s 0 :CC be a nonexpansive semigroup with Fix(S). Let the sequence { x n } be generated iteratively by the following explicit algorithm:

x n + 1 = P C [ α n ( γ n x n ) + ( 1 α n ) 1 λ n 0 λ n T ( s ) x n d s ] ,n0,
(3.10)

where { α n }, { β n } and { γ n } are sequences of real numbers in [0,1] and { λ n } is a sequence of positive real numbers. Suppose that the following conditions are satisfied:

  1. (i)

    lim n α n =0, n = 0 α n = and lim n γ n =1;

  2. (ii)

    lim n λ n = and lim n λ n 1 λ n =1.

Then the sequence { x n } generated by (3.10) strongly converges to a point x Fix(S).

Remark 3.5

It is known that the algorithm

x n + 1 =(1 β n ) x n + β n P C [ α n x n + ( 1 α n ) 1 λ n 0 λ n T ( s ) x n d s ] ,n0,

has only weak convergence. However, our similar algorithm (3.6) (with γ n 1) has strong convergence.

References

  1. 1.

    Browder FE: Convergence of approximation to fixed points of nonexpansive nonlinear mappings in Hilbert spaces. Arch. Ration. Mech. Anal. 1967, 24: 82–90.

  2. 2.

    Browder FE: Convergence theorems for sequences of nonlinear operators in Banach spaces. Math. Z. 1967, 100: 201–225. 10.1007/BF01109805

  3. 3.

    Halpern B: Fixed points of nonexpansive maps. Bull. Am. Math. Soc. 1967, 73: 957–961. 10.1090/S0002-9904-1967-11864-0

  4. 4.

    Lions PL: Approximation de points fixes de contractions. C. R. Acad. Sci. Paris Sér. A-B 1977, 284: 1357–1359.

  5. 5.

    Baillon JB: Un théoréme de type ergodique pour les contractions non linéaires dans un espace de Hilbert. C. R. Acad. Sci. Paris Sér. A-B 1975, 280: 1511–1514.

  6. 6.

    Brézis H, Browder FE: Nonlinear ergodic theorems. Bull. Am. Math. Soc. 1976, 82: 959–961. 10.1090/S0002-9904-1976-14233-4

  7. 7.

    Brézis H, Browder FE: Remarks on nonlinear ergodic theory. Adv. Math. 1977, 25: 165–177. 10.1016/0001-8708(77)90003-2

  8. 8.

    Geobel K, Kirk WA Cambridge Studies in Advanced Mathematics 28. In Topics in Metric Fixed Point Theory. Cambridge University Press, Cambridge; 1990.

  9. 9.

    Marino G, Xu HK: A general iterative method for nonexpansive mappings in Hilbert spaces. J. Math. Anal. Appl. 2006, 318: 43–52. 10.1016/j.jmaa.2005.05.028

  10. 10.

    Moudafi A: Viscosity approximation methods for fixed-points problems. J. Math. Anal. Appl. 2000, 241: 46–55. 10.1006/jmaa.1999.6615

  11. 11.

    Chang SS: Viscosity approximation methods for a finite family of nonexpansive mappings in Banach spaces. J. Math. Anal. Appl. 2006, 323: 1402–1416. 10.1016/j.jmaa.2005.11.057

  12. 12.

    Xu HK: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. 2002, 66: 240–256. 10.1112/S0024610702003332

  13. 13.

    Suzuki T: Strong convergence theorems for infinite families of nonexpansive mappings in general Banach spaces. Fixed Point Theory Appl. 2005, 2005: 103–123.

  14. 14.

    Mainge PE: Approximation methods for common fixed points of nonexpansive mappings in Hilbert spaces. J. Math. Anal. Appl. 2007, 325: 469–479. 10.1016/j.jmaa.2005.12.066

  15. 15.

    Shioji N, Takahashi W: Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces. Proc. Am. Math. Soc. 1997, 125: 3641–3645. 10.1090/S0002-9939-97-04033-1

  16. 16.

    Shimizu T, Takahashi W: Strong convergence to common fixed points of families of nonexpansive mappings. J. Math. Anal. Appl. 1997, 211: 71–83. 10.1006/jmaa.1997.5398

  17. 17.

    Shioji N, Takahashi W: Strong convergence theorems for asymptotically nonexpansive mappings in Hilbert spaces. Nonlinear Anal. TMA 1998, 34: 87–99. 10.1016/S0362-546X(97)00682-2

  18. 18.

    Takahashi W: A nonlinear ergodic theorem for an amenable semigroup of nonexpansive mappings in a Hilbert space. Proc. Am. Math. Soc. 1981, 81: 253–256. 10.1090/S0002-9939-1981-0593468-X

  19. 19.

    Rodé G: An ergodic theorem for semigroups of nonexpansive mappings in a Hilbert space. J. Math. Anal. Appl. 1982, 85: 172–178. 10.1016/0022-247X(82)90032-4

  20. 20.

    Zeng LC, Yao JC: Implicit iteration scheme with perturbed mapping for common fixed points of a finite family of nonexpansive mappings. Nonlinear Anal. TMA 2006, 64: 2507–2515. 10.1016/j.na.2005.08.028

  21. 21.

    Yao Y, Chen R, Liou YC: A unified implicit algorithm for solving the triple-hierarchical constrained optimization problem. Math. Comput. Model. 2012, 55: 1506–1515. 10.1016/j.mcm.2011.10.041

  22. 22.

    Yao Y, Shahzad N: Strong convergence of a proximal point algorithm with general errors. Optim. Lett. 2012. doi:10.1007/s11590–011–0286–2

  23. 23.

    Yao Y, Shahzad N: New methods with perturbations for non-expansive mappings in Hilbert spaces. Fixed Point Theory Appl. 2011., 2011: Article ID 79

  24. 24.

    Yao Y, Yao JC: On modified iterative method for nonexpansive mappings and monotone mappings. Appl. Math. Comput. 2007, 186: 1551–1558. 10.1016/j.amc.2006.08.062

  25. 25.

    Plubtieng S, Wangkeeree R: Strong convergence of modified Mann iterations for a countable family of nonexpansive mappings. Nonlinear Anal. TMA 2009, 70: 3110–3118. 10.1016/j.na.2008.04.014

  26. 26.

    Cho YJ, Qin X: Convergence of a general iterative method for nonexpansive mappings in Hilbert spaces. J. Comput. Appl. Math. 2009, 228: 458–465. 10.1016/j.cam.2008.10.004

  27. 27.

    Petrusel A, Yao JC: Viscosity approximation to common fixed points of families of nonexpansive mappings with generalized contractions mappings. Nonlinear Anal. TMA 2008, 69: 1100–1111. 10.1016/j.na.2007.06.016

  28. 28.

    Lau ATM, Shioji N, Takahashi W: Existence of nonexpansive retractions for amenable semigroups of nonexpansive mappings and nonlinear ergodic theorems in Banach spaces. J. Funct. Anal. 1999, 161: 62–75. 10.1006/jfan.1998.3352

  29. 29.

    Chen R, Song Y: Convergence to common fixed point of nonexpansive semigroups. J. Comput. Appl. Math. 2007, 200: 566–575. 10.1016/j.cam.2006.01.009

  30. 30.

    Cianciaruso F, Marino G, Muglia L: Iterative methods for equilibrium and fixed point problems for nonexpansive semigroups in Hilbert spaces. J. Optim. Theory Appl. 2010, 146: 491–509. 10.1007/s10957-009-9628-y

  31. 31.

    Chen R, He H: Viscosity approximation of common fixed points of nonexpansive semigroups in Banach space. Appl. Math. Lett. 2007, 20: 751–757. 10.1016/j.aml.2006.09.003

  32. 32.

    Zegeye H, Shahzad N: Strong convergence theorems for a finite family of nonexpansive mappings and semigroups via the hybrid method. Nonlinear Anal. TMA 2010, 72: 325–329. 10.1016/j.na.2009.06.056

  33. 33.

    Buong N: Strong convergence theorem for nonexpansive semigroups in Hilbert space. Nonlinear Anal. TMA 2010, 72: 4534–4540. 10.1016/j.na.2010.02.031

  34. 34.

    Lau AT, Takahashi W: Fixed point properties for semigroup of nonexpansive mappings on Fréchet spaces. Nonlinear Anal. TMA 2009, 70: 3837–3841. 10.1016/j.na.2008.07.041

  35. 35.

    Lau ATM, Nishiura K, Takahashi W: Nonlinear ergodic theorems for semigroups of nonexpansive mappings and left ideals. Nonlinear Anal. TMA 1996, 26: 1411–1427. 10.1016/0362-546X(94)00347-K

  36. 36.

    Lau ATM: Semigroup of nonexpansive mappings on a Hilbert space. J. Math. Anal. Appl. 1985, 105: 514–522. 10.1016/0022-247X(85)90066-6

  37. 37.

    Lau ATM, Miyake H, Takahashi W: Approximation of fixed points for amenable semigroups of nonexpansive mappings in Banach spaces. Nonlinear Anal. TMA 2007, 67: 1211–1225. 10.1016/j.na.2006.07.008

  38. 38.

    Lau ATM, Takahashi W: Invariant means and semigroups of nonexpansive mappings on uniformly convex Banach spaces. J. Math. Anal. Appl. 1990, 153: 497–505. 10.1016/0022-247X(90)90228-8

  39. 39.

    Lau ATM, Takahashi W: Invariant submeans and semigroups of nonexpansive mappings on Banach spaces with normal structure. J. Funct. Anal. 1996, 142: 79–88. 10.1006/jfan.1996.0144

  40. 40.

    Lau ATM, Zhang Y: Fixed point properties of semigroups of non-expansive mappings. J. Funct. Anal. 2008, 254: 2534–2554. 10.1016/j.jfa.2008.02.006

  41. 41.

    Katchang P, Kumam P: An iterative algorithm for common fixed points for nonexpansive semigroups and strictly pseudo-contractive mappings with optimization problems. J. Glob. Optim. 2012. doi:10.1007/s10898–012–9927-y

  42. 42.

    Kumam P, Wattanawitoon K: A general composite explicit iterative scheme of fixed point solutions of variational inequalities for nonexpansive semigroups. Math. Comput. Model. 2011, 53: 998–1006. 10.1016/j.mcm.2010.11.057

  43. 43.

    Sunthrayuth P, Kumam P: A general iterative algorithm for the solution of variational inequalities for a nonexpansive semigroup in Banach spaces. J. Nonlinear Anal. Optim. 2010, 1: 139–150.

  44. 44.

    Chang SS, Cho YJ, Joseph Lee HW, Chan CK: Strong convergence theorems for Lipschitzian demi-contraction semigroups in Banach spaces. Fixed Point Theory Appl. 2011., 2011: Article ID 583423

  45. 45.

    Cho YJ, Ćirić LB, Wang SH:Convergence theorems for nonexpansive semigroups in CAT(0) spaces. Nonlinear Anal. 2011, 74: 6050–6059. 10.1016/j.na.2011.05.082

  46. 46.

    Yao Y, Cho YJ, Liou YC: Hierarchical convergence of an implicit double-net algorithm for nonexpansive semigroups and variational inequality problems. Fixed Point Theory Appl. 2011., 2011: Article ID 101

Download references

Acknowledgements

The third author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant Number: 2012-0008170).

Author information

Correspondence to Jung Im Kang or Yeol Je Cho.

Additional information

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors carried out the proof. All authors conceived of the study, and participated in its design and coordination. All authors read and approved the final manuscript.

Rights and permissions

Reprints and Permissions

About this article

Keywords

  • nonexpansive semigroup
  • common fixed point
  • algorithms
  • projection