Skip to main content

An implicit method for finding a common fixed point of a representation of nonexpansive mappings in Banach spaces

An Erratum to this article was published on 10 November 2015

Abstract

We introduce an implicit method for finding an element of the set of common fixed points of a representation of nonexpansive mappings. Then we prove the strong convergence of the proposed implicit scheme to the common fixed point of a representation of nonexpansive mappings.

MSC:90C33, 47H10.

1 Introduction

Let C be a nonempty closed and convex subset of a Banach space E and E be the dual space of E. Let , denote the pairing between E and E . The normalized duality mapping J:E E is defined by

J(x)= { f E : x , f = x 2 = f 2 }

for all xE. In the sequel, we use j to denote the single-valued normalized duality mapping. Let U={xE:x=1}. E is said to be smooth or to have a Gâteaux differentiable norm if the limit

lim t 0 x + t y x t

exists for each x,yU. E is said to have a uniformly Gâteaux differentiable norm if for each yU, the limit is attained uniformly for all xU. E is said to be uniformly smooth or is said to have a uniformly Féchet differentiable norm if the limit is attained uniformly for x,yU. 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. A Banach space E is smooth if the duality mapping J of E is single-valued. We know that if E is smooth, then J is norm to weak-star continuous; for more details, see [1].

Let C be a nonempty closed and convex subset of a Banach space E. A mapping T of C into itself is called nonexpansive if TxTyxy for all x,yC, and a mapping f is an α-contraction on E if f(x)f(y)αxy, x,yE such that 0α<1.

In this paper, motivated by Lashkarizadeh Bami and Soori [2] and Hussain and Takahashi [3], we introduce the following general implicit algorithm for finding a common element of the set of fixed points of a representation S={ T t :tS} of a semigroup S as nonexpansive mappings from C into itself, with respect to a left regular sequence of means defined on an appropriate subspace of bounded real-valued functions of the semigroup. On the other hand, our goal is to prove that there exists a sunny nonexpansive retraction P of C onto Fix(S) and xC such that the following sequence { z n } converges strongly to Px:

z n = ϵ n f( z n )+(1 ϵ n ) T μ n z n (nN).

2 Preliminaries

Let S be a semigroup. We denote by B(S) the Banach space of all bounded real-valued functions defined on S with supremum norm. For each sS and fB(S), we define l s and r s in B(S) by

( l s f)(t)=f(st),( r s f)(t)=f(ts)(tS).

Let X be a subspace of B(S) containing 1, and let X be its topological dual. An element μ of X is said to be a mean on X if μ=μ(1)=1. We often write μ t (f(t)) instead of μ(f) for μ X and fX. Let X be left invariant (resp. right invariant), i.e., l s (X)X (resp. r s (X)X) for each sS. A mean μ on X is said to be left invariant (resp. right invariant) if μ( l s f)=μ(f) (resp. μ( r s f)=μ(f)) for each sS and fX. X is said to be left (resp. right) amenable if X has a left (resp. right) invariant mean. X is amenable if X is both left and right amenable. As is well known, B(S) is amenable when S is a commutative semigroup (see p.29 of [1]). A net { μ α } of means on X is said to be left regular if

lim α l s μ α μ α =0

for each sS, where l s is the adjoint operator of l s .

Let f be a function of the semigroup S into a reflexive Banach space E such that the weak closure of {f(t):tS} is weakly compact, and let X be a subspace of B(S) containing all the functions tf(t), x with x E . We know from [4] that for any μ X , there exists a unique element f μ in E such that f μ , x = μ t f(t), x for all x E . We denote such f μ by f(t)dμ(t). Moreover, if μ is a mean on X, then from [5], f(t)dμ(t) co ¯ {f(t):tS}.

Let C be a nonempty closed and convex subset of E. Then a family S={ T s :sS} of mappings from C into itself is said to be a representation of S as a nonexpansive mapping on C into itself if S satisfies the following:

  1. (1)

    T s t x= T s T t x for all s,tS and xC;

  2. (2)

    for every sS, the mapping T s :CC is nonexpansive.

We denote by Fix(S) the set of common fixed points of S, that is, Fix(S)= s S {xC: T s x=x}.

Theorem 2.1 [6]

Let S be a semigroup, let C be a closed, convex subset of a reflexive Banach space E, S={ T s :sS} be a representation of S as a nonexpansive mapping from C into itself such that weak closure of { T t x:tS} is weakly compact for each xC, and let X be a subspace of B(S) such that 1X and the mapping tT(t)x, x be an element of X for each xC and x E, and μ be a mean on X. If we write T μ x instead of T t xdμ(t), then the following hold.

  1. (i)

    T μ is a nonexpansive mapping from C into C.

  2. (ii)

    T μ x=x for each xFix(S).

  3. (iii)

    T μ x co ¯ { T t x:tS} for each xC.

  4. (iv)

    If X is r s -invariant for each sS and μ is right invariant, then T μ T t = T μ for each tS.

Remark From Theorem 4.1.6 in [1], every uniformly convex Banach space is strictly convex and reflexive.

Let D be a subset of B, where B is a subset of a Banach space E, and let P be a retraction of B onto D, that is, Px=x for each xD. Then P is said to be sunny if for each xB and t0 with Px+t(xPx)B, P(Px+t(xPx))=Px. A subset D of B is said to be a sunny nonexpansive retract of B if there exists a sunny nonexpansive retraction P of B onto D. We know that if E is smooth and P is a retraction of B onto D, then P is sunny and nonexpansive if and only if for each xB and zD, xPx,J(zPx)0. For more details, see [1].

Lemma 2.2 [7]

Let S be a semigroup, and let C be a compact convex subset of a real strictly convex and smooth Banach space E. Suppose that S={ T s :sS} is a representation of S as a nonexpansive mapping from C into itself. Let X be a left invariant subspace of B(S) such that 1X, and the function t T t x, x is an element of X for each xC and x E . If μ is a left invariant mean on X, then Fix( T μ )= T μ C=Fix(S) and there exists a unique sunny nonexpansive retraction from C onto Fix(S).

Throughout the rest of this paper, the open ball of radius r centered at 0 is denoted by B r . Let C be a nonempty closed convex subset of a Banach space E. For ϵ>0 and a mapping T:CC, we let F ϵ (T) be the set of ϵ-approximate fixed points of T, i.e., F ϵ (T)={xC:xTxϵ}.

3 Main result

In this section, we deal with a strong convergence approximation scheme for finding a common element of the set of common fixed points of a representation of nonexpansive mappings.

Theorem 3.1 Let S be a semigroup. Let C be a nonempty compact convex subset of a real strictly convex and reflexive and smooth Banach space E. Suppose that S={ T s :sS} is a representation of S as a nonexpansive mapping from C into itself such that Fix(S). Let X be a left invariant subspace of B(S) such that 1X, and the function t T t x, x is an element of X for each xC and x E . Let { μ n } be a left regular sequence of means on X. Suppose that f is an α-contraction on C. Let ϵ n be a sequence in (0,1) such that lim n ϵ n =0. Then there exists a unique sunny nonexpansive retraction P of C onto Fix(S) and xC such that the following sequence { z n } generated by

z n = ϵ n f( z n )+(1 ϵ n ) T μ n z n (nN)
(1)

strongly converges to Px.

Proof By Proposition 1.7.3 and Theorem 1.9.21 in [8], any compact subset C of a reflexive Banach space E is weakly compact, and from Proposition 1.9.18 in [8], any closed convex subset of a weakly compact subset C of a Banach space E is itself weakly compact, and by Proposition 1.9.13 in [8], any convex subset C of a normed space E is weakly closed if and only if C is closed. Therefore, weak closure of { T t x:tS} is weakly compact for each xC.

We divide the proof into five steps.

Step 1. The existence of z n which satisfies (1).

This follows immediately from the fact that for every nN, the mapping N n given by

N n x:= ϵ n f(x)+(1 ϵ n ) T μ n x(xC)

is a contraction. To see this, put β n =(1+ ϵ n (α1)), then 0 β n <1 (nN). Then we have

N n x N n y ϵ n f ( x ) f ( y ) + ( 1 ϵ n ) T μ n x T μ n y ϵ n α x y + ( 1 ϵ n ) x y = ( 1 + ϵ n ( α 1 ) ) x y = β n x y .

Therefore, by the Banach contraction principle [1], there exists a unique point z n C such that N n z n = z n .

Step 2. lim n z n T t z n =0 for all tS.

Consider tS and let ϵ>0. By Lemma 1 in [9], there exists δ>0 such that co ¯ F δ ( T t )+2 B δ F ϵ ( T t ). By Corollary 2.8 in [10], there also exists a natural number N such that

1 N + 1 i = 0 N T t i s y T t ( 1 N + 1 i = 0 N T t i s y ) δ
(2)

for all sS and yC. Let pFix(S) and M 0 be a positive number such that sup y C y M 0 . Let tS, since { μ n } is strongly left regular, there exists N 0 N such that μ n l t i μ n δ ( 3 M 0 ) for n N 0 and i=1,2,,N. Then we have

sup y C T μ n y 1 N + 1 i = 0 N T t i s y d μ n ( s ) = sup y C sup x = 1 | T μ n y , x 1 N + 1 i = 0 N T t i s y d μ n ( s ) , x | = sup y C sup x = 1 | 1 N + 1 i = 0 N ( μ n ) s T s y , x 1 N + 1 i = 0 N ( μ n ) s T t i s y , x | 1 N + 1 i = 0 N sup y C sup x = 1 | ( μ n ) s T s y , x ( l t i μ n ) s T s y , x | max i = 1 , 2 , , N μ n l t i μ n ( M 0 + 2 p ) max i = 1 , 2 , , N μ n l t i μ n ( 3 M 0 ) δ ( n N 0 ) .
(3)

By Theorem 2.1 we have

1 N + 1 i = 0 N T t i s yd μ n (s) co ¯ { 1 N + 1 i = 0 N T t i ( T s y ) : s S } .
(4)

It follows from (2)-(4) that

T μ n y co ¯ { 1 N + 1 i = 0 N T t i s y : s S } + B δ co ¯ F δ ( T t ) + 2 B δ F ϵ ( T t )

for all yC and n N 0 . Therefore, lim sup n sup y C T t ( T μ n y) T μ n yϵ. Since ϵ>0 is arbitrary, we have

lim sup n sup y C T t ( T μ n y ) T μ n y =0.
(5)

Let tS and ϵ>0, then there exists δ>0 which satisfies (2). Take L 0 =(1+α)2 M 0 +f(p)p. Now, from the condition lim n ϵ n =0 and from (5), there exists a natural number N 1 such that T μ n y F δ ( T t ) for all yC and ϵ n < δ 2 L 0 for all n N 1 . Since pFix(S), we have

ϵ n f ( z n ) T μ n z n ϵ n ( f ( z n ) f ( p ) + f ( p ) p + T μ n p T μ n z n ) ϵ n ( α z n p + f ( p ) p + A z n p ) ϵ n ( α z n p + f ( p ) p + z n p ) ϵ n ( ( 1 + α ) z n p + f ( p ) p ) ϵ n ( ( 1 + α ) 2 M 0 + f ( p ) p ) = ϵ n L 0 δ 2

for all n N 1 . Observe that

z n = ϵ n f ( z n ) + ( 1 ϵ n ) T μ n z n = T μ n z n + ϵ n ( f ( z n ) T μ n z n ) F δ ( T t ) + B δ 2 F δ ( T t ) + 2 B δ F ϵ ( T t )

for all n N 1 . This shows that

z n T t z n ϵ(n N 1 ).

Since ϵ>0 is arbitrary, we get lim n z n T t z n =0.

Step 3. S{ z n }Fix(S), where S{ z n } denotes the set of strongly limit points of { z n }.

Let zS{ z n }, and let { z n j } be a subsequence of { z n } such that z n j z,

T t z z T t z T t z n j + T t z n j z n j + z n j z 2 z n j z + T t z n j z n j ,

then by Step 2,

T t zz2 lim j z n j z+ lim j T t z n j z n j =0,

therefore zFix(S).

Step 4. There exists a unique sunny nonexpansive retraction P of C onto Fix(S) and xC such that

Γ:= lim sup n x P x , J ( z n P x ) 0.
(6)

By Lemma 2.2 there exists a unique sunny nonexpansive retraction P of C onto Fix(S). The Banach contraction mapping principle guarantees that fP has a unique fixed point xC. We show that

Γ:= lim sup n x P x , J ( z n P x ) 0.

Note that from the definition of Γ and the fact that C is a compact subset of E, we can select a subsequence { z n j } of { z n } with the following properties:

  1. (i)

    lim j xPx,J( z n j Px)=Γ;

  2. (ii)

    { z n j } converges strongly to a point z.

By Step 3, we have zFix(S). Since E is smooth, we have

Γ= lim j x P x , J ( z n j P x ) = x P x , J ( z P x ) 0.

Since fPx=x, we have (fI)Px=xPx. From Theorem 4.2.1(v) in [1], for x,yE and fJ(y), x 2 y 2 2(xy,f). Therefore, for each nN, we have

ϵ n ( α 1 ) z n P x 2 [ ϵ n α z n P x + ( 1 ϵ n ) z n P x ] 2 z n P x 2 [ ϵ n f ( z n ) f ( P x ) + ( 1 ϵ n ) T μ n z n P x ] 2 z n P x 2 2 ϵ n ( f ( z n ) f ( P x ) ) + ( 1 ϵ n ) ( T μ n z n P x ) ( z n P x ) , J ( z n P x ) = 2 ϵ n ( f I ) P x , J ( z n P x ) = 2 ϵ n x P x , J ( z n P x ) .

Hence

z n P x 2 2 1 α x P x , J ( z n P x ) .
(7)

Step 5. { z n } strongly converges to Px.

Indeed, from (6), (7) and PxFix(S), we conclude

lim sup n z n P x 2 2 1 α lim sup n x P x , J ( z n P x ) 0.

That is, z n Px. □

Remark 3.2 It would be an interesting problem to prove Theorem 3.1 for continuous representations instead of nonexpansive.

References

  1. Takahashi W: Nonlinear Functional Analysis: Fixed Point Theory and Its Applications. Yokohama Publishers, Yokohama; 2000.

    MATH  Google Scholar 

  2. Lashkarizadeh Bami M, Soori E: Strong convergence of a general implicit algorithm for variational inequality problems and equilibrium problems and a continuous representation of nonexpansive mappings. Bull. Iran. Math. Soc. 2014, 40: 977–1001.

    MathSciNet  Google Scholar 

  3. Hussain N, Takahashi W: Weak and strong convergence theorems for semigroups of mappings without continuity in Hilbert spaces. J. Nonlinear Convex Anal. 2013, 14(4):769–783.

    MathSciNet  Google Scholar 

  4. Hirano N, Kido K, Takahashi W: Nonexpansive retractions and nonlinear ergodic theorems in Banach spaces. Nonlinear Anal. 1988, 12: 1269–1281. 10.1016/0362-546X(88)90059-4

    Article  MathSciNet  Google Scholar 

  5. Kido K, Takahashi W: Mean ergodic theorems for semigroups of linear operators. J. Math. Anal. Appl. 1984, 103: 387–394. 10.1016/0022-247X(84)90136-7

    Article  MathSciNet  Google Scholar 

  6. Saeidi S: Existence of ergodic retractions for semigroups in Banach spaces. Nonlinear Anal. 2008, 69: 3417–3422. 10.1016/j.na.2007.09.031

    Article  MathSciNet  Google Scholar 

  7. Saeidi S: A note on ‘Approximation of fixed points for amenable semigroups of nonexpansive mappings in Banach spaces’. Nonlinear Anal. 2010, 72: 546–548. 10.1016/j.na.2009.06.010

    Article  MathSciNet  Google Scholar 

  8. Agarwal RP, Oregan D, Sahu DR Topological Fixed Point Theory and Its Applications 6. In Fixed Point Theory for Lipschitzian-Type Mappings with Applications. Springer, New York; 2009.

    Google Scholar 

  9. Shioji N, Takahashi W: Strong convergence of averaged approximants for asymptotically nonexpansive mappings in Banach spaces. J. Approx. Theory 1999, 97: 53–64. 10.1006/jath.1996.3251

    Article  MathSciNet  Google Scholar 

  10. Atsushiba S, Takahashi W: A nonlinear ergodic theorem for nonexpansive mappings with compact domain. Math. Jpn. 2000, 52: 183–195.

    MathSciNet  Google Scholar 

Download references

Acknowledgements

This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The first author acknowledges with thanks DSR, KAU for financial support. The authors would like to thank the referee of the paper for his helpful comments and invaluable suggestions. This research was supported by the Center of Excellence for Mathematics and the Office of Graduate Studies of the Lorestan University and the University of Isfahan.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Ebrahim Soori.

Additional information

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

An erratum to this article is available at http://dx.doi.org/10.1186/s13663-015-0450-y.

Rights and permissions

Open Access  This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.

The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hussain, N., Lashkarizadeh Bami, M. & Soori, E. An implicit method for finding a common fixed point of a representation of nonexpansive mappings in Banach spaces. Fixed Point Theory Appl 2014, 238 (2014). https://doi.org/10.1186/1687-1812-2014-238

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/1687-1812-2014-238

Keywords