Skip to main content

Relaxed Halpern-type iteration method for countable families of totally quasi-ϕ-asymptotically nonexpansive mappings

Abstract

Based on an original idea, namely, a specific way of choosing the indexes of involved mappings, we propose a relaxed Halpern-type iterative algorithm for approximating some common fixed point of a kind of nonlinear mappings and obtain a strong convergence theorem under suitable conditions. Since the involved mappings need no assumption of being uniformly totally quasi-ϕ-asymptotically nonexpansive, and there is no need to compute projections onto intersections of countably many closed and convex sets, the results improve those of the authors with related interest.

MSC:47H09, 47H10, 47J25.

1 Introduction

Throughout this paper, we assume that E is a real Banach space with its dual E , C is a nonempty closed convex subset of E, and J:E 2 E is the normalized duality mapping defined by

Jx= { f E : x , f = x 2 = f 2 } ,xE.

In the sequel, we use F(T) to denote the set of fixed points of a mapping T.

Definition 1.1 [1]

A mapping T:CC is said to be totally quasi-ϕ-asymptotically nonexpansive if F(T) and there exist nonnegative real sequences { ν n }, { μ n } with ν n , μ n 0 (as n) and a strictly increasing continuous function ζ: R + R + with ζ(0)=0 such that

ϕ ( p , T n x ) ϕ(p,x)+ ν n ζ ( ϕ ( p , x ) ) + μ n ,n1,xC,pF(T),
(1.1)

where ϕ:E×E R + denotes the Lyapunov functional defined by

ϕ(x,y)= x 2 2x,Jy+ y 2 ,x,yE.
(1.2)

It is obvious from the definition of ϕ that

( x y ) 2 ϕ(x,y) ( x + y ) 2 .
(1.3)

Definition 1.2 [1]

  1. (1)

    A countable family of mappings { T i }:CC is said to be totally uniformly quasi-ϕ-asymptotically nonexpansive if F:= i = 1 F( T i ) and there exist nonnegative real sequences { ν n }, { μ n } with ν n , μ n 0 (as n) and a strictly increasing continuous function ζ: R + R + with ζ(0)=0 such that

    ϕ ( p , T i n x ) ϕ(p,x)+ ν n ζ ( ϕ ( p , x ) ) + μ n ,n1,i1,xC,pF(T).
    (1.4)
  2. (2)

    A mapping T:CC is said to be uniformly L-Lipschitz continuous if there exists a constant L>0 such that

    T n x T n y Lxy,n1,x,yC.
    (1.5)

In 2012, Chang et al. [1] used the following modified Halpern-type iteration algorithm for totally quasi-ϕ-asymptotically nonexpansive mappings to have the strong convergence under a limit condition only in the framework of Banach spaces.

{ x 1 C ; C 1 = C , y n , m = J 1 [ α n J x 1 + ( 1 α n ) J T m n x n ] , m 1 , C n + 1 = { z C n : sup m 1 ϕ ( z , y n , m ) α n ϕ ( z , x 1 ) + ( 1 α n ) ϕ ( z , x n ) + ξ n } , x n + 1 = Π C n + 1 x 1 , n 1 ,
(1.6)

where { T i }:CC is a countable family of closed and uniformly totally quasi-ϕ-asymptotically nonexpansive mappings; and ξ n = ν n sup p F ζ(ϕ(p, x n ))+ μ n , Π C n + 1 is the generalized projection (see (2.1)) of E onto C n + 1 . Their results extended and improved the corresponding results of Qin et al. [2, 3], Wang et al. [4], Martinez-Yanes and Xu [5] and others.

However, it is obviously a very strong condition that the involved mappings are assumed to be uniformly totally quasi-ϕ-asymptotically nonexpansive. Additionally, the assumption conditions imposed on all C n (n2) are not weak at all since each one is in fact an intersection of countably many sets. This fact makes the projection very hard to compute, and therefore the method proposed in their paper does not seem to be valuable in practice.

Inspired and motivated by the studies mentioned above, in this article, we introduce a relaxed iterative algorithm for approximating some common fixed point of a countable family of totally quasi-ϕ-asymptotically nonexpansive mappings and obtain a strong convergence theorem.

2 Preliminaries

Following Alber [6], the generalized projection Π C :EC is defined by

Π C =arg inf y C ϕ(y,x),xE.
(2.1)

Lemma 2.1 [6]

Let E be a smooth, strictly convex and reflexive Banach space, and let C be a nonempty closed convex subset of E. Then the following conclusions hold:

  1. (1)

    ϕ(x, Π C y)+ϕ( Π C y,y)ϕ(x,y) for all xC and yE;

  2. (2)

    If xE and zC, then z= Π C xzy,JxJz0, yC;

  3. (3)

    For x,yE, ϕ(x,y)=0 if and only if x=y.

Remark 2.2 The following basic properties for a Banach space E can be found in Cioranescu [7].

  1. (i)

    If E is uniformly smooth, then J is uniformly continuous on each bounded subset of E;

  2. (ii)

    If E is reflexive and strictly convex, then J 1 is norm-weak-continuous;

  3. (iii)

    If E is a smooth, strictly convex and reflexive Banach space, then the normalized duality mapping J:E 2 E is single-valued, one-to-one and onto;

  4. (iv)

    A Banach space E is uniformly smooth if and only if E is uniformly convex;

  5. (v)

    Each uniformly convex Banach space E has the Kadec-Klee property, i.e., for any sequence { x n }E, if x n xE and x n x, then x n x as n.

Lemma 2.3 [1]

Let E be a real uniformly smooth and strictly convex Banach space with the Kadec-Klee property, and let C be a nonempty closed convex subset of E. Let { x n } and { y n } be two sequences in C such that x n p and ϕ( x n , y n )0, where ϕ is the function defined by (1.2), then y n p.

Lemma 2.4 [1]

Let E and C be the same as in Lemma 2.3. Let T:CC be a closed and totally quasi-ϕ-asymptotically nonexpansive mapping with nonnegative real sequences { ν n }, { μ n } and a strictly increasing continuous function ζ: R + R + such that ν n , μ n 0 and ζ(0)=0. If μ 1 =0, then the fixed point set F(T) of T is a closed and convex subset of C.

Lemma 2.5 [8]

The unique solutions to the positive integer equation

n=i+ ( m 1 ) m 2 ,mi,n=1,2,3,,
(2.2)

are

i=n ( m 1 ) m 2 ,m= [ 1 2 2 n + 1 4 ] ,n=1,2,3,,
(2.3)

where [x] denotes the maximal integer that is not larger than x.

3 Main results

Theorem 3.1 Let E be a real uniformly smooth and strictly convex Banach space with the Kadec-Klee property, let C be a nonempty closed convex subset of E, and let T i :CC, i=1,2, , be a countable family of closed and totally quasi-ϕ-asymptotically nonexpansive mappings with nonnegative real sequences { ν n ( i ) }, { μ n ( i ) } satisfying ν n ( i ) 0 and μ n ( i ) 0 (as n and for each i1) and a strictly increasing and continuous function ζ: R + R + satisfying condition (1.1) and each T i is uniformly L i -Lipschitz continuous. Let { α n } be a sequence in [0,1] with α n 0. Let { x n } be the sequence generated by

{ x 1 C ; C 1 = C , y n = J 1 [ α n J x 1 + ( 1 α n ) J T i n m n x n ] , C n + 1 = { z C n : ϕ ( z , y n ) α n ϕ ( z , x 1 ) + ( 1 α n ) ϕ ( z , x n ) + ξ n } , x n + 1 = Π C n + 1 x 1 , n 1 ,
(3.1)

where ξ n := ν m n ( i n ) sup p F ζ(ϕ(p, x n ))+ μ m n ( i n ) , Π C n + 1 is the generalized projection of E onto C n + 1 , and i n and m n are the solutions to the positive integer equation: n=i+ ( m 1 ) m 2 (mi, n=1,2,), that is, for each n1, there exist unique i n and m n such that

i 1 = 1 , i 2 = 1 , i 3 = 2 , i 4 = 1 , i 5 = 2 , i 6 = 3 , i 7 = 1 , i 8 = 2 , ; m 1 = 1 , m 2 = 2 , m 3 = 2 , m 4 = 3 , m 5 = 3 , m 6 = 3 , m 7 = 4 , m 8 = 4 , .

If F:= i = 1 F( T i ) is bounded and μ 1 ( i ) =0 for each i1, then { x n } converges strongly to Π F x 1 .

Proof We divide the proof into several steps.

  1. (I)

    F and C n (n1) both are closed and convex subsets in C.

In fact, it follows from Lemma 2.4 that each F( T i ) is a closed and convex subset of C, so is F. In addition, with C 1 (=C) being closed and convex, we may assume that C n is closed and convex for some n2. In view of the definition of ϕ, we have that

C n + 1 = { z C : φ ( z ) a } C n ,

where φ(z)=2 α n z,J x 1 +2(1 α n )z,J x n 2z,J y n and a= α n x 1 2 +(1 α n ) x n 2 y n 2 + ξ n . This shows that C n + 1 is closed and convex.

  1. (II)

    F is a subset of n = 1 C n .

It is obvious that F C 1 . Suppose that F C n for some n2. Then, for any pF C n , we have

ϕ ( p , y n ) = ϕ ( p , J 1 [ α n J x 1 + ( 1 α n ) J T i n m n x n ] ) = p 2 2 p , α n J x 1 + ( 1 α n ) J T i n m n x n + α n J x 1 + ( 1 α n ) J T i n m n x n 2 p 2 2 α n p , J x 1 2 ( 1 α n ) p , J T i n m n x n + α n x 1 2 + ( 1 α n ) T i n m n x n 2 = α n ϕ ( p , x 1 ) + ( 1 α n ) ϕ ( p , T i n m n x n ) α n ϕ ( p , x 1 ) + ( 1 α n ) [ ϕ ( p , x n ) + ν m n ( i n ) ζ ( ϕ ( p , x n ) ) + μ m n ( i n ) ] α n ϕ ( p , x 1 ) + ( 1 α n ) [ ϕ ( p , x n ) + ν m n ( i n ) sup p F ζ ( ϕ ( p , x n ) ) + μ m n ( i n ) ] = α n ϕ ( p , x 1 ) + ( 1 α n ) ϕ ( p , x n ) + ξ n .
(3.2)

This implies that p C n + 1 , and so F C n + 1 .

  1. (III)

    x n x C as n.

In fact, since x n = Π C n x 1 , from Lemma 2.1(2) we have x n y,J x 1 J x n 0, y C n . Again since F n = 1 C n , we have x n p,J x 1 J x n 0, pF. It follows from Lemma 2.1(1) that for each pF and for each n1,

ϕ( x n , x 1 )=ϕ( Π C n x 1 , x 1 )ϕ(p, x 1 )ϕ(p, x n )ϕ(p, x 1 ),

which implies that {ϕ( x n , x 1 )} is bounded, so is { x n }. Since for all n1, x n = Π C n x 1 and x n + 1 = Π C n + 1 x 1 C n + 1 C n , we have ϕ( x n , x 1 )ϕ( x n + 1 , x 1 ). This implies that {ϕ( x n , x 1 )} is nondecreasing, hence the limit

lim n ϕ( x n , x 1 ) exists.

Since E is reflexive, there exists a subsequence { x n i } of { x n } such that x n i x C as i. Since C n is closed and convex and C n + 1 C n , this implies that C n is weakly closed and x C n for each n1. In view of x n i = Π C n i x 1 , we have

ϕ( x n i , x 1 )ϕ ( x , x 1 ) ,i1.

Since the norm is weakly lower semi-continuous, we have

lim inf i ϕ ( x n i , x 1 ) = lim inf i ( x n i 2 2 x n i , J x 1 + x 1 2 ) x 2 2 x , J x 1 + x 1 2 = ϕ ( x , x 1 ) ,

and so

ϕ ( x , x 1 ) lim inf i ϕ( x n i , x 1 ) lim sup i ϕ( x n i , x 1 )ϕ ( x , x 1 ) .

This implies that lim i ϕ( x n i , x 1 )=ϕ( x , x 1 ), and so x n i x as i. Since x n i x , by virtue of the Kadec-Klee property of E, we obtain that

lim i x n i = x .

Since {ϕ( x n , x 1 )} is convergent, this, together with lim i ϕ( x n i , x 1 )=ϕ( x , x 1 ), shows that lim n ϕ( x n , x 1 )=ϕ( x , x 1 ). If there exists some subsequence { x n j } of { x n } such that x n j y as j, then from Lemma 2.1(1) we have that

ϕ ( x , y ) = lim i , j ϕ ( x n i , x n j ) = lim i , j ϕ ( x n i , Π C n j x 1 ) lim i , j ( ϕ ( x n i , x 1 ) ϕ ( Π C n j x 1 , x 1 ) ) = lim i , j ( ϕ ( x n i , x 1 ) ϕ ( x n j , x 1 ) ) = ϕ ( x , x 1 ) ϕ ( x , x 1 ) = 0 ,

that is, x =y and so

lim n x n = x .
(3.3)
  1. (IV)

    x is a member of F.

Set K i ={kN:k=i+ ( m 1 ) m 2 ,mi,mN} for each i1. For example, by Lemma 2.5 and the definition of K 1 , we have K 1 ={1,2,4,7,11,16,} and i 1 = i 2 = i 4 = i 7 = i 11 = i 16 ==1. Then we have

ξ k = ν m k ( i ) sup p F ζ ( ϕ ( p , x k ) ) + μ m k ( i ) ,k K i .
(3.4)

Note that { m k } k K i ={i,i+1,i+2,}, i.e., m k as K i k. It follows from (3.3) and (3.4) that

lim K i k ξ k =0.
(3.5)

Since x n + 1 C n + 1 , it follows from (3.1), (3.3) and (3.5) that

ϕ( x k + 1 , y k ) α k ϕ( x k + 1 , x 1 )+(1 α k )ϕ( x k + 1 , x k )+ ξ k 0
(3.6)

as K i k. Since x k x as K i k, it follows from (3.6) and Lemma 2.3 that

lim K i k y k = x .
(3.7)

Note that T i k m k = T i m k whenever k K i for each i1. Since { x k } k K i is bounded, so is { T i m k x k } k K i . In view of α k 0, hence from (3.1) we have that

lim K i k J y k J T i m k = lim K i k α k J x 1 J T i m k =0.
(3.8)

In addition, J y k J x implies that lim K i k J T i m k =J x . Remark 2.2(ii) yields that, as K i k,

T i m k x k x ,i1.
(3.9)

Again, since for each i1, as K i k,

| T i m k x k x | = | J ( T i m k x k ) J x | J ( T i m k x k ) J x 0 ,
(3.10)

this, together with (3.10) and the Kadec-Klee property of E, shows that

lim K i k T i m k x k = x ,i1.
(3.11)

On the other hand, by the assumptions that for each i1, T i is uniformly L i -Lipschitz continuous, and noting that m k + 1 1= m k for all k K i , we then have

T i m k + 1 x k T i m k x k T i m k + 1 x k T i m k + 1 x k + 1 + T i m k + 1 x k + 1 x k + 1 + x k + 1 x k + x k T i m k x k ( L i + 1 ) x k + 1 x k + T i m k + 1 x k + 1 x k + 1 + x k T i m k x k .
(3.12)

From (3.11) and x k x ( K i k), we have that lim K i k T i m k + 1 x k T i m k x k =0 and lim K i k T i m k + 1 x k = x , i.e., lim K i k T i ( T i m k + 1 1 x k )= x . It then follows that, for each i1,

lim K i k T i ( T i m k x k ) = x .
(3.13)

In view of the closedness of T i , it follows from (3.11) that T i x = x , namely, for each i1, x F( T i ) and hence x F.

  1. (V)

    x = Π F x 1 , and so x n Π F x 1 as n.

Put u= Π F x 1 . Since uF C n and x n = Π C n x 1 , we have ϕ( x n , x 1 )ϕ(u, x 1 ), n1. Then

ϕ ( x , x 1 ) = lim n ϕ( x n , x 1 )ϕ(u, x 1 ),
(3.14)

which implies that x =u since u= Π F x 1 , and hence x n x = Π F x 1 . This completes the proof. □

Remark 3.2 Note that algorithm (3.1) just depends on the projection onto a single closed and convex set for each fixed n. An example [9] of how to compute such a projection is given as follows.

Dykstra’s algorithm Let Ω 1 , Ω 2 ,, Ω p be closed and convex subsets of R n . For any i=1,2,,p and x 0 R n , the sequences { x i k } are defined by the following recursive formulae:

{ x 0 k = x p k 1 , x i k = P Ω i ( x i 1 k y i k 1 ) , i = 1 , 2 , , p , y i k = x i k ( x i 1 k y i k 1 ) , i = 1 , 2 , , p ,
(3.15)

for k=1,2, with initial values x p 0 = x 0 and y i 0 =0 for i=1,2,,p. If Ω:= i = 1 p Ω i , then { x i k } converges to x = P Ω ( x 0 ), where P Ω (x):=arg inf y Ω y x 2 , x R n .

We now give a nontrivial example of the calculation of common fixed points for specific mappings.

Example 3.3 Let E= R 1 with the standard norm =|| and C=[0,1]. Let { T i }:CC be a sequence of nonexpansive mappings defined by T i x= x i i . Consider the following iteration sequence generated by

{ x 1 C ; C 1 = C , y n = J 1 [ α n J x n + ( 1 α n ) J z n ] , z n = J 1 [ β n J x n + ( 1 β n ) J T i n x n ] , C n + 1 = { v C n : ϕ ( v , y n ) ϕ ( v , x n ) } , x n + 1 = Π C n + 1 x 1 , n 1 ,
(3.16)

where { α n }={ 2 3 1 4 n }, { β n }={ 4 5 1 2 n } and Π C n + 1 (x):=arg inf y C n + 1 |yx|. Note that J=I and ϕ(x,y)= | x y | 2 for all x,yE since E is a Hilbert space. Moreover, it is not difficult to obtain that C n + 1 =[0, x n + y n 2 ] for all n1. Then (3.16) is reduced to

{ x 1 C ; C 1 = C , y n = ( 2 3 1 4 n ) x n + ( 1 3 + 1 4 n ) z n , z n = ( 4 5 1 2 n ) x n + ( 1 5 + 1 2 n ) T i n x n , C n + 1 = { v C n : | v y n | | v x n | } , x n + 1 = x n + y n 2 , n 1 ,
(3.17)

where i n is the solution to the positive integer equation: n=i+ ( m 1 ) m 2 (mi, n=1,2,). It is clear that { T i } is a sequence of closed and totally quasi-ϕ-asymptotically nonexpansive mappings with a common fixed point zero. It then can be shown by a similar way of Theorem 3.1 that { x n } converges strongly to zero. The numerical experiment outcome obtained by using MATLAB 7.10.0.499 shows that as x 1 =1, the computations of x 100 , x 200 , x 300 and x 400 are 0.025141746, 0.00078472044, 0.000025282198 and 0.00000082531714, respectively. This example illustrates the effectiveness of the introduced algorithm for countable families of totally quasi-ϕ-asymptotically nonexpansive mappings.

4 Applications

The so-called convex feasibility problem for a family of mappings { T i } i = 1 is to find a point in the nonempty intersection i = 1 F( T i ), which exactly illustrates the importance of finding common fixed points of infinite families. The following example also clarifies the same thing.

Example 4.1 Let E be a smooth, strictly convex and reflexive Banach space, let C be a nonempty and closed convex subset of E, and let { f i } i = 1 :C×CR be a sequence of bifunctions satisfying the conditions: for each i1,

(A1) f i (x,x)=0;

(A2) f i is monotone, i.e., f i (x,y)+ f i (y,x)0;

(A3) lim sup t 0 f i (x+t(zx),y) f i (x,y);

(A4) the mapping y f i (x,y) is convex and lower semicontinuous.

A system of equilibrium problems for { f i } i = 1 is to find an x C such that

f i ( x , y ) 0,yC,i1,

whose set of common solutions is denoted by EP:= i = 1 EP( f i ), where EP( f i ) denotes the set of solutions to the equilibrium problem for f i (i=1,2,). It is shown in [[10], Theorem 4.3] that such a system of problems can be reduced to the approximation of some fixed point of a sequence of nonlinear mappings.

References

  1. Chang S-s, Lee HWJ, Chan CK, Zhang WB: A modified Halpern-type iterative algorithm for totally quasi- ϕ -asymptotically nonexpansive mappings with applications. Appl. Math. Comput. 2012, 218(11):6489–6497. 10.1016/j.amc.2011.12.019

    MathSciNet  Article  MATH  Google Scholar 

  2. Qin XL, Su YF: Strong convergence theorems for relatively nonexpansive mappings in a Banach space. Nonlinear Anal. 2007, 67: 1958–1965. 10.1016/j.na.2006.08.021

    MathSciNet  Article  MATH  Google Scholar 

  3. Qin XL, Cho YJ, Kang SM, Zhou HY: Convergence of a modified Halpern-type iterative algorithm for quasi- ϕ -nonexpansive mappings. Appl. Math. Lett. 2009, 22: 1051–1055. 10.1016/j.aml.2009.01.015

    MathSciNet  Article  MATH  Google Scholar 

  4. Wang ZM, Su YF, Wang DX, Dong YC: A modified Halpern-type iteration algorithm for a family of hemi-relative nonexpansive mappings and systems of equilibrium problems in Banach spaces. J. Comput. Appl. Math. 2011, 235: 2364–2371. 10.1016/j.cam.2010.10.036

    MathSciNet  Article  MATH  Google Scholar 

  5. Martinez-Yanes C, Xu HK: Strong convergence of the CQ method for fixed point iteration processes. Nonlinear Anal. 2006, 64: 2400–2411. 10.1016/j.na.2005.08.018

    MathSciNet  Article  MATH  Google Scholar 

  6. 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: Kartosator AG. Dekker, New York; 1996:15–50.

    Google Scholar 

  7. Cioranescu I: Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems. Kluwer Academic, Dordrecht; 1990.

    Book  MATH  Google Scholar 

  8. Deng W-Q, Bai P: An implicit iteration process for common fixed points of two infinite families of asymptotically nonexpansive mappings in Banach spaces. J. Appl. Math. 2013., 2013: Article ID 602582

    Google Scholar 

  9. Boyle JP, Dykstra RL: A method for finding projections onto the intersections of convex sets in Hilbert spaces. Lecture Notes in Statistics 37. Advances in Order Restricted Statistical Inference 1986, 28–47.

    Chapter  Google Scholar 

  10. Deng W-Q: A new approach to the approximation of common fixed points of an infinite family of relatively quasinonexpansive mappings with applications. Abstr. Appl. Anal. 2012., 2012: Article ID 437430

    Google Scholar 

Download references

Acknowledgements

This work is supported by the National Natural Science Foundation of China (Grant No. 11061037).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Wei-Qi Deng.

Additional information

Competing interests

The author declares that they have no competing interests.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and Permissions

About this article

Cite this article

Deng, WQ. Relaxed Halpern-type iteration method for countable families of totally quasi-ϕ-asymptotically nonexpansive mappings. J Inequal Appl 2013, 367 (2013). https://doi.org/10.1186/1029-242X-2013-367

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/1029-242X-2013-367

Keywords

  • Halpern-type iteration
  • totally quasi-ϕ-asymptotically nonexpansive mappings
  • generalized projection