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A modified Picard-Mann hybrid iterative algorithm for common fixed points of countable families of nonexpansive mappings

Abstract

An up-to-date method is used for approximating common fixed points of countable families of nonlinear mappings. A modified Picard-Mann hybrid iterative algorithm is introduced with the help of our method for the class of nonexpansive mappings. Strong convergence and weak convergence theorems are established in the framework of uniformly convex Banach spaces. Our results extend the corresponding ones announced by Khan (Fixed Point Theory Appl. 2013:69, 2013, doi:10.1186/1687-1812-2013-69) to the case of countable families of nonexpansive mappings.

MSC:47H09, 47J25.

1 Introduction

Let K be a nonempty closed convex subset of a real uniformly convex Banach space E. A mapping T:KK is said to be nonexpansive if

TxTyxy,x,yK.

Iterative techniques for approximating fixed points of nonexpansive mappings have been studied by various authors (see, e.g., [15]) who used the Mann iteration process or the Ishikawa process. In 2013, Khan [6] introduced the following Picard-Mann hybrid iterative process for a single nonexpansive mapping T. For any initial point x 0 K:

{ x n + 1 = T y n , y n = ( 1 α n ) x n + α n T x n , n N ,
(1.1)

where { α n } is a real sequence in (0,1). He showed that the new process converges faster than all of Picard, Mann and Ishikawa iterative processes in the sense of Berinde [7] for contractions. He also proved strong convergence and weak convergence theorems with the help of his process for the class of nonexpansive mappings in general Banach spaces and apply it to obtain a result in uniformly convex Banach spaces.

Inspired and motivated by the studies mentioned above, in this paper, we use an up-to-date method for the approximation of common fixed points of countable families of nonlinear operators. We introduce a modified Picard-Mann hybrid iterative algorithm with the help of our method for the class of nonexpansive mappings. We prove strong convergence and weak convergence theorems in the framework of Banach spaces. Our results extend the corresponding ones for one map in [6].

2 Preliminaries

Throughout this paper we assume that E is a real Banach space with its dual E , K 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.

We say that E is strictly convex if the following implication holds for x,yE:

x=y=1,xy x + y 2 <1.
(2.1)

It is also said to be uniformly convex if, for any ϵ>0, there exists a δ>0 such that

x=y=1,xyϵ x + y 2 1δ.
(2.2)

It is well known that if E is a uniformly convex Banach space, then E is reflexive and strictly convex. A Banach space E is said to be smooth if

lim t 0 x + t y x t
(2.3)

exists for each x,yS(E):={xE:x=1}. In this case, the norm of E is said to be Gâteaux differentiable. The space E is said to have uniformly Gâteaux differentiable norm if for each yS(E), the limit (2.3) is attained uniformly for xS(E). The norm of E is said to be Fréchet differentiable if for each xS(E), the limit (2.3) is attained uniformly for yS(E). The norm of E is said to be uniformly Fréchet differentiable (and E is said to be uniformly smooth) if the limit (2.3) is attained uniformly for x,yS(E).

Note The readers can find all the definitions and concepts mentioned above in [8].

A Banach space E is said to satisfy Opial’s condition if, for any sequence { x n } in E, x n x implies that

lim sup n x n x< lim sup n x n y
(2.4)

for all yE with yx, where x n x denotes that { x n } converges weakly to x.

A mapping T with domain D(T) and range R(T) in E is said to be demi-closed at p if whenever { x n } is a sequence in D(T) such that { x n } converges weakly to x D(T) and {T x n } converges strongly to p, then T x =p.

Remark 2.1 The following basic properties for a Banach space E can be found in [9].

  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.

We need the following lemmas for our main results.

Lemma 2.2 [10]

Let E be a real uniformly convex Banach space and let a, b be two constant with 0<a<b<1. Suppose that { t n }[a,b] is a real sequence and { x n }, { y n } are two sequences in E. Then the conditions

lim n t n x n + ( 1 t n ) y n =d, lim sup n x n d, lim sup n y n d
(2.5)

imply that li m n x n y n =0, where d0 is a constant.

Lemma 2.3 [11]

Let E be a real uniformly convex Banach space, let K be a nonempty closed convex subset of E, and let T:KK be a nonexpansive mapping. Then IT is demi-closed at zero.

Lemma 2.4 [12]

The unique solutions to the positive integer equation

n=i+ ( m 1 ) m 2 ,mi,n=1,2,3,
(2.6)

are

i=n ( m 1 ) m 2 ,m= [ 1 2 2 n + 1 4 ] ,n=1,2,3,,
(2.7)

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

3 Main results

Lemma 3.1 Let E be a real uniformly convex Banach space and K a nonempty closed convex subset of E. Let { T i } i = 1 be a sequence of nonexpansive mappings from K to itself. For an arbitrary initial point x 1 K, the modified Picard-Mann hybrid iterative scheme { x n } is defined as follows:

{ x n + 1 = T i n y n , y n = ( 1 α n ) x n + α n T i n x n , n N ,
(3.1)

where { α n } is a sequence in [ϵ,1ϵ] for some ϵ(0,1) and i n is the solution to the positive integer equation: n=i+ ( m 1 ) m 2 (mi, n=1,2,), that is, for each n1, there exists a unique i 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 , i 9 = 3 , i 10 = 4 , i 11 = 1 , .

If F:={xK: T i x=x,i1}, then

  1. (1)

    lim n x n q exists, qF;

  2. (2)

    lim n d( x n ,F) exists, where d( x n ,F)= inf q F x n q;

  3. (3)

    lim n x n T i x n =0, i1.

Proof (1) For any qF, by (3.1), we have

y n q = ( 1 α n ) ( x n q ) + α n ( T i n x n q ) ( 1 α n ) x n q + α n T i n x n T i n q x n q ,
(3.2)

and hence

x n + 1 q= T i n y n q y n q x n q.
(3.3)

This shows that { x n q} is decreasing and hence lim n x n q exists.

(2) This conclusion can easily be shown by taking the infimum in (3.3) for all qF.

(3) Assume, by the conclusion of (1), lim n x n q=d. We then claim that lim n y n q=d, that is,

lim n ( 1 α n ) ( x n q ) + α n ( T i n x n q ) =d.
(3.4)

In fact, noting that x n + 1 q< y n q, we have

d= lim inf n x n + 1 q lim inf n y n q;

on the other hand, it follows from (3.2) that

lim sup n y n qd,

which implies that lim n y n q=d.

Next, T i n x n q x n q implies that

lim sup n T i n x n q lim sup n x n q=d,
(3.5)

and hence, it follows from (3.4), (3.5), and Lemma 2.2 that

lim n x n T i n x n =0.
(3.6)

On the other hand, since

x n + 1 x n x n + 1 T i n x n + T i n x n x n = T i n y n T i n x n + T i n x n x n y n x n + T i n x n x n α n T i n x n x n + T i n x n x n ,

we have, from (3.6),

lim n x n + 1 x n =0.
(3.7)

By induction, for any nonnegative integer p, we also have

lim n x n + p x n =0.
(3.8)

For each p0, since

x n T i n + p x n x n x n + p + x n + p T i n + p x n x n x n + p + x n + p T i n + p x n + p + T i n + p x n + p T i n + p x n 2 x n x n + p + x n + p T i n + p x n + p ,
(3.9)

it follows from (3.6) and (3.8) that

lim n x n T i n + p x n =0.
(3.10)

Now, for each i1, we claim that

lim n x n T i x n =0.
(3.11)

As a matter of fact, setting n= k m +i, where k m = ( m 1 ) m 2 , mi, we obtain

x n T i x n x n x k m + x k m T i x n x n x k m + x k m T i k m + i x k m + T i k m + i x k m T i x n = x n x k m + x k m T i k m + i x k m + T i x k m T i x n 2 x n x k m + x k m T i k m + i x k m = 2 x n x n i + x k m T i k m + i x k m .
(3.12)

Note that k m as n. It then follows from (3.8) and (3.10) that (3.11) holds obviously. This completes the proof. □

Remark 3.2 The key point of the proof of Lemma 3.1 lies in the use of a special way of choosing the indices of involved mappings, which makes the generalization of finite families of nonlinear mappings to infinite ones possible. Moreover, with the help of our method, some known results on the common fixed points of countable families of nonexpansive mappings have been improved. We now give an example to show why our work, compared with that of others, is an improvement.

In 2011, for the approximation of common fixed points of a countable family of nonexpansive mappings { T n }, Zhang et al. [13] introduced in his iterative algorithm a mapping T defined by a convex linear combination of { T n }, i.e., T= n = 1 λ n T n , λ n 0 (n=1,2,) with n = 1 λ n =1. However, it is easy to see that the accurate computation of T x n at each step of the iteration process is not easily attainable, which will leads to gradually increasing errors. By using a special way of choosing the indices of involved mappings, Deng [14] recently improved the corresponding results announced by Zhang et al. [13]. Since the strong convergence theorems for solving some variational inequality problems and hierarchical fixed point problems are obtained without the aid of the convex linear combination of a countable family of nonexpansive mappings, our results are more applicable than those of other authors with related research interest.

Theorem 3.3 Let E be a real uniformly convex Banach space and K a nonempty closed convex subset of E. Let { T i } i = 1 be a sequence of nonexpansive mappings from K to itself. Suppose that { x n } is a sequence defined by (3.1). If F:={xK: T i x=x,i1} and there exist T i 0 { T i } i = 1 and a nondecreasing function f:[0,)[0,) with f(0)=0 and f(r)>0 for all r(0,) such that f(d( x n ,F)) x n T i 0 x n for all n1, then { x n } converges strongly to some common fixed point of { T i } i = 1 .

Proof Since

f ( d ( x n , F ) ) x n T i 0 x n ,

by taking lim sup as n on both sides in the inequality above, we obtain

lim n f ( d ( x n , F ) ) =0,

which implies lim n d( x n ,F)=0 by the definition of the function f.

Now we show that { x n } is a Cauchy sequence. Since lim n d( x n ,F)=0, then for any ϵ>0, there exists a positive integer N such that d( x n ,F)< ϵ 2 for all nN. On the other hand, there exists a pF such that x N p=d( x N ,F)< ϵ 2 , because d( x N ,F)= inf q F x N q and F is closed.

Thus, for any n,mN, it follows from (3.3) that

x n x m x n p+ x m p2 x N p<ϵ.

This implies that { x n } is a Cauchy sequence, and hence there exists an xK such that x n x as n. Then lim n d( x n ,F)=0 yields d(x,F)=0. Further, it follows from the closedness of F that xF. This completes the proof. □

Theorem 3.4 Let E be a real uniformly convex Banach space satisfying Opial’s condition and K a nonempty closed convex subset of E. Let { T i } i = 1 be a sequence of nonexpansive mappings from K to itself. Suppose that { x n } is a sequence defined by (3.1). If F:={xK: T i x=x,i1}, then { x n } converges weakly to some common fixed point of { T i } i = 1 .

Proof For any qF, by Lemma 3.1, we know that lim n x n q exists. We now prove that { x n } has a unique weakly subsequential limit in F. First of all, Lemmas 2.3 and 3.1 guarantee that each weakly subsequential limit of { x n } is a common fixed point of { T i } i = 1 . Secondly, Opial’s condition guarantees that the weakly subsequential limit of x n is unique. Consequently, { x n } converges weakly to a common fixed point of { T i } i = 1 . This completes the proof. □

Remark 3.5 The results presented in this paper extend those of Khan [6], whose research areas are limited to the situation of a single nonexpansive mapping.

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Acknowledgements

The author is very grateful to the referees for their useful suggestions, by which the contents of this article has been improved. This work is supported by the National Natural Science Foundation of China (Grant No. 11061037).

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Deng, W. A modified Picard-Mann hybrid iterative algorithm for common fixed points of countable families of nonexpansive mappings. Fixed Point Theory Appl 2014, 58 (2014). https://doi.org/10.1186/1687-1812-2014-58

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Keywords

  • Picard-Mann hybrid iteration
  • common fixed points
  • countable families of nonexpansive mappings
  • strong and weak convergence