Strong and weak convergence theorems for an infinite family of nonexpansive mappings and applications

In this paper, let E be a reflexive and strictly convex Banach space which either is uniformly smooth or has a weakly continuous duality map. We consider the hybrid viscosity approximation method for finding a common fixed point of an infinite family of nonexpansive mappings in E. We prove the strong convergence of this method to a common fixed point of the infinite family of nonexpansive mappings, which solves a variational inequality on their common fixed point set. We also give a weak convergence theorem for the hybrid viscosity approximation method involving an infinite family of nonexpansive mappings in a Hilbert space.

MSC: 47H17, 47H09, 47H10, 47H05.

Let C be a nonempty closed convex subset of a (real) Banach space E, and let be a nonlinear mapping. Denote by the set of fixed points of Ti.e.. Recall that T is nonexpansive if

A self-mapping is said to be a contraction on C if there exists a constant α in such that

As in [1], we use the notation to denote the collection of all contractions on Ci.e.

Note that each f in has a unique fixed point in C.

One classical way to study a nonexpansive mapping is to use contractions to approximate T[2-4]. More precisely, for each t in we define a contraction by

where u in C is an arbitrary but fixed point. Banach’s contraction mapping principle guarantees that has a unique fixed point in C. It is unclear, in general, how behaves as , even if T has a fixed point. However, in the case a Hilbert space and T having a fixed point, Browder [2] proved that converges strongly to a fixed point of T. Reich [3] extends Browder’s result and proves that if E is a uniformly smooth Banach space, then converges strongly to a fixed point of T and the limit defines the (unique) sunny nonexpansive retraction from C onto . Xu [4] proved that Browder’s results hold in reflexive Banach spaces with weakly continuous duality mappings. See Section 2 for definitions and notations.

Recall that the original Mann’s iterative process was introduced in [5] in 1953. Let be a map of a closed and convex subset C of a Hilbert space. The original Mann’s iterative process generates a sequence in the following manner:

where the sequence lies in the interval . If T is a nonexpansive mapping with a fixed point and the control sequence is chosen so that , then the sequence generated by original Mann’s iterative process (1.1) converges weakly to a fixed point of T (this is also valid in a uniformly convex Banach space with a Frechet differentiable norm [6]). In an infinite-dimensional Hilbert space, the original Mann’s iterative process guarantees only the weak convergence. Therefore, many authors try to modify the original Mann’s iterative process to ensure the strong convergence for nonexpansive mappings (see [3,7-13] and the references therein).

Kim and Xu [14] proposed the following simpler modification of the original Mann’s iterative process: Let C be a nonempty closed convex subset of a Banach space E and a nonexpansive mapping such that . For an arbitrary in C, define in the following way:

where u in C is an arbitrary but fixed element in C, and and are two sequences in . The modified Mann’s Iteration scheme (1.2) is a convex combination of a particular point u in C and the original Mann’s iterative process (1.1). There is no additional projection involved in iteration scheme (1.2). They proved a strong convergence theorem for the iteration scheme (1.2) under some control conditions on the parameters ’s and ’s.

Recently, Yao, Chen and Yao [12] combined the viscosity approximation method [1] and the modified Mann’s iteration scheme [14] to develop the following hybrid viscosity approximation method. Let C be a nonempty closed convex subset of a Banach space E, let a nonexpansive mapping such that , and let . For any arbitrary but fixed point in C, define in the following way:

where and are two sequences in . They proved under certain different control conditions on the sequences and that converges strongly to a fixed point of T. Their result extends and improves the main results in Kim and Xu [14].

Under the assumption that no parameter sequence converges to zero, Ceng and Yao [15] proved the strong convergence of the sequence generated by (1.3) to a fixed point of T, which solves a variational inequality on .

Theorem 1.1 (See [15], Theorem 3.1])

LetCbe a nonempty closed convex subset of a uniformly smooth Banach spaceE. Letbe a nonexpansive mapping with, and letwith a contractive constantαin. Given sequencesandinsuch that the following control conditions are satisfied:

(C1) , for some integer, and;

(C2) ;

(C3) .

For an arbitraryinC, letbe defined by (1.3). Then,

In this case, solves the variational inequality

On the other hand, a similar problem concerning a family of nonexpansive mappings has also been considered by many authors. The well-known convex feasibility problem reduces to finding a common fixed point of a family of nonexpansive mappings; see, e.g., [16,17]. The problem of finding an optimal point that minimizes a given cost function over the common fixed point set of a family of nonexpansive mappings is of wide interdisciplinary interest and practical importance; see, e.g., [18-20]. In particular, a simple algorithm solving the problem of minimizing a quadratic function over the common fixed point set of a family of nonexpansive mappings is of extreme value in many applications including set theoretic signal estimation; see, e.g., [20,21].

Let be nonexpansive mappings of a nonempty closed and convex subset C of a Banach space E into itself. Let be real numbers in . Qin, Cho, Kang and Kang [22] considered the nonexpansive mapping defined by

Motivated by [7,8,11,12,14,23], they proposed the following iterative algorithm:

where u in C is a given point. They proved

Theorem 1.2 (See [22], Theorem 2.1 and its proof])

LetCbe a nonempty closed convex subset of a reflexive and strictly convex Banach spaceEwith a weakly continuous duality mapwith gaugeφ. Letbe a nonexpansive mapping fromCinto itself for . Assume that. Givenand given sequences, andinsatisfying

(i) and;

(ii) ;

(iii) , for somebin.

Then the sequencedefined by (1.5) converges strongly to some pointinF. Here, thus defined is the unique sunny nonexpansive retraction of Reich type fromContoF, that is, solves the variational inequality

In this paper, let E be a reflexive and strictly convex Banach space which either is uniformly smooth or has a weakly continuous duality map with gauge φ. Combining two iterative methods (1.3) and (1.5), we give the following hybrid viscosity approximation scheme. Let C be a nonempty closed convex subset of E, let be a nonexpansive mapping for each  , such that , and let . Define in the following way:

where is defined by (1.4), is a sequence in , and and are two sequences in . It is proved under some appropriate control conditions on the sequences and that converges strongly to a common fixed point of the infinite family of nonexpansive mappings  , which solves a variational inequality on . Such a result includes Theorem 1.2 as a special case. Furthermore, we also give a weak convergence theorem for the hybrid viscosity approximation method (1.6) involving an infinite family of nonexpansive mappings in a Hilbert space H. The results presented in this paper can be viewed as supplements, improvements and extensions of some known results in the literature, e.g., [1,7,8,11-15,22-24].

Let E be a (real) Banach space with the Banach dual space in pairing . We write to indicate that the sequence converges weakly to x, and to indicate that converges strongly to x. The unit sphere of E is denoted by .

The norm of E is said to be Gateaux differentiable (and E is said to be smooth) if

exists for every in U. Recall that if E is reflexive, then E is smooth if and only if is strictly convex, i.e., for every distinct in of norm one, there holds . The norm of E is said to be uniformly Frechet differentiable (and E is said to be uniformly smooth) if the limit in (2.1) is attained uniformly for in . Every uniformly smooth Banach space E is reflexive and smooth.

The normalized duality mappingJ from E into the family of nonempty (by Hahn-Banach theorem) weak* compact subsets of is defined by

If E is smooth then J is single-valued and norm-to-weak^∗ continuous. It is also well known that if E is uniformly smooth, then J is uniformly norm-to-norm continuous on bounded subsets of E.

In order to establish new strong and weak convergence theorems for hybrid viscosity approximation method (1.6), we need the following lemmas. The first lemma is a very well-known (subdifferential) inequality; see, e.g., [25].

Lemma 2.1 ([25])

LetEbe a real Banach space andJthe normalized duality map onE. Then, for any giveninE, the following inequality holds:

Lemma 2.2 ([26], Lemma 2])

Letandbe bounded sequences in a Banach spaceE, and letbe a sequence insuch that. Supposefor all integersand. Then, .

Lemma 2.3 ([27])

Letbe a sequence of nonnegative real numbers satisfying the condition

where, are sequences of real numbers such that

(i) and, or equivalently,

(ii) , oris convergent.

Then, .

Recall that, if are nonempty subsets of a Banach space E such that C is nonempty, closed and convex, then a mapping is sunny[28] provided for all x in C and whenever . A sunny nonexpansive retraction is a sunny retraction, which is also nonexpansive. Sunny nonexpansive retractions play an important role; see, e.g., [1,22]. They are characterized as follows [28]: if E is a smooth Banach space, then is a sunny nonexpansive retraction if and only if there holds the inequality

Lemma 2.4 ([1], Theorem 4.1])

LetEbe a uniformly smooth Banach space, Cbe a nonempty closed convex subset ofE, be a nonexpansive mapping with, and. Thendefined by

converges strongly to a point in. Defineby

Then, solves the variational inequality

In particular, ifis a constant, then the mapis reduced to the sunny nonexpansive retraction of Reich type fromConto, i.e.,

Recall that a gauge is a continuous strictly increasing function such that and as . Associated to gauge φ is the duality map defined by

Following Browder [29], we say that a Banach space E has a weakly continuous duality map if there exists gauge φ for which the duality map is single-valued and weak-to-weak^∗ sequentially continuous. It is known that has a weakly continuous duality map with gauge for all . Set

Then

where ∂ denotes the subdifferential in the sense of convex analysis; see [25,30] for more details.

The first part of the following lemma is an immediate consequence of the subdifferential inequality, and the proof of the second part can be found in [31].

Lemma 2.5Assume thatEhas a weakly continuous duality mapwith gaugeφ.

(i) For all, there holds the inequality

(ii) Assume a sequenceinEis weakly convergent to a pointx. Then there holds the identity

Xu [4] showed that, if E is a reflexive Banach space and has a weakly continuous duality map with gauge φ, then there is a sunny nonexpansive retraction from C onto . Further this result is extended to the following general case.

Lemma 2.6 ([32], Theorem 3.1 and its proof])

LetEbe a reflexive Banach space and have a weakly continuous duality mapwith gaugeφ, letCbe a nonempty closed convex subset ofE, letbe a nonexpansive mapping with, and let. Thendefined by

converges strongly to a point inas. Defineby

Then, solves the variational inequality

In particular, ifis a constant, then the mapis reduced to the sunny nonexpansive retraction of Reich type fromConto, i.e.,

Recall that E satisfies Opial’s property [33] provided, for each sequence in E, the condition implies

It is known in [33] that each () enjoys this property, while does not unless . It is known in [34] that every separable Banach space can be equivalently renormed so that it satisfies Opial’s property. We denote by the weakω-limit set of i.e.

Finally, recall that in a Hilbert space H, there holds the following equality

See, e.g., Takahashi [35].

We will also use the following elementary lemmas in the sequel.

Lemma 2.7 ([36])

Letandbe the sequences of nonnegative real numbers such thatandfor all. Thenexists.

Lemma 2.8 (Demiclosedness Principle [25,30])

Assume thatTis a nonexpansive self-mapping of a nonempty closed convex subsetCof a Hilbert spaceH. IfThas a fixed point, thenis demiclosed. That is, wheneverinCandinH, it follows that. Here, Iis the identity operator ofH.

Lemma 3.1 ([24])

LetCbe a nonempty closed convex subset of a strictly convex Banach spaceE. Letbe nonexpansive mappings fromCinto itself such thatand letbe real numbers such thatfor all. Then, for everyxinCand, the limitexists.

Using Lemma 3.1, one can define the mapping W from C into itself as follows.

Such a mapping W is called the W-mapping generated by and  . Throughout this paper, we always assume that for some real constant b and for all .

Lemma 3.2 ([24])

LetCbe a nonempty closed convex subset of a strictly convex Banach spaceE. Letbe nonexpansive mappings ofCinto itself such thatand letbe real numbers such thatfor any. Then, .

Here comes the main result of this paper.

Theorem 3.3LetCbe a nonempty closed convex subset of a reflexive and strictly convex Banach spaceE. Assume, in addition, Eeither is uniformly smooth or has a weakly continuous duality mapwith gaugeφ. Letbe a nonexpansive mapping for eachsuch that, andwith contractive constantαin. Given sequences, andin, the following conditions are satisfied:

(C1) , for some, and;

(C2) ;

(C3) ;

(C4) , for some constantbin.

For an arbitrary, letbe generated by

Then,

In this case,

(i) ifEis uniformly smooth, thensolves the variational inequality

(ii) ifEhas a weakly continuous duality mapwith gaugeφ, thensolves the variational inequality

Proof First, let us show that is bounded. Indeed, taking an element p in arbitrarily, we obtain that for all . It follows from the nonexpansivity of that

Observe that

By simple induction, we have

Hence is bounded, and so are the sequences , and .

Suppose that as . Then for all . From (3.2) it follows that

that is, . Again from (3.2) we obtain that

Conversely, suppose that (). Put

Then, it follows from (C1) and (C2) that

and hence

Define by

Observe that

It follows that

Since and are nonexpansive, from (1.4) we have

Since is a bounded sequence and all are nonexpansive with a common fixed point p, there is such that

Substituting (3.6) into (3.5), we have

From conditions (C3), (C4) and the boundedness of and , it follows that

Hence by Lemma 2.2 we have

It follows from (3.3) and (3.4) that

From (3.2), we have

This implies that

Since and , we get

Observe that

It follows from (C2), (3.7) and (3.8) that

Also, note that

From [37], Remark 2.2] (see also [38], Remark 3.1]), we have

It follows

In terms of (3.1) and Lemma 3.2, is a nonexpansive mapping such that . In the following, we discuss two cases.

(i) Firstly, suppose that E is uniformly smooth. Let be the unique fixed point of the contraction mapping given by

By Lemma 2.4, we can define

and solves the variational inequality

Let us show that

where . Note that

Applying Lemma 2.1 we derive

where

The last inequality implies

It follows that

where is a constant such that for all and small enough t in . Taking the limsup as in (3.11) and noticing the fact that the two limits are interchangeable due to the fact that the duality map J is uniformly norm-to-norm continuous on any bounded subset of E, we obtain (3.10).

Now, let us show that as . Indeed, observe

Then, utilizing Lemma 2.1 we get

It follows that, for all , we have

due to (C1). For every , put

and

Since , we have . Now, we have

It is readily seen from (C1) and (3.10) that

Therefore, applying Lemma 2.3 to (3.12), we conclude that as .

(ii) Secondly, suppose that E has a weakly continuous duality map with gauge φ. Let be the unique fixed point of the contraction mapping given by

By Lemma 2.6, we can define , and solves the variational inequality

Let us show that

where . We take a subsequence of such that

Since E is reflexive and is bounded, we may further assume that for some in C. Since is weakly continuous, utilizing Lemma 2.5, we have

Put

It follows that

From (3.9), we have

Furthermore, observe that

Combining (3.16) with (3.17), we obtain

Hence and (by Lemma 3.2). Thus, from (3.13) and (3.15), it is easy to see that

Therefore, we deduce that (3.14) holds.

Now, let us show that as . Indeed, observe that

Therefore, by applying Lemma 2.5, we have

Applying Lemma 2.3, we get

which implies that , i.e., . This completes the proof. □

Corollary 3.4The conclusion in Theorem 3.3 still holds, provided the conditions (C1)-(C4) are replaced by the following:

(D1) , for some integer;

(D2) and;

(D3) and;

(D4) , for somebin.

Proof Observe that

Since and , it follows that

Consequently, all conditions of Theorem 3.3 are satisfied. So, utilizing Theorem 3.3, we obtain the desired result. □

Corollary 3.5LetCbe a nonempty closed convex subset of a reflexive and strictly convex Banach spaceE. Assume, in addition, Eeither is uniformly smooth or has a weakly continuous duality mapwith gaugeφ. Letbe a nonexpansive mapping for eachsuch that, and letwith contractive constantαin. Given sequences, andin, the following conditions are satisfied:

(E1) and;

(E2) ;

(E3) for some.

Then. for an arbitrary but fixedinC, the sequencedefined by (3.2) converges strongly to a common fixed pointinF. Moreover,

(i) ifEis uniformly smooth, thensolves the variational inequality

(ii) ifEhas a weakly continuous duality mapwith gaugeφ, thensolves the variational inequality

Proof Repeating the arguments in the proof of Theorem 3.3, we know that is bounded, and so are the sequences , and . Since , it is easy to see that there hold the following:

(i) ();

(ii) , for some integer ;

(iii) .

Therefore, all conditions of Theorem 3.3 are satisfied. So, utilizing Theorem 3.3, we obtain the desired result. □

To end this paper, we give a weak convergence theorem for hybrid viscosity approximation method (3.2) involving an infinite family of nonexpansive mappings in a Hilbert space H.

Theorem 3.6LetCbe a nonempty closed convex subset of a Hilbert spaceH. Letbe a nonexpansive mapping for eachsuch thatand. Given sequences, andin, the following conditions are satisfied:

(F1) ;

(F2) ;

(F3) , for some.

Then, for an arbitrary but fixedinC, the sequencedefined by (3.2) converges weakly to a common fixed point of the infinite family of nonexpansive mappings .

Proof Take an arbitrary p in . Repeating the arguments in the proof of Theorem 3.3, we know that is bounded, and so are the sequences , and .

It follows from (2.3) that

Since and is bounded, we get . Utilizing Lemma 2.7, we conclude that exists. Furthermore, it follows from (3.18) that for all , we have

Since and , it follows from (3.19) that . Also, observe that

From [37], Remark 2.2] (see also [38], Remark 3.1]), we have

This implies immediately that

Now, let us show that (see (2.2)). Indeed, let . Then there exists a subsequence of such that . Since , by Lemma 2.8, .

Finally, let us show that is a singleton. Indeed, let be another subsequence of such that . Then also lies in F. If , by Opial’s property of H, we reach the following contradiction:

This implies that is a singleton. Consequently, converges weakly to an element of F. □

Remark 3.7 As pointed out in [22], Remark 2.1], the mild conditions are imposed on the parameter sequence , which are different from those in [8,11,18,23]. Theorem 2.1 in [22] is a supplement to Remark 5 of Zhou, Wei and Cho [23] in reflexive Banach spaces. Moreover, it extends Theorem 1 in [14] from the case of a single nonexpansive mapping to that of an infinite family of nonexpansive mappings, and relaxes the restrictions imposed on the parameters in [14], Theorem 1]. Compared with Theorem 2.1 in [22] (i.e., Theorem 1.2), our Theorems 3.3 and 3.6 supplement, improve and extend them in the following aspects:

(1) The hybrid viscosity approximation method (3.2) includes their modified Mann’s iterative process (1.5) as a special case.

(2) We relax the restrictions imposed on the parameters in [22], Theorem 2.1]; for instance, there can be no parameter sequence convergent to zero in our Theorem 3.3.

(3) In Theorem 3.3, the problem of finding a common fixed point of an infinite family of nonexpansive mappings is also considered in the framework of uniformly smooth Banach space.

(4) In order to show the strong convergence of the hybrid viscosity approximation method (3.2), we use the techniques very different from those in the proof of [22], Theorem 2.1]; for instance, we use Theorem 4.1 in [1] and Theorem 3.1 in [32].

(5) Theorem 3.3 shows that the hybrid viscosity approximation method (3.2) converges strongly to a common fixed point of an infinite family of nonexpansive mappings, which solves a variational inequality on their common fixed point set.

(6) In Theorem 3.6, the conditions imposed on and are very different from those in [22], Theorem 2.1].

(7) In the proof of Theorem 3.6, we use the techniques very different from those in the proof of [22], Theorem 2.1]; for instance, we use Opial’s property of Hilbert space and Tan and Xu’s lemma in [36].

The authors declare that they have no competing interests.

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

LCC was partially supported by the National Science Foundation of China (11071169), Innovation Program of Shanghai Municipal Education Commission (09ZZ133) and Leading Academic Discipline Project of Shanghai Normal University (DZL707). NCW was partially supported by the grant NSC 99-2115-M-110-007-MY3. JC was partially supported by the grant NSC 99-2115-M-037-002-MY3.

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