Let be a real Banach space and a nonempty closed convex subset of . A mapping is said to be asymptotically nonexpansive [1] if there exists a sequence of positive real numbers with such that

for all and all .

The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [1] in 1972. They proved that if is a nonempty bounded closed convex subset of a uniformly convex Banach space , then every asymptotically nonexpansive self-mapping of has a fixed point. Further, the set of fixed points of is closed and convex. Since 1972, a host of authors have studied the weak and strong convergence problems of the iterative algorithms for such a class of mappings (see, e.g., [1–3] and the references therein).

It is well known that in an infinite-dimensional Hilbert space, the normal Mann's iterative algorithm has only weak convergence, in general, even for nonexpansive mappings. Consequently, in order to obtain strong convergence, one has to modify the normal Mann's iteration algorithm; the so-called hybrid projection iteration method is such a modification.

The hybrid projection iteration algorithm (HPIA) was introduced initially by Haugazeau [4] in 1968. For 40 years, (HPIA) has received rapid developments. For details, the readers are referred to papers in [5–11] and the references therein.

In 2003, Nakajo and Takahashi [6] proposed the following modification of the Mann iteration method for a nonexpansive mapping in a Hilbert space :

where is a closed convex subset of , denotes the metric projection from onto a closed convex subset of . They proved that if the sequence is bounded above from one then the sequence generated by (1.2) converges strongly to , where denote the fixed points set of .

In 2006, Kim and Xu [12] proposed the following modification of the Mann iteration method for asymptotically nonexpansive mapping in a Hilbert space :

where is bounded closed convex subset and

They proved that if the sequence is bounded above from one, then the sequence generated by (1.3) converges strongly to .

They also proposed the following modification of the Mann iteration method for asymptotically nonexpansive semigroup in a Hilbert space :

where is bounded closed convex subset and

and is nonincreasing in and bounded measurable function such that, for all as , and for each ,

They proved that if the sequence is bounded above from one, then the sequence generated by (1.5) converges strongly to , where denote the common fixed points set of .

In 2006, Martinez-Yanes and Xu [7] proposed the following modification of the Ishikawa iteration method for nonexpansive mapping in a Hilbert space :

where is a closed convex subset of . They proved that if the sequence is bounded above from one and , then the sequence generated by (1.8) converges strongly to .

Martinez-Yanes and Xu [7] proposed also the following modification of the Halpern iteration method for nonexpansive mapping in a Hilbert space :

where is a closed convex subset of . They proved that if the sequence , then the sequence generated by (1.9) converges strongly to .

In 2005, Matsushita and Takahashi [8] proposed the following hybrid iteration method with generalized projection for relatively nonexpansive mapping in a Banach space :

They proved the following convergence theorem.

Theorem MT..

Let be a uniformly convex and uniformly smooth Banach space, let be a nonempty closed convex subset of , let be a relatively nonexpansive mapping from into itself, and let be a sequence of real numbers such that and . Suppose that is given by (1.10), where is the duality mapping on . If is nonempty, then converges strongly to , where is the generalized projection from onto .

In 2009, Zhou et al. [11] proposed the following modification of the hybrid iteration method with generalized projection for a family of closed and quasi--asymptotically nonexpansive mappings in a Banach space :

They proved the following convergence theorem.

Theorem ZGT.

Let be a nonempty bounded closed convex subset of a uniformly convex and uniformly smooth Banach space , and let be a family of -asymptotically nonexpansive mappings such that . Assume that every , is asymptotically regular on . Let be a real sequence in such that . Define a sequence as given by ( 1 ), then converges strongly to , where , for all , , and is the generalized projection from onto .

Very recently, Kimura and Takahashi [13] established strong convergence theorems by the hybrid method for a family of relatively nonexpansive mappings as follows.

Theorem KT.

Let be a strictly convex reflexive Banach space having the Kadec-Klee property and a Fréchet differentiable norm, and let be a nonempty and closed convex subset of and a family of relatively nonexpensive mappings of into itself having a common fixed point. Let be a sequence in such that . For an arbitrarily chosen point , generate a sequence by the following iterative scheme: , and

for every , then converges strongly to , where is the set of common fixed points of and is the metric projection of onto a nonempty closed convex subset of .

Motivated by these results above, the purpose of this paper is to propose a Modified hybrid projection algorithm and prove strong convergence theorems for a family of --asymptotically nonexpansive mappings which are asymptotically regular on . In order to get the strong convergence theorems for such a family of mappings, the classical hybrid projection iteration algorithm is modified and then is used to approximate the common fixed points of such a family of mappings. In the meantime, the method of the proof is different from the original one. Our results improve and extend the corresponding results announced by Zhou et al. [11], and Kimura and Takahashi [13], and some others.

Let be a Banach space with dual . Denote by the duality product. The normalize duality mapping from to is defined by

for all , where denotes the dual space of and the generalized duality pairing between and . It is well known that if is uniformly convex, then is uniformly continuous on bounded subsets of .

It is also very well known that if is a nonempty closed convex subset of a Hilbert space and is the metric projection of onto , then is nonexpansive. This fact actually characterizes Hilbert spaces , and consequently, it is not available in more general Banach spaces. In this connection, Alber [14] recently introduced a generalized projection operator in a Banach space which is an analogue of the metric projection in Hilbert spaces.

Next, we assume that is a real smooth Banach space. Let us consider the functional defined by [7, 8] as

for all . Observe that, in a Hilbert space , (2.2) reduces to , ,.

The generalized projection is a map that assigns to an arbitrary point , the minimum point of the functional , that is, , where is the solution to the minimization problem

Existence and uniqueness of the operator follow from the properties of the C functional and strict monotonicity of the mapping (see, e.g., [14–18]). In Hilbert spaces, . It is obvious from the definition of function that

for all .

Remark 2.1.

If is a reflexive strictly convex and smooth Banach space, then for , if and only if . It is sufficient to show that if , then . From (2.4), we have . This implies that From the definitions of , we have . That is, see [17, 18] for more details.

Let be a closed convex subset of and a mapping from into itself. is said to be -asymptotically nonexpansive if there exists some real sequence with and such that for all and . is said to be -asymptotically nonexpansive [9] if there exists some real sequence with and and such that for all , , and . is said to be asymptotically regular on if, for any bounded subset of , there holds the following equality:

We remark that a -asymptotically nonexpansive mapping with a nonempty fixed point set is a quasi--asymptotically nonexpansive mapping, but the converse may be not true.

We present some examples which are closed and quasi--asymptotically nonexpansive.

Example 2.2.

Let be a real line. We define a mapping by

Then is continuous quasi-nonexpansive, and hence it is closed and nonexpansive with the constant sequence but not asymptotically nonexpansive.

Example 2.3.

Let be a uniformly smooth and strictly convex Banach space, and is a maximal monotone mapping such that is nonempty. Then, is a closed and quasi--asymptotically nonexpansive mapping from onto , and .

Example 2.4.

Let be the generalized projection from a smooth, strictly convex, and reflexive Banach space onto a nonempty closed convex subset of . Then, is a closed and quasi--asymptotically nonexpansive mapping from onto with .

Let be a sequence of nonempty closed convex subsets of a reflexive Banach space . We denote two subsets and as follows: if and only if there exists such that converges strongly to and that for all . Similarly, if and only if there exists a subsequence of and a sequence such that converges weakly to and that for all . We define the Mosco convergence [19] of as follows. If satisfies that , it is said that converges to in the sense of Mosco, and we write . For more details, see [20].

The following theorem plays an important role in our results.

Theorem 2.5 (see Ibaraki et al. [21]).

Let be a smooth, reflexive, and strictly convex Banach space having the Kadec-Klee property. Let be a sequence of nonempty closed convex subsets of . If exists and is nonempty, then converges strongly to for each .

We also need the following lemmas for the proof of our main results.

Lemma 2.6 (Kamimura and Takahashi [16]).

Let be a uniformly convex and smooth Banach space, and let , be two sequences of if and either or is bounded, then .

Lemma 2.7 (Alber [14]).

Let be a reflexive, strictly convex and smooth Banach space, let be a nonempty closed convex subset of , and let . Then

for all .

Lemma 2.8.

Let be a uniformly convex and smooth Banach space, let be a closed convex subset of , and let be a closed and -asympotically nonexpansive mapping from into itself. Then is a closed convex subset of .

Now we are in a proposition to prove the main results of this paper. In the sequel, we use the letter to denote an index set.

Theorem 3.1.

Let be a nonempty closed convex subset of a uniformly convex and uniformly smooth Banach space , and let be a family of closed and -asymptotically nonexpansive mappings such that . Assume that every , is asymptotically regular on . Let , , be real sequences in such that , . Define a sequence in in the following manner:

Then converges strongly to , where is the generalized projection from onto .

Proof.

Firstly, we show that is closed and convex for each .

From the definition of , it is obvious that is closed for each . We show that is convex for each . Observe that the set

can be written as

For and , denote , , and by noting that is convex, we have

So we obtain

which infers that , so we get that is convex for each . Thus is closed and convex for every .

Secondly, we prove that , for all .

Indeed, by noting that is convex and using (2.2), we have, for any and all , that

which infers that , for all and , and hence . This proves that , for all and .

Thirdly, we will show that .

Since is a decreasing sequence of closed convex subsets of such that is nonempty, it follows that

By Theorem 2.5, converges strongly to .

Fourthly, we prove that .

Since , from the definition of , we get

From , one obtains as , and it follows from , for every that we have

and hence as by Lemma 2.6. It follows that as . Since is uniformly norm-to-norm continuous on any bounded sets of , we conclude that

for every . By the definition of and the assumption on , we deduce that

for every and . So we get

Since , we have as .

Since is also uniformly norm-to-norm continuous on any bounded sets of , we conclude that

Noting that as , we have

as . Observe that

By using (3.14), (3.15), and the asymptotic regularity of , we have

as , that is, . Now the closedness property of gives that is a common fixed point of the family , thus .

Finally, since and is a nonempty closed convex subset of , we conclude that . This completes the proof.

Remark 3.2.

The boundedness assumption on in Theorem ZGT can be dropped.

Remark 3.3.

The asymptotic regularity assumption on in Theorem 3.1 can be weakened to the assumption that as .

Recall that is called uniformly Lipschitzian continuous if there exists some such that

for all and .

Remark 3.4.

The assumption that as can be replaced by the uniform Lipschitz continuity of .

With above observations, we have the following convergence result.

Corollary 3.5.

Let be a nonempty closed convex subset of a uniformly convex and uniformly smooth Banach space , and let be a family of uniformly Lipschitzian continuous and - -asymptotically nonexpansive mappings such that . Let , , be real sequences in such that , . Define a sequence in in the following manner:

Then converges strongly to , where is the generalized projection from onto .

Proof.

Following the proof lines of Theorem 3.1, we can prove that is nonempty closed convex, is closed convex, for all and . At this point, it is sufficient to show that as . Again, from the proof lines of Theorem 3.1, we have the following conclusions:

Observe that

so that as . By Theorem 3.1, we have the desired conclusion. This completes the proof.

When in Theorem 3.1, we obtain the following result.

Corollary 3.6.

Let be a nonempty closed convex subset of a uniformly convex and uniformly smooth Banach space , and let be a family of closed and -asymptotically nonexpansive mappings such that . Assume that every , is asymptotically regular on . Let be a real sequence in such that . Define a sequence in in the following manner:

Then converges strongly to , where is the generalized projection from onto .

When in Theorem 3.1, we obtain the following result.

Corollary 3.7.

Let be a nonempty closed convex subset of a uniformly convex and uniformly smooth Banach space , and let be a family of closed and -asymptotically nonexpansive mappings such that . Assume that every , is asymptotically regular on . Let be a real sequence in such that . Define a sequence in in the following manner:

Then converges strongly to , where is the generalized projection from onto .

In the spirit of Theorem 3.1, we can prove the following strong convergence theorem.

Theorem 3.8.

Let be a nonempty closed convex subset of a uniformly convex and uniformly smooth Banach space , and let be a family of closed and -nonexpansive mappings such that . Let , , be real sequences in such that , . Define a sequence in in the following manner:

Then converges strongly to , where is the generalized projection from onto .

Proof.

Following the proof lines of Theorem 3.1, we have the following conclusions:

(1) is a nonempty closed convex subset of ;

(2) is closed covex for all ;

(3) , for all ;

(4) ;

(5) for all .

The closedness property of together with (4) and (5) implies that converges strongly to a common fixed point of the family . As shown in Theorem 3.1, . This completes the proof.

When in Theorem 3.8, we obtain the following result.

Corollary 3.9.

Let be a nonempty closed convex subset of a uniformly convex and uniformly smooth Banach space, and let be a family of closed and -nonexpansive mappings such that . Let be a real sequence in such that . Define a sequence in in the following manner:

Then converges strongly to , where is the generalized projection from onto .

When in Theorem 3.8, we obtain the following result.

Corollary 3.10.

Let be a nonempty closed convex subset of a uniformly convex and uniformly smooth Banach space , and let be a family of closed and nonexpansive mappings such that . Let be a real sequence in such that . Define a sequence in in the following manner:

Then converges strongly to , where is the generalized projection from onto .