Generalized Mann Iterations for Approximating Fixed Points of a Family of Hemicontractions

Let be a real Banach space and a nonempty closed subset of . A mapping is said to be pseudocontractive (see, e.g., [1]) if

(1.1)

holds for all . is said to be strictly pseudocontractive if, for all , there exists a constant such that

(1.2)

Denote by the set of fixed points of . A map is called hemicontractive if and for all , , the following inequality holds:

(1.3)

It is easy to see that the class of pseudocontractive mappings with fixed points is a subset of the class of hemicontractions.

There are many papers in the literature dealing with the approximation of fixed points for several classes of nonlinear mappings (see, e.g., [1–11], and the reference therein). In these works, there are two iterative methods to be used to find a point in . One is explicit and one is implicit.

The explicit one is the following well-known Mann iteration.

Let be a nonempty closed convex subset of . For any , the sequence is defined by

(1.4)

where is a real sequence in satisfying some assumptions.

It has been applied to many classes of nonlinear mappings to find a fixed point. However, for hemicontractive mappings and strictly pseudocontractive mappings, the iteration process of convergence is in general not strong (see a counterexample given by Chidume and Mutangadura [3]). Most recently, Marino and Xu [6] proved that the Mann iterative sequence converges weakly to a fixed point for strictly pseudocontractive mappings in a Hilbert space, while the real sequence satisfying (i) and (ii) .

In order to get strong convergence for fixed points of hemicontractive mappings and strictly pseudocontractive mappings, the following Mann-type implicit iteration scheme is introduced.

Let be a nonempty closed convex subset of with . For any , the sequence is generated by

(I)

where is a real sequence in satisfying suitable conditions.

Recently, in the setting of a Hilbert space, Rafiq [12] proved that the Mann-type implicit iterative sequence converges strongly to a fixed point for hemicontractive mappings, under the assumption that the domain of is a compact convex subset of a Hilbert space, and for some .

In this paper, we will study the strong convergence of the generalized Mann-type iteration scheme (see Definition 2.1) for hemicontractive and, respectively, pseudocontractive mappings. As we will see, our theorems extend the corresponding results in [12] in four aspects. (1) The space setting is a more general one: uniformly convex Banach space, which could not be a Hilbert space. (2) The requirement of the compactness on the domain of the mapping is dropped. (3) A single mapping is replaced by a family of mappings. (4) The Mann-type implicit iteration is replaced by the generalized Mann iteration. Moreover, we give answers to a question asked in [13].

Definition 2.1 (generalized Mann iteration).

Let be a fixed integer, , and a nonempty closed convex subset of satisfying the condition . Let be a family of mappings. For each , the sequence is defined by

(II)

where , , and are three sequences in with and is bounded.

The modulus of convexity of is the function defined by

(2.1)

is called uniformly convex if and only if, for all such that . is called -uniformly convex if there exists a constant , such that . It is well known (see [10]) that

Let be a Banach space, and . Then, we denote .

Definition 2.2 (see [4]).

Let be a nondecreasing function with and , for all .

Lemma 2.3 (see [8]).

Let be a real uniformly convex Banach space with the modulus of convexity of power type . Then, for all in and , there exists a constant such that

(2.2)

where .

Remark 2.4.

If in the previous lemma, then we denote .

Lemma 2.5.

Let be a real Banach space and the normalized duality mapping. Then for any in and , such that

(2.3)

Lemma 2.6 (see [7]).

Let and be three nonnegative real sequences, satisfying

(2.4)

with and . Then, exists. In addition, if has a subsequence converging to zero, then .

Proposition 2.7.

If is a strict pseudocontraction, then satisfies the Lipschitz condition

(2.5)

Proof.

By the definition of the strict pseudocontraction, we have

(2.6)

A simple computation shows the conclusion.

Lemma 3.1.

Let be a uniformly convex Banach space with the convex modulus of power type , a nonempty closed convex subset of satisfying , and hemicontractive mappings with . Let , , , and be the sequences in (II) and

(3.1)

where is the constant in Remark 2.4. Then,

(1) exists for all ,

(2) exists,

1. (3)

   if is continuous, then , for all .

    

Proof. (1) Let . By the boundedness assumption on , there exists a constant , for any , such that . From the definition of hemicontractive mappings, we have

(3.2)

Using Lemmas 2.3, 2.5, and (3.2), we obtain

(3.3)

Hence,

(3.4)

It follows from (II) and Lemma 2.5 that

(3.5)

By the condition , we may assume that

(3.6)

Therefore,

(3.7)

Substituting (3.7) into (3.4), we get

(3.8)

Assumptions (i) and (ii) imply that there exists a positive integer such that for every ,

(3.9)

Hence, for all ,

(3.10)

where

(3.11)

From (3.9) and conditions (i) and (ii), it follows that

(3.12)

By Lemma 2.6, we see that exists and the sequence is bounded.

(2) It is easy to verify that exists.

(3) By the boundedness of , there exists a constant such that , for all . From (3.10), we get, for ,

(3.13)

which implies

(3.14)

Thus,

(3.15)

It implies that

(3.16)

Therefore, by (3.7), we have

(3.17)

Using (II), we obtain

(3.18)

By a combination with the continuity of ( , we get

(3.19)

It is clear that for each , there exists such that . Consequently,

(3.20)

This completes the proof.

Theorem 3.2.

Let the assumptions of Lemma 3.1 hold, and let be continuous. Then, converges strongly to a common fixed point of if and only if .

Proof.

The necessity is obvious.

Now, we prove the sufficiency. Since , it follows from Lemma 3.1 that .

For any , we have

(3.21)

Hence, we get

(3.22)

So, is a Cauchy sequence in . By the closedness of , we get that the sequence converges strongly to . Let a sequence , for some , be such that converges strongly to . By the continuity of , we obtain

(3.23)

Therefore, . This implies that is closed. Therefore, is closed. By , we get . This completes the proof.

Theorem 3.3.

Let the assumptions of Lemma 3.1 hold. Let be continuous and satisfy condition . Then, converges strongly to a common fixed point of .

Proof.

Since satisfies condition , and for each , it follows from the existence of that . Applying the similar arguments as in the proof of Theorem 3.2, we conclude that converges strongly to a common fixed point of . This completes the proof.

As a direct consequence of Theorem 3.3, we get the following result.

Corollary 3.4 (see [12, Theorem 3]).

Let H be a real Hilbert space, a nonempty closed convex subset of satisfying , and continuous hemicontractive mapping which satisfies condition (A). Let be a real sequence in with . For any , the sequence is defined by

(3.24)

Then, converges strongly to a fixed point of .

Proof.

Employing the similar proof method of Lemma 3.1, we obtain by (3.10)

(3.25)

This implies

(3.26)

By , we have . Equation (3.7) implies that . Since satisfies condition (A) and the limit exists, we get . The rest of the proof follows now directly from Theorem 3.2. This completes the proof.

Remark 3.5.

Theorems 3.2 and 3.3 extend [12, Theorem 3] essentially since the following hold.

Moreover, if , then is well defined by (II). Hence, Theorems 3.2 and 3.3 are also answers to the question proposed by Qing [13].

Theorem 3.6.

Let and be as the assumptions of Lemma 3.1. Let be strictly pseudocontractive mappings with being nonempty. Let , , , , and be the sequences in (II) and

(3.27)

where is the constant in Remark 2.4. Then,

(1) converges strongly to a common fixed point of if and only if .

1. (2)

   If satisfies condition ( ) , then converges strongly to a common fixed point of .

    

Remark 3.7.

Theorem 3.6 extends the corresponding result [6, Theorem 3.1].