#### Abstract

Let be a real Hilbert space and a nonempty closed convex subset of . Suppose is a multivalued Lipschitz pseudocontractive mapping such that . An Ishikawa-type iterative algorithm is constructed and it is shown that, for the corresponding sequence , under appropriate conditions on the iteration parameters, holds. Finally, convergence theorems are proved under approximate additional conditions. Our theorems are significant improvement on important recent results of Panyanak (2007) and Sastry and Babu (2005).

#### 1. Introduction

Let be a nonempty subset of a normed space . The set is called proximinal (see, e.g., ) if for each there exists such that where for all . It is known that every nonempty closed convex subset of a real Hilbert space is proximinal. Let and denote the families of nonempty closed bounded subsets and nonempty proximinal bounded subsets of , respectively. The Hausdorff metric on is defined by for all . Let be a multivalued mapping on . A point is called a fixed point of if and only if . The set is called the fixed point set of .

A multivalued mapping is called Lipschitzian if there exists such that

In (3), if , is said to be a contraction, and is called nonexpansive if .

Existence theorem for fixed point of multivalued contractions and nonexpansive mappings using the Hausdorff metric have been proved by several authors (see, e.g., Nadler Jr. , Markin , and Lim ). Later, an interesting and rich fixed point theory for such maps and more general maps was developed which has applications in control theory, convex optimization, differential inclusion, and economics (see Gorniewicz  and references cited therein).

Several theorems have been proved on the approximation of fixed points of multivalued nonexpansive mappings (see, e.g., [14, 9, 10] and the references therein) and their generalizations (see, e.g., [11, 12]).

Sastry and Babu  introduced the following iterative scheme. Let be a multivalued mapping and let be a fixed point of . The sequence of iterates is given for by where is a real sequence in (0,1) satisfying the following conditions:(i);(ii).

They also introduced the following sequence: where , are real sequences satisfying the following conditions:(i);(ii);(iii).

Sastry and Babu called the process defined by (4) a Mann iteration process and the process defined by (5) where the iteration parameters satisfy conditions (i), (ii), and (iii) an Ishikawa iteration process. They proved in  that the Mann and Ishikawa iteration schemes for a multivalued map with fixed point converge to a fixed point of under certain conditions. More precisely, they proved the following result for a multivalued nonexpansive map with compact domain.

Theorem  SB (Sastry and Babu ). Let be real Hilbert space, a nonempty compact convex subset of , and a multivalued nonexpansive map with a fixed point . Assume that (i)  ; (ii)  ; and (iii)  . Then, the sequence defined by (5) converges strongly to a fixed point of .

Panyanak  extended the above result of Sastry and Babu  to uniformly convex real Banach spaces. He proved the following result.

Theorem  P1 (Panyanak ). Let be a uniformly convex real Banach space, a nonempty compact convex subset of , and a multivalued nonexpansive map with a fixed point . Assume that (i)  ; (ii)  ; and (iii)  . Then, the sequence defined by (5) converges strongly to a fixed point of .

Panyanak  also modified the iteration schemes of Sastry and Babu . Let be a nonempty closed convex subset of a real Banach space and let be a multivalued map with a nonempty proximinal subset of .

The sequence of Mann iterates is defined by , where is such that and is such that .

The sequence of Ishikawa iterates is defined by , where is such that and is such that . Consider where is such that and is such that . Before we state his theorem, we need the following definition.

A mapping is said to satisfy condition (I) if there exists a strictly increasing function with for all such that

Theorem  P2 (Panyanak ). Let be a uniformly convex real Banach space, a nonempty closed bounded convex subset of , and a multivalued nonexpansive map that satisfies condition (I). Assume that (i)   and (ii)  . Suppose that is a nonempty proximinal subset of . Then, the sequence defined by (6) converges strongly to a fixed point of .

Panyanak  then asked the following question.

Question (P). Is Theorem  P2 true for the Ishikawa iterates defined by (7) and (8)?

As remarked by Nadler Jr. , the definition of the Hausdorff metric on gives the following useful result.

Lemma 1. Let and . For every , there exists such that

Song and Wang [3, 4] modified the iteration process by Panyanak  and improved the results therein. They gave their iteration scheme as follows.

Let be a nonempty closed convex subset of a real Banach space and let be a multivalued map. Let , and be such that . Choosing , where , are such that

They then proved the following result.

Theorem  SW (Song and Wang [3, 4]). Let be a nonempty compact convex subset of a uniformly convex real Banach space . Let be a multivalued nonexpansive mapping with satisfying for all . Assume that (i)  ; (ii)  ; and (iii)  . Then, the Ishikawa sequence defined by (11) converges strongly to a fixed point of .

Shahzad and Zegeye  extended and improved the results of Sastry and Babu , Panyanak , and Son and Wang [3, 4] to multivalued quasi-nonexpansive maps. Also, in an attempt to remove the restriction in Theorem  SW, they introduced a new iteration scheme as follows.

Let be a nonempty closed convex subset of a real Banach space, a multivalued map, and . Let . Choose , and define as follows: where , . They then proved the following result.

Theorem  SZ (Shahzad and Zegeye ). Let be a uniformly convex real Banach space, a nonempty convex subset of , and a multivalued map with such that is nonexpansive. Let be the Ishikawa iterates defined by (13). Assume that satisfies condition (I) and . Then, converges strongly to a fixed point of .

Remark 2. In recursion formula (4), the authors take such that . The existence of satisfying this condition is guaranteed by the assumption that is proximinal. In general such a is extremely difficult to pick. If is proximinal, it is not difficult to prove that it is closed. If, in addition, it is a convex subset of a real Hilbert space, then is unique and is characterized by
One can see from this inequality that it is not easy to pick satisfying at every step of the iteration process. So, recursion formula (4) is not convenient to use in any possible application. Also, the recursion formulas defined in (7) and (8) are not convenient to use in any possible application. The sequences and are not known precisely. The restrictions , , , and , , make them difficult to use. These restrictions on and depend on , the fixed points set. So, the recursions formulas (7) and (8) are not easily useable.

Definition 3. Let be a nonempty subset of a real Hilbert space . A map is called -strictly pseudocontractive if there exists such that
If in (16), the map is said to be pseudocontractive.

Browder and Petryshyn  introduced and studied the class of strictly pseudocontractive maps as an important generalization of the class of nonexpansive maps (mappings satisfying ). It is trivial to see that every nonexpansive map is strictly pseudocontractive.

Motivated by this, Chidume et al.  introduced the class of multivalued strictly pseudocontractive maps defined on a real Hilbert space as follows.

Definition 4. A multivalued map is called -strictly pseudocontractive if there exists such that, for all ,
If in (17), the map is said to be pseudocontractive.

We observe from (17) that every nonexpansive mapping is strict pseudocontractive and hence the class of pseudocontractive mappings is a more general class of mappings.

Then, they proved strong convergence theorems for this class of mappings. The recursion formula used is of the Krasnoselskii-type .

Theorem  CA1 (Chidume et al. ). Let be a nonempty closed convex subset of a real Hilbert space . Suppose that is a multivalued -strictly pseudocontractive mapping such that . Assume that . Let be a sequence defined by ,where and . Then, .

Theorem  CA2 (Chidume et al. ). Let be a nonempty compact convex subset of a real Hilbert space and let be a multivalued -strictly pseudocontractive mapping with such that . Suppose that is continuous. Let be a sequence defined by ,where and . Then, the sequence converges strongly to a fixed point of .

Remark 5. We note that, for the more general situation of approximating a fixed point of a multivalued Lipschitz pseudocontractive map in a real Hilbert space, an example of Chidume and Mutangadura  shows that, even in the single-valued case, the Mann iteration method does not always converge in the setting of Theorem  CA2.

We now give an example of multivalued pseudocontractive map (Definition 4) which is not nonexpansive.

Example 6. Let be the multivalued map defined by (i) satisfies, for all , (ii) is not nonexpansive.

Proof of (i). Inequality (21) is obvious for . Now for , we proceed as follows.
Case  1. Assume that . In this case, and . Therefore, we have
Hence, for all , we have
Case  2. Assume that and . In this case, we have and .
Therefore,
On the other hand, let and .(i)If , then (ii)If , we have
Therefore,
This completes the proof of (i).

Proof of (ii). If and , we have that and . So, This proves that is not nonexpansive.

It is our purpose in this paper to prove strong convergence theorems for the class of multivalued Lipschitz pseudocontractive maps in real Hilbert spaces. We use the recursion formula (11), dispensing with the second restriction on the sequences and : , . This class of maps is much more larger than that of multivalued nonexpansive maps used in Theorem  SW. So, in the setting of real Hilbert spaces, our theorem improves and extends the result of Song and Wang [3, 4].

#### 2. Preliminaries

In the sequel, we will need the following results.

Lemma 7 (Daffer and Kaneko ). Let and be sequences of nonnegative real numbers satisfying the following relation: where is a nonnegative integer. If , then exists.

Lemma 8. Let be a real Hilbert space. Then for all , and .

#### 3. Main Results

We use the following iteration scheme.

Let be a nonempty closed convex subset of a real Hilbert space and , , and real sequences in . Let be the sequence defined from arbitrary by where , are such that We first prove the following theorem.

Theorem 9. Let be a nonempty closed convex subset of a real Hilbert space and a multivalued -Lipschitz pseudocontractive mapping with and . Let be the sequence defined by (31) and (32). Assume that (i)  ; (ii)   ; (iii)   and . Then, .

Proof. Let . Using Lemma 8, the fact that is pseudocontractive, and the assumption , we have
Observing that then, from inequality (33) and identity (34), we have that
Using again Lemma 8, the fact that is pseudocontractive, and the assumption , we obtain the following estimates:
Therefore, inequalities (35) and (36) and condition (i) imply that
Using inequality (32), the fact that is -Lipschitzian, and the recursion formula (31), we have
Therefore, from inequalities (37) and (38), we obtain
Observing that condition (ii) yields that , for all for some , it then follows that which implies, by condition (iii), that . Since , it follows that . Therefore, .

We now prove the following corollaries of Theorem 9.

Corollary 10. Let be a nonempty closed convex subset of a real Hilbert space and a multivalued Lipschitz pseudocontractive mapping with and . Let be the sequence defined by (31) and (32). Assume that is hemicompact, and (i)  ; (ii)  ; (iii)  ; and (iv)  . Then, converges strongly to a fixed point of .

Proof. From Theorem 9, we have that . So there exists a subsequence of such that . Using the fact that is hemicompact, the sequence has a subsequence denoted again by that converges strongly to some . Since is continuous, we have . Therefore, and so . Now setting in inequality (39) and using condition (ii) we have that for all for some . Therefore, Lemma 7 implies that exists. Since , it then follows that converges strongly to , completing the proof.

We can easily observe that if is nonexpansive, then it is Lipschitzian and pseudocontractive. Therefore, the following corollary generalizes Theorem  SW of Song and Wang [3, 4] in the setting of Hilbert spaces.

Corollary 11. Let be a nonempty compact convex subset of a real Hilbert space and a multivalued Lipschitz pseudocontractive mapping with and . Let be the sequence defined by (31) and (32). Assume that (i)  ; (ii)  ; (iii)   and . Then, converges strongly to a fixed point of .

Proof. Since is compact, it follows that is hemicompact. So, the proof follows from Corollary 10.

Corollary 12. Let be a nonempty closed convex subset of a real Hilbert space , and let be a multivalued Lipschitz pseudocontractive mapping with and . Let be the sequence defined by (31) and (32). Assume that satisfies condition (I) and (i)  ; (ii)  ; (iii)  ; and (iv)  . Then, converges strongly to a fixed point of .

Proof. From Theorem 9, we have that . So there exists a subsequence of such that . Since satisfies condition (I), we have . Thus there exist a subsequence of denoted again by and a sequence such that
By setting in inequality (39) and using the condition (ii), we have for all , for some .
We now show that is a Cauchy sequence in . Notice that, for ,
From condition (iv), it follows that is a Cauchy sequence in and thus converges strongly to some . Using the fact that is -Lipschitzian and , we have so that and thus . Therefore, and converges strongly to . Now setting in inequality (39) and using condition (ii) we have that for all for some . Therefore, Lemma 7 implies that exists. Since , it then follows that converges strongly to , completing the proof.

Remark 13. Our theorem and corollaries improve convergence theorems for multivalued nonexpansive mappings in [14, 12, 13] in the following sense.
(i) In our algorithm, do not have to satisfy the restrictive conditions and in the recursion formula (5) and similar restrictions in the recursion formulas (7) and (8). These restrictions on and depend on , a fixed point that is being approximated or the fixed points set . Also, in our algorithm, the second restriction on the sequences and , , in the recursion formula (11), is removed.
(ii) Our theorem and corollaries are proved for the class of multivalued Lipschitz pseudocontractive mappings which is much more general than that of multivalued nonexpansive mappings

Remark 14. Corollary 11 is an extension of the Theorem of Ishikawa  from single-valued to multivalued Lipschitz pseudocontractive mappings.

Remark 15. Real sequences that satisfy the hypotheses of Theorems 9 are , , and .

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgment

This paper is dedicated to the memory of Professor Galaye DIA (1941-2013). The authors thank the referee for his work and his valuable suggestion that helped to improve the presentation of this paper.