Abstract and Applied Analysis

Volume 2013 (2013), Article ID 629468, 10 pages

http://dx.doi.org/10.1155/2013/629468

Research Article

Convergence Theorems for Fixed Points of Multivalued Strictly Pseudocontractive Mappings in Hilbert Spaces

C. E. Chidume,¹ C. O. Chidume,² N. Djitté,³ and M. S. Minjibir1,4

¹Mathematics Institute, African University of Science and Technology, PMB 681, Garki, Abuja, Nigeria

²Department of Mathematics and Statistics, Auburn University, Auburn, AL 36849, USA

³Université Gaston Berger, 234 Saint Louis, Senegal

⁴Department of Mathematical Sciences, Bayero University, PMB 3011, Kano, Nigeria

Received 10 September 2012; Accepted 15 April 2013

Academic Editor: Josef Diblík

Copyright © 2013 C. E. Chidume et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Let be a nonempty, closed, and convex subset of a real Hilbert space . Suppose that is a multivalued strictly pseudocontractive mapping such that . A Krasnoselskii-type iteration sequence is constructed and shown to be an approximate fixed point sequence of ; that is, holds. Convergence theorems are also proved under appropriate additional conditions.

1. Introduction

For several years, the study of fixed point theory of multivalued nonlinear mappings has attracted, and continues to attract, the interest of several well-known mathematicians (see, e.g., Brouwer [1], Kakutani [2], Nash [3, 4], Geanakoplos [5], Nadler [6], and Downing and Kirk [7]).

Interest in such studies stems, perhaps, mainly from the usefulness of such fixed point theory in real-world applications, such as in Game Theory and Market Economy, and in other areas of mathematics, such as in Nonsmooth Differential Equations. We describe briefly the connection of fixed point theory of multivalued mappings and these applications.

Game Theory and Market Economy. In game theory and market economy, the existence of equilibrium was uniformly obtained by the application of a fixed point theorem. In fact, Nash [3, 4] showed the existence of equilibria for noncooperative static games as a direct consequence of Brouwer [1] or Kakutani [2] fixed point theorem. More precisely, under some regularity conditions, given a game, there always exists a multivalued mapping whose fixed points coincide with the equilibrium points of the game. A model example of such an application is the Nash equilibrium theorem (see, e.g., [3]).

Consider a game with players denoted by , , where is the set of possible strategies of the th player and is assumed to be nonempty, compact, and convex, and is the payoff (or gain function) of the player and is assumed to be continuous. The player can take individual actions, represented by a vector . All players together can take a collective action, which is a combined vector . For each , and , we will use the following standard notations:

A strategy permits the ’th player to maximize his gain under the condition that the remaining players have chosen their strategies if and only if

Now, let be the multivalued mapping defined by

Definition 1. A collective action is called a Nash equilibrium point if, for each , is the best response for the ’th player to the action made by the remaining players. That is, for each ,

or, equivalently,

This is equivalent to that is a fixed point of the multivalued mapping defined by

From the point of view of social recognition, game theory is perhaps the most successful area of application of fixed point theory of multivalued mappings. However, it has been remarked that the applications of this theory to equilibrium are mostly static: they enhance understanding conditions under which equilibrium may be achieved but do not indicate how to construct a process starting from a nonequilibrium point and convergent to equilibrium solution. This is part of the problem that is being addressed by iterative methods for fixed point of multivalued mappings.

Nonsmooth Differential Equations. The mainstream of applications of fixed point theory for multivalued mappings has been initially motivated by the problem of differential equations (DEs) with discontinuous right-hand sides which gave birth to the existence theory of differential inclusion (DIs). Here is a simple model for this type of application.

Consider the initial value problem

If is discontinuous with bounded jumps, measurable in , one looks for solutions in the sense of Filippov [8, 9] which are solutions of the differential inclusion

where

Now set and let be the multivalued NemyTskii operator defined by

Finally, let be the multivalued mapping defined by , where is the inverse of the derivative operator given by

One can see that problem (8) reduces to the fixed point problem: .

Finally, a variety of fixed point theorems for multivalued mappings with nonempty and convex values is available to conclude the existence of solution. We used a first-order differential equation as a model for simplicity of presentation, but this approach is most commonly used with respect to second-order boundary value problems for ordinary differential equations or partial differential equations. For more about these topics, one can consult [10–13] and references therein as examples.

We have seen that a Nash equilibrium point is a fixed point of a multivalued mapping , that is, a solution of the inclusion for some nonlinear mapping . This inclusion can be rewritten as , where and is the identity mapping on .

Many problems in applications can be modeled in the form , where, for example, is a monotone operator, for all , , . Typical examples include the equilibrium state of evolution equations and critical points of some functionals defined on Hilbert spaces . Let be a proper, lower-semicontinuous convex function; then it is known (see, e.g., Rockafellar [14] or Minty [15]) that the multivalued mapping , the subdifferential of , is maximal monotone, where for ,

In this case, the solutions of the inclusion , if any, correspond to the critical points of , which are exactly its minimizer points.

Also, the proximal point algorithm of Martinet [16] and Rockafellar [17] studied also by a host of authors is connected with iterative algorithm for approximating a solution of where is a maximal monotone operator on a Hilbert space.

In studying the equation , Browder introduced an operator defined by where is the identity mapping on . He called such an operator pseudocontractive. It is clear that solutions of now correspond to fixed points of . In general, pseudocontraactive mappings are not continuous. However, in studying fixed point theory for pseudocontractive mappings, some continuity condition (e.g., Lipschitz condition) is imposed on the operator. An important subclass of the class of Lipschitz pseudocontractive mappings is the class of nonexpansive mappings, that is, mappings such that for all . Apart from being an obvious generalization of the contraction mappings, nonexpansive mappings are important, as has been observed by Bruck [18], mainly for the following two reasons.

(i)Nonexpansive mappings are intimately connected with the monotonicity methods developed since the early 1960s and constitute one of the first classes of nonlinear mappings for which fixed point theorems were obtained by using the fine geometric properties of the underlying Banach spaces instead of compactness properties. (ii)Nonexpansive mappings appear in applications as the transition operators for initial value problems of differential inclusions of the form , where the operators are, in general, set-valued and are accretive or dissipative and minimally continuous.

The class of strictly pseudocontractive mappings defined in Hilbert spaces which was introduced in 1967 by Browder and Petryshyn [19] is a superclass of the class of nonexpansive mappings and a subclass of the class of Lipschitz pseudocontractions. While pseudocontractive mappings are generally not continuous, the strictly pseudocontractive mappings inherit Lipschitz property from their definitions. The study of fixed point theory for strictly pseudocontractive mappings may help in the study of fixed point theory for nonexpansive mappings and for Lipschitz pseudocontractive mappings. Consequently, the study by several authors of iterative methods for fixed point of multivalued nonexpansive mappings has motivated the study in this paper of iterative methods for approximating fixed points of the more general strictly pseudocontractive mappings. Part of the novelty of this paper is that, even in the special case of multivalued nonexpansive mappings, convergence theorems are proved here for the Krasnoselskii-type sequence which is known to be superior to the Mann-type and Ishikawa-type sequences so far studied. It is worth mentioning here that iterative methods for approximating fixed points of nonexpansive mappings constitute the central tools used in signal processing and image restoration (see, e.g., Byrne [20]).

Let be a nonempty subset of a normed space . The set is called proximinal (see, e.g., [21–23]) if for each there exists such that

where for all . Every nonempty, closed, and convex subset of a real Hilbert space is proximinal. Let and denote the families of nonempty, closed, and bounded subsets and of nonempty, proximinal, and 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 . The fixed point set of is denoted by .

A multivalued mapping is called -Lipschitzian if there exists such that

When in (15), we say that is a contraction, and is called nonexpansive if .

Several papers deal with the problem of approximating fixed points of multivalued nonexpansive mappings (see, e.g., [21–26] and the references therein) and their generalizations (see, e.g., [27, 28]).

Sastry and Babu [21] introduced the following iterative schemes. Let be a multivalued mapping, and let be a fixed point of . Define iteratively the sequence from by

where is a real sequence in (0,1) satisfying the following conditions:

(i),(ii).

They also introduced the following scheme:

where and are sequences of real numbers satisfying the following conditions:

(i),(ii),(iii) .

Sastry and Babu called a process defined by (16) a Mann iteration process and a process defined by (17) where the iteration parameters and satisfy conditions (i), (ii), and (iii) an Ishikawa iteration process. They proved in [21] that the Mann and Ishikawa iteration schemes for a multivalued mapping with fixed point converge to a fixed point of under certain conditions. More precisely, they proved the following result for a multivalued nonexpansive mapping with compact domain.

Theorem SB (Sastry and Babu [21]). Let be real Hilbert space, let be a nonempty, compact, and convex subset of , and let be a multivalued nonexpansive mapping with a fixed point . Assume that (i) , , (ii) , and (iii) . Then, the sequence defined by (17) converges strongly to a fixed point of .

Panyanak [22] extended the above result of Sastry and Babu [21] to uniformly convex real Banach spaces. He proved the following result.

Theorem P1 (Panyanak [22]). Let be a uniformly convex real Banach space, and let be a nonempty, compact, and convex subset of and a multivalued nonexpansive mapping with a fixed point . Assume that (i) , , (ii) , and (iii) . Then, the sequence defined by (17) converges strongly to a fixed point of .

Panyanak [22] also modified the iteration schemes of Sastry and Babu [21]. Let be a nonempty, closed, and convex subset of a real Banach space, and let be a multivalued mapping such that is 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 . The sequence is defined iteratively by the following way

where is such that and is such that . Before we state his theorem, we need the following definition.

Definition 2. A mapping is said to satisfy condition (I) if there exists a strictly increasing function with , for all such that

Theorem P2 (Panyanak [22]). Let be a uniformly convex real Banach space, let be a nonempty, closed, bounded, and convex subset of , and let be a multivalued nonexpansive mapping that satisfies condition (I). Assume that (i) and (ii) . Suppose that is a nonempty proximinal subset of . Then, the sequence defined by (18) converges strongly to a fixed point of .

Panyanak [22] then asked the following question.

Question (P). Is Theorem P2 true for the Ishikawa iteration defined by (19) and (20)?

For multivalued mappings, the following lemma is a consequence of the definition of Hausdorff metric, as remarked by Nadler [6].

Lemma 3. Let and . For every , there exists such that

Recently, Song and Wang [23] modified the iteration process due to Panyanak [22] and improved the results therein. They gave their iteration scheme as follows.

Let be a nonempty, closed, and convex subset of a real Banach space, and let be a multivalued mapping. Let and be such that . Choose ,

where and are such that

They then proved the following result.

Theorem SW (Song and Wang [23]). Let be a nonempty, compact and 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 (23) converges strongly to a fixed point of .

More recently, Shahzad and Zegeye [29] extended and improved the results of Sastry and Babu [21], Panyanak [22], and Son and Wang [23] to multivalued quasi-nonexpansive mappings. Also, in an attempt to remove the restriction for all in Theorem SW, they introduced a new iteration scheme as follows.

Let be a nonempty, closed, and convex subset of a real Banach space, and let be a multivalued mapping and . Let . Choose , and define as follows:

where and . They then proved the following result.

Theorem SZ (Shahzad and Zegeye [29]). Let be a uniformly convex real Banach space, let be a nonempty, closed, and convex subset of , and let be a multivalued mapping with such that is nonexpansive. Let be the Ishikawa iterates defined by (25). Assume that satisfies condition (I) and . Then, converges strongly to a fixed point of .

Remark 4. In recursion formula (16), 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 (16) is not convenient to use in any possible application. Also, the recursion formula defined in (23) is not convenient to use in any possible application. The sequences and are not known precisely. Only their existence is guaranteed by Lemma 3. Unlike as in the case of formula (16), characterizations of and guaranteed by Lemma 3 are not even known. So, recursion formulas (23) are not really useable.

It is our purpose in this paper to first introduce the important class of multivalued strictly pseudocontractive mappings which is more general than the class of multivalued nonexpansive mappings. Then, we prove strong convergence theorems for this class of mappings. The recursion formula used in our more general setting is of the Krasnoselskii type [30] which is known to be superior (see, e.g., Remark 20) to the recursion formula of Mann [31] or Ishikawa [32]. We achieve these results by means of an incisive result similar to the result of Nadler [6] which we prove in Lemma 7.

2. Preliminaries

In the sequel, we will need the following definitions and results.

Definition 5. Let be a real Hilbert space and let be a multivalued mapping. The multivalued mapping is said to be strongly demiclosed at 0 (see, e.g., [27]) if for any sequence such that converges strongly to and converges strongly to 0, then .

Definition 6. Let be a real Hilbert space. A multivalued mapping is said to be -strictly pseudocontractive if there exist such that for all one has

If in (28), the mapping is said to be pseudocontractive.

We now prove the following lemma which will play a central role in the sequel.

Lemma 7. Let be a reflexive real Banach space and let . Assume that is weakly closed. Then, for every , there exists such that

Proof. Let and let be a sequence of positive real numbers such that as . From Lemma 3, for each , there exists such that

It then follows that the sequence is bounded. Since is reflexive and is weakly closed, there exists a subsequence of that converges weakly to some . Now, using inequality (30), the fact that converges weakly to and , as , it follows that

This proves the lemma.

Proposition 8. Let be a nonempty subset of a real Hilbert space and let be a multivalued -strictly pseudocontractive mapping. Assume that for every , the set is weakly closed. Then, is Lipschitzian.

Proof. Let and . From Lemma 7, there exists such that

Using the fact that is -strictly pseudocontractive, and inequality (32), we obtain the following estimates:

so that

Hence,

Therefore, is -Lipschitzian with .

Remark 9. We note that for a single-valued mapping , for each , the set is always weakly closed.

We now prove the following lemma which will also be crucial in what follows.

Lemma 10. Let be a nonempty and closed subset of a real Hilbert space and let be a -strictly pseudocontractive mapping. Assume that for every , the set is weakly closed. Then, is strongly demiclosed at zero.

Proof. Let be such that and as . Since is closed, we have that . Since, for every , is proximinal, let such that . Using Lemma 7, for each , there exists such that

We then have

Observing that , it then follows that

Taking the limit as , we have that . Therefore, , completing the proof.

3. Main Results

We prove the following theorem.

Theorem 11. Let be a nonempty, closed, and convex subset of a real Hilbert space . Suppose that is a multivalued -strictly pseudocontractive mapping such that . Assume that for all . Let be a sequence defined by ,

where and . Then, .

Proof. Let . We have the following well-known identity:

which holds for all and for all . Using inequality (28) and the assumption that for all , we obtain the following estimates:

It then follows that

which implies that

Hence, . Since , we have that .

A mapping is called hemicompact if, for any sequence in such that as , there exists a subsequence of such that . We note that if is compact, then every multivalued mapping is hemicompact.

We now prove the following corollaries of Theorem 11.

Corollary 12. Let be a nonempty, closed, and convex subset of a real Hilbert space , and let be a multivalued -strictly pseudocontractive mapping with such that for all . Suppose that is hemicompact and continuous. Let be a sequence defined by ,

where and . Then, the sequence converges strongly to a fixed point of .

Proof. From Theorem 11, we have that . Since is hemicompact, there exists a subsequence of such that as for some . Since is continuous, we also have as . Therefore, and so . Setting in the proof of Theorem 11, it follows from inequality (42) that exists. So, converges strongly to . This completes the proof.

Corollary 13. Let be a nonempty, compact, and convex subset of a real Hilbert space , and let be a multivalued -strictly pseudocontractive mapping with such that for all . Suppose that is continuous. Let be a sequence defined by ,

where and . Then, the sequence converges strongly to a fixed point of .

Proof. Observing that if is compact, every mapping is hemicompact, the proof follows from Corollary 12.

Corollary 14. Let be a nonempty, closed, and convex subset of a real Hilbert space , and let be a multivalued nonexpansive mapping such that for all . Suppose that is hemicompact. Let be a sequence defined by ,

where and . Then, the sequence converges strongly to a fixed point of .

Proof. Since is nonexpansive and hemicompact, then it is strictly pseudocontractive, hemicompact, and continuous. So, the proof follows from Corollary 12.

Remark 15. In Corollary 12, the continuity assumption on can be dispensed with if we assume that for every , is proximinal and weakly closed. In fact, we have the following result.

Corollary 16. Let be a nonempty, closed, and convex subset of a real Hilbert space , and let be a multivalued -strictly pseudocontractive mapping with such that for every , is weakly closed and for all . Suppose that is hemicompact. Let be a sequence defined by ,

where and . Then, the sequence converges strongly to a fixed point of .

Proof. Following the same arguments as in the proof of Corollary 12, we have and . Furthermore, from Lemma 10, is strongly demiclosed at zero. It then follows that . Setting and following the same computations as in the proof of Theorem 11, we have from inequality (42) that exists. Since converges strongly to , it follows that converges strongly to , completing the proof.

Corollary 17. Let be a nonempty, closed, and convex subset of a real Hilbert space , and let be a multivalued -strictly pseudocontractive mapping with such that for every , is weakly closed and for all . Suppose that satisfies condition (I). Let be a sequence defined by ,

where and . Then, the sequence converges strongly to a fixed point of .

Proof. From Theorem 11, we have that . Using the fact that satisfies condition (I), it follows that . Thus there exist a subsequence of and a sequence such that

By setting and following the same arguments as in the proof of Theorem 11, we obtain from inequality (42) that

We now show that is a Cauchy sequence in . Notice that

This shows 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 . Setting in the proof of Theorem 11, it follows from inequality (42) that exists. So, converges strongly to . This completes the proof.

Corollary 18. Let be a nonempty compact convex subset of a real Hilbert space , and let be a multivalued -strictly pseudocontractive mapping with such that for every , is weakly closed and for all . Let be a sequence defined by ,

where and . Then, the sequence converges strongly to a fixed point of .

Proof. From Theorem 11, we have that . Since and is compact, has a subsequence that converges strongly to some . Furthermore, from Lemma 10, is strongly demiclosed at zero. It then follows that . Setting and following the same arguments as in the proof of Theorem 11, we have from inequality (42) that exists. Since converges strongly to , it follows that converges strongly to . This completes the proof.

Corollary 19. Let be a nonempty, compact, and convex subset of a real Hilbert space , and let be a multivalued nonexpansive mapping. Assume that for all . Let be a sequence defined by ,

where and . Then, the sequence converges strongly to a fixed point of .

Remark 20. Recursion formula (39) of Theorem 11 is the Krasnoselskii type (see, e.g., [30]) and is known to be superior than the recursion formula of the Mann algorithm (see, e.g., Mann [31]) in the following sense.

(i)Recursion formula (39) requires less computation time than the Mann algorithm because the parameter in formula (39) is fixed in , whereas in the algorithm of Mann, is replaced by a sequence in satisfying the following conditions: and . The must be computed at each step of the iteration process.(ii)The Krasnoselskii-type algorithm usually yields rate of convergence as fast as that of a geometric progression, whereas the Mann algorithm usually has order of convergence of the form .

Remark 21. Any consideration of the Ishikawa iterative algorithm (see, e.g., [32]) involving two parameters (two sequences in ) for the above problem is completely undesirable. Moreover, the rate of convergence of the Ishikawa-type algorithm is generally of the form and the algorithm requires a lot more computation than even the Mann process. Consequently, the question asked in [22], Question (P) above, whether an Ishikawa-type algorithm will converge (when it was already known that a Mann-type process converges) has no merit.

Remark 22. Our theorem and corollaries improve convergence theorems for multivalued nonexpansive mappings in [21–23, 25, 26, 28] in the following sense.

(i)In our algorithm, is arbitrary and does not have to satisfy the very restrictive condition in recursion formula (16), and similar restrictions in recursion formula (17). These restrictions on depend on , a fixed point that is being approximated. (ii)The algorithms used in our theorem and corollaries which are proved for the much larger class of multivalued strict pseudocontractions are of the Krasnoselskii type.

Remark 23. In [29], the authors replace the condition for all with the following two restrictions: (i) on the sequence :, for example, and . We observe that if is a closed convex subset of a real Hilbert space, then is unique and is characterized by

(ii) on : the authors demand that be nonexpansive. So, the first restriction makes the recursion formula difficult to use in any possible application, while the second restriction reduces the class of mappings to which the results are applicable. This is the price to pay for removing the condition for all .

Remark 24. Corollary 12 is an extension of Theorem 12 of Browder and Petryshyn [19] from single-valued to multivalued strictly pseudocontractive mappings.

Remark 25. A careful examination of our proofs in this paper reveals that all our results have carried over to the class of multivalued quasinonexpansive mappings.

Remark 26. The addition of bounded error terms to the recursion formula (39) leads to no generalization.

We conclude this paper with examples where for each , is proximinal and weakly closed.

Example 27. Let be an increasing function. Define by

where and . For every , is either a singleton or a closed and bounded interval. Therefore, is always weakly closed and convex. Hence, for every , the set is proximinal and weakly closed.

Example 28. Let be a real Hilbert space, and let be a convex continuous function. Let be the multivalued mapping defined by

where is the subdifferential of at which is defined by

It is well known that for every , is nonempty, weakly closed, and convex. Therefore, since is a real Hilbert space, it then follows that for every , the set is proximinal and weakly closed.

The condition for all which is imposed in all our theorems of this paper is not crucial. Our emphasis in this paper is to show that a Krasnoselskii-type sequence converges. It is easy to construct trivial examples for which this condition is satisfied. We do not do this. Instead, we show how this condition can be replaced with another condition which does not assume that the multivalued mapping is single-valued on the nonempty fixed point set. This can be found in the paper by Shahzad and Zegeye [29].

Let be a nonempty, closed, and convex subset of a real Hilbert space, let be a multivalued mapping, and let be defined by

We will need the following result.

Lemma 29 (Song and Cho [33]). Let be a nonempty subset of a real Banach space, and let be a multivalued mapping. Then, the following are equivalent:

(i); (ii); (iii).

Moreover, .

Remark 30. We observe from Lemma 29 that if  is any multivalued mapping with , then the corresponding multivalued mapping satisfies for all , condition imposed in all our theorems and corollaries. Consequently, examples of multivalued mappings satisfying the condition for all abound.

Furthermore, we now prove the following theorem where we dispense with the condition for all .

Theorem 31. Let be a nonempty, closed, and convex subset of a real Hilbert space , and let be a multivalued mapping such that . Assume that is -strictly pseudocontractive. Let be a sequence defined iteratively from arbitrary point by

where and . Then, .

Proof. Let . We have the following well-known identity:

which holds for all and for all . Using recursion formula (61), the identity (62), the fact that is -strictly pseudocontractive, and Lemma 29, we obtain the following estimates:

It then follows that

which implies that

Hence, . Since (and hence, ), we have that , completing the proof.

We conclude this paper with examples of multivalued mappings for which is strictly pseudocontractive, a condition assumed in Theorem 31. Trivially, every nonexpansive mapping is strictly pseudocontractive.

Example 32. Let , with the usual metric and be the multivalued mapping defined by

Then is strictly pseudocontractive. In fact, for all .

Example 33. The following example is given in Shahzad and Zegeye [29]. Let be nonempty subset of a normed space . A multivalued mapping is called -nonexpansive (see, e.g., [34]) if for all and with , there exists with such that

It is clear that if is -nonexpansive, then is nonexpansive and hence, strictly pseudocontractive. We also note that -nonexpansiveness is different from nonexpansiveness for multivalued mappings. Let K = , and let be defined by for . Then, for and thus it is nonexpansive and hence strictly pseudocontractive. Note also that is -nonexpansive but is not nonexpansive (see [35]).

Acknowledgment

The authors thank the referees for their comments and remarks that helped to improve the presentation of this paper.

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