Journal of Applied Mathematics

Volume 2013 (2013), Article ID 541079, 8 pages

http://dx.doi.org/10.1155/2013/541079

Research Article

Strong Convergence Theorems for Solutions of Equations of Hammerstein Type

Chih-Sheng Chuang

Department of Mathematics, National Cheng Kung University, Tainan 701, Taiwan

Received 11 March 2013; Accepted 18 April 2013

Academic Editor: Erdal Karapınar

Copyright © 2013 Chih-Sheng Chuang. 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.

We consider an auxiliary operator, defined in a real Hilbert space in terms of and , that is, monotone and Lipschitz mappings (resp., monotone and bounded mappings). We use an explicit iterative process that converges strongly to a solution of equation of Hammerstein type. Furthermore, our results improve related results in the literature.

1. Introduction

Let be a real Hilbert space. A mapping is said to be monotone if for every . is called maximal monotone if it is monotone and the , the range of , for each , where is the identity mapping on . is said to satisfy the range condition if for each . For monotone mappings, there are many related equations of evolution. Several problems that arise in differential equations, for instance, elliptic boundary value problems whose linear parts possess Green's function, can be put in operator form as

where and are monotone mappings. In fact, (1) comes from the following integral equation of Hammerstein type [1]:

where is a -finite measure on the measure space ; the real kernel is defined by , is a real-valued function defined on and is, in general, nonlinear, and is a given function on . If we now define an operator by

and the so-called superposition or Nemytskii operator by , then (2) can be put in (1) (without loss of generality, we may assume that ).

Note that equations of Hammerstein type play a crucial role in the theory of optimal control systems and in automation and network theory, and several existence and uniqueness theorems have been proved for equations of the Hammerstein type. For details, one can refer to [2–7].

In 2005, Chidume and Zegeye [8] constructed an iterative process as follows:

where is a real Hilbert space, and : are bounded monotone mappings satisfying the range condition, , and and are sequences in . Chidume and Zegeye [8] show that this sequence converges strongly to the solution of (1) under suitable conditions.

In 2011, Chidume and Ofoedu [9] introduced a coupled explicit iterative process as follows:

where is a uniformly smooth real Banach space, and : are bounded and monotone mappings, and , , and are sequences in . Chidume and Ofoedu [9] gave a strong convergence theorem for approximation of the solution of (1) under suitable conditions.

In 2012, Chidume and Djitté [10] consider the following iterative process:

where is a real Hilbert space, and are bounded and maximal monotone mappings, and are sequences in and Chidume and Djitté [10] show that this iterative process converges to an approximate solution of nonlinear equations of Hammerstein type under suitable conditions.

Motivated by the previous works, in this paper, we consider an auxiliary operator, defined in a real Hilbert space in terms of and , that is monotone and Lipschitz mapping, or monotone and bounded mappings. We use an explicit iterative process that converges strongly to a solution of equation of Hammerstein type. Furthermore, our results improve related results in the literature.

2. Preliminaries

Throughout this paper, let be the set of positive integers and let be the set of real numbers. Let be a (real) Hilbert space with inner product and norm , respectively.

Lemma 1. Let be a real Hilbert space. One has for all .

Lemma 2 (see [11]). Let be a sequence of nonnegative real numbers, a sequence of real numbers in with , and a sequence of nonnegative real numbers with , a sequence of real numbers with . Suppose that for each . Then, .

Let be the Banach space of bounded sequences with the supremum norm. A linear functional on is called a mean if , where . For , the value is also denoted by . A mean on is called a Banach limit if it satisfies . If is a Banach limit on , then for ,

In particular, if and , then we have . For details, we can refer to [12].

Lemma 3 (see [13]). Let be a real number and such that for all Banach limit on . If , then, .

Lemma 4 (see [14]). Let be a nonempty closed convex subset of a Hilbert space , let be a bounded sequence in , and let be a Banach limit on . Let be defined by for each . Then there exists a unique such that .

Lemma 5 (see [15]). Let be a Hilbert space, let be a bounded sequence in , and let be a mean on . Then, there exists a unique point such that for each . Indeed, .

Let be a real Hilbert space. Let with norm

Hence, is a real Hilbert space with inner product for all , [8].

Lemma 6. Let be a real Hilbert space, and let . Let , be two mappings, and let be defined by

(i)If and are monotone mappings, then is a monotone mapping [8, Lemma 3.1].(ii)If and are bounded mappings, then is a bounded mapping [8, Lemma 3.1].(iii)If and are Lipschitz mappings with Lipschitz constants and , respectively, then is a Lipschitz mapping. Indeed, the Lipschitz constant of is , where [16, Remark 13.6].

3. Main Results (I)

Let be a real Hilbert space. Let , be two mappings, and let be defined by for each . Then, we observe that is a solution of if and only if   is a solution of in for .

Theorem 7. Let be a real Hilbert space. Let , be Lipschitz and monotone mappings. Suppose that has a solution in . Let be a sequence in . Let and be sequences in . Let and be sequences in defined iteratively from arbitrary , by

Suppose that one of the following conditions holds:

Then, the sequences , , , and are bounded.

Proof. Since and are Lipschitz mappings, we may assume that the Lipschitz constants of and are and , respectively. Let

Let with the norm for each . Take any such that is solution of , and let be fixed. Let and . We observe that . For each , let .

For each , it follows from Lemma 1 that

Similarly, we have

For each , by (13) and (14), we know that

For each , since is monotone and , we know that

Hence, for each , it follows from (15) and (16) that

For conditions (i)–(iii), we only need to consider one case since the proof is similar. Now, we assume that . Then, there exists such that for each . Choose such that and . Let .

Now, we want to show that for each . Clearly, . Suppose that for some . Then, . Indeed, if not, then we have

Hence, by (17) and (18), we get

By (19), we have

This implies that

This leads to a contradiction. So, . Hence, by mathematical induction, we know that . Therefore, and are bounded sequences. Furthermore, and are bounded sequences since and are Lipschitz mappings. For conditions (ii) and (iii), the proof is similar. Therefore, the proof is completed.

Remark 8. (i) Theorem 7 improves the conditions of [17, Theorem 3.1] if the space in [17] is reduced to a real Hilbert space. Indeed, [17, Theorem 3.1] assumes that .

(ii) Furthermore, we know that it is impossible to assume that in [17, Theorem 3.1]. However, we can choose in our result. Indeed, if and , then we have the following result as a special case of Theorem 7.

Corollary 9. Let be a real Hilbert space. Let , be Lipschitz and monotone mappings. Suppose that has a solution in . Let be a sequence in . Let and be sequences in defined iteratively from arbitrary , by

If , then the sequences , , , and are bounded.

In fact, following the same argument as the proof of Theorem 7, we can get the following result.

Theorem 10. Let be a real Hilbert space. Let , be Lipschitz and monotone mappings. Suppose that has a solution in . Let and be sequences in . Let and be sequences in defined iteratively from arbitrary , by

Suppose that one of the following conditions holds:

Then, the sequences , , , and are bounded.

Remark 11. Corollary 9 is also a special case of Theorem 10.

Theorem 12. Let be a real Hilbert space. Let , be Lipschitz and monotone mappings. Suppose that has a solution in . Let be a sequence in . Let and be sequences in . Let and be sequences in defined iteratively from arbitrary , by

Assume that

(i); ;(ii) one of the following conditions holds:

(iii) one of the following conditions holds:

Then, there exists a subset of such that if with , then the sequence converges strongly to .

Proof. Let and be the same as the proof of Theorem 7. Let be a Banach limit on . Let be a sequence with and for each . Clearly, . By Lemma 4, there is a unique such that

Let , and we assume that . Let . Then, . Hence, for each , it follows from Lemma 1 that

By Lemma 5, there exists such that for each . By (29), for each ,

In (30), letting , we get

Clearly,

By (31), (32), and Lemma 3, we know that .

Hence,

By (17), (33), and Lemma 2, we know that . Therefore, converges strongly to .

Remark 13. (i) The conclusion of Theorem 12 is still true if , , and .

(ii) The conclusion of Theorem 12 is still true if , , and .

Remark 14. Following the same argument as in Remark 8, we know that Theorem 12 improves the conditions of [17, Theorem 3.2] if the space is reduced to a real Hilbert space.

In Theorem 12, if and for each , then we have the following result.

Corollary 15. Let be a real Hilbert space. Let , be Lipschitz and monotone mappings. Suppose that has a solution in . Let be a sequence in . Let and be sequences in defined iteratively from arbitrary , by

Assume that and . Then there exists a subset of such that if with , then the sequence converges strongly to .

In Theorem 12, if for each , then we have the following result.

Corollary 16. Let be a real Hilbert space. Let , be Lipschitz and monotone mappings. Suppose that has a solution in . Let be a sequence in . Let be a sequence in . Let and be sequences in defined iteratively from arbitrary , by

Assume that , , and . Then, there exists a subset of such that if with , then the sequence converges strongly to .

In fact, following the same argument as the proof of Theorem 12, we get the following result.

Theorem 17. Let be a real Hilbert space. Let , be Lipschitz and monotone mappings. Suppose that has a solution in . Let and be sequences in . Let and be sequences in defined iteratively from arbitrary , by

Assume that , , and . Then, there exists a subset of such that if with , then the sequence converges strongly to .

Remark 18. Corollary 15 is also a special case of Theorem 17.

4. Main Results (II)

In this section, we consider that and are bounded mappings.

Theorem 19. Let be a real Hilbert space. Let , be bounded and monotone mappings. Suppose that has a solution in . Let be a sequence in . Let and be sequences in . Let and be sequences in defined iteratively from arbitrary , by

Suppose that one of the following conditions holds:

Then, the sequences , , , and are bounded.

Proof. Since and are bounded, we know that is bounded. Hence, . From the proof of Theorem 7, we know that

for each .

Choose such that . For conditions (i)–(iv), we only need to consider one case since the proof is similar. Now, we assume that . Then, there exists such that

for each . Choose such that and . Let . Clearly, .

Now, we want to show that for each . Clearly, . Suppose that for some . Then, . Indeed, if not, then we have

Clearly, . Hence, by (39) and (41), we get

This implies that

Furthermore, we get

This leads to a contradiction. So, . Hence, by mathematical induction, we know that . Therefore, and are bounded sequences. Furthermore, and are bounded sequences since and are bounded mappings.

Remark 20. (i) Theorem 19 improves the conditions of [9, Theorem 3.1] if the space is reduced to a real Hilbert space. Indeed, [9, Theorem 3.1] assumes that .

(ii) Furthermore, we know that it is impossible to assume that in [9, Theorem 3.1]. However, we can choose in our result. Indeed, if and , then we have the following result as special case of Theorem 19.

Corollary 21. Let be a real Hilbert space. Let , be bounded and monotone mappings. Suppose that has a solution in . Let be a sequence in . Let and be sequences in defined iteratively from arbitrary , by

If , then the sequences , , , and are bounded.

Following the same argument as the proof of Theorem 19, we get the following result. Note that Corollary 21 is also a special case of the following result.

Theorem 22. Let be a real Hilbert space. Let , be bounded and monotone mappings. Suppose that has a solution in . Let and be sequences in . Let and be sequences in defined iteratively from arbitrary , by

Suppose that one of the following conditions holds:

Then, the sequences , , , and are bounded.

Following the similar argument as the proof of Theorem 12, we get the following result.

Theorem 23. Let be a real Hilbert space. Let , be bounded and monotone mappings. Suppose that has a solution in . Let be a sequence in . Let and be sequences in . Let and be sequences in defined iteratively from arbitrary , by

Assume that

(i); ;(ii)one of the following conditions holds:

(iii)one of the following conditions holds:

Then, there exists a subset of such that if with , then the sequence converges strongly to .

Remark 24. (i) The conclusion of Theorem 23 is still true if , , and . Furthermore, the conclusion of Theorem 23 is still true if , , and .

(ii) Theorem 23 improves the conditions of [9, Theorem 3.2] if the space is reduced to a real Hilbert space. Indeed, [9, Theorem 3.2] assumes that .

The following is a special case of Theorem 23.

Corollary 25. Let be a real Hilbert space. Let , be bounded and monotone mappings. Suppose that has a solution in . Let be a sequence in . Let and be sequences in defined iteratively from arbitrary , by

Assume that and . Then, there exists a subset of such that if with , then the sequence converges strongly to .

The following is also a special case of Theorem 23.

Corollary 26. Let be a real Hilbert space. Let , be bounded and monotone mappings. Suppose that has a solution in . Let be a sequence in . Let be a sequence in . Let and be sequences in defined iteratively from arbitrary , by

Assume that , , and . Then, there exists a subset of such that if with , then the sequence converges strongly to .

Furthermore, we get the following result. Note that Corollary 25 is also a special case of the following result.

Theorem 27. Let be a real Hilbert space. Let , be bounded and monotone mappings. Suppose that has a solution in . Let and be sequences in . Let and be sequences in defined iteratively from arbitrary , by

Assume that , , and . Then, there exists a subset of such that if with , then the sequence converges strongly to .

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