Journal of Applied Mathematics

Volume 2013 (2013), Article ID 284937, 5 pages

http://dx.doi.org/10.1155/2013/284937

Research Article

On the Convergence of Implicit Picard Iterative Sequences for Strongly Pseudocontractive Mappings in Banach Spaces

Shin Min Kang,¹ Arif Rafiq,² and Sun Young Cho³

¹Department of Mathematics and RINS, Gyeongsang National University, Jinju 660-701, Republic of Korea

²School of CS and Mathematics, Hajvery University, 43-52 Industrial Area, Gulberg-III, Lahore 54660, Pakistan

³Department of Mathematics, Gyeongsang National University, Jinju 660-701, Republic of Korea

Received 14 December 2012; Accepted 26 March 2013

Academic Editor: Luigi Muglia

Copyright © 2013 Shin Min Kang 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.

We study the convergence of implicit Picard iterative sequences for strongly accretive and strongly pseudocontractive mappings. We have also improved the results of Ćirić et al. (2009).

1. Introduction and Preliminaries

Let be a real Banach space with dual . The symbol stands for the domain of .

Let be a mapping.

Definition 1. The mapping is said to be Lipschitzian if there exists a constant such that

for all .

Definition 2. The mapping is called strongly pseudocontractive if there exists such that

for all and . If in inequality (2), then is called pseudocontractive.

We will denote by the normalized duality mapping from to defined by

where denotes the generalized duality pairing. It follows from inequality (2) that is strongly pseudocontractive if and only if there exists such that

for all , where . Consequently, from inequality (4) it follows easily that is strongly pseudocontractive if and only if

for all and .

Closely related to the class of pseudocontractive maps is the class of accretive operators.

Let be an operator.

Definition 3. The operator is called accretive if

for all and .

Also, as a consequence of Kato [1], this accretive condition can be expressed in terms of the duality mapping as follows.

For each , there exists such that

Consequently, inequality (2) with yields that is accretive if and only if is pseudocontractive. Furthermore, from setting , it follows from inequality (5) that is strongly pseudocontractive if and only if is accretive, and, using (7), this implies that is strongly pseudocontractive if and only if there exists such that

for all . The operator satisfying inequality (8) is called strongly accretive. It is then clear that is strongly accretive if and only if is strongly pseudocontractive. Thus, the mapping theory for strongly accretive operators is closely related to the fixed point theory of strongly pseudocontractive mappings. We will exploit this connection in the sequel.

The notion of accretive operators was introduced independently in 1967 by Kato [1] and Browder [2]. An early fundamental result in the theory of accretive operators, due to Browder, states that the initial value problem

is solvable if is locally Lipschitzian and accretive on . If is independent of , then and the solution of this equation corresponds to the equilibrium points of the system (9). Consequently, considerable research efforts have been devoted, especially within the past 15 years or so, to developing constructive techniques for the determination of the kernels of accretive operators in Banach spaces (see, e.g., [3–19]. Two well-known iterative schemes, the Mann iterative method (see, e.g., [20]) and the Ishikawa iterative scheme (see, e.g., [21]), have successfully been employed.

The Mann and Ishikawa iterative schemes are global and their rate of convergence is generally of the order . It is clear that if, for an operator , the classical iterative sequence of the form, , (the so-called Picard iterative sequence) converges, then it is certainly superior and preferred to either the Mann or the Ishikawa sequence since it requires less computations and, moreover, its rate of convergence is always at least as fast as that of a geometric progression.

In [22, 23], Chidume proved the following results.

Theorem 4. Let be an arbitrary real Banach space and Lipschitz with constant and strongly accretive with a strong accretive constant . Let denote a solution of the equation . Set and define by for each .

For arbitrary , define the sequence in by

Then converges strongly to with

where . Moreover, is unique.

Corollary 5. Let be an arbitrary real Banach space and a nonempty convex subset of . Let be Lipschitz (with constant ) and strongly pseudocontractive (i.e., satisfies inequality (5) for all ). Assume that has a fixed point . Set and define by for each . For arbitrary , define the sequence in by

Then converges strongly to with

where . Moreover, is unique.

Recently, Ćirić et al. [24] improved the results of Chidume [22, 23], Liu [14], and Sastry and Babu [18] as in the following results.

Theorem 6. Let be an arbitrary real Banach space and a Lipschitz (with constant ) and strongly accretive with a strong accretive constant . Let denote a solution of the equation . Set , and define by for each . For arbitrary , define the sequence in by

Then converges strongly to with

where . Thus the choice yields . Moreover, is unique.

Corollary 7. Let be an arbitrary real Banach space and a nonempty convex subset of . Let be Lipschitz (with constant ) and strongly pseudocontractive (i.e., satisfies inequality (5) for all ). Assume that has a fixed point . Set , and define by for each . For arbitrary , define the sequence in by

Then converges strongly to with

where . Moreover, is unique.

In this paper, we study the convergence of implicit Picard iterative sequences for strongly accretive and strongly pseudocontractive mappings. We have also improved the results of Ćirić et al. [24].

2. Main Results

In the following theorems, will denote the Lipschitz constant of the operator and will denote the strong accretive constant of the operator as in inequality (8). Furthermore, is defined by

With these notations, we prove the following theorem.

Theorem 8. Let be an arbitrary real Banach space and Lipschitz and strongly accretive with a strong accretive constant . Let denote a solution of the equation . Define by for each . For arbitrary , define the sequence in by

Then converges strongly to with

where . Thus the choice yields . Moreover, is unique.

Proof. Existence of follows from [5, Theorem 13.1]. Define where denotes the identity mapping on . Observe that if and only if is a fixed point of . Moreover, is strongly pseudocontractive since is strongly accretive, and so also satisfies inequality (5) for all and . Furthermore, the recursion formula becomes

Observe that

and from the recursion formula (21)

which implies that

This implies using inequality (5) with and that

Observe that

and so

so that from (25) we obtain

Therefore

where

From (29) and (30), we get

as . Hence as . Uniqueness follows from the strong accretivity property of .

The following is an immediate corollary of Theorem 8.

Corollary 9. Let be an arbitrary real Banach space and a nonempty convex subset of . Let be Lipschitz (with constant ) and strongly pseudocontractive (i.e., satisfies inequality (5) for all ). Assume that has a fixed point . Set , and define by for each . For arbitrary , define the sequence in by

Then converges strongly to with

where . Thus the choice yields . Moreover, is unique.

Proof. Observe that is a fixed point of if and only if it is a fixed point of . Furthermore, the recursion formula (32) is simplified to the formula

which is similar to (21). Following the method of computations as in the proof of the Theorem 8, we obtain

Set . Then from (35) we obtain

as . This completes the proof.

Remark 10. Since and , we have

So we can easily obtain

Now

Thus the relation between Ćirić et al. [24] and our parameter of convergence, that is, between and , respectively, is the following:

Our convergence parameter shows the overall improvement for , and consequently the results of Ćirić et al. [24] are improved.

Acknowledgment

The authors would like to thank the referees for useful comments and suggestions.

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