Research Article | Open Access

Shin Min Kang, Arif Rafiq, Sun Young Cho, "On the Convergence of Implicit Picard Iterative Sequences for Strongly Pseudocontractive Mappings in Banach Spaces", *Journal of Applied Mathematics*, vol. 2013, Article ID 284937, 5 pages, 2013. https://doi.org/10.1155/2013/284937

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

**Academic Editor:**Luigi Muglia

#### 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.

#### References

- T. Kato, “Nonlinear semigroups and evolution equations,”
*Journal of the Mathematical Society of Japan*, vol. 19, pp. 508–520, 1967. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - F. E. Browder, “Nonlinear mappings of nonexpansive and accretive type in Banach spaces,”
*Bulletin of the American Mathematical Society*, vol. 73, pp. 875–882, 1967. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - C. E. Chidume, “Iterative approximation of fixed points of Lipschitzian strictly pseudocontractive mappings,”
*Proceedings of the American Mathematical Society*, vol. 99, no. 2, pp. 283–288, 1987. View at: Publisher Site | Google Scholar | MathSciNet - C. E. Chidume, “An iterative process for nonlinear Lipschitzian strongly accretive mappings in ${L}_{p}$ spaces,”
*Journal of Mathematical Analysis and Applications*, vol. 151, no. 2, pp. 453–461, 1990. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - K. Deimling,
*Nonlinear Functional Analysis*, Springer, Berlin, Germany, 1985. View at: MathSciNet - L. Deng, “On Chidume's open questions,”
*Journal of Mathematical Analysis and Applications*, vol. 174, no. 2, pp. 441–449, 1993. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - L. Deng, “An iterative process for nonlinear Lipschitzian and strongly accretive mappings in uniformly convex and uniformly smooth Banach spaces,”
*Acta Applicandae Mathematicae*, vol. 32, no. 2, pp. 183–196, 1993. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - L. Deng, “Iteration processes for nonlinear Lipschitzian strongly accretive mappings in ${L}_{p}$ spaces,”
*Journal of Mathematical Analysis and Applications*, vol. 188, no. 1, pp. 128–140, 1994. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - L. Deng and X. P. Ding, “Iterative approximation of Lipschitz strictly pseudocontractive mappings in uniformly smooth Banach spaces,”
*Nonlinear Analysis*, vol. 24, no. 7, pp. 981–987, 1995. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - N. Hussain, L. B. Ćirić, Y. J. Cho, and A. Rafiq, “On Mann-type iteration method for a family of hemicontractive mappings in Hilbert spaces,”
*Journal of Inequalities and Applications*, vol. 2013, article 41, 11 pages, 2013. View at: Publisher Site | Google Scholar | MathSciNet - N. Hussain, A. Rafiq, and L. B. Ćirić, “Stability of the Ishikawa iteration scheme with errors for two strictly hemicontractive operators in Banach spaces,”
*Fixed Point Theory and Applications*, vol. 2012, article 160, 14 pages, 2012. View at: Publisher Site | Google Scholar | MathSciNet - N. Hussain, A. Rafiq, L. B. Ćirić, and S. Al-Mezel, “Almost stability of the Mann type iteration method with error term involving strictly hemicontractive mappings in smooth Banach spaces,”
*Journal of Inequalities and Applications*, vol. 2012, article 207, 11 pages, 2012. View at: Publisher Site | Google Scholar | MathSciNet - N. Hussain, A. Rafiq, B. Damjanović, and R. Lazović, “On rate of convergence of various iterative schemes,”
*Fixed Point Theory and Applications*, vol. 2011, article 45, 6 pages, 2011. View at: Google Scholar | MathSciNet - L. Liu, “Approximation of fixed points of a strictly pseudocontractive mapping,”
*Proceedings of the American Mathematical Society*, vol. 125, no. 5, pp. 1363–1366, 1997. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Q. H. Liu, “The convergence theorems of the sequence of Ishikawa iterates for hemicontractive mappings,”
*Journal of Mathematical Analysis and Applications*, vol. 148, no. 1, pp. 55–62, 1990. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - A. Rafiq, “On Mann iteration in Hilbert spaces,”
*Nonlinear Analysis*, vol. 66, no. 10, pp. 2230–2236, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - A. Rafiq, “Implicit fixed point iterations for pseudocontractive mappings,”
*Kodai Mathematical Journal*, vol. 32, no. 1, pp. 146–158, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - K. P. R. Sastry and G. V. R. Babu, “Approximation of fixed points of strictly pseudocontractive mappings on arbitrary closed, convex sets in a Banach space,”
*Proceedings of the American Mathematical Society*, vol. 128, no. 10, pp. 2907–2909, 2000. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - X. L. Weng, “Fixed point iteration for local strictly pseudo-contractive mapping,”
*Proceedings of the American Mathematical Society*, vol. 113, no. 3, pp. 727–731, 1991. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - W. R. Mann, “Mean value methods in iteration,”
*Proceedings of the American Mathematical Society*, vol. 4, pp. 506–510, 1953. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - S. Ishikawa, “Fixed points by a new iteration method,”
*Proceedings of the American Mathematical Society*, vol. 44, pp. 147–150, 1974. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - C. E. Chidume, “Picard iteration for strongly accretive and strongly pseudocontractive Lipschitz maps,”
*International Centre for Theoretical Physics*. In press. View at: Google Scholar - C. E. Chidume, “Iterative algorithms for nonexpansive mappings and some of their generalizations,” in
*Nonlinear Analysis and Applications: to V. Lakshmikantham on His 80th Birthday*, vol. 1,2, pp. 383–429, Kluwer Academic, Dordrecht, The Netherlands, 2003. View at: Google Scholar | Zentralblatt MATH | MathSciNet - L. Ćirić, A. Rafiq, and N. Cakić, “On Picard iterations for strongly accretive and strongly pseudo-contractive Lipschitz mappings,”
*Nonlinear Analysis*, vol. 70, no. 12, pp. 4332–4337, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet

#### Copyright

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.