Journal of Applied MathematicsVolume 2012 (2012), Article ID 341953, 11 pagesdoi:10.1155/2012/341953

Research Article

An Iterative Algorithm on Approximating Fixed Points of Pseudocontractive Mappings

Youli Yu

School of Mathematics and Information Engineering, Taizhou University, Linhai 317000, China

Received 4 September 2011; Accepted 16 September 2011

Academic Editor: Yonghong Yao

Copyright © 2012 Youli Yu. 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 E be a real reflexive Banach space with a uniformly Gâteaux differentiable norm. Let K be a nonempty bounded closed convex subset of E, and every nonempty closed convex bounded subset of K has the fixed point property for non-expansive self-mappings. Let a contractive mapping and be a uniformly continuous pseudocontractive mapping with . Let be a sequence satisfying the following conditions: (i) ; (ii) . Define the sequence in K by , , for all . Under some appropriate assumptions, we prove that the sequence converges strongly to a fixed point which is the unique solution of the following variational inequality: , for all .

1. Introduction

Let be a real Banach space with dual . We denote by the normalized duality mapping from to defined by where denotes the generalized duality pairing.

It is well known that, if is smooth, then is single-valued. In the sequel, we will denote the single-valued normalized duality mapping by . We use to denote the domain and range of , respectively.

An operator is called pseudocontractive if there exists such that

A point is a fixed point of provided . Denote by the set of fixed points of , that is, .

Within the past 40 years or so, many authors have been devoted to the iterative construction of fixed points of pseudocontractive mappings (see [1–10]).

In 1974, Ishikawa [11] introduced an iterative scheme to approximate the fixed points of Lipschitzian pseudocontractive mappings and proved the following result.

Theorem 1.1 (see [11]). If is a compact convex subset of a Hilbert space is a Lipschitzian pseudocontractive mapping. Define the sequence in by where are sequences of positive numbers satisfying the conditions  , , . Then, the sequence converges strongly to a fixed point of .

In connection with the iterative approximation of fixed points of pseudo-contractions, in 2001, Chidume and Mutangadura [12] provided an example of a Lipschitz pseudocontractive mapping with a unique fixed point for which the Mann iterative algorithm failed to converge. Chidume and Zegeye [13] introduced a new iterative scheme for approximating the fixed points of pseudocontractive mappings.

Theorem 1.2 (see [13]). Let be a real reflexive Banach space with a uniformly Gâteaux differentiable norm. Let be a nonempty closed convex subset of . Let be a -Lipschitzian pseudocontractive mapping such that . Suppose that every nonempty closed convex bounded subset of has the fixed point property for nonexpansive self-mappings. Let and be two sequences in satisfying the following conditions:   ,, . For given arbitrarily, let the sequence be defined iteratively by Then, the sequence defined by (1.4) converges strongly to a fixed point of .

Prototypes for the iteration parameters are, for example, and for and . But we observe that the canonical choices of and are impossible. This bring us a question.

Question 1. Under what conditions, and are sufficient to guarantee the strong convergence of the iterative scheme (1.4) to a fixed point of ?

In this paper, we explore an iterative scheme to approximate the fixed points of pseudocontractive mappings and prove that, under some appropriate assumptions, the proposed iterative scheme converges strongly to a fixed point of , which solves some variational inequality. Our results improve and extend many results given in the literature.

2. Preliminaries

Let be a nonempty closed convex subset of a real Banach space . Recall that a mapping is called contractive if there exists a constant such that

Let be a continuous linear functional on and . We write instead of . We call a Banach limit if satisfies and for all .

If is a Banach limit, then we have the following.  For all , implies . for any fixed positive integer . for all . If with , then for any Banach limit .

For more details on Banach limits, we refer readers to [14]. We need the following lemmas for proving our main results.

Lemma 2.1 (see [15]). Let be a Banach space. Suppose that is a nonempty closed convex subset of and is a continuous pseudocontractive mapping satisfying the weakly inward condition: is the closure of for each , where . Then, for each , there exists a unique continuous path for all , satisfying the following equation Furthermore, if is a reflexive Banach space with a uniformly Gâteaux differentiable norm and every nonempty closed convex bounded subset of has the fixed point property for nonexpansive self-mappings, then, as , converges strongly to a fixed point of .

Lemma 2.2 (see [16]). (1) If is smooth Banach space, then the duality mapping is single valued and strong-weak* continuous. (2) If is a Banach space with a uniformly Gâteaux differentiable norm, then the duality mapping is single valued and norm to weak star uniformly continuous on bounded sets of .

Lemma 2.3 (see [17]). Let be a sequence of nonnegative real numbers satisfying for all , where , and two sequences of real numbers such that and . Then converges to zero as .

Lemma 2.4 (see [18]). Let be a real Banach space, and let be the normalized duality mapping. Then, for any given ,

Lemma 2.5 (see [14]). Let be a real number, and let such that for all Banach limits. If   , then .

3. Main Results

Now, we are ready to give our main results in this paper.

Theorem 3.1. Let be a real reflexive Banach space with a uniformly Gâteaux differentiable norm. Let be a nonempty bounded closed convex subset of , and every nonempty closed convex bounded subset of has the fixed point property for nonexpansive self-mappings. Let a contractive mapping and be a uniformly continuous pseudocontractive mapping with . Let be a sequence satisfying the conditions:   , . Define the sequence in by If , then the sequence converges strongly to a fixed point , which is the unique solution of the following variational inequality:

Proof. Take , and let . Then, we have From (3.1), we obtain By (3.4), we have Combining (3.3) and (3.5), we have Next, we prove that . Indeed, taking in Lemma 2.1, we have and, hence, Therefore, we have where is some constant such that for all and . Letting , we have From Lemma 2.1, we know as . Since the duality mapping is norm to weak star uniformly continuous from Lemma 2.2, we have From (3.6), we have where is a constant such that It follows that where Note that By the uniformly continuity of , we have Hence, it is clear that and .Finally, applying Lemma 2.3 to (3.14), we can conclude that . This completes the proof.

From Theorem 3.1, we can prove the following corollary.

Corollary 3.2. Let be a real reflexive Banach space with a uniformly Gâteaux differentiable norm. Let be a nonempty bounded closed convex subset of , and every nonempty closed convex bounded subset of has the fixed point property for nonexpansive self-mappings. Let be a uniformly continuous pseudocontractive mapping with . Let be a sequence satisfying the conditions:   , . Define the sequence in by Then, the sequence converges strongly to a fixed point of if and only if .

Theorem 3.3. Let be a uniformly smooth Banach space and a nonempty bounded closed convex subset of . Let be a contractive mapping and a uniformly continuous pseudocontractive mapping with . Let be a sequence satisfying the conditions:  , . If , then the sequence defined by (3.1) converges strongly to a fixed point , which is the unique solution of the following variational inequality:

Proof. Since every uniformly smooth Banach space is reflexive and whose norm is uniformly Gâteaux differentiable, at the same time, every closed convex and bounded subset of has the fixed point property for nonexpansive mappings. Hence, from Theorem 3.1, we can obtain the result. This completes the proof.

From Theorem 3.3, we can prove the following corollary.

Corollary 3.4. Let be a uniformly smooth Banach space and a nonempty bounded closed convex subset of . Let be a uniformly continuous pseudocontractive mapping with . Let be a sequence satisfying the conditions:  , . Define the sequence in by Then, the sequence converges strongly to a fixed point of if and only if .

Theorem 3.5. Let be a nonempty bounded closed convex subset of a real reflexive Banach space with a uniformly Gâteaux differentiable norm. Let a contractive mapping and be a uniformly continuous pseudocontractive mapping. Let be a sequence satisfying the conditions:  , . If , where is defined as (3.22) below, then the sequence defined by (3.1) converges strongly to a fixed point , which is the unique solution of the following variational inequality:

Proof. First, we note that the sequence is bounded. Now, if we define , then is convex and continuous. Also, we can easily prove that as . Since is reflexive, there exists such that . So the set Clearly, is closed convex subset of .Now, we can take and . By the convexity of , we have that . It follows that By Lemma 2.4, we have Taking the Banach limit in (3.24), we have This implies Therefore, it follows from (3.23) and (3.26) that Since the normalized duality mapping is single valued and norm-weak* uniformly continuous on bounded subset of , we have This implies that, for any , there exists such that, for all and , Taking the Banach limit and noting that (3.27), we have By the arbitrariness of , we obtain At the same time, we note that Since are bounded and the duality mapping is single valued and norm topology to weak star topology uniformly continuous on bounded sets in Banach space with a uniformly Gâteaux differentiable norm, it follows that From (3.31), (3.33), and Lemma 2.5, we conclude that Finally, by the similar arguments as that the proof in Theorem 3.1, it is easy prove that the sequence converges to a fixed point of . This completes the proof.

From Theorem 3.5, we can easily to prove the following result.

Corollary 3.6. Let be a nonempty bounded closed convex subset of a real reflexive Banach space with a uniformly Gâteaux differentiable norm. Let be a contractive mapping and a uniformly continuous pseudocontractive mapping. Let be a sequence satisfying the conditions:  , . Define the sequence in by If , where is defined as (3.22), then the sequence defined by (3.35) converges strongly to a fixed point .

Acknowledgment

This research was partially supported by Youth Foundation of Taizhou University (2011QN11).

References

S. S. Chang, “On the convergence problems of Ishikawa and Mann iterative processes with error for Φ-pseudo contractive type mappings,” Applied Mathematics and Mechanics, vol. 21, no. 1, pp. 1–12, 2000.

C. E. Chidume and C. Moore, “The solution by iteration of nonlinear equations in uniformly smooth Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 215, no. 1, pp. 132–146, 1997.

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.

W. R. Mann, “Mean value methods in iteration,” Proceedings of the American Mathematical Society, vol. 4, pp. 506–510, 1953.

M. O. Osilike, “Iterative solution of nonlinear equations of the Φ-strongly accretive type,” Journal of Mathematical Analysis and Applications, vol. 200, no. 2, pp. 259–271, 1996.

S. Reich, “Iterative methods for accretive sets,” in Nonlinear Equations in Abstract Spaces, pp. 317–326, Academic Press, New York, NY, USA, 1978.

Y. Yao, G. Marino, and Y. C. Liou, “A hybrid method for monotone variational inequalities involving pseudocontractions,” Fixed Point Theory and Applications, vol. 2011, Article ID 180534, 8 pages, 2011.

Y. Yao, Y. C. Liou, and S. M. Kang, “Iterative methods for k-strict pseudo-contractive mappings in Hilbert spaces,” Analele Stiintifice ale Universitatii Ovidius Constanta, vol. 19, no. 1, pp. 313–330, 2011.

Y. Yao, Y. C. Liou, and J.-C. Yao, “New relaxed hybrid-extragradient method for fixed point problems, a general system of variational inequality problems and generalized mixed equilibrium problems,” Optimization, vol. 60, no. 3, pp. 395–412, 2011.

Y. Yao, Y. J. Cho, and Y. C. Liou, “Algorithms of common solutions for variational inclusions, mixed equilibrium problems and fixed point problems,” European Journal of Operational Research, vol. 212, no. 2, pp. 242–250, 2011.

S. Ishikawa, “Fixed points and iteration of a nonexpansive mapping in a Banach space,” Proceedings of the American Mathematical Society, vol. 59, no. 1, pp. 65–71, 1976.

C. E. Chidume and S. A. Mutangadura, “An example of the Mann iteration method for Lipschitz pseudocontractions,” Proceedings of the American Mathematical Society, vol. 129, no. 8, pp. 2359–2363, 2001.

C. E. Chidume and H. Zegeye, “Approximate fixed point sequences and convergence theorems for Lipschitz pseudocontractive maps,” Proceedings of the American Mathematical Society, vol. 132, no. 3, pp. 831–840, 2004.

N. Shioji and W. Takahashi, “Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces,” Proceedings of the American Mathematical Society, vol. 125, no. 12, pp. 3641–3645, 1997.

C. H. Morales and J. S. Jung, “Convergence of paths for pseudocontractive mappings in Banach spaces,” Proceedings of the American Mathematical Society, vol. 128, no. 11, pp. 3411–3419, 2000.

S. S. Chang, Y. J. Cho, and H. Zhou, Iterative Methods for Nonlinear Operator Equations in Banach Spaces, Nova Science, Huntington, NY, USA, 2002.

T. H. Kim and H. K. Xu, “Strong convergence of modified Mann iterations,” Nonlinear Analysis, Theory, Methods & Applications, vol. 61, no. 1-2, pp. 51–60, 2005.

S. S. Chang, Y. J. Cho, B. S. Lee, J. S. Jung, and S. M. Kang, “Iterative approximations of fixed points and solutions for strongly accretive and strongly pseudo-contractive mappings in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 224, no. 1, pp. 149–165, 1998.