Research Article | Open Access

Wei Xu, Yuanheng Wang, "A General Iterative Algorithm with Strongly Positive Operators for Strict Pseudo-Contractions", *Abstract and Applied Analysis*, vol. 2013, Article ID 757986, 6 pages, 2013. https://doi.org/10.1155/2013/757986

# A General Iterative Algorithm with Strongly Positive Operators for Strict Pseudo-Contractions

**Academic Editor:**Claudia Timofte

#### Abstract

This paper deals with a new iterative algorithm with a strongly positive operator *A* for a *k*-strict pseudo-contraction *T* and a non-self-Lipschitzian mapping *S* in Hilbert spaces. Under certain appropriate conditions, the sequence converges strongly to a fixed point of *T*, which solves some variational inequality. The results here improve and extend some recent related results.

#### 1. Introduction

Let be a closed convex subset of Hilbert space with inner product and norm , be a nonlinear mapping. The fixed point set of is denoted by ; that is, . Fixed point problem is very general in the sense that it includes, as spacial cases, optimization problems, variational inequalities, minimax problems, the Nash equilibrium problem in noncooperative games, and others.

Recall that a mapping is said to be nonexpansive if for all . A mapping is said to be strongly positive, if there exists a constant such that for all . In 2000, Moudafi [1] investigated the fixed point problem of nonexpansive mapping with viscosity approximation method. Let be a contraction on ; that is, there exists a constant such that for all ; define a sequence by where is an arbitrary starting point in and is a sequence in . In 2004 Xu [2] proved that if the parameter satisfies some approximate conditions, the sequence generated by (1) converges strongly to not only a fixed point of but also the unique solution of the variational inequality

In 2010, Tian [3] considered a general hybrid steepest-descent method: where is a Lipschitzian and strongly monotone operator. Under certain conditions, he proved that the sequence generated by (3) converges strongly to the unique solution of the variational inequality On the other hand, Marino and Xu [4] introduced the following iterative scheme: where is a strongly positive bounded linear operator. It was proven that under certain conditions on the parameters, the sequence generated by (5) converges strongly to the unique solution of the variational inequality

It is well known that a typical convex minimization is that of minimizing a quadratic function on the sets of the fixed points of a nonexpansive mapping: where is a given point of . The solution is also the optimality condition for the minimization problem where is a potential function for ; that is, . Some authors investigated each iterative method for nonexpansive mappings for solving convex minimization problems and got some convergence results; see, for example [5–7].

In 2011, Ceng et al. [8] introduced a general iterative algorithm with strongly positive operators for nonexpansive mappings: and proved that under certain conditions on the parameters the sequence generated by (9) converges strongly to a fixed point of , which also solves the variational inequality

Recently the problems of the approximation of the common fixed points of nonexpansive mappings were extended to the case of a family of finite or infinite pseudo-contractions; see, for example, [9–11].

Motivated and inspired by the above research works, we consider some fixed point problems with non-self mappings and introduce a new general iterative algorithm with strongly positive operators for -strict pseudo-contractions which is a wider map class then the nonexpansive map class where is the metric projection, is a non-self-Lipschitzian mapping, is a -strict pseudo-contraction, and is a strongly positive bounded linear operator. Under certain conditions on the parameters, we prove that the sequence generated by (11) converges strongly to a fixed point of , which solves the variational inequality

#### 2. Preliminaries

In this section, we recall some useful definitions and lemmas for the proof of the main results.

*Definition 1. *A mapping is said to be -Lipschitzian, if there exists a constant such that
A mapping is said to be -strict pseudo-contraction, if there exists a constant such that

It is clear that a Lipschitzian map is a contractive map when and is a nonexpansive map when . If ; then a -strict pseudo-contraction map is a nonexpansive map.

*Definition 2. *A mapping is said to be the metric projection, if for any , there exists a unique nearest point in *C* denoted by such that
And it is well known that if is a nonempty closed convex subset of , then the exists (e.g., see [12]).

Lemma 3 (see [13]). *Let and be any points. There holds
**
And if and only if there holds
**
and if and only if there holds the relation
*

Lemma 4 (see [9], Demiclosedness princple). *Let be a nonempty closed convex subset of a real Hilbert space and let be a nonexpansive mapping with . If is a sequence in weakly converging to and if converges strongly to , then ; in particular if , then .*

Lemma 5 (see [14]). *Let be a number in and . Let be a t-Lipschitzian and -strongly monotone operator on a Hilbert space. Associate with a nonexpansive mapping and define the mapping by
**
Then is a contraction provided ; that is,
*

Lemma 6 (see [4]). *Assume that is a strongly positive bounded linear operator on a Hilbert space with coefficient and ; then .*

Lemma 7 (see [15]). *Let be a nonempty closed convex subset of a real Hilbert space . Let be a -strict pseudo-contractive mapping. Let and be two nonnegative real numbers such that ; then
*

Lemma 8 (see [16]). *Let be a Hilbert space and a nonempty convex subset of . Let be a -strict pseudo-contractive mapping. Define a mapping for all . Then as , is a nonexpansive mapping such that .*

Lemma 9 (see [17]). *Let be a sequence of nonnegative real numbers satisfying the following relation: , where (i) , ; (ii) or ; then .*

#### 3. Main Results

In this section, we prove the strong convergence results on the iterative algorithm for -strict pseudo-contractions.

Theorem 10. *Let be a nonempty closed convex subset of a real Hilbert space , a non-self-L-Lipschitzian mapping, and a -strict pseudo-contractive mapping such that . Let be a t-Lipschitzian and -strongly monotone mapping and a -strongly positive bounded linear operator. For a given , let the sequences and generated by (11), where , , , satisfy the following conditions:*(i)*,
,
;
*(ii)*, , , , ;*(iii)*, , , .**Then the sequence converges strongly to a fixed point of , which solves the variational inequality
*

*Proof. *The proof is divided into five steps. *Step 1*. We first show that the sequences , are bounded. Take , own to be a -strict pseudo-contractive mapping, and define . By Lemma 8 is nonexpansive and ; therefore :

Thus we immediately get that is a -Lipschitzian mapping. Then we estimate :

On the other hand, notice that , ; without loss of generality, we may assume that ; thus
Together with (24), we have
By conditions (ii) and (iii), we get that is bounded, and so are ,,,.*Step 2*. Now we prove that as . Denote :
where is a constant such that
By the conditions (i), (ii), and (iii) and Lemma 9, we get as .*Step 3*. Now we prove that as :
On the other hand,
Thus we have as . Observe that
we immediately get as .*Step 4*. Now we show that , where is the unique solution of the variational inequality. Take a subsequence of such that

Observe that the sequence is bounded; without loss of generality we may assume that . By Lemma 4, we get . Therefore by Lemma 3, we have
*Step 5*. Next we prove that as :

Notice that
thus
By the conditions (ii), (iii) and Lemma 9, we conclude that as , which solves the variational inequality . This completes the proof.

*Remark 11. *The iterative algorithm in Theorem 10 here is a new approximating method, and Lemma 7 plays a key role in the proof of the main results which makes the proof simple.

*Remark 12. *The results in this paper improve and extend some recent related results. For example, Theorem 10 here improves and extends Theorem 3.2 in [8] in the following ways:(i)the nonexpansive mapping in [8] is extended to the case of -strict pseudo-contractions ;(ii)the self-contraction in [8] is extended to the case of a (possiblly non-self) Lipschitzian mapping .

#### Acknowledgments

The authors would like to thank editors and referees for many useful comments and suggestions for the improvement of the paper. This work is partially supported by the Natural Science Foundation of Zhejiang Province (Y6100696, Y6110270) and the National Natural Science Foundation (11071169, 11271330).

#### References

- A. Moudafi, “Viscosity approximation methods for fixed-points problems,”
*Journal of Mathematical Analysis and Applications*, vol. 241, no. 1, pp. 46–55, 2000. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - H.-K. Xu, “Viscosity approximation methods for nonexpansive mappings,”
*Journal of Mathematical Analysis and Applications*, vol. 298, no. 1, pp. 279–291, 2004. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - M. Tian, “A general iterative algorithm for nonexpansive mappings in Hilbert spaces,”
*Nonlinear Analysis: Theory, Methods & Application A*, vol. 73, no. 3, pp. 689–694, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - G. Marino and H.-K. Xu, “A general iterative method for nonexpansive mappings in Hilbert spaces,”
*Journal of Mathematical Analysis and Applications*, vol. 318, no. 1, pp. 43–52, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - F. Deutsch and I. Yamada, “Minimizing certain convex functions over the intersection of the fixed point sets of nonexpansive mappings,”
*Numerical Functional Analysis and Optimization*, vol. 19, no. 1-2, pp. 33–56, 1998. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - H. K. Xu, “An iterative approach to quadratic optimization,”
*Journal of Optimization Theory and Applications*, vol. 116, no. 3, pp. 659–678, 2003. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - I. Yamada, “The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings,” in
*Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications*, D. Butnariu, Y. Censor, and S. ReichS, Eds., vol. 8, pp. 473–504, North-Holland, Amsterdam, The Netherland, 2001. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - L.-C. Ceng, S.-M. Guu, and J.-C. Yao, “A general composite iterative algorithm for nonexpansive mappings in Hilbert spaces,”
*Computers & Mathematics with Applications*, vol. 61, no. 9, pp. 2447–2455, 2011. View at: Publisher Site | Google Scholar | MathSciNet - Y. H. Wang and Y. H. Xia, “Strong convergence for asymptotically pseudo-contractions with the demiclosedness principle in Banach spaces,”
*Fixed Point Theory and Applications*, vol. 2012, article 45, 2012. View at: Google Scholar - Y. L. Song, H. Y. Hu, Y. Q. Wang et al., “Strong convergence of a new general iterative method for variational inequality problems in Hilbert spaces,”
*Fixed Point Theory and Applications*, vol. 2012, article 46, 2012. View at: Google Scholar - W. Xu and Y. H. Wang, “Strong convergence of the iterative methods for hierarchical fixed point problems of an infinite family strictly non-self pseudo-contractions,”
*Abstract and Applied Analysis*, vol. 2012, Article ID 457024, 11 pages, 2012. View at: Publisher Site | Google Scholar - N. Young,
*An Introduction to Hilbert Space*, Cambridge University Press, Cambridge, UK, 1988. View at: MathSciNet - Y. H. Yao, M. A. Noor, and Y. C. Liou, “Strong convergence of a modified extragradient method to the miniumnorm solution of variational inequalities,”
*Abstract and Applied Analysis*, vol. 2012, Article ID 817436, 9 pages, 2012. View at: Publisher Site | Google Scholar - H. K. Xu and T. H. Kim, “Convergence of hybrid steepest-descent methods for variational inequalities,”
*Journal of Optimization Theory and Applications*, vol. 119, no. 1, pp. 185–201, 2003. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Y. Yao, Y.-C. Liou, and S. M. Kang, “Approach to common elements of variational inequality problems and fixed point problems via a relaxed extragradient method,”
*Computers & Mathematics with Applications*, vol. 59, no. 11, pp. 3472–3480, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - F. E. Browder and W. V. Petryshyn, “Construction of fixed points of nonlinear mappings in Hilbert space,”
*Journal of Mathematical Analysis and Applications*, vol. 20, pp. 197–228, 1967. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Y. Wang and L. Yang, “Modified relaxed extragradient method for a general system of variational inequalities and nonexpansive mappings in Banach spaces,”
*Abstract and Applied Analysis*, vol. 2012, Article ID 818970, 14 pages, 2012. View at: Google Scholar | Zentralblatt MATH | MathSciNet

#### Copyright

Copyright © 2013 Wei Xu and Yuanheng Wang. 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.