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
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.
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  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  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  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  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.  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
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 ).
Lemma 3 (see ). 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 , 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 ). 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 ). Assume that is a strongly positive bounded linear operator on a Hilbert space with coefficient and ; then .
Lemma 7 (see ). 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 ). 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 ). 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 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  in the following ways:(i)the nonexpansive mapping in  is extended to the case of -strict pseudo-contractions ;(ii)the self-contraction in  is extended to the case of a (possiblly non-self) Lipschitzian mapping .
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).
- A. Moudafi, “Viscosity approximation methods for fixed-points problems,” Journal of Mathematical Analysis and Applications, vol. 241, no. 1, pp. 46–55, 2000.
- H.-K. Xu, “Viscosity approximation methods for nonexpansive mappings,” Journal of Mathematical Analysis and Applications, vol. 298, no. 1, pp. 279–291, 2004.
- 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.
- 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.
- 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.
- H. K. Xu, “An iterative approach to quadratic optimization,” Journal of Optimization Theory and Applications, vol. 116, no. 3, pp. 659–678, 2003.
- 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.
- 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.
- 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.
- 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.
- 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.
- N. Young, An Introduction to Hilbert Space, Cambridge University Press, Cambridge, UK, 1988.
- 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.
- 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.
- 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.
- 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.
- 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.
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.