- Open Access
Strong convergence of viscosity approximation methods for the fixed-point of pseudo-contractive and monotone mappings
© Tang; licensee Springer. 2013
Received: 15 October 2012
Accepted: 25 September 2013
Published: 8 November 2013
In this paper, we introduce a viscosity iterative process, which converges strongly to a common element of the set of fixed points of a pseudo-contractive mapping and the set of solutions of a monotone mapping. We also prove that the common element is the unique solution of certain variational inequality. The strong convergence theorems are obtained under some mild conditions. The results presented in this paper extend and unify most of the results that have been proposed for this class of nonlinear mappings.
MSC:47H09, 47H10, 47L25.
Obviously, the class of monotone mappings includes the class of the α-inverse strongly monotone mappings.
Clearly, the class of pseudo-contractive mappings includes the class of strict pseudo-contractive mappings and non-expansive mappings. We denote by the set of fixed points of T, that is, .
Viscosity approximation methods are very important, because they are applied to convex optimization, linear programming, monotone inclusions and elliptic differential equations. In a Hilbert space, many authors have studied the fixed points problems of the fixed points for the non-expansive mappings and monotone mappings by the viscosity approximation methods, and obtained a series of good results, see [3–18].
Suppose that A is a monotone mapping from C into H. The classical variational inequality problem is formulated as finding a point such that , . The set of solutions of variational inequality problems is denoted by .
where T is a non-expansive mapping, A is an α-inverse strong monotone operator.
where are asymptotically non-expansive mappings, and , are non-expansive mappings.
Our concern is now the following: Is it possible to construct a new sequence that converges strongly to a common element of the intersection of the set of fixed points of a pseudo-contractive mapping and the solution set of a variational inequality problem for a monotone mapping?
Lemma 2.1 
Lemma 2.2 
and are single-valued;
and are firmly non-expansive mappings, i.e., , ;
and are closed convex.
Lemma 2.3 
3 Main results
Therefore, is bounded. Consequently, we get that , and , are bounded.
Now, we show that .
Since sequence is bounded, then there exists a sub-sequence of and such that . Next, we show that .
If , by the continuity of A, we have , . Thus, .
Let , we have , thus, . Consequently, we conclude that .
Let , . According to Lemma 2.1 and formula (3.23), we have that , i.e., the sequence converges strongly to .
According to formula (3.23), we conclude that is the solution of the variational inequality (3.2). Now, we show that is the unique solution of the variational inequality (3.2).
Because , hence we conclude that , the uniqueness of the solution is obtained. □
Proof Putting in Theorem 3.1, we can obtain the result. □
If in Theorem 3.1 and Theorem 3.2, let be a constant mapping, we have the following theorems.
This work was supported by the Natural Science Foundation Project of Chongqing (CSTC, 2012jjA00039) and the Science and Technology Research Project of Chongqing Municipal Education Commission (KJ130712, KJ130731).
- Halpern B: Fixed points of nonexpanding maps. Bull. Am. Math. Soc. 1967, 73: 957–961. 10.1090/S0002-9904-1967-11864-0View ArticleMathSciNetMATHGoogle Scholar
- Moudafi A: Viscosity approximation methods for fixed point problems. J. Math. Anal. Appl. 2000, 241: 46–55. 10.1006/jmaa.1999.6615MathSciNetView ArticleMATHGoogle Scholar
- Yao Y: A general iterative method for a finite family of nonexpansive mappings. Nonlinear Anal. 2007, 66: 2676–2687. 10.1016/j.na.2006.03.047MathSciNetView ArticleMATHGoogle Scholar
- Yao Y, Chen R, Yao JC: Strong convergence and certain conditions for modified Mann iteration. Nonlinear Anal. 2008, 68: 1687–1693. 10.1016/j.na.2007.01.009MathSciNetView ArticleMATHGoogle Scholar
- Chang SS, Lee HWJ, Chan CK: On Reich’s strong convergence theorem for asymptotically nonexpansive mappings in Banach spaces. Nonlinear Anal. 2007, 66: 2364–2374. 10.1016/j.na.2006.03.025MathSciNetView ArticleMATHGoogle Scholar
- Xu HK: Viscosity approximation methods for nonexpansive mappings. J. Math. Anal. Appl. 2004, 298: 279–291. 10.1016/j.jmaa.2004.04.059MathSciNetView ArticleMATHGoogle Scholar
- Matsushita S, Takahashi W: Strong convergence theorems for nonexpansive nonself-mappings without boundary conditions. Nonlinear Anal. 2008, 68: 412–419. 10.1016/j.na.2006.11.007MathSciNetView ArticleMATHGoogle Scholar
- Deng L, Liu Q: Iterative scheme for nonself generalized asymptotically quasi-nonexpansive mappings. Appl. Math. Comput. 2008, 205: 317–324. 10.1016/j.amc.2008.08.027MathSciNetView ArticleMATHGoogle Scholar
- Song Y: A new sufficient condition for the strong convergence of Halpern type iterations. Appl. Math. Comput. 2008, 198: 721–728. 10.1016/j.amc.2007.09.010MathSciNetView ArticleMATHGoogle Scholar
- Shioji N, Takahashi W: Strong convergence of approximated sequences for non-expansive mappings in Banach spaces. Proc. Am. Math. Soc. 1997, 125: 3641–3645. 10.1090/S0002-9939-97-04033-1MathSciNetView ArticleMATHGoogle Scholar
- Chang SS: On Chidume’s open questions and approximation solutions of multivalued strongly accretive mappings equations in Banach spaces. J. Math. Anal. Appl. 1997, 216: 94–111. 10.1006/jmaa.1997.5661MathSciNetView ArticleMATHGoogle Scholar
- Xu HK: An iterative approach to quadratic optimization. J. Optim. Theory Appl. 2003, 116: 659–678. 10.1023/A:1023073621589MathSciNetView ArticleMATHGoogle Scholar
- Takahashi W: Non-Linear Functional Analysis-Fixed Point Theory and Its Applications. Yokohama Publishers, Yokohama; 2000. (in Japanese)MATHGoogle Scholar
- Lou J, Zhang L, He Z: Viscosity approximation methods for asymptotically nonexpansive mappings. Appl. Math. Comput. 2008, 203: 171–177. 10.1016/j.amc.2008.04.018MathSciNetView ArticleMATHGoogle Scholar
- Ceng LC, Xu HK, Yao JC: The viscosity approximation method for asymptotically nonexpansive mappings in Banach spaces. Nonlinear Anal. 2008, 69: 1402–1412. 10.1016/j.na.2007.06.040MathSciNetView ArticleMATHGoogle Scholar
- Song Y, Chen R: Strong convergence theorems on an iterative method for a family of finite non-expansive mappings. Appl. Math. Comput. 2006, 180: 275–287. 10.1016/j.amc.2005.12.013MathSciNetView ArticleMATHGoogle Scholar
- Zegeye H, Shahzad N: Strong convergence for monotone mappings and relatively weak nonexpansive mappings. Nonlinear Anal. 2009, 70: 2707–2716. 10.1016/j.na.2008.03.058MathSciNetView ArticleMATHGoogle Scholar
- Wen DJ: Projection methods for generalized system of nonconvex variational inequalities with different nonlinear operators. Nonlinear Anal. 2010, 73: 2292–2297. 10.1016/j.na.2010.06.010MathSciNetView ArticleMATHGoogle Scholar
- Takahashi W, Toyoda M: Weak convergence theorems for nonexpansive mappings and monotone mappings. J. Optim. Theory Appl. 2003, 118: 417–428. 10.1023/A:1025407607560MathSciNetView ArticleMATHGoogle Scholar
- Takahashi W, Ueda Y: On Reich’s strong convergence theorems for resolvents of accretive operators. J. Math. Anal. Appl. 1984, 104: 546–553. 10.1016/0022-247X(84)90019-2MathSciNetView ArticleMATHGoogle Scholar
- Zegeye H, Shahzad N: Strong convergence of an iterative method for pseudo-contractive and monotone mappings. J. Glob. Optim. 2012, 54: 173–184. 10.1007/s10898-011-9755-5MathSciNetView ArticleMATHGoogle Scholar
- Suzuki T: Strong convergence of Krasnoselskii and Mann’s type sequences for one-parameter nonexpansive semigroups without Bochner integrals. J. Math. Anal. Appl. 2005, 305: 227–239. 10.1016/j.jmaa.2004.11.017MathSciNetView ArticleMATHGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.