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.
Keywordspseudo-contractive mappings monotone mappings fixed point variational inequalities viscosity approximation
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).
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