Strong convergence of viscosity approximation methods for the fixed-point of pseudo-contractive and monotone mappings
- PDF / 255,837 Bytes
- 11 Pages / 595.276 x 793.701 pts Page_size
- 105 Downloads / 207 Views
RESEARCH
Open Access
Strong convergence of viscosity approximation methods for the fixed-point of pseudo-contractive and monotone mappings Yan Tang* * Correspondence: [email protected] College of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing, 400067, China
Abstract 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 Keywords: pseudo-contractive mappings; monotone mappings; fixed point; variational inequalities; viscosity approximation
1 Introduction Let C be a closed convex subset of a real Hilbert space H. A mapping A : C → H is called monotone if and only if x – y, Ax – Ay ≥ ,
∀x, y ∈ C.
(.)
A mapping A : C → H is called α-inverse strongly monotone if there exists a positive real number α > such that x – y, Ax – Ay ≥ αAx – Ay ,
∀x, y ∈ C.
(.)
Obviously, the class of monotone mappings includes the class of the α-inverse strongly monotone mappings. A mapping T : C → H is called pseudo-contractive if ∀x, y ∈ C, we have Tx – Ty, x – y ≤ x – y .
(.)
A mapping T : C → H is called κ-strict pseudo-contractive, if there exists a constant ≤ κ ≤ such that x – y, Tx – Ty ≤ x – y – κ (I – T)x – (I – T)y ,
∀x, y ∈ C.
(.)
©2013 Tang; licensee Springer. 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.
Tang Fixed Point Theory and Applications 2013, 2013:273 http://www.fixedpointtheoryandapplications.com/content/2013/1/273
Page 2 of 11
A mapping T : C → C is called non-expansive if Tx – Ty ≤ x – y,
∀x, y ∈ C.
(.)
Clearly, the class of pseudo-contractive mappings includes the class of strict pseudocontractive mappings and non-expansive mappings. We denote by F(T) the set of fixed points of T, that is, F(T) = {x ∈ C : Tx = x}. A mapping f : C → C is called contractive with a contraction coefficient if there exists a constant ρ ∈ (, ) such that f (x) – f (y) ≤ ρx – y,
∀x, y ∈ C.
(.)
For finding an element of the set of fixed points of the non-expansive mappings, Halpern [] was the first to study the convergence of the scheme in xn+ = αn+ u + ( – αn+ )T(xn ).
(.)
In , Moudafi [] introduced the viscosity approximation methods and proved the strong convergence of the following iterative algorithm under some suitable conditions xn+ = αn f (xn ) + ( – αn )T(xn ).
(.)
Viscosity approximation methods are very i
Data Loading...
