Iterative common solutions for monotone inclusion problems, fixed point problems and equilibrium problems

作者：Takahashi, Wataru; Wong, Ngai Ching; Yao, Jen Chih^*

来源：Fixed Point Theory and Applications, 2012, 181.

DOI：10.1186/1687-1812-2012-181

• maximal monotone operator
• resolvent
• inverse-strongly monotone mapping
• generalized hybrid mapping
• fixed point
• iteration procedure
• equilibrium problem

Let H be a real Hilbert space, and let C be a nonempty closed convex subset of H. Let alpha > 0, and let A be an alpha-inverse strongly-monotone mapping of C into H. Let T be a generalized hybrid mapping of C into H. Let B and W be maximal monotone operators on H such that the domains of B and W are included in C. Let 0 < k < 1, and let g be a k-contraction of H into itself. Let V be a (gamma) over bar -strongly monotone and L-Lipschitzian continuous operator with (gamma) over bar > 0 and L > 0. Take mu, gamma is an element of R as follows:
0 < mu < 2 (gamma) over bar /L-2, 0 < gamma < (gamma) over bar -L-2 mu/2/k.
Suppose that F(T) boolean AND (A B)(-1) 0 boolean AND W-1 0 not equal circle divide, where F(T) and (A B)(-1) 0, W-1 0 are the set of fixed points of T and the sets of zero points of A B and W, respectively. In this paper, we prove a strong convergence theorem for finding a point z(0) of F(T) boolean AND (A B)(-1) 0 n W-1 0, where z(0) is a unique fixed point of P-F(T)boolean AND(A B)(0 boolean AND W)0(-1)-1(I -V gamma g). This point z(0) is an element of F(T) boolean AND (A B)(-1) 0 boolean AND W-1 0 is also a unique solution of the variational inequality
<(V - gamma g)z(0),q-z(0)> >= 0, for all q is an element of F(T) boolean AND (A B)(-1) 0 boolean AND W(-1)0.
Using this result, we obtain new and well-known strong convergence theorems in a Hilbert space. In particular, we solve a problem posed by Kurokawa and Takahashi (Nonlinear Anal. 73: 1562-1568, 2010).

• 出版日期2012
• 单位中山大学