Abstract
In this paper, we investigate the problem of finding common solutions of variational inclusions, variational inequalities and fixed point problems in real Hilbert spaces. Motivated by Nadezhkina and Takahashi's extragradient method [N. Nadezhkina, W. Takahashi, Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl. 128 (2006) 191-201], we propose and analyze a modified extragradient algorithm for finding common solutions. It is proven that three sequences generated by this algorithm converge weakly to the same common solution under very mild conditions by virtue of the Opial condition of Hilbert spaces, the demi-closedness principle for nonexpansive mappings and the coincidence of solutions of variational inequalities with zeros of maximal monotone operators.
| Original language | American English |
|---|---|
| Pages (from-to) | 21-31 |
| Journal | Journal of Nonlinear and Convex Analysis |
| Volume | 14 |
| Issue number | 1 |
| State | Published - 2013 |
Keywords
- INCLUSIONS
- MONOTONE MAPPINGS
- NONEXPANSIVE-MAPPINGS
- OPERATORS
- Variational inclusion
- inverse strongly monotone mapping
- maximal monotone mapping
- nonexpansive mapping
- variational inequality
- weak convergence