An iterative method for variational inequality problems and fixed point problems in Hilbert spaces. (English) Zbl 1199.47285

Qin, Xiaolong; Shang, Meijuan; Su, Yongfu

Mat. Vesn. 60, No. 2, 107-120 (2008).

The authors provide an iterative method to compute the common element of the set of fixed points of a nonexpansive mapping. Precisely, the authors consider a real Hilbert space \(H\) with the inner product \(\langle\cdot,\cdot\rangle\), and a nonempty closed convex subset \(C\) of \(H\). The projection of \(H\) onto \(C\) is denoted by \(P_C\). The mapping \(A:C\to H\) is relaxed \((\gamma,r)\)-cocoercive and \(\mu\)-Lipschitz continuous. The classical variational inequality is to find \(u\in C\) such that \(\langle Au,v-u\rangle\geq0\), for all \(v\in C\). The set of solutions of the previous variational inequality is denoted by \(VI(C,A)\). The mapping \(S:C\to C\) is nonexpansive, and the set of all fixed points of \(S\) is denoted by \(F(S)\). Under the condition \(F(S)\cap VI(C,A)\neq\emptyset\), the authors construct the iterative sequence \((x_n)_n\) in \(H\), such that \(x_n\) tends strongly to \(P_{F(S)\cap VI(C,A)}u\), where \(u\in H\) is given.

Reviewer: Dragan Đorđević (Niš)

MSC:

47J25	Iterative procedures involving nonlinear operators
47H09	Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47J20	Variational and other types of inequalities involving nonlinear operators (general)

Keywords:

projection method; relaxed cocoercive mapping; nonexpansive mapping