Convergence theorems on an iterative method for variational inequality problems and fixed point problems (Q2655438)

Let \(H\) be a Hilbert space with inner product \(\langle \cdot , \cdot\rangle\) and norm \(\|\cdot,\cdot\|\) and \(C\) be a nonempty closed convex subset of \(H\). A nonlinear mapping \(A:C\rightarrow H\) is said to be \(\alpha\)-inverse-strongly monotone if there exists a positive real number \(\alpha>0\) such that \[ \langle Ax-Ay,x-y\rangle \geq \alpha\|Ax-Ay\|^2 \quad\text{for all }x,y\in C. \] A mapping \(T:C\rightarrow C\) is said to be strictly pseudo-contractive with coefficient \(k\in(0,1)\) if \[ \|Tx-Ty\|^2 \leq \|x-y\|^2 + k\|(I-T)x-(I-T)y\|^2 \quad\text{for all }x,y \in C. \] The main result of the paper is Theorem 2.1, which constructs a viscosity iterative method for approximating the (unique) common element of the (nonempty) set of fixed points of a strict pseudo-contraction and of the set of solutions to variational inequalities with inverse strongly monotone mappings. Corollary 2.1 refers to the (nonempty) set of fixed points for nonexpansive mappings. The paper also contains a section of applications of Theorem 2.1. Theorem 3.1 is a similar result for strict pseudo-contractions instead of inverse-strongly monotone mappings. Theorem 3.2 extends Corollary 2.1 by considering \(C=H\). Theorem 3.3 extends Theorem 2.1 by considering a finite family of inverse-strongly monotone mappings.