General iterative algorithms for hierarchical fixed points approach to variational inequalities
Summary: This paper deals with new methods for approximating a solution to the fixed point problem; find \(\tilde{x} \in F(T)\), where \(H\) is a Hilbert space, \(C\) is a closed convex subset of \(H\), \(f\) is a \(\rho\)-contraction from \(C\) into \(H, 0 < \rho < 1, A\) is a strongly positive linear-bounded operator with coefficient \(\bar{\gamma} > 0, 0 < \gamma < \bar{\gamma}/\rho, T\) is a nonexpansive mapping on \(C\), and \(P_{F(T)}\) denotes the metric projection on the set of fixed point of \(T\). Under a suitable different parameter, we obtain strong convergence theorems by using the projection method which solves the variational inequality \(\langle (A - \gamma f)\tilde{x} + \tau(I - S)\tilde{x}, x - \tilde{x}\rangle \geq 0\) for \(x \in F(T)\), where \(\tau \in [0, \infty)\). Our results generalize and improve the corresponding results of \textit{Y. Yao}, \textit{Y. J. Cho} and \textit{ \textit{Y.-Ch. Liou} [Math. Comput. Modelling 52, No. 9--10, 1697--1705 (2010; Zbl 1205.65192)] and some authors. Furthermore, we give an example which supports our main theorem in the last part.}
scientific article; zbMATH DE number 6106129