Properties of a class of approximately shrinking operators and their applications (Q2925792)

Let \(H\) be a Hilbert space, \(C\) a closed convex subset of \(H\), and \(F: H\rightarrow H\) a strongly monotone and Lipschitz continuous operator. In order to solve the variational inequality problem \(\mathrm{VIP}(F,C)\): NEWLINE\[NEWLINE \text{ find } \overline{u}\in C \text{ such that } \langle F(\overline{u}), x-\overline{u}\rangle \geq 0 \text{ for all } x\in C, NEWLINE\]NEWLINE the authors introduce a class of approximately shrinking operators, prove the closedness of this class with respect to compositions and convex combinations, and then use these operators to construct a generalised hybrid steepest descent method (GHSD) for solving the \(\mathrm{VIP}(F,C)\) above.NEWLINENEWLINESufficient conditions for the convergence of the GHSD method as well as several examples of methods which satisfy these conditions are also presented.NEWLINENEWLINEIn particular, the results are used in the case \(C=\bigcap_{i=1}^{m} \mathrm{Fix}\,(U_i)\), where \(U_i: H\rightarrow H\), \(i=1,2,\dots,m\), are quasi-nonexpansive operators having a common fixed point.