STRONG CONVERGENCE THEOREMS FOR ASYMPTOTICALLY QUASI-NONEXPANSIVE MAPPINGS AND INVERSE-STRONGLY MONOTONE MAPPINGS
Xin-Feng He, Yongchun Xu, Zhen He · East Asian Mathematical Journal · 2011
Abstract. In this paper, we consider an iterative scheme for ndinga common element of the set of xed points of a asymptotically quasi-nonexpansive mapping and the set of solutions of the variational inequal-ity for an inverse strongly monotone mapping in a Hilbert space. Thenwe show that the sequence converges strongly to a common element oftwo sets. Using this result, we consider the problem of nding a com-mon xed point of a asymptotically quasi-nonexpansive mapping and astrictly pseudocontractive mapping and the problem of nding a commonelement of the set of xed points of a asymptotically quasi-nonexpansivemapping and the set of zeros of an inverse-strongly monotone mapping. 1. Introduction and preliminariesLet Cbe a closed convex subset of a real Hilbert space Hand let P C be themetric projection of Honto C.A mapping Aof Cinto His called monotone if for all x;y2C,hx y;AxAyi0:The variational inequality problem is to nd a u2Csuch that hv u;Aui0 for all v2C; see [1,2,4,6,11]. The set of solutions of the variational inequalityis denoted by VI(C;A).A mapping Aof Cinto His called inverse-strongly monotone if there existsa positive real number such that hx y;Ax Ayi kAx Ayk