Fixed Point Theory and Applications
Volume 2007 (2007), Article ID 28619, 8 pages
doi:10.1155/2007/28619

An iteration method for nonexpansive mappings in Hilbert spaces

Abstract

In real Hilbert space H, from an arbitrary initial point x0∈H, an explicit iteration scheme is defined as follows: xn+1=αnxn+(1−αn)Tλn+1xn,n≥0, where Tλn+1xn=Txn−λn+1μF(Txn), T:H→H is a nonexpansive mapping such that F(T)={x∈K:Tx=x} is nonempty, F:H→H is a η-strongly monotone and k-Lipschitzian mapping, {αn}⊂(0,1), and {λn}⊂[0,1). Under some suitable conditions, the sequence {xn} is shown to converge strongly to a fixed point of T and the necessary and sufficient conditions that {xn} converges strongly to a fixed point of T are obtained.