General alternative regularization methods for nonexpansive mappings in Hilbert spaces
Yang Cai-ping, Songnian He · Fixed Point Theory and Applications · 2014
Let C be a nonempty closed convex subset of a real Hilbert space H with the inner product •, • and the norm • .Let T : C → C be a nonexpansive mapping with a nonempty set of fixed points Fix(T) and let h : C → C be a Lipschitzian strong pseudo-contraction.We first point out that the sequence generated by the usual viscosity approximation methodstrongly to a fixed point of T, which also solves the variational inequality problem: finding an element x * such that h(x * ) -x * , x -x * ≤ 0 for all x ∈ Fix(T).Furthermore, we prove that if T is replaced with the sequence of average mappings (1where β * and β * are two positive constants, then the same convergence result holds provided conditions (i) and (ii) are satisfied.Finally, an algorithm for finding a common fixed point of a family of finite nonexpansive mappings is also proposed and its strong convergence is proved.Our results in this paper extend and improve the alternative regularization methods proposed by HK Xu.