General alternative regularization methods for nonexpansive mappings in Hilbert spaces

Let C be a nonempty closed convex subset of a real Hilbert space H with the inner product <center dot,center dot > and the norm parallel to center dot parallel to. 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 method x(n+1) = lambda(n)h(x(n)) + (1 -lambda(n))Tx(n) may not converge to a fixed point of T, even not bounded. Secondly, we prove that if the sequence (lambda(n)) subset of (0, 1) satisfies the conditions: (i) lambda(n) -> 0, (ii) Sigma(infinity)(n=0) lambda(n) = infinity and (iii) Sigma(infinity)(n=0) vertical bar lambda(n+1) - lambda(n)vertical bar < infinity or lim(n ->infinity) lambda(n+1)/lambda(n) = 1, then the sequence (X-n) generated by a general alternative regularization method: Xn+1 = T(lambda(n)h(X-n) + (1 - lambda(n))X-n) converges strongly 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 is an element of Fix(T). Furthermore, we prove that if T is replaced with the sequence of average mappings (1 - beta(n))/ + beta T-n (n >= 0) such that 0 < beta(*) <= beta(n) <= beta* < 1, where beta* and beta* 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.