A GENERAL ITERATIVE ALGORITHM FOR A FINITE FAMILY OF NONEXPANSIVE MAPPINGS IN A HILBERT SPACE

다국어 초록 (Multilingual Abstract)

Let C be a nonempty closed convex subset of a real Hilbert space H. Consider the following iterative algorithm given by x0 2 C arbitrarily chosen,xn+1 = αnγf(Wnxn)+βnxn+((1−βn)I−αnA)WnPC(I−snB)xn, ∀n≥0,where γ > 0, B : C → H is a β-inverse-strongly monotone mapping, f is a contraction of H into itself with a coefficient α (0 < α < 1), Pc is a projection of H onto C, A is a strongly positive linear bounded operator on H and Wn is the W-mapping generated by a finite family of nonexpansive mappings T1, T2, ... TN and {λn,1}, {λn,2}, . . . , {λn,N}. Nonexpansivity of each Ti ensures the nonexpansivity of Wn. We prove that the sequence {χn} generated by the above iterative algorithm converges strongly to a common fixed point q ∈ F := ∩Ni=1F(Ti)∩VＩ(C,B) which solves the variational inequality ＜(γf − A)q, p − q＞≤ 0 for all p ∈ F. Using this result, we consider the problem of finding a common fixed point of a finite family of nonexpansive mappings and a strictly pseudocontractive mapping and the problem of finding a common element of the set of common fixed points of a finite family of nonexpansive mappings and the set of zeros of an inverse-strongly monotone mapping. The results obtained in this paper extend and improve the several recent results in this area.

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