Strong Convergence Of Modified Iterative Process For Nonexpansive Mappings

In this paper we deal with some strong convergence theorems of mod-ified iterative process for nonexpansive mappings.In the first chapter,we introduce some definitions,lemmas and recent results of iterative process for nonexpansive mappings,see[1-17]In the second chapter,we use a modified iterative processx_n+1=α_f(x_n)+β_nx_n+γ_nT_xn instead of x_n+1=α_nf(x_n)+(1-α_n)Tx_n to study the fixed points problem for nonexpansive mapping T on a closed convex subsetΩof a Hilbert space H.When {α_n},{β_n},{γ_n}(?)[0,1]satisfy:α_n+β_n +γ_n = 1, lim_{n→∞}α_n = 0,∑_n=1^∞=1α_n=∞,0＜lim inf_{n→∞}β_n≤lim sup_{n→∞}β_n＜1 and some other appropriate conditions,{x_n} converge strongly to a fixed point of T which solves the variational inequality:<(I-f)x^*,x^* - x>≤0,x∈Fix(T). We delete the condition∑_n=0^∞+|α_n+1-α_n|＜∞or lim_{n→∞}α_n+1/α_n=1 in Theorem 3.1 of Hong-Kun Xu[1].In the third chapter,we use a nonexpansive mapping S_x:=(1-δ)x+δTx instead of nonexpansive mapping T,introduce a modified Ishikawa iterative process x_n+1=α_nu +(1 -α_n)Sy_n, { y_n=β_nx_n +(1 -β_n)Sx_n,(?)n≥0. In a real Banach space,When {α_n},{β_n}(?)[0,1]satisfy:lim_{n→∞}α_n = 0,∑_n=0^∞α_n =∞,lim_{n→∞}β_n = 1 and some other appropriate conditions, {x_n} converge strongly to a fixed point of T.We delete the conditions ||x_n-Tx_n||→0 and {y_n} are bounded in Theorem 3 of S.S.Chang, H.W.Joseph Lee and ChiKan Chan[3].Theorem 3.1 of C.E.Chidume, C.O.Chidume[2]is a special case of our results whenβ_n=1.We also give a partly affirmative answer to Reich's open question in Banach space with a uniformly Gateaux differentiable norm.In the fourth chapter,we consider the stability of iterative processX_n+1=α_nu+(1-α_n)SP_Ω(x_n-λ_nTx_n),x₀=u∈Ω, introduced in H.Iiduk,W.Takahashi[4]with respect to perturbations of constrain set.LetΩ_n,S_n,P_Ωnare perturbations of closed convex compact setΩ,nonexpansive mapping S,and metric projection P_Ωrespectively,we get a perturbed iterative processx_n+1=α_nu +(1 -α_n)S_n+1P_Ωn+1(x_n -λ_nTx_n),x₀ = u∈Ω, Where {a_n}(?)[0,1),{λ_n} satisfy:lim_{n→∞}α_n = 0,∑_n=1^∞α_n =∞,∑_n=1^∞|α_n+1-α_n|＜∞,∑_n=1^∞|λ_n+1-λ_n|＜∞and some other appro-priate conditions,{x_n} also converge to P_{F(S)∩VI(Ω,T)u}strongly.We also study a new perturbed iterative processx_n+1=α_nu +β_nx_n +γ_nS_n+1P_Ωn+1(x_n -λ_nTx_n),x₀ = u∈Ω. We delete the condition∑_n=1^∞|α_n+1-α_n|＜∞and use a weaker condition lim_{n→∞}|λ_n+1-λ_n| = 0 instead of∑_n=1^∞|λ_n+1-λ_n|＜∞in the above results.