Global Iterative Schemes for Accretive Operators

Let E be a real q-uniformly smooth Banach space and A:E→2^E be an m-accretive operator which satisfies a linear growth condition of the form Ax≤c(1+x) for some constant c0 and for all x∈E. It is proved that if two real sequences {λ_n} and {θ_n} satisfy appropriate conditions, the sequence {x_n} generated from arbitrary x₀∈E by x_n+1=x_n-λ_n(u_n+θ _n(x_n-z)); u_n∈Ax_nn≥0, converges strongly to some x*∈A^-1(0). Furthermore, if E is a reflexive Banach space with a uniformly Gâteaux differentiable norm, and if every weakly compact convex subset of E has the fixed-point property for nonexpansive mappings and A:D(A)E→2^E is m-accretive, then for arbitrary, z,x₀∈E the sequence {x_n} defined by x_n+1+λ_n(u_n+1+θ_n(x _n+1-z))=x_n+e_n, for u_n∈Ax_n, where e_n∈E is such that ∑e_n<∞∀n≥0, converges strongly to some x*∈A^-1(0).