Strong convergence of paths for nonexpansive semigroups in Banach spaces
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Let E be a uniformly convex Banach space with a uniformly Gˆateaux differentiable norm, C be a nonempty closed convex subset of E and f : C → C be a fixed bounded continuous strong pseudocontraction with the coefficient α ∈ (0,1). Let {λ_t}(_0<t<1) be a net of positive real numbers such that lim(_t→0) λ_t = ∞ and S = {T(s) : 0 ≤ s < ∞} be a nonexpansive semigroup on C such that F(S) ≠ Φ, where F(S) denotes the set of fixed points of the semigroup. Then sequence {x_t} defined by x_t = tf(x_t) + (1-t)[수식]∫_0(^λt)T(s)x_tds converges strongly as t → 0 to x ∈ F(S), which solves the following variational inequality <(f-I)x, p - x> ≤ 0 for all p ∈ F(S).
Abstract
Let E be a uniformly convex Banach space with a uniformly Gˆateaux differentiable norm, C be a nonempty closed convex subset of E and f : C → C be a fixed bounded continuous strong pseudocontraction with the coefficient α ∈ (0,1). Let {λ_t}(_0<t<1) be a net of positive real numbers such that lim(_t→0) λ_t = ∞ and S = {T(s) : 0 ≤ s < ∞} be a nonexpansive semigroup on C such that F(S) ≠ Φ, where F(S) denotes the set of fixed points of the semigroup. Then sequence {x_t} defined by x_t = tf(x_t) + (1-t)[수식]∫_0(^λt)T(s)x_tds converges strongly as t → 0 to x ∈ F(S), which solves the following variational inequality <(f-I)x, p - x> ≤ 0 for all p ∈ F(S).
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