Convergence of a Viscosity Iterative Method for Multivalued Nonself-Mappings in Banach Spaces. (19th February 2013) - Jung, Jong Soo
Abstract : Let E be a reflexive Banach space having a weakly sequentially continuous duality mapping J φ with gauge function φ, C a nonempty closed convex subset of E, and T : C → 𝒦 ( E ) a multivalued nonself-mapping such that P T is nonexpansive, where P T ( x ) = u x ∈ T x : ∥ x - u x ∥ = d ( x, T x ) . Let f : C → C be a contraction with constant k . Suppose that, for each v ∈ C and t ∈ ( 0, 1 ), the contraction defined by S t x = t P T x + ( 1 - t ) v has a fixed point x t ∈ C . Let α n , β n , and γ n be three sequences in ( 0, 1 ) satisfying approximate conditions. Then, for arbitrary x 0 ∈ C, the sequence x n generated by x n ∈ α n f ( x n - 1 ) + β n x n - 1 + γ n P T ( x n ) for all n ≥ 1 converges strongly to a fixed point of T .