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 .