Convergence of a Viscosity Iterative Method for Multivalued Nonself-Mappings in Banach Spaces

Let E be a reflexive Banach space having a weakly sequentially continuous duality mapping J(phi) with gauge function phi, 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) is an element of Tx : parallel to x - u(x)parallel to = d(x, Tx)}. Let f : C --> C be a contraction with constant k. Suppose that, for each v is an element of C and l is an element of (0,1), the contraction defined by S(t)x = tP(T)x + (1 - t)v has a fixed point x(t) is an element of C. Let {alpha(n)}, {beta(n)}, and {gamma(n)} be three sequences in (0, 1) satisfying approximate conditions. Then, for arbitrary x(0) is an element of C, the sequence {x(n)} generated by x(n) is an element of alpha(n)f(x(n-1)) + beta(n)x(n-1) + gamma P-n(T)(x(n)) for all n >= 1 converges strongly to a fixed point of T.