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

Abstract EN

Let E be a reflexive Banach space having a weakly sequentially continuous duality mapping Jφ with gauge function φ, C a nonempty closed convex subset of E, and T:C→?(E) a multivalued nonself-mapping such that PT is nonexpansive, where PT(x)={ux∈Tx:∥x-ux∥=d(x,Tx)}.

Let f:C→C be a contraction with constant k.

Suppose that, for each v∈C and t∈(0,1), the contraction defined by Stx=tPTx+(1-t)v has a fixed point xt∈C.

Let {αn}, {βn}, and {γn} be three sequences in (0,1) satisfying approximate conditions.

Then, for arbitrary x0∈C, the sequence {xn} generated by xn∈αnf(xn-1)+βnxn-1+γnPT(xn) for all n≥1 converges strongly to a fixed point of T.