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

Jong Soo Jung

Abstract and Applied Analysis, 2013, vol. 2013, issue 1

Abstract: 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 → 𝒦(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.

Date: 2013
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