Abstract
Let be a reflexive Banach space having a weakly sequentially continuous duality mapping with gauge function , a nonempty closed convex subset of , and a multivalued nonself-mapping such that is nonexpansive, where . Let be a contraction with constant . Suppose that, for each and , the contraction defined by has a fixed point . Let , and be three sequences in satisfying approximate conditions. Then, for arbitrary , the sequence generated by for all converges strongly to a fixed point of .
Highlights
Let E be a Banach space and C a nonempty closed subset of E
Suppose that C is a nonexpansive retract of E and that for each v ∈ C and t ∈ (0, 1) the contraction St defined by Stx = tPTx + (1 − t)v has a fixed point xt ∈ C
Suppose that for each v ∈ C and t ∈ (0, 1), the contraction St defined by Stx = tPTx+(1−t)v has a fixed point xt ∈ C
Summary
Let E be a Banach space and C a nonempty closed subset of E. Let E be a uniformly convex Banach space E having a uniformly Gateaux differentiable norm, C a nonempty closed convex subset of E, and T : C → K(E) a multivalued nonself-mapping such that PT is nonexpansive. In this paper, inspired and motivated by the abovementioned results, we consider a viscosity iterative method for a multivalued nonself-mapping in a reflexive Banach space having a weakly sequentially continuous duality mapping and establish the strong convergence of the sequence generated by the proposed iterative method.
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