Viscosity Approximation Methods for Multivalued Mappings in Banach Spaces
Yunan Cui, Zuo Zhan Fei, Henryk Hudzik · Numerical Functional Analysis and Optimization · 2012
Let K be a nonempty closed and convex subset of a real reflexive Banach space X that has weakly sequentially continuous duality mapping J. Let T: K → K be a multivalued non-expansive non-self-mapping satisfying the weakly inwardness condition as well as the condition T(y) = {y} for any y ∈ F(T) and such that for a contraction f: K → K and any t ∈ (0, 1), there exists x t ∈ K satisfying x t ∈ tf(x t ) + (1 − t)Tx t . Then it is proved that {x t } ⊂ K converges strongly to a fixed point of T, which is also a solution of certain variational inequality. Moreover, the convergence of two explicit methods are also investigated.