Viscosity approximation scheme for a family multivalued mapping

Let KK be a nonempty closed convex subset of a real reflexive Banach space XX that has weakly sequentially continuous duality mapping JφJφ for some gauge φφ. Let Ti:K→KTi:K→K be a family of multivalued nonexpansive mappings with F:=∩i=0∞F(Ti)≠0̸ which is a sunny nonexpansive retract of KK with QQ a nonexpansive retraction. It is our purpose in this paper to prove the convergence of two viscosity approximation schemes to a common fixed point x̄=Qf(x̄) of a family of multivalued nonexpansive mappings in Banach spaces. Moreover, x̄ is the unique solution in FF to a certain variational inequality.

► I construct two viscosity approximation sequences for multivalued mappings.
► I prove the convergence of the viscosity approximation scheme to a common fixed point.
► The fixed point is the unique solution to a certain variational inequality.