Viscosity methods of approximation for a common fixed point of a family of quasi-nonexpansive mappings Original Research Article

Let KK be a nonempty closed convex subset of a real reflexive Banach space EE that has weakly continuous duality mapping JφJφ for some gauge φφ. Let Ti:K→K,i=1,2,…Ti:K→K,i=1,2,…, be a family of quasi-nonexpansive mappings with F≔∩i≥1F(Ti)≠0̸F≔∩i≥1F(Ti)≠0̸ which is a sunny nonexpansive retract of KK with QQ a nonexpansive retraction. For given x0∈Kx0∈K, let {xn}{xn} be generated by the algorithm xn+1≔αnf(xn)+(1−αn)Tn(xn),n≥0xn+1≔αnf(xn)+(1−αn)Tn(xn),n≥0, where f:K→Kf:K→K is a contraction mapping and {αn}⊆(0,1){αn}⊆(0,1) a sequence satisfying certain conditions. Suppose that {xn}{xn}satisfies condition (A). Then it is proved that {xn}{xn} converges strongly to a common fixed point View the MathML sourcex̄=Qf(x̄) of a family Ti,i=1,2,…Ti,i=1,2,…. Moreover, View the MathML sourcex̄ is the unique solution in FF to a certain variational inequality.