Fixed point theorems of semigroup of isometry mappings and α-nonexpansive mappings on weak⁎ compact convex sets

A.T.-M. Lau raised the question “Does a nonempty weak^⁎ compact convex subset X of a dual Banach space $B{}^{⁎}$ have the fixed point property for a left reversible semigroup of nonexpansive mappings whenever X has the weak^⁎ normal structure?” Fendler et al. proved that a dual Banach space $B{}^{⁎}$ has the weak^⁎ fixed point property for a left reversible semigroup of nonexpansive mappings in the presence of the asymptotic center property. In this paper, we prove that a left reversible semigroup of isometry mappings has a common fixed point in the Chebyshev center of X whenever X is a nonempty weak^⁎ compact convex subset of a dual Banach space $B{}^{⁎}$ and $B{}^{⁎}$ has the asymptotic center property. In 2010, A.T.-M. Lau et al. proved that a group $G$ of isometric self-maps on a weak^⁎ compact convex subset X of a dual Banach space $B{}^{⁎}$ has a common fixed point in X whenever X has the weak^⁎ normal structure. In this paper, we prove that a group $G$ of isometry mappings from X into itself has a common fixed point in the Chebyshev center of X. Moreover, we prove that if X is a nonempty weak^⁎ compact convex set having the weak^⁎ normal structure in a dual Banach space $B{}^{⁎}$ and $J$ is the set of all surjective isometry mappings on X then $J$ has a common fixed point in the Chebyshev center of X. In 2011, Aoyama and Kohsaka introduced the class of α-nonexpansive mappings in Banach spaces. In 2018, Amini et al. proved the weak fixed point property of an α-nonexpansive mapping in the presence of the normal structure. In this paper, we prove the weak^⁎ fixed point property of an α-nonexpansive mapping in the presence of the weak^⁎ normal structure. Further, we extend the fixed point result of Amini et al. for a commuting family of α-nonexpansive mappings.