The Uniform Convexity and Uniform Smoothness of Variable Exponent Lebesgue Spaces and Contractive Semigroups

In this article, we investigate the geometrical structure of the variable exponent Lebesgue spaces, their uniform convexity, and uniform smoothness. Employing uniform convexity, we establish the asymptotic properties of semigroups of nonlinear operators. The existence of uniquely defined common fixed points is established under the Lipschitzian growth condition.