M.Sc Thesis

M.Sc StudentWallwater Aya
SubjectAlmost Convergence and a Dual Ergodic Theorem for
Nonlinear Semigroups in Banach Spaces
DepartmentDepartment of Mathematics
Supervisor PROFESSOR EMERITUS Simeon Reich
Full Thesis textFull thesis text - English Version


This thesis presents a method for examining (ergodic) convergence properties of semigroups of nonlinear operators. Our method is based on a classical tool called Banach limits. Banach limits are a family of special linear functionals that arose as a result of the Hahn-Banach theorem. Although these functionals were intensively used in the discrete case, they were rarely (if ever) used in the continuous context. In his book, “Théorie des Opérations Linéaires” (1932), after presenting the concept of these functionals, Banach shows, inter alia, how to construct Banach limits for the space of bounded real-valued functions defined on the non-negative half real line.

In Chapter 2 of the thesis we prove some basic properties of Banach limits in this case. Chapter 3 focuses on functions which are almost convergent. Chapter 4 deals with selected topics from Banach space theory, such as duality mappings and the subdifferential. In Chapter 5 some results regarding fixed points and the common fixed point property are reviewed. In Chapter 6 the Bochner integral is constructed. Chapter 7 is dedicated to ergodicity. Our main theorem, Theorem 42, is stated and established in this chapter. It is followed by a brief discussion of open