Marek Galewski

This textbook gives an introduction to optimization tools which arise around the Weierstrass theorem about the minimum of a lower semicontinuous function. Starting from a Euclidean space, it moves further into the infinite dimensional setting towards the direct variational method, going through differentiation and introducing relevant background information on the way. Exercises accompany the text and include observations, remarks, and examples that help understand the presented material. Although some basic knowledge of functional analysis is assumed, covering Hilbert and Banach spaces and the Lebesgue integration, the required background material is covered throughout the text, and literature suggestions are provided. For less experienced readers, a summary of some optimization techniques is also included. The book will appeal to both students and instructors in specialized courses on optimization, wishing to learn more about variational methods.

This textbook introduces some basic tools from the theory of monotone operators together with some of their applications. Examples that work for ordinary differential equations are provided. The illustrating material is kept relatively simple, while at the same time offering inspiring applications to the reader. The material will appeal to graduate students in mathematics who want to learn some basics in the theory of monotone operators. Furthermore, it offers a smooth transition to studying more advanced topics pertaining to more refined applications by shifting to pseudomonotone operators, and next, to multivalued monotone operators.