An Introduction to Nonsmooth Analysis

Nonsmooth Analysis is a relatively recent area of mathematical analysis. The literature about this subject consists mainly in research papers and books. The purpose of this book is to provide a handbook for undergraduate and graduate students of mathematics that introduce this interesting area in detail.

• Includes different kinds of sub and super differentials as well as generalized gradients
• Includes also the main tools of the theory, as Sum and Chain Rules or Mean Value theorems

• Content is introduced in an elementary way, developing many examples, allowing the reader to understand a theory which is scattered in many papers and research books

  Inhalt

  Chapter 1. Basic concepts and results: Upper and lower limits. Semicontinuity. Differentiability. Two important Theorems.
  Chapter 2. Convex Functions: Convex sets and convex functions. Continuity of convex functions. Separation Results. Convexity and Differentiability.
  Chapter 3. The subdifferential of a Convex function: Subdifferential properties. Examples.
  Chapter 4. The subdifferential. General case: Definition and basic properties. Geometrical meaning of the subdifferential. Density of subdifferentiability points. Proximal subdifferential
  Chapter 5. Calculus: Sum Rule. Constrained minima. Chain Rule. Regular functions: Elementary properties. Mean Value results. Decreasing Functions
  Chapter 6. Lipschitz functions and the generalized gradient: Lipschitz regular functions. The generalized gradient. Generalized Jacobian. Graphical derivative
  Chapter 7. Applications: Flow invariant sets. Viscosity solutions. Solving equations.