Nonsmooth Analysis is a relatively recent area of mathematical analysis. The literature about this subject consists mainly in res...
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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
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.