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Handbook of Applied Analysis

Part of the Advances in Mechanics and Mathematics Book Series (volume 19)

DOI: 10.1007/b120946

ISBN: 978-0-387-78906-4

Publisher: Springer US - Springer Journals

May 29, 2009

1 - 10 of 10 Chapters

LitStream

Chapter 1Smooth and Nonsmooth Calculus

Papageorgiou, Nikolaos S.; Kyritsi-Yiallourou, Sophia Th.

Pages: 1 –61

[*In this chapter we provide an overview of the main aspects of the mooth and nonsmooth calculus. We start with the smooth calculus and we introduce the notions of Gâteaux and Fréchet derivatives. We develop all the important calculus rules and we present the implicit and inverse function...
LitStream

Chapter 2Extremal Problems and Optimal Control

Papageorgiou, Nikolaos S.; Kyritsi-Yiallourou, Sophia Th.

Pages: 63 –145

[*This chapter is devoted to the study of various kinds of variational problems and of the mathematical tools needed to deal with them. We start with the so-called “direct method”, that leads to a systematic study of lower semicontinuous functionals. Then we consider optimization problems with...
LitStream

Chapter 3Nonlinear Operators and Fixed Points

Papageorgiou, Nikolaos S.; Kyritsi-Yiallourou, Sophia Th.

Pages: 147 –265

[*In this chapter, first we study certain broad classes of nonlinear operators that arise naturally in applications and then we develop some related degree theories and fixed point theorems. We start with compact operators, we conduct a detailed study of both linear and nonlinear operators. Then...
LitStream

Chapter 4Critical Point Theory and Variational Methods

Papageorgiou, Nikolaos S.; Kyritsi-Yiallourou, Sophia Th.

Pages: 267 –350

[*The variational method in the study of nonlinear boundary value problems is based on the critical point theory, that provides minimax characterizations of the critical values over certain homotopically stable families of sets. Using the deformation method, we derive the main results of the...
LitStream

Chapter 5Boundary Value Problems–Hamiltonian Systems

Papageorgiou, Nikolaos S.; Kyritsi-Yiallourou, Sophia Th.

Pages: 351 –454

[*Having the general abstract theories developed in Chapter 4, in this chapter we deal with concrete boundary value problems (both for ordinary and partial differential equations). In the first section, we illustrate the uses of the so-called “variational method“, which is based on the critical...
LitStream

Chapter 6Multivalued Analysis

Papageorgiou, Nikolaos S.; Kyritsi-Yiallourou, Sophia Th.

Pages: 455 –527

[*Multivalued analysis deals with the study of maps whose values are sets. Multivalued analysis is closely related to nonsmooth analysis and a symbiotic relationship exists between them which feeds both with new ideas and directions to grow. This chapter presents in detail the main aspects of...
LitStream

Chapter 7Economic Equilibrium and Optimal Economic Planning

Papageorgiou, Nikolaos S.; Kyritsi-Yiallourou, Sophia Th.

Pages: 529 –608

[*This chapter is devoted to mathematical economics. We start with the static model of an exchange economy with a measure space of agents (an abstract device to model perfect competition). We introduce the notions of core allocation and of Walras allocations. We show that in the context of...
LitStream

Chapter 8Game Theory

Papageorgiou, Nikolaos S.; Kyritsi-Yiallourou, Sophia Th.

Pages: 609 –650

[*Game-theoretic models provide a substantial amount of generalization of the basic notions of mathematical economics, such as core, equilibria, saddle points and intertempolar optimum. In this chapter, we deal with different models in game theory. We start with noncooperative games which lead to...
LitStream

Chapter 9Uncertainty, Information, Decision Making

Papageorgiou, Nikolaos S.; Kyritsi-Yiallourou, Sophia Th.

Pages: 651 –690

[*In this chapter, we study how information can be incorporated as a variable in various decision models. In a decision model with uncertainty, it is natural to model information by sub-σ-fields of a probability space which represents the set of all possible states of the “world”. For this reason...
LitStream

Chapter 10Evolution Equations

Papageorgiou, Nikolaos S.; Kyritsi-Yiallourou, Sophia Th.

Pages: 691 –751

[*Evolution equations are the abstract formulation of dynamic partial differential equations. In this chapter, we examine both semilinear and nonlinear evolution equations. We start by developing the mathematical tools which are necessary in this study. Among them, the notion of evolution triple...

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