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E-book: Leray-Schauder Type Alternatives, Complementarity Problems and Variational Inequalities

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Complementarity theory, a relatively new domain in applied mathematics, has deep connections with several aspects of fundamental mathematics and also has many applications in optimization, economics and engineering. The study of variational inequalities is another domain of applied mathematics with many applications to the study of certain problems with unilateral conditions. This book is the first to discuss complementarity theory and variational inequalities using Leray-Schauder type alternatives. The ideas and method presented in this book may be considered as a starting point for new developments.

This book is the first to discuss complementarity theory and variational inequalities using Leray-Schauder type alternatives. Complementarity theory, a relatively new domain in applied mathematics, has deep connections with several aspects of fundamental mathematics. The ideas and method presented in this book may be considered as a starting point for new developments. The book presents a new kind of application for the Leray-Schauder principle.

Reviews

From the reviews:









"The author of this book is one of the leading specialists in both the theory and applications of complementarity problems . The book will be of interest to specialists in applied nonlinear analysis, variational inequalities, complementarity theory, equilibrium theory, and operations research. It may also be used to get a glimpse of the diversity of the directions in which current research in this field is still moving." (Jürgen Appell, Zentralblatt MATH, Vol. 1095 (21), 2006)



"The reviewed monograph studies various classes of complementarity problems and variational inequalities using a unified approach based upon the Leray-Schauder alternative and the concept of an exceptional family of elements (EFE). The book is written in a very clear and mathematically rigorous manner, and it is strongly recommended to researchers, postgraduate and graduate students interested in the variational inequality and complementarity problems." (Vyacheslav V. Kalashnikov, Mathematical Reviews, Issue 2007 b)

Preface xi
Preliminary Notions
1(48)
Topological spaces. Some fundamental notions
1(3)
Metric spaces
4(2)
Some classes of topological vector spaces
6(7)
Compactness and compact operators
13(1)
Measures of noncompactness and condensing operators
14(5)
Topological degrees
19(10)
Zero-epi mappings
29(6)
Convex cones
35(5)
Projection operators
40(9)
Complementarity Problems and Variational Inequalities
49(22)
Complementarity problems
49(10)
Variational inequalities
59(3)
Complementarity problems, variational inequalities, equivalences and equations
62(9)
Leray--Schauder Alternatives
71(38)
The Leray--Schauder alternative by topological degree
72(2)
The Leray--Schauder alternative by the fixed point theory
74(2)
The Leray-Schauder alternative by the topological transversality theory
76(5)
Some classes of mappings and Leray--Schauder type alternatives
81(9)
An implicit Leray--Schauder alternative
90(5)
Leray--Schauder type alternatives for set-valued mappings
95(14)
The Origin of the Notion of Exceptional Family of Elements
109(28)
Exceptional family of elements, topological degree and nonlinear complementarity problems in Rn
109(9)
Exceptional family of elements, topological degree and implicit complementarity problems in Rn
118(3)
A general notion of an exceptional family of elements for continuous mappings
121(6)
An exceptional family of elements, zero-epi mappings and nonlinear complementarity problems in Hilbert spaces
127(4)
Two applications
131(6)
Leray--Schauder Type Alternatives. Existence Theorems
137(88)
Nonlinear complementarity problems in arbitrary Hilbert spaces
138(33)
Implicit complementarity problems
171(9)
Set-valued complementarity problems
180(14)
Exceptional family of elements and monotonicity
194(7)
Semi-definite complementarity problems
201(2)
Feasibility and an exceptional family of elements
203(12)
Path of ε-solutions and exceptional families of elements
215(10)
Infinitesimal Exceptional Family of Elements
225(22)
Scalar derivatives
225(101)
Infinitesimal exceptional family of elements
326
Applications to complementarity theory
238(6)
Infinitesimal interior-point-ε-exceptional family of elements
244(3)
More About the Notion of Exceptional Family of Elements
247(32)
EFE-acceptable mappings
247(9)
Skrypnik topological degree and exceptional families of elements
256(10)
A necessary and sufficient condition for the non-existence of an exceptional family of elements for a given mapping
266(5)
Exceptional family of elements. Generalization to Banach spaces
271(8)
Exceptional Family of Elements and Variational Inequalities
279(34)
Explicit Leray--Schauder type alternatives and variational inequalities
279(13)
Implicit Leray--Schauder type alternatives and variational inequalities
292(11)
Asymptotic Minty's variational inequalities and condition (θ)
303(3)
Complementarity problems and variational inequalities with integral operators
306(6)
Comments
312(1)
Bibliography 313(22)
Index 335


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