Topological Fixed Point Principles for Boundary Value Problems

ISBN-13: 9781402013805 / Angielski / Twarda / 2003 / 761 str.

Jan Andres; Lech Gorniewicz; J. Andres

Opis

Our book is devoted to the topological fixed point theory both for single-valued and multivalued mappings in locally convex spaces, including its application to boundary value problems for ordinary differential equations (inclusions) and to (multivalued) dynamical systems. It is the first monograph dealing with the topo logical fixed point theory in non-metric spaces. Although the theoretical material was tendentially selected with respect to ap plications, we wished to have a self-consistent text (see the scheme below). There fore, we supplied three appendices concerning almost-periodic and derivo-periodic single-valued {multivalued) functions and (multivalued) fractals. The last topic which is quite new can be also regarded as a contribution to the fixed point theory in hyperspaces. Nevertheless, the reader is assumed to be at least partly famil iar in some related sections with the notions like the Bochner integral, the Au mann multivalued integral, the Arzela-Ascoli lemma, the Gronwall inequality, the Brouwer degree, the Leray-Schauder degree, the topological (covering) dimension, the elemens of homological algebra, ... Otherwise, one can use the recommended literature. Hence, in Chapter I, the topological and analytical background is built. Then, in Chapter II (and partly already in Chapter I), topological principles necessary for applications are developed, namely: the fixed point index theory (resp. the topological degree theory), the Lefschetz and the Nielsen theories both in absolute and relative cases, periodic point theorems, topological essentiality, continuation-type theorems.

Recenzje

From the reviews:

"This book is the most complete and well written text so far on the applications of topological fixed point principles to boundary value problems for ordinary differential equations and differential inclusions. It is a unique monograph dealing with topological fixed point theory in the framework of non-metric spaces, and part of the material focuses on recent results of one author, or both of them." -- MATHEMATICAL REVIEWS

"The monograph is devoted to the topological fixed point theory ... . The book is self-contained and every chapter concludes by a section of Remarks and Comments ... . I believe that this monumental monograph will be extremely useful to postgraduates students and researchers in topological fixed point theory nonlinear analysis, nonlinear differential equations and inclusions ... . This book should stimulate a great deal of interest and research in topological methods in general and in their applications in particular." (Radu Precup, Studia universitatis Babes-Bolyai Mathematica, Vol. XLIX (1), 2004)

Spis treści

Preface. Scheme for the relationship of single sections. I: Theoretical Background. I.1. Structure of locally convex spaces. I.2. ANR-spaces and AR-spaces. I.3. Multivalued mappings and their selections. I.4. Admissible mappings. I.5. Special classes of admissible mappings. I.6. Lefschetz fixed point theorem for admissible mappings. I.7. Lefschetz fixed point theorem for condensing mappings. I.8. Fixed point index and topological degree for admissible maps in locally convex spaces. I.9. Noncompact case. I.10. Nielsen number. I.11. Nielsen number: Noncompact case. I.12. Remarks and comments. II: General Principles. II.1 Topological structure of fixed point sets: Aronszajn Browder Gupta-type results. II.2. Topological structure of fixed point sets: inverse limit method. II.3. Topological dimension of fixed point sets. II.4. Topological essentiality. II.5. Relative theories of Lefschetz and Nielsen. II.6. Periodic point principles. II.7. Fixed point index for condensing maps. II.8. Approximation method for the fixed point theory of multivalued mappings. II.9. Topological degree defined by means of approximation methods. II.10. Continuation principles based on a fixed point index. II.11. Continuation principles based on a coincidence index. II.12. Remarks and comments. III: Application to Differential Equations and Inclusions. III.1. Topological approach to differential equations and inclusions. III.2. Topological structure of solution sets: initial value problems. III.3. Topological structure of solution sets: boundary value problems. III.4. Poincaré operators. III.5. Existence results. III.6. Multiplicity results. III.7. Wazewski-type results. III.8. Bounding and guiding functions approach. III.9. Infinitely many subharmonics. III.10. Almost-periodic problems. III.11. Some further applications. III.13.Remarks and comments. Appendices. A.1. Almost-periodic single-valued and multivalued functions. A.2. Derivo-periodic single-valued and multivalued functions. A.3. Fractals and multivalued fractals. References. Index.