**Mathematical Surveys and Monographs**

Volume: 11;
1964;
198 pp;
Softcover

MSC: Primary 34;
Secondary 57

**Print ISBN: 978-0-8218-1511-3
Product Code: SURV/11**

List Price: $57.00

AMS Member Price: $45.60

MAA Member Price: $51.30

**Electronic ISBN: 978-1-4704-1239-5
Product Code: SURV/11.E**

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# Fixed points and topological degree in nonlinear analysis

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*J Cronin*

The topological methods based on fixed-point theory and on local
topological degree which have been developed by Leray, Schauder, Nirenberg,
Cesari and others for the study of nonlinear differential equations are here
described in detail, beginning with elementary considerations. The reader is
not assumed to have any knowledge of topology beyond the theory of point sets
in Euclidean n-space which ordinarily forms part of a course in advanced
calculus.

The methods are first developed for Euclidean n-space and applied to the
study of existence and stability of periodic and almost-periodic solutions of
systems of ordinary differential equations, both quasi-linear and with
“large” nonlinearities. Then, after being extended to
infinite-dimensional “function-spaces”, these methods are applied
to integral equations, partial differential equations and further problems
concerning periodic solutions of ordinary differential equations.

#### Table of Contents

# Table of Contents

## Fixed points and topological degree in nonlinear analysis

- TABLE OF CONTENTS xi12 free
- CHAPTER I. TOPOLOGICAL TECHNIQUES IN EUCLIDEAN n-SPACE 114 free
- 0. Introduction 114
- 1. The fixed point theorem 114
- 2. The order of a point relative to a cycle: cells, chains, and cycles; orientation of R[sup(n)]; intersection numbers; order of a point relative to a cycle 215
- 3. The order of a point relative to a continuous image of z[sup(n–1)] 1629
- 4. Properties of v[φ, K, p] 2538
- 5. The local degree relative to a complex 2639
- 6. The local degree relative to the closure of a bounded open set 3043
- 7. The local degree as a lower bound for the number of solutions 3245
- 8. A product theorem for local degree 3649
- 9. Computation of the local degree 3750
- 10. A reduction theorem and an in-the-large implicit function theorem 5063
- 11. A proof of the fixed point theorem 5265
- 12. The index of a fixed point 5265
- 13. The index of a vector field 5366
- 14. Generalizations 5467

- CHAPTER II. APPLICATIONS TO ORDINARY DIFFERENTIAL EQUATIONS 5669
- 1. Some existence theorems for differential equations 5669
- 2. Linear systems 6376
- 3. Existence of periodic solutions of nonautonomous quasilinear systems 6477
- 4. Some stability theory 7083
- 5. Stability of periodic solutions of nonautonomous quasilinear systems 7588
- 6. Some examples of nonautonomous quasilinear systems 8598
- 7. Almost periodic solutions of nonautonomous quasilinear systems 96109
- 8. Periodic solutions of autonomous quasilinear systems 105118
- 9. Periodic solutions of systems with a "large" nonlinearity 109122

- CHAPTER III. TOPOLOGICAL TECHNIQUES IN FUNCTION SPACE 119132
- 1. Introduction 119132
- 2. Some linear space theory 120133
- 3. Examples which show that a fixed point theorem and a definition of local degree cannot be obtained for arbitrary continuous transformations from a Banach space into a Banach space: Kakutani's example; Leray's example 124137
- 4. Compact transformations: properties of compact transformations; Schauder Fixed Point Theorem; Schaefer's Theorem 130143
- 5. Definition and properties of the Leray-Schauder degree 134147
- 6. Proof of the Schauder Theorem using the Leray-Schauder degree 139152
- 7. Computation of the Leray-Schauder degree 139152
- 8. A partially analytic approach: contraction mappings; Banach Fixed Point Theorem; some further Banach space theory; local study 140153

- CHAPTER IV. APPLICATIONS TO INTEGRAL EQUATIONS, PARTIAL DIFFERENTIAL EQUATIONS AND ORDINARY DIFFERENTIAL EQUATIONS WITH LARGE NONLINEARITIES 151164
- 1. Introduction 151164
- 2. Integral equations 152165
- 3. Problems in partial differential equations 156169
- Elliptic differential equations 156169
- 4. Statement of the Leray-Schauder-Nirenberg result 156169
- 5. The Banach spaces in which the Leray-Schauder-Nirenberg result is formulated 157170
- 6. The Schauder Existence Theorem . 161174
- 7. The Leray-Schauder method 162175
- 8. The Nirenberg method 167180
- 9. Some other work on elliptic equations 170183
- 10. Local study of elliptic differential equations 171184

- Parabolic differential equations 176189
- Hyperbolic differential equations 180193
- Ordinary differential equations 180193

- BlBLlOGRAPHY 186199
- INDEX 195208