Table of contents

CONTENTS Introduction Chapter I. Linear algebra in superspaces § 1. Linear superspaces § 2. Modules over superalgebras § 3. Matrix algebra § 4. Free modules § 5. Bilinear forms § 6. The supertrace § 7. The Berezinian (Berezin function) § 8. Tensor algebras § 9. Lie superalgebras and derivations of superalgebras Chapter II. Analysis in superspaces and superdomains § 1. Definition of superspaces and superdomains § 2. Vector fields and Taylor series § 3. The inverse function theorem and the implicit function theorem § 4. Integration in superdomains Chapter III. Supermanifolds § 1. Definition of a supermanifold § 2. Subsupermanifolds § 3. Families Notes References

CONTENTS Introduction Chapter I. Approximative methods in the fixed-point theory of multi-valued maps § 1.1. Multi-valued maps and single-valued approximations § 1.2. The rotation of multi-valued vector fields with convex images and fixed-point theorems § 1.3. Obstruction theory and single-valued approximations of multi-valued maps § 1.4. Guide to the literature in Chapter I Chapter II. Homological methods in the fixed-point theory of multi-valued maps. The finite-dimensional case § 2.1. Formulation of a version of the Vietoris-Begle-Sklyarenko theorem § 2.2. The topological characteristic of a multi-valued vector field in a finite-dimensional space § 2.3. The rotation and the topological characteristic of -acyclic and generalized -acyclic multi-valued vector fields § 2.4. Some theorems on the computation of the topological characteristic § 2.5. Fixed-point theorems § 2.6. The Lefschetz theorem § 2.7. Guide to the literature in Chapter II Chapter III. Homological methods in the fixed-point theory of multi-valued maps. The infinite-dimensional case § 3.1. Partitions and the cohomology defined by them § 3.2. The topological characteristic of a multi-valued vector field in a Banach space § 3.3. The rotation of almost acyclic multi-valued vector fields § 3.4. Computation of the topological characteristic and fixed-point theorems § 3.5. Guide to the literature in Chapter III Appendix. Some applications References

CONTENTS Introduction CHAPTER I. Generalization of the second Lyapunov method § 1. Lyapunov functions that are positive-definite in part of the variables § 2. Equipotential surfaces of the perturbed Lyapunov function § 3. Lemma on the proximity of solutions of the systems (1) and (3) § 4. Investigation of stability by means of a perturbed Lyapunov function defined on an annular domain § 5. Investigation of stability over a finite interval § 6. Investigation of stability in higher approximations § 7. Theorems on instability in the "neutral" case CHAPTER II. Investigation of the stability of resonance problems § 1. Statement of the problem § 2. Investigation of the stability of systems of equations of the form (1.1) having an asymptotically stable averaged system in the single frequency case § 3. Investigation of the stability of systems of equations of the form (1.1) having an asymptotically stable averaged system in the multi-frequency case § 4. Investigation of the stability of a multi-frequency system for a finite time interval CHAPTER III. Investigation of the stability of orbits in the three-body problem § 1. Canonical variables, equations of motion, and integrals of motion in the three-body problem § 2. Resonance curves and the choice of new variables § 3. Construction of a perturbed Lyapunov function and investigation of stability in the three-body problem References