Geometrical Methods of Nonlinear Analysis

Description

The methods we deal with in the present book originated long ago. Theydate back to Kronecker, Poincare, Brouwer, and Hopf who developed thetopological theory of continuous mappings in finite dimensional spaces. Asecond stream of ideas originated from the investigations of Birkhoff andKellogg, Banach, Cacciopoli, Schauder, and A.N. Tihonov on fixed pointtheory. A major contribution was supplied by the work of A.M. Ljapunov,E. Schmidt, A.I. Nekrasov, P.S. Uryson, Hammerstein, and Golomb on thetheory of various classes of integral equations as well as by L.A. Ljusternik'swork on variational problems. The contributions by N.N. Bogoljubov andN.M. Krylov, M.V. Keldys, G.I. Petrov, and A.I. Lur'e to projection methodsfor approximate solutions of equations particularly influenced the development of methods in nonlinear operator equations. These methods arefundamental to many schemes of construction of topological invariants formappings in infinite dimensional spaces. In the course of constructing thesemethods, ideas from L.V. Kantorovic's approximation theory proved veryuseful. The last fifteen to twenty years, finally, brought novel approachesand solutions to new problems.In our time, qualitative geometric methods for the investigation of nonlinearoperator equations are forming a well established theory with manyapplications to integral equations, various boundary value problems, andproblems from physics and mechanics.In the first chapter we study vector fields in finite dimensional spaces.Here, the notion of rotation of a vector field on the boundary of a domainturns out to be an important concept. A detailed discussion of this conceptcan be found in many textbooks on topology. In our context, the mainemphasis will be on theorems and methods allowing us to effectively computeor estimate the rotation of concrete vector fields. In order to understandthe first chapter, the reader is not assumed to have any specialknowledge of topological facts. This is because in the main constructions,we shall use only three main properties of rotation which are easy toformulate.Rotation for various classes of vector fields in function spaces is discussedin Chapters 2 and 4. The most detailed description of the theory ofcompletely continuous vector fields has originally been given by Leray andSchauder. We shall also consider various other classes of fields involvingFredholm operators or operators which are positive, condensing, or multivaluedor which have other special properties. The main emphasis willalways be on the problem of how to actually compute or estimate therotation.To compute or estimate rotations of vector fields, one uses theorems onrelatedness or invariance of rotation. Chapter 3 is devoted to this topic.Theorems on relatedness or invariance of rotation contain formulae relatingrotations of different vector fields (on the boundaries of the respectivedomains) describing one and the same problem in different function spaces.The idea underlying the theorem on relatedness may, for instance, beillustrated by problems on forced vibrations in systems described by ordinarydifferential equations. The initial values of periodic solutions are thensingular points of the vector field of translation (by the period) alongtrajectories of the vector field. These fields are defined in finite dimensionalphase space. Although the fields are not explicitly known they can beanalyzed if one uses well established methods from the qualitative theory ofdifferential equations. On the other hand, the existence problem for periodicsolutions can be transformed into a problem in nonlinear integral equations;periodic solutions then correspond to singular points of a vector fieldin some function space with a completely continuous integral operator givenby an explicit formula. In the problem on forced vibrations, the theorem onrelatedness provides a formula relating the rotation of the vector field of