Contents 4
 Preface 6
 Chapter 1 10
 Introduction 10
 1.1    General assumptions and basic concepts 11
 1.2    Some new results 15
 1.3    Historical remarks 17
 Chapter 2 29
 Bifurcation from simple    eigenvalues 29
 2.1    Simple eigenvalues and transversality 30
 2.2    The theorem of M. G. Crandall and P. H. Rabi-    nowitz 35
 2.3   Local bifurcation diagrams 40
 2.4    The exchange stability principle 42
 ft-Hn 44
 2.5    Applications 51
 A. 67
 Chapter 3 77
 First general bifurcation    results 77
 3.1    Lyapunov-Schmidt reductions 79
 3.2    The theorem of J. Ize 83
 3.3   The global alternative of P. H. Rabinowitz 90
 3.4    The theorem of D. Westreich 92
 Chapter 4 96
 The algebraic multiplicity 96
 4.1    Motivating the concept of transversality 98
 4.2    Transversal eigenvalues 101
 4.3   Algebraic eigenvalues 114
 4.4    Analytic families 128
 4.5   Simple degenerate eigenvalues 134
 Chapter 5 138
 Other fundamental properties    of the multiplicity 138
 5.1    The multiplicity of R. J. Magnus 139
 5.3    The fundamental theorem 154
 5.4   The classical algebraic multiplicity 157
 5.5   Finite dimensional characterizations 163
 5.6    The parity of the crossing number 166
 Chapter 6 172
 Global bifurcation theory 172
 6.1    Preliminaries 175
 6.2   Local bifurcation 177
 6.3   Global behavior of the bounded components 181
 6.4   Unilateral global bifurcation 188
 6.5   Unilateral bifurcation for positive operators 199
 Chapter 7 205
 Applications 205
 7.1   Positive solutions of semilinear elliptic problems 205
 7.2   Coexistence states for elliptic systems 224
 7.3    Examples 239
 -P9 252
 7.4    A further application 254
 References 260