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 !=«*,«)                                  P.4) 31 f £o«i = 0 34 2.2 The theorem of M. G. Crandall and P. H. Rabi- nowitz 35 2.3 Local bifurcation diagrams 40 v° 42 2.4 The exchange stability principle 42 <&(4 43 £W3(o,o,o) : Rxr^y 44 ft-Hn 44 2.5 Applications 51 £(A,d)(M = (j5) 58 A. 67 \i 73 Chapter 3 77 First general bifurcation results 77 3.1 Lyapunov-Schmidt reductions 79 3.2 The theorem of J. Ize 83 <—^<x.'0 86 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 0£*(H iV[£f ]) © i*[£*] = V 135 Chapter 5 138 Other fundamental properties of the multiplicity 138 5.1 The multiplicity of R. J. Magnus 139 ^o-+i}( 143 (5.9) 143 (5.14) 145 5.3 The fundamental theorem 154 5.4 The classical algebraic multiplicity 157 i>o, 160 5.5 Finite dimensional characterizations 163 (5.69) 165 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 £:=e;n{(A,o) : Ags} 182 6.4 Unilateral global bifurcation 188 enQ-„ndfl*(Ao,o)^0 195 (A^Oeer^Ao^) 197 6.5 Unilateral bifurcation for positive operators 199 Chapter 7 205 Applications 205 7.1 Positive solutions of semilinear elliptic problems 205 \] 221 7.2 Coexistence states for elliptic systems 224 r£ u rj = 90 224 7.3 Examples 239 0i*,0,v)e£ 241 -P9 252 A<af[-A,D] 253 7.4 A further application 254 References 260