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Knowing that we have used a lot of mathematical codes in the main text, you will no doubt accept another one for lecture notes series and publishers quoted most frequently. In cases where an English text does the same job as a text written in different language we have preferred the English one. This remark applies, in particular, to translations of Russian references if available. Finally, as in many other books, you will notice that the author quotes himself too often; in our case this is simply explained by the fact that we have contributed many exercises.

Publishers: Acad. Press = Academic Press, New York; AMS = American Mathematical Society, Providence; Benjamin = W. A. Benjamin, Inc., New York; LMS = London Mathe­matical Society, London; North-Holland = North-Holland Publ. Company, Amsterdam; Pitman = Pitman Publishing Ltd., London; Prentice Hall = Prentice-Hall, Inc., Englewood Cliffs, New Jersey; Springer = Springer-Verlag, Berlin or New York; Univ. Press = Princeton University Press, Princeton; Wiley = J. Wiley&Sons, New York

Lecture Notes: LNM = Lecture Notes in Mathematics, Springer; RNM = Research Notes in Mathematics, Pitman

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1982 Browder, F. E.: [1] On the spectral theory of elliptic differential operators 1. Math. Ann. 142,

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Symbols

ct-MNC, fJ-MNC 41 a.c. = absolutely continuous a.e. = almost everywhere ANR 354 AR 354 catM 354 codimX 51 conv(D) 17 (c)o 43 C(B) 5 C(B; ]Rm) 5 Cx(D) 42 Ck(Q) 6 Ck(Q) 6 Coo (Q) 6 CL(X) 61 Cy(Q) 69 diamB 41 DI ~ Dz 288 ocp(xo) 281 DLS 72 epi(cp) 282 Iflo 5 f IA = restriction of f to A 7 Fx 38 .'?x 115 .'?(Q, Y) 55 P,,(xo, Yo) 46 grad cp 32 graph(T) 51 I = id = identity ind cp 32

i(F, Q, K) = i(F, Q) 238 ind(T) 78 int(A) = A 50 If (x) 6 %(Q, Y) 55 lsc = lower semicontinuous [P 44 [00 46 IJ'(J) 44 L(X, Y), L(X) 38 Ln(X, Y) 46 .IN 51 N(T) 51 (9 (x) 187 o(lhl) 6 r(x) 79 rq,(T) 80 Q(x) 79 R(T) 51 Sf(Q) 6 0" (x) 79 O"q,(T) 80 SCy(Q) 69 supp cp 7 T* 51 X-I 50

X<c 79 X* 38 X.l 51 X-I 50 x* 38 (x, x*) 38

446

(x, y)± 123 X ~ Y 218 usc = upper semicontinuous w, w*-topology 88 cp 323 <p 328

<1> (X) 78 <1>+ (X) 78 --' weak convergence ~ weak star convergence 2A 280

Symbols

Index

approximations Galerkin 257 successive 39 Yosida 104

a priori bound 60 atlas 358

ball-measure of noncompactness 41 Banach algebra 50

lattice 237 barrel 297 base (Schauder) 43, 237, 258 bifurcation

supercritical, subcritical 385 trans critical 385 vertical 385

bifurcation point 380 asymptotic 387 inverse 390 positive 402

Borel a-algebra 306 bundle

tangent 361 cotangent 361

category (Lusternik-Schnirelmann) 354 chart 358 cluster point of a net 89 complexification of a Banach space 79 commutativity property of degree 214 component (connected) 17 cone 218

dual 221 minihedral (strongly) 219 normal 219 plastering 219 regular (fully) 219

reproducing 219 total 219

conical shell 239 contraction

directional 202 generalized 201 k-l'- 69 strict 69

convergence discrete 266 weak 88 weak star 88

covering structure 357 critical point 6, 349, 374

deformation 350 derivative

directional 46 Frechet 46 Gateaux 46 weak Gateaux 154

diameter of a set 41 differential 361 direct sum (topological) 51 dual problem 324

eigenvalue algebraic multiplicity 64 geometric multiplicity 64 inverse problem 20

epigraph 282 e-chain 400 equation

Abel-Liouville integral 201 approximation solvable 261 auxiliary 1 76 Bellman functional 346 bifurcation 176

448

equation Euler 339 evolution 137 Fredholm integral 82 functional differential 76 Hamilton-Jacobi 293 Hammerstein integral 101, 106, 111,

119, 156, 377 Ricatti 341 Schroder functional 164 Theodorsen integral 202 Volterra integral (first kind) 125, 246 wave 181,273 Wiener-Hopf integral 82, 177

equicontinuity 42 equidifferentiability 46 exponential formula 138

Fenchel transform 289 fibration 357 Finsler

metric 363 structure 362

fixed point asymptotic theory 213 index 238 property 213

function absolutely continuous 138 Lebesgue measurable 306 quasi-periodic 169

functional 38 closure of 328 conjugate 289, 323 continuous linear 38 convex 279, 281 lower semicontinuous 283 Minkowski 88, 135 positive linear 221 second conjugate 327 strictly positive 221 sublinear 52 support 322 uniformly positive 221

Gelfand formula 79 generator (infinitesimal) 128, 141 genus 355 gradient 32, 46, 347

Hamiltonian system 169,373 Hausdorff distance 41 hyperplane

non vertical 282 real 52 separating 52 supporting 52

index fixed point 238 isolated solution 30 Fredholm operator 78 gradient map 32 Morse 374

indicator 283 inequality

Holder 48 quasivariational 346 variational 346

Jacobian 6

kone (dual) 334 Kronecker symbol 5

Index

Kuratowski measure of noncompactness 41

Lagrange multiplier 333 Landau symbol 6 Lemma

Du Bois-Reymond 347 Gronwall 68 Sard 9

Leray-Schauder boundary condition 204 Lipschitz condition 39 locally Lipschitz 53 Lyapunov-Schmidt method 176

majorant 232 manifold 358

class Ck , Ck - 359 codimension k 358 Finsler 363

map (see also operator) analytic 150 bounded 69 Dunford 80 duality 115 gradient 32 multivalued (see multi) odd 21 tangent 361 weakly inward 207, 245

measure of noncompactness 41, 85 method of moments 266 minorant 232 model

chemical reactor 243, 248, 422 population growth 240 thermostat 310

mollifier 7 monomial (n-th order) 54 monotonic sequence 218 Morse theory 374 multi

accretive 280

Index

multi cyclically monotone 297 lower semicontinuous 299 monotone 280 Poincare 304 upper semicontinuous 299 s-w*-upper semicontinuous 281

net 89 Newton method 157 norm

equivalent 38 monotonic 219 semimonotonic 219

operator 38 accretive (strictly, strongly) 124 adjoint 51 almost strongly positive 238 angle-bounded 133 A-proper 259 asymptotically linear 66, 225 closed 55 coercive 100, 117 compact 55 n-compact 269 completely continuous 55 y-condensing 69 L-y-condensing 173 demicontinuous 98 dissipative 97, 124 finite-dimensional 55 Fredholm 78, 91 hemicontinuous 98 hyperaccretive 126 hypermaximal monotone 102, 119 increasing (strongly) 224 y-Lipschitz 69 limit compact 85 linear 38 locally compact 87 maximal accretive 126 maximal monotone 102, 119 monotone 97, 117 Nemytskii 48 nonexpansive 69 normally solvable 198 order convex 252 Poincare 19 positive (strongly) 226 potential 47 proper 55 pseudo-A-proper 267 pseudo-contractive 202 pseudomonotone 110, 120 quasibounded 67 quasicompact linear 234 quasi monotone 252

semicondensing 130 semi-Fredholm 78 semimonotone 11 0 strictly monotone 97, 117 strongly monotone 97 type (S) 123

order bounded 224 interval 224 topology 229 unit 234

paracompactness 47 partition of unity 54 potential 32, 47 Pontryagin maximum principle 345 programming

convex 330 dynamic 346 mathematical 330

projection 51 metric 67 scheme 257

property finite union 40 finite intersection 40 reduction of degree 29, 59, 74

pseudo inverse 168

quasi-interior point 234

relaxed problem 329 retract 45, 66

absolute 354 absolute neighbourhood 354

saddle point 329 sandwich problem 24 scale of Banach spaces 159 semigroup 128 semi-inner product 96,115, 123 seminorm 41, 52 set

absorbant 87 balanced 87 bounded 40, 87 bounded from above 218 Chebyshev 67 compact 40, 87 contractible 353 convex 17 dense 40 normal structure (asymptotic) 195 precompact 40 relatively compact 40 resolvent 79 separable 40

449

450

set spectral 80 totally bounded 40

small divisor problem 164 solution

DS 140 integral 139 nonnegative 219

space Banach 38 cotangent 359 Frechet 88 locally convex 88 locally uniformly convex 111 measurable 305 oriented 29 Orlicz 135 projective 356 reflexive 92 Sobolev 109 strictly convex 111 tangent 359 topological vector 87 uniformly convex 111

spectral radius 79 spectral theory 79 spectrum 79

essential 80 Fredholm 80

stability of equilibrium 416 linearized 416 orbital 421

subditTerential 281 subgradient 281 subnet 89 sun 67 support 7

theorem Alaoglu 93 Arens-Eells 54 Baire's category 50 Banach fixed point 39 Bohnenblust-Karlin fixed point Borsuk 21, 74, 90 Borsuk-Ulam 22 Brouwer fixed point 17 closed graph 51

315

closed range 51 Darbo fixed point 69, 71 domain invariance 23, 58, 74, 90 Dubovitskii-Milyutin 343 Dugundji extension 44 Eberlein-Smulyan 92 Fan fixed point 315 Fan saddle point 329 Fenchel duality 323 Hadamard 153 Hahn-Banach 52 hedgehog 19 Hille-Y osida 140 Hopf 29 implicit function ('hard') 148,162 inverse function 7, 149 Jordan's separation 26, 59, 75 Kakutani fixed point 315 Kolmogorov 88 Krasnoselskii fixed point 71 Krein-Rutman 226 Lusin 103 Lusternik 345 Lusternik -Schnirelmann-Borsuk 22 Mazur 52 mod p 214 v. Neumann intersection 317 open mapping 51, 154 Perron-Frobenius 18 Pringsheim 227 Schauder fixed point 60 Smulyan 113 spectral (compact operators) 63 Trojanski 113 Tychonov fixed point 90 uniform boundedness 50

Toricelli point 322, 326

value critical 349 regular 6, 91 singular 6

vector field 361 essential 92 inessential 92 pseudo-gradient 363 tangent 20, 361

winding number 2, 30

Index

Springer-Verlag Berlin Heidelberg New York Tokyo

M.A.Krasnosel'skii, P.P.Zabrelko

Geometrical Methods of Nonlinear Analysis Translated from the Russian by C. Fenske 1984. XIX, 409 pages. (Grundlehren der mathema­tischen Wissenschaften, Band 263) ISBN 3-540-12945-6

Contents: Vector Fields in Finite Dimensional Spaces. - Completely Continuous Vector Fields. - Principles of Relatedness. - Fields With Noncompact Operators. - Solvability of Nonlinear Equations. - Equations With Many Solutions. - Construction of Solutions. -Small Perturbations of Nonlinear Equations. - Refe­rences. - List of Symbols. - Index.

M.Golubitsky, D.G.Schaeffer

Singularities and Groups in Bifurcation Theory Volume I 1985. 114 figures. XVII, 463 pages. (Applied Mathe­matical Sciences, Volume 51) ISBN 3-540-90999-0

Contents: A Brief Introduction to the Central Ideas of the Theory. - The Recognition Problem. - Unfolding Theory. - Classification by Codimension. - An Example of Moduli. - Bifurcation with Z2-Symmetry. - The Liapunov-Schmidt Reduction. - The Hopf Bifurcation. - Two Degrees of Freedom Without Symmetry. - Two Degrees of Freedom with (Z2Ef)ZJ­Symmetry. - Bibliography. - Symbol Index. - Index.

Springer-Verlag Berlin Heidelberg New York Tokyo

E.Zeidler

Nonlinear Functional Analysis and Its Applications Part I: Fixed Point Theorems

Translated from the German by P. Wadsack 1984. ISBN 3-540-90914-1

From the contents: The two basic ftxed point theorems of Banach and Schauder. - Calculus in Banach spaces. - The implicit function theorem. - Newton's method. - The conti­nuation method. - Positive operators. - Analytic bifurcation theory. - Fixed point theorems for multivalued mappings. -Nonexpansive and condensing operators. - Mapping degree and ftxed point index and their applications. - Analytic maps. - Topological bifurcation theory. - Essential maps. -Asymptotic ftxed point theorems. - Numerous applications to ordinary and partial differential equations, integral equa­tions, game theory etc. - Comprehensive bibliography.

E.Zeidler

Nonlinear Functional Analysis and its Applications Part III: Variational Methods and Optimization

Translated from the German by L. F. Boron 1985. 111 ftgures. XXII, 662 pages. ISBN 3-540-90915-X

Contents: Introduction to the Subject. - Two Fundamental Existence and Uniqueness Principles. - Extremal Problems Without Side Conditions. - Extremal Problems with Smooth Side Conditions. - Extremal Problems with General Side Conditions. - Saddle Points and Duality. -Variational Inequalities. - Appendix. - References. - List of Symbols. - List of Theorems. - List of the Most Important Deftnitions. - Index.

In preparation:

E. Zeidler, Nonlinear Functional Analysis and Its Applications II ISBN 3-540-90913-1

E. Zeidler, Nonlinear Functional Analysis and Its Applications IV