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Zeidler E. — Nonlinear Functional Analysis and Its Applications: Part 1: Fixed-Point Theorems
Zeidler E. — Nonlinear Functional Analysis and Its Applications: Part 1: Fixed-Point Theorems



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Название: Nonlinear Functional Analysis and Its Applications: Part 1: Fixed-Point Theorems

Автор: Zeidler E.

Аннотация:

This is the first of a five-volume exposition of the main principles of nonlinear functional analysis and
its applications to the natural sciences, economics, and numerical analysis. The presentation is self-contained and
accessible to the nonspecialist. Among the topics of Volume I are the two basic fixed-point theorems of Banach and
Schauder, calculus in Banach spaces, the implicit function theorem, Newton`s method, analytic bifurcation theory,
fixed-point theorems for multivalued mappings, nonexpansive and condensing operators, mapping degree and fixed-point
index and their applications, analytic maps, and asymptotic fixed-point theorems.
The book contains numerous applications to such areas as ordinary and partial differential equations,
integral equations, and game theory. Many exercises and a comprehensive bibliography complement the text.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1986

Количество страниц: 897

Добавлена в каталог: 22.12.2011

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
Fixed-point, trick for semi-linear equations      399
Formal languages      510
Formal languages, and the Tarski fixed-point theorem      511
Foundations of mathematics      746ff 806
Fourier series      787
Fr$\acute{e}$chet derivative      see F-derivative
Fr$\acute{e}$chet space (F-space)      784
Fredholm alternatives, closed range theorem      111
Fredholm alternatives, compact operators      373
Fredholm alternatives, differential equations      259ff 792
Fredholm alternatives, dual pairs      788ff
Fredholm alternatives, Fredholm operators      366 370
Fredholm alternatives, integral equations      788ff
Fredholm integral equations of first kind      805
Fredholm integral equations of second kind      805
Fredholm mappings, linear      365
Fredholm mappings, nonlinear      366
Fredholm mappings, properties of      365ff
Functional      see Operator
Functional calculus      798 (see also Part II)
Functions of operators      797
Fundamental theorem of algebra and mapping degree      520 617
Fundamental theorem of algebra, generalized      639
Fundamental theorem of algebra, historical remarks      675
G$\hat{a}$teaux-derivative      see G-derivative
G-derivative, basic definition      135
G-derivative, basic ideas      131
G-derivative, nth order      136
G-differential, basic definition      135
G-differential, basic ideas      131
Game theory      461 469
Gauge field theory      67 114 728;
Generating order cone      276
Generic bifurcation      381 391 416 428
Generic finiteness of the solution set      189
Genericity      106 802
Genericity, as proof strategy      188 253ff 525
Gerschgorin disk      796
Graph norm      778
Graph of a map      448 748
GREATEST      see Element
Green's function      791
Green's function, and reduction of differential equations to integral equations      59 791
Green's function, generalized solution operator and      238
Green's function, positivity of      338
H$\ddot{o}$lder continuous functions      230ff
H$\ddot{o}$lder continuous functions, embedding theorems for      232
H$\ddot{o}$lder continuous functions, properties of      248
H-isomorphism      785
Haar measure      67 466
Hard implicit function theorem      218ff
Hausdorff dimension      see Part V
Hausdorff measure of noncompactness      611
Hausdorff metric      449
Hausdorff space      754
Hilbert space (H-space)      785
Hilbert space (H-space), important theorems in      787
Hilbert's problems      228 589 749
Historical remarks on Baire spaces      801
Historical remarks on Banach spaces      768
Historical remarks on bifurcation theory      350 395
Historical remarks on dimension theory      593ff
Historical remarks on foundations of mathematics in the 19th century      582ff
Historical remarks on Fredholm alternatives and Fredholm operators      370ff
Historical remarks on functional analysis, linear      750 752
Historical remarks on functional analysis, nonlinear      see Part V
Historical remarks on fundamental theorem of algebra      675
Historical remarks on Hilbert spaces      784
Historical remarks on iterative solutions of differential and integral equations in the 19th century      15
Historical remarks on linear spaces      763
Historical remarks on locally convex spaces      779
Historical remarks on mapping degree      519 555 542
Historical remarks on metric spaces      761
Historical remarks on partial differential equations      226ff 236ff
Historical remarks on positivity      269ff
Historical remarks on set theory      589 745ff
Historical remarks on topology      585 592ff 692 750 752
Homeomorphisms      755
Homeomorphisms, $e$-homeomorphisms      755
Homeomorphisms, existence of global      174 649 705 706
Homeomorphisms, local      174
Homeomorphisms, local index of      577
Homeomorphisms, sign of      577
Homology      728; see Part V
Homotopy invariance of essential maps      698
Homotopy invariance of fixed-point index      529
Homotopy invariance of mapping degree      570
Homotopy, basic definitions      527
Homotopy, classes      714
Homotopy, extension of a      698
Homotopy, groups      714
Homotopy, mod $y$      569
Homotopy, stable      717
Homotopy, theory      714
Hopf bifurcation      411 438 655 663
Hull closed convex      764
Hull convex      764
Hull linear      764
Immersion      152
Implicit function theorem      150
Implicit function theorem, basic applications of the      151ff
Implicit function theorem, basic ideas of the      150
Implicit function theorem, hard      218
Independence of axiom of choice      748
Independence of continuum hypothesis      749
Independence of parallel axiom      749
Index jump) proof strategies, fixed-point theory      10 61 188 253
Index of elliptic differential operators      259
Index of fixed points      529 532
Index of Fredholm operators      366
Index of integral operators and mapping degree      371
Index of linear operators      767
Index of zeros      524 533
Index theorem of Atiyah - Singer      372 727;
Index theorem of Leray - Schauder      619
Induced topology (identical with subspace topology)      755
Inequality, abstract      281
Inequality, differential      887(see also Maximum principle)
Inequality, Fan      449; see Part IV
Inequality, Garding      794
Inequality, integral      82 281
Inequality, Poincar$\acute{e}$ - Friedrichs      794
Inequality, quasivariational      448; see Part IV
Inequality, Variational      3 453
Infimum in an ordered set      503
Infinite-dimensional      765
Inner product      see Scalar product
Integrability condition      166 169ff 204
Integral equations,      see Applications to
Integral equations, and fixed-point theory      10 58 61
Integral equations, Fredholm      805
Integral equations, Hammerstein      805
Integral equations, Urysohn      805
Integral of vector functions of one real variable      75
Interior      751
Interior point      751
Intersection number      661
Interval mathematics      508
Interval mathematics, Newton's method in      513
Invariant manifold      112 412
Invariant set      112 158
Invariant subspaces of linear operators      66
Invariant tori      104 222 412
Inverse mappings, differentiation of      172
Inverse mappings, global theorem      174
Inverse mappings, local theorem      172
Isometry      761
Isomorphic B-spaces      771
Isomorphic H-spaces      785
Isomorphic linear spaces      764
Isomorphic manifolds      755
Isomorphic metric spaces      761
Isomorphic topological spaces      755
Isomorphic topological vector spaces      768
Iterative methods      16 21 481
Iterative methods for linear equations      30ff 35 40 343
Iterative methods for linear equations, block iteration      42
Iterative methods for linear equations, optimal relaxation      42
Iterative methods for linear equations, relaxation      41
Iterative methods for linear equations, single step      41
Iterative methods for linear equations, total step      41
Iterative methods, accelerated convergence      25
Iterative methods, basic ideas of      19 157
Iterative methods, for bifurcation problems      393
Iterative methods, for differential equations      28 240 324 327 330
Iterative methods, for eigenvalue problems      40 215
Iterative methods, for fixed-point problems      16 157 159 199 282 481
Iterative methods, for implicit functions      151
Iterative methods, for integral equations      28 58
Iterative methods, for regular semi-linear equations      398
Iterative methods, Newton's method and      206
Iterative methods, positivity and      282 343
Iterative methods, possible complex structure of      164 723 742
Iterative methods, rate of convergence      see Rate
Iterative methods, regularization and      see Hard implicit function theorem
Iterative methods, stable      31
Jets      804; see Part IV
Jordan curve      119 592
Jordan curve theorem      119 581 592
Jordan normal form      373
Kolmogorov - Arnold - Moser theory (KAM-theory)      104 220
Kronecker elimination method      434 435
Kronecker existence principle      520 529 569
Kronecker integral and mapping degree      519 535
Lattice      503
Lattice, complete      504
Lebesgue integral      see Part II
Lebesgue measure      see Part II
Lebesgue space      771
Lemma      see Theorem
Leray - Schauder principle      see Continuation method
Limit cycle      98 119 408
Linear hull      764
Linear operator      764
Linear space      764
Linear subspace      764
Linearization may fail      5 359 409 736
Linearization principle      5 87 95 99 131 150 159 161 172 176 178 183 283 353 358 398 424 619 626 666
Linearization technique in differential calculus      131
Linearly independent      765
Linking number      727; see Part V
Ljapunov - Schmidt method in bifurcation theory      375
Ljapunov center theorem      655; see Part IV
Ljapunov function      84
Ljapunov stability      87
local      171
Locally convex space      779
Loop      758
Lower limit      761
Lower semi-continuous functional      456 760
Lower semi-continuous multivalued mapping      450
Lowerbound in an ordered set      503
M - S - Cauchy sequence in locally convex spaces      780
M - S - Cauchy sequence in uniform spaces      801
M - S-sequence      759
M - S-sequentially closed      759
M - S-sequentially compact      759
M - S-sequentially continuous      759
M - S-subsequence      760
Magnetic monopoles      111 729
Manifold      see Banach manifold
Manifold center, stable, and unstable      112; see Part V
MAP      see Operator
Map, multivalued      447
Map, single-valued      748
Mapping degree      see Fixed-point index
Mapping degree for special classes of operators, $i$-compact operators on closed convex sets      604
Mapping degree for special classes of operators, analytic Fredholm maps      648
Mapping degree for special classes of operators, analytic type      645
Mapping degree for special classes of operators, basic situation (compact perturbations of identity)      569 608
Mapping degree for special classes of operators, classical analytic functions      616
Mapping degree for special classes of operators, compact perturbations of homeomorphisms      578
Mapping degree for special classes of operators, condensing perturbations of identity      568 607
Mapping degree for special classes of operators, continuous operators in $\mathbb{R}^{N}$      535 569
Mapping degree for special classes of operators, operators which lower dimension      526 681 695 716
Mapping degree for special classes of operators, overview of generalizations in Parts II and V      600
Mapping degree for special classes of operators, real functions      521 535
Mapping degree, basic definition via fixed-point index      531
Mapping degree, basic ideas      519ff
Mapping degree, basic properties      568ff 574ff
Mapping degree, coincidence degree      580
Mapping degree, construction of the      521 551
Mapping degree, local      532
Mapping degree, uniqueness of the      570
Markov chain      343
Mathematical biology      162 733
Mathematical economics      255 469
Mathematical logic      749
Matrix, adjoint      796
Matrix, norm      796
Matrix, positive (Frobenius — Perron theory)      342
Matrix, transposed      796
Maximal      see Element
Maximal chain      511
Maximum norm      772
Maximum principle, basic ideas      274
Maximum principle, for elliptic equations      334ff
Maximum principle, for ordinary differential equations      334ff
Maximum principle, for parabolic equations      337
Method, approximation      see Approximation
Method, averaging      203
Method, continuation      see Continuation method
Method, for computing the fixed-point index      see Fixed-point index
Method, iterative      see Iterative method
Method, Kronecker elimination      434 435
Method, line      124
Method, majorant      210 214 363 395
Method, Newton (see Newton's method) method, nonlinear Fourier method      124
Method, of a priori estimates      see a priori estimates
Method, of discretization      124; see Part II
Method, of Green's function      59 791
Method, of Ljapunov - Schmidt in bifurcation theory      376
Method, of Ljapunov function      84
Method, of Newton diagram in bifurcation theory      432
Method, of Poincar$\acute{e}$ map      86
Method, of projection      see Part II
Method, of projection in bifurcation theory      379
Method, of quasilinearization      217
Method, of regula falsi      218
Method, of retraction      50 51 563
Method, of shift operator      86
Method, of subsolutions and supersolutions      271 282 286
Method, overview of important methods in nonlinear functional analysis      2ff
Method, Rothe      124
Method, shooting      216
Metric      761
Metric space      761
Metrization      596 762
Metrization theorem      596 597
Minimal surfaces      250
Moore-Smith sequence      see M-S- sequence
Multiplicity of a characteristic number      374
Multiplicity of a characteristic number, algebraic      374
Multiplicity of a fixed point      617
Multivalued map      447
Multivalued map, continuous      451
Multivalued map, lower semi-continuous      450
Multivalued map, upper semi-continuous      450
Nested interval principle      762
Nested interval principle, generalized      495 783
1 2 3 4 5 6
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