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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
$C^{k}$-diffeomorphism      171
$C^{k}$-map      147
$i$-compact (image compact)      604
$r$- weakly stable      160 324
$r$-stable      160
$r$-unstable      160 324
A priori estimates, abstract forms of      265
A priori estimates, and existence      227
A priori estimates, and global solutions for ordinary differential equations      79
A priori estimates, basic ideas of      227ff
A priori estimates, coerciveness and      173 (see also Part II)
A priori estimates, for linear elliptic equations      234 257ff
A priori estimates, for ordinary differential equations      79 794
A priori estimates, for parabolic equations      252
A priori estimates, for quasi-linear elliptic equations      246
A priori estimates, Fredholm operators, regularization and      263
A priori estimates, Galerkin method and      see Part II
A priori estimates, Gronwall's lemma and      83
A priori estimates, Leray - Schauder principle and      245
A priori estimates, linear operators and      230
A priori estimates, Ljapunov function and      84
Abstract Hammerstein equations      307ff
Admissible homotopy      569
Admissible map      569
Algebraic geometry and bifurcation theory      352 442 639 690
Algebraic topology      726ff; see Part V
Analytic major ant method      363
Analytic operator      362
Analytic set      646
Analytic space      646
ANE (absolute neighborhood extensor)      598
ANR (absolute neighborhood retract)      598 608
Antipodal theorem      700
Antipodal theorem with general symmetries      721
Antipodal theorem, basic ideas of the      694
Applications to celestial mechanics      65 126 222 736
Applications to differential inclusions      468
Applications to eigenvalue problems      see Eigenvalue
Applications to formal languages      510
Applications to foundations of set theory      489 511
Applications to functional differential equations      733
Applications to game theory      461 469
Applications to Haar measure      67 466
Applications to heat conduction      301
Applications to Hopf bifurcation      411 438
Applications to implicit functions      155 195
Applications to integral equations      10 21 36 57 58 60 156 289 316 387 633 634 670 788 790
Applications to integral inequalities      82 281
Applications to interval arithmetic      508 513
Applications to linear and nonlinear oscillations      91ff 101ff 201 401ff
Applications to linear elliptic partial differential equations      233
Applications to mathematical biology      162 733
Applications to mathematical economics      255 469
Applications to mathematical logic (G$\ddot{o}$del's incompleteness theorem)      749
Applications to nonlinear boundary-value problems      326
Applications to nonlinear harmonic oscillator      59
Applications to ordinary differential equations      see Ordinary
Applications to periodic solutions      see Periodic
Applications to Prandtl's boundary-layer equation      341
Applications to quasi-linear elliptic partial differential equations      246 250 251
Applications to quasi-periodic solutions      104 221
Applications to semi-linear elliptic partial differential equations      60 239 317 344 347 390 634 671
Applications to semi-linear hyperbolic partial differential equations      60 64
Applications to semi-linear parabolic partial differential equations      60 329
Applications to systems of real equations      10 22 35 40 52 155 177 195 209 256 257 430ff 657
Applications to total differential equations      166ff
Applications to variational inequalities      453
Applications to von Neumann algebras      486
Approximation as a proof strategy      55 56 187 188 497 500 525 537 542
Approximation methods (overview)      8 (see also Section 37.29 in Part III)
Approximation of compact operators      55
Approximation of continuous functions (Weierstrass theorem)      770
Approximation of continuous operators by smooth generic operators      533
Approximation of nonsmooth functions      549
Asymptotic linear operators      296 603 706
Asymptotically stable equilibrium point      87
Asymptotically stable fixed point      158
Asymptotically stable solution of an elliptic equation      323
Asymptotically stable, weakly bifurcation solution      424
Autocombustion      314 346
Axiom of Choice      747
Axiom of choice, equivalent forms of the      512
Axiom of choice, independence of the      748ff
Axiomatic systems, consistency      750
Axiomatic systems, independence      750
Axiomatic systems, undecidability      750
Axiomatic systems, unsolvability      750
Baire category      802
Baire category, and iterative methods      33
Baire space      801
Banach manifold      180; see Parts IV and V
Banach manifold, one-dimensional      254
Banach space (B-space)      752 768
Banach space (B-space), $\varepsilon$-neighborhood      769
Banach space (B-space), basic concepts      768ff
Banach space (B-space), bounded linear operator      773
Banach space (B-space), bounded set      769
Banach space (B-space), Cauchy sequence      769
Banach space (B-space), closed ball      see List of symbols
Banach space (B-space), closed set      769
Banach space (B-space), compact operator      53
Banach space (B-space), compact set      769
Banach space (B-space), completion      111
Banach space (B-space), complexification      770
Banach space (B-space), continuous linear functional      774
Banach space (B-space), continuous linear operator      764 773
Banach space (B-space), continuous operator      770
Banach space (B-space), convergence      769 775
Banach space (B-space), convergence principles      480 776
Banach space (B-space), dense set      769
Banach space (B-space), direct sum      765
Banach space (B-space), dual operator      775
Banach space (B-space), dual space      774
Banach space (B-space), duality      774
Banach space (B-space), equivalent norms      772
Banach space (B-space), examples      771ff
Banach space (B-space), factor space      765 770
Banach space (B-space), geometry of A-spaces      see Part III
Banach space (B-space), graph closed operator ($g$-closed)      778
Banach space (B-space), important theorems      776ff 782
Banach space (B-space), infinite series      773
Banach space (B-space), norm      768
Banach space (B-space), norm convergence      see Convergence
Banach space (B-space), norm isomorphism      771
Banach space (B-space), open ball      see List of Symbols
Banach space (B-space), open set      769
Banach space (B-space), operator norm      773
Banach space (B-space), product space      755 770
Banach space (B-space), projection operator      766
Banach space (B-space), reflexive      11
Banach space (B-space), relatively compact set      769
Banach space (B-space), separable      770
Banach space (B-space), separation theorem      see Part III
Banach space (B-space), spectral theory      794
Banach space (B-space), splitting property      766
Banach space (B-space), strictly convex      486
Banach space (B-space), strong convergence      see Convergence
Banach space (B-space), strong${}^{*}$ topology      781
Banach space (B-space), topological direct sum      766
Banach space (B-space), triangle inequality      769
Banach space (B-space), uniformly convex      474
Banach space (B-space), weak (weak${}^{*}$) convergence      775
Banach space (B-space), weak (weak${}^{*}$) topology      781
Basic ideas of approximation methods      8 19
Basic ideas of bifurcation theory      108 350ff 653 663
Basic ideas of differential calculus      131
Basic ideas of fixed-point index and mapping degree      519ff
Basic ideas of fixed-point theory      9ff 61
Basic ideas of functional analysis, linear      768ff 787ff
Basic ideas of functional analysis, nonlinear      2ff
Basic ideas of implicit function theorem      150
Basic ideas of modern mathematical physics      100ff 656 726ff;
Basic ideas of noncompact operators      489
Basic ideas of positivity      269ff
Basic ideas of problems, quasi-linear      246
Basic ideas of problems, semi-linear      60 239 398
Basic ideas of stability      87 91 99 117 157 282 423 663
Basic ideas of the implicit function theorem and its applications      151
Basic ideas of the relation between classical analysis and nonlinear functional analysis      6
Basis of a linear space      765
Basis of a topology      751
Basis of neighborhoods      754
Bieberbach conjecture      257
Bifurcation at infinity      315 673
Bifurcation diagram      386 419
Bifurcation Feigenbaum      738ff
Bifurcation generic      381 391 416 428
Bifurcation global      667ff
Bifurcation Hopf      411 438 655 663
Bifurcation local      381 391 416 428 665
Bifurcation point      428
Bifurcation preventing secondary      681
Bifurcation stability and      423 655
Bifurcation subcritical      386
Bifurcation supercritical      386
Bifurcation theory, approach to, analytical      350ff
Bifurcation theory, approach to, topological      653ff
Bifurcation theory, approach to, variational      see Chapter 45 in Part III
Bifurcation theory, basic ideas of      350ff 654ff
Bifurcation theory, basic technique of      see Branching equations
Bifurcation theory, basic topological principle (index jump)      657
Bifurcation theory, blowing-up of singularities      353 357 397 440
Bifurcation theory, implicit function theorem and      397 418
Bifurcation theory, rescaling technique      419
Bifurcation transcritical      386
Bifurcation transversality and      382
big bang      113 728;
Biorthogonal system      786
Black holes in cosmos      113 730;
Blowing-up of singularities      353 357 397 440
Blowing-up of solutions of differential equations      113
Boundary      751
Boundary point      751
Bounded operator      757
Bounded operator, linear      773
Bounded set in B-spaces      769
Bounded set in locally convex spaces      780
Bounded set in topological vector spaces      768
Branching equations)      657
Branching equations, basic ideas of      352
Branching equations, method of Ljapunov - Schmidt      375
Branching equations, solution methods for      352 378 418ff 430ff 440ff 689
Brouwer fixed-point theorem      51
Brouwer fixed-point theorem, constructive proof      254
Brouwer fixed-point theorem, elementary proof via the Knaster- Kuratowski - Mazurkiewicz lemma      see Part IV
Brouwer fixed-point theorem, numerous propositions which are equivalent to the      see Chapter 77 in Part IV
Brouwer fixed-point theorem, overview of other proofs      68
Brouwer fixed-point theorem, proof via the fixed-point index      530
Bundle      see Parts IV and V
Bundle fiber      729
Bundle tangent      183
Bundle vector      728
Cantor continuum hypothesis      749
Cantor discontinuum      583 594
Cantor discovery of set theory      591ff
Cantor set      583 758
cardinal number      749
Cauchy sequence      762
Cauchy sequencein B-spaces      769
Cauchy sequencein locally convex spaces      780
Cauchy sequencein uniform spaces      801
Chain in ordered sets      503
Chain in ordered sets, $\varepsilon$-chain and connectivity      636
Chain rule      138
Chain rule for higher derivatives      192
Characteristic class      728; see Part V
Characteristic number      269
Chemical reactions      119
Classification of differential equations      804
classification of integral equations      804
Classification of spaces in functional analysis      752 753
Closed set      751 754
Closed set characterization by M - S-sequences      759 760
Closed set characterization by sequences      759 760
Closed set in B-spaces      769
Cobordism theory      526 722;
Codimension      765
Coercive      173
Coercive weakly      173
Cohomology      517 728;
Coincidence degree      580
Collection (class)      746
Collection (class), nonset      746
Collection (class), set      746
Common fixed point of an operator family      66 486
Common zero of polynomials and resultants      434
Compact operator      53
Compact operator, $i$-compact      604
Compact operator, characterization by sequences      769 770
Compact perturbation of identity      173
Compact set      756
Compact set, characterization by finite $\varepsilon$-nets      762
Compact set, characterization by M - S-sequences      759 760
Compact set, characterization by sequences      759 760
Compact set, in B-spaces      769
Compactness theorem of Arzela - Ascoli      772
Complementing condition      258 262
Complete locally convex space      780
Complete metric space      762
Complete orthonormal system      786 787
Complete uniform space      801
Component      757
Cone      276
Cone order      276
Continuation method      227 241 243 253
Continuation method, geometric heart of the      661
Continuation method, Leray - Schauder principle      245 556 628 631 658
Continuation method, of Poincar$\acute{e}$ for periodic solutions      203
Continuous operator      755
Continuous operator, basic properties      755
Continuous operator, characterization by M - S-sequences      759 760
Continuous operator, characterization by sequences      759 760
Continuous operator, in B-spaces      770
Continuous operator, in locally convex spaces      780
Continuum      564
Continuum Hypothesis      749
Continuum hypothesis, independence of the      749
Continuum of fixed points      564
Continuum of solutions      629 631 668
Convergence      801
Convergence in B-spaces      769
Convergence in B-spaces, norm convergence (identical with convergence) convergence in B-spaces, strong convergence (identical with convergence) convergence in B-spaces, weak      775
Convergence in B-spaces, weak${}^{*}$      775
Convergence in metric spaces      761
Convergence in topological spaces      758
Convergence of iterative methods      see Rate of convergence
Convergence of M - S-sequences      760
Convergence of sequences      759
Convex functional      765
Convex hull      764
Convex operator      298
Convex set      764
Critical points of functionals      642 (see also Part III)
Critical points of functionals, existence of three      644
Critical points of functionals, regular      642
Cusp catastrophe      437
Cycles      736 739
Cycles and fixed points of iterated maps      739
Dense set      751
Dense set characterization by sequences      769
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