Lyapunov-Schmidt methods in nonlinear analysis & applications / by Nikolay Sidorov ... [and others].

Format:

• Book

Contributor:

• Sidorov, Nikolaĭ.
• Rosengarten Family Fund.

Series:

• Mathematics and its applications (Kluwer Academic Publishers) ; v. 550.
• Mathematics and its applications ; v. 550

Language:

• English
• Russian

Subjects (All):

• Bifurcation theory.
• Lyapunov functions.
• Mathematical analysis.

Physical Description:

xx, 548 pages : illustrations ; 25 cm.

Other Title:

Lyapunov-Schmidt methods in nonlinear analysis and applications

Place of Publication:

Dordrecht ; Boston : Kluwer Academic Publishers, 2002.

Contents:

• 1. On Regularization of Linear Equations on the Basis of Perturbation Theory 1
• 1. Generalised Jordan chains, sets and root numbers of linear operators 1
• 2. Regularization of linear equations with Fredholm operators 6
• 3. Principal theorem on regularization of linear equations by the perturbation method 14
• 4. Regularization of linear equations on the basis of perturbation theory in Hilbert spaces 21
• 5. Computation of eigenvalues and eigenvectors of linear operators by pseudo-perturbation method 28
• 2. Investigation of Bifurcation Points of a Nonlinear Equations 43
• 1. Lyapunov-Schmidt BEq in the problem of a bifurcation point 45
• 2. General existence theorems for the bifurcation points 50
• 3. Construction of asymptotics in a neighborhood of a bifurcation point 63
• 4. Asymptotic bifurcation points 87
• 5. On perturbation of the branch points of nonlinear equations 92
• 3. Regularization of Computation of Solutions in a Neighborhood of the Branch Point 99
• 1. Construction of the regularizing equation in the problem at a branch point 103
• 2. Definition and properties of simple solutions 121
• 3. Regularization of calculations of simple solutions of nonlinear equations 134
• 4. Regularization of method for continuation along parameter in a neighborhood of a branch point 142
• 4. Iterations, Interlaced Equations and Lyapunov Convex Majorants in Nonlinear Analysis 151
• 1. Iterations and uniformization of branching solutions in nonlinear analysis 151
• 1.1 BEq and the selection of the initial approximation 152
• 1.2 On the role of supporting lines and Newton diagrams in the construction of the initial approximation 157
• 1.3 A one-step iteration method 159
• 1.4 An N-step iteration method 163
• 1.5 On regularization in the sense of Tikhonov, modifications and possible generalizations of an N-step method 168
• 2. Interlaced and potential BEq 172
• 2.1 The property of (S, K)-interlacing of an equation and its inheritance by the BEq 174
• 2.2 (T, M)-interlaced and (T[superscript 2], M)-interlaced BEq 178
• 2.3 [alpha]-parametric interlaced BEq 180
• 2.4 Interlaced BEq of potential type 183
• 2.5 Surface bundle of a domain of free parameters 189
• 2.6 Parametrization of solutions and the method of successive approximations 195
• 3. On the role of Lyapunov convex majorants in the nonlocal existence theorems of implicit functions 198
• 3.1 Majorants independent of parameters 200
• 3.2 Majorants depending on a parameter 209
• 3.3 Investigation of the existence domain of the solution of equation F(u, [varepsilon]) = 0. 213
• 5. Methods of Representation Theory and Group Analysis in Bifurcation Theory 217
• 1. Nonlinear equations invariant under transformation groups 218
• 1.1 Lyapounov-Schmidt BEq and some methods of their reduction 218
• 1.2 Some applications 223
• 2. Hereditary symmetry of branching equations and resolving systems 226
• 2.1 Invariance properties of BEq 226
• 2.2 Resolving systems for differential equations with Fredholm operator at the derivative and their symmetry 232
• 2.3 On the Grobman-Hartman theorem for equations with degenerate operator at the derivative 246
• 3. Construction and investigation of the branching equation by group analysis methods 252
• 3.1 BEq of solutions invariant relative to subgroups of the original equation group symmetry 252
• 3.2 Potential BEq 256
• 4. Direct methods of BEq group invariance usage for its general form construction by admitted group symmetry 259
• 4.1 Applications of Lie
• Ovsyannikov theorem about invariant manifolds for the construction of BEq general form by allowing group symmetry 270
• 5. Non-linearly perturbed Helmholtz equations 279
• 5.1 Domain symmetry and bifurcational solutions asymptotics 279
• 5.2 Periodic solutions 286
• 6. Capillary
• gravity waves in fluid layers 295
• 6.1 Capillary
• gravity waves in a floating fluid spatial layer 296
• 6.2 Capillary
• gravity waves at the interface of two fluids flow 299
• 6.3 Capillary
• gravity waves on a cylindrical surface 299
• 6.4 Ferrofluid layer in a magnetic field 301
• 7. Fluid phase state crystallization problem in statistical crystal theory 302
• 7.1 The statement of the problem 302
• 7.2 Subspaces N(B[subscript s]). Their expansions on irreducible subspaces relative to O[subscript h] 306
• 7.3 The BEq construction 311
• 7.4 Asymptotics of small solutions families for n[subscript s] = 1, 3 315
• 7.5 Solutions invariant relative to normal divisors O[subscript h] 317
• 8. Andronov
• Hopf bifurcation under group symmetry conditions 320
• 8.1 BEq derivation in non-stationary bifurcation 321
• 8.2 Symmetry inheritance theorem 323
• 8.3 BEq construction by group analysis methods 325
• 8.4 On the asymptotics of small solutions 328
• 9. Stability of the bifurcating solutions 330
• 6. Singular Differential Equations in Banach Spaces 337
• 1. Fundamental operator functions 338
• 1.1 Generalized functions in Banach spaces 339
• 1.2 Fundamental operator functions of singular differential operators 343
• 1.3 Fundamental operator functions of singular integral and integro-differential operators 357
• 2. The initial value problem for a differential equation having a Noetherian operator at the derivative. Periodic solutions and the property of convergence 362
• 2.1 Auxiliary information on Jordan sets of Noetherian operators 362
• 2.2 The initial value problem for a linear differential equation 364
• 2.3 The initial value problem for a nonlinear differential equation 368
• 2.4 Periodic solutions 373
• 2.5 Integral pseudo-solutions 376
• 3. Non-stationary differential equations with singularities 383
• 3.1 The initial value problem for a non-stationary linear equation and systems of 1st kind integral Volterra equations 383
• 3.2 The initial value problem for a non-stationary linear differential equation and a system of integral Volterra equations with a singularity 393
• 3.3 Branching differential equations of the initial value problem with singularity 397
• 3.4 The initial value problem for a nonlinear differential equation and equations with a singular point 404
• 4. Partial differential equations with the Fredholm operator in the main part 413
• 5. The theory of semigroups and groups of operators with kernels 419
• 5.1 Relative resolvents. Relatively adjoint elements 419
• 5.2 Relatively spectrally bounded operators and analytical groups of operators with kernells 421
• 5.3 Relatively sectorial operators and analytical semigroups of operators with kernels 428
• 7. Steady-State Solutions of the Vlasov
• Maxwell System 431
• 2. Stationary solutions of a VM system 434
• 2.1 The reduction of VM system to the resolving elliptic system (2.28), (2.29) 435
• 2.2 The reduction of the resolving system to the unique resolving equation 439
• 2.3 Existence of solutions of the boundary value problem (2.40)-(2.42) 443
• 2.4 Applications of the reduction theorems 449
• 2.5 Normalized solutions for a one-component distribution function 462
• 3. Non-stationary problem 468
• 4. Bifurcation points and nontrivial solutions of the stationary VM system 474
• 4.2 Statement of the boundary value problem and the problem at a bifurcation point 475
• 4.3 Resolution of the bifurcation equation 485
• 4.4 The existence theorem for bifurcation points and the construction of asymptotic solutions 487
• A Positive solutions of the nonlinear singular boundary value problem of magnetic insulation 497.

Notes:

Includes bibliographical references (pages 513-545) and index.

Local Notes:

Acquired for the Penn Libraries with assistance from the Rosengarten Family Fund.

ISBN:

1402009410

OCLC:

50725318

Online:

The Rosengarten Family Fund Home Page