| Foreword |
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vii | |
| Introduction |
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xv | |
| Acknowledgments |
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xxi | |
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Part I Operator and Differential-Operator Equations |
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1 | (132) |
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1 Auxiliary Information on the Theory of Linear Operators |
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3 | (36) |
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1.1 Generalized Jordan Chains, Sets and Root Numbers of Linear Operators |
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3 | (6) |
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1.2 Regularization of Linear Equations with Fredholm Operators |
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9 | (7) |
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1.3 Principal Theorem of Regularization of Linear Equations by the Perturbation Method |
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16 | (6) |
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1.4 Regularization of Linear Equations Based on the Perturbation Theory in Hilbert Spaces |
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22 | (9) |
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1.5 Regularization with Vector Regularizing Parameter for the First Kind Equations |
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31 | (8) |
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2 Volterra Operator Equations with Piecewise Continuous Kernels: Solvability and Regularized Approximate Methods |
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39 | (44) |
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2.1 Theory of the Volterra Operator Equations with Piecewise Continuous Kernels |
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39 | (23) |
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62 | (21) |
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3 Nonlinear Differential Equations Near Branching Points |
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83 | (14) |
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84 | (9) |
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3.2 Open Problems and Generalizations |
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93 | (2) |
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3.3 Magnetic Insulation Model Example |
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95 | (2) |
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4 Nonlinear Operator Equations with a Functional Perturbation of the Argument |
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97 | (8) |
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4.1 Nonlinear Operator Equations |
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97 | (6) |
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103 | (2) |
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5 Nonlinear Systems' Equilibrium Points: Stability, Branching, Blow-Up |
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105 | (12) |
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5.1 Reduction of a Nonlinear System in the Neighborhood of an Equilibrium Point to a Single Differential Equation |
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108 | (3) |
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5.2 The Construction of a Solution of a Nonlinear System by the Successive Approximations Method |
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111 | (3) |
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114 | (3) |
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6 Nonclassic Boundary Value Problems in the Theory of Irregular Systems of Equations with Partial Derivatives |
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117 | (14) |
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6.1 Skeleton Chains of Linear Operators |
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120 | (2) |
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6.2 Abstract Irregular Equation Reduction to the Sequence of Regular Equations |
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122 | (7) |
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6.3 Skeleton Decomposition in the Theory of Irregular Ordinary Differential Equation in Banach Space |
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129 | (2) |
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131 | (2) |
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Part II Lyapunov Methods in Theory of Nonlinear Equations with Parameters |
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133 | (154) |
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8 Lyapunov Convex Majorants in the Existence Theorems |
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135 | (24) |
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8.1 Parameter-Independent Majorants |
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137 | (10) |
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8.2 Majorants Depending on a Parameter |
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147 | (6) |
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8.3 Solution Existence Domain |
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153 | (6) |
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9 Investigation of Bifurcation Points of Nonlinear Equations |
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159 | (10) |
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9.1 Lyapunov--Schmidt Method in the Problem of a Bifurcation Point |
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161 | (3) |
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164 | (5) |
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10 General Existence Theorems for the Bifurcation Points |
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169 | (14) |
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179 | (4) |
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11 Construction of Asymptotics in a Neighborhood of a Bifurcation Point |
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183 | (22) |
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11.1 Analytic Lyapunov--Schmidt Method in the Study of Branching Equations |
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183 | (4) |
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11.2 Variational Methods in the Study of Branching Equations |
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187 | (18) |
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12 Regularization of Computation of Solutions in a Branch Point Neighborhood |
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205 | (8) |
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12.1 Construction of the Regularization Equation in the Problem at a Branch Point |
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206 | (1) |
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12.2 Definition and Properties of Simple Solutions |
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207 | (3) |
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12.3 Construction of Regularization Equation of Simple Solutions |
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210 | (3) |
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13 Iteration Methods, Analytical Initial Approximations, Interlaced Equations |
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213 | (8) |
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13.1 Iterations and Uniformization of Branching Solutions |
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213 | (1) |
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13.2 Branching Equation and the Selection of Initial Approximation |
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214 | (7) |
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14 Iterative Methods Using Newton Diagrams |
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221 | (20) |
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14.1 One-Step Iteration Method |
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224 | (4) |
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14.2 N-step Iteration Method |
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228 | (6) |
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14.3 Iteration Method for Nonlinear Equation Invariant Under Transformation Groups |
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234 | (7) |
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15 Small Solutions of Nonlinear Equations with Vector Parameter in Sectorial Neighborhoods |
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241 | (16) |
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15.1 Construction of the Minimal Branch of Solutions of Equation with Fredholm Operator |
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243 | (6) |
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15.2 Sufficient Conditions of the Minimal Branch Existence |
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249 | (8) |
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16 Successive Approximations to the Solutions to Nonlinear Equations with a Vector Parameter |
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257 | (8) |
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16.1 Existence Theorem and Successive Approximations |
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258 | (7) |
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17 Interlaced and Potential Branching Equation |
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265 | (20) |
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17.1 Property of (S, K)-interlacing of an Equation and Its Inheritance by Branching Equation |
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266 | (5) |
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17.2 (T, M)-interlaced and (T2, M)-interlaced Branching Equation |
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271 | (3) |
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17.3 α-Parametric Interlaced Branching Equation |
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274 | (3) |
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17.4 Interlaced Branching Equation of Potential Type |
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277 | (8) |
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285 | (2) |
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287 | (172) |
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19 The Family of Steady-State Solutions of Vlasov--Maxwell System |
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289 | (24) |
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19.1 Ansatz of the Distribution Function and Reduction of Stationary Vlasov--Maxwell Equations to Elliptic System |
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289 | (24) |
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20 Boundary Value Problems for the Vlasov--Maxwell System |
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313 | (10) |
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313 | (6) |
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20.2 Collisionless Kinetic Models (Classical and Relativistic Vlasov--Maxwell Systems) |
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319 | (1) |
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20.3 Quantum Models: Wigner--Poisson and Schrodinger--Poisson Systems |
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320 | (1) |
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20.4 Mixed Quantum-Classical Kinetic Systems |
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320 | (3) |
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21 Stationary Solutions of Vlasov--Maxwell System |
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323 | (12) |
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21.1 Problem Reduction to the System of Nonlinear Elliptic Equations |
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324 | (5) |
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329 | (6) |
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22 Existence of Solutions for the Boundary Value Problem |
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335 | (14) |
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22.1 Existence of Solution for Nonlocal Boundary Value Problem |
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342 | (7) |
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23 Nonstationary Solutions of the Vlasov--Maxwell System |
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349 | (14) |
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23.1 Reduction of the Vlasov--Maxwell System to Nonlinear Wave Equation |
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349 | (8) |
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23.2 Existence of Nonstationary Solutions of the Vlasov--Maxwell System in the Bounded Domain |
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357 | (6) |
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24 Linear Stability of the Stationary Solutions of the Vlasov--Maxwell System |
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363 | (10) |
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25 Bifurcation of Stationary Solutions of the Vlasov--Maxwell System |
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373 | (14) |
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25.1 Bifurcation of Solutions of Nonlinear Equations in Banach Spaces |
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377 | (9) |
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386 | (1) |
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26 Statement of the Boundary Value Problem and the Bifurcation Problem |
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387 | (14) |
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27 Resolving Branching Equation |
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401 | (14) |
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27.1 The Existence Theorem for Bifurcation Points and the Construction of Asymptotic Solutions |
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404 | (11) |
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28 Numerical Modeling of the Limit Problem for the Magnetically Noninsulated Diode |
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415 | (38) |
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415 | (1) |
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28.2 Description of Vacuum Diode |
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416 | (1) |
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28.3 Shot Noise in a Diode |
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417 | (2) |
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28.4 Description of the Mathematical Model |
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419 | (5) |
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28.5 Solution Trajectory, Upper and Lower Solutions |
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424 | (10) |
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28.6 Second Lower Solution Hypothesis |
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434 | (4) |
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438 | (10) |
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448 | (5) |
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453 | (6) |
| Bibliography |
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459 | (12) |
| Index |
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471 | |
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