Approximate Solution of Operator Equations

1 Successive approximations.- §1. Existence of the fixed point of a contraction operator.- 1.1. Contraction operators.- 1.2. Use of an equivalent norm.- 1.3. Relative uniqueness of the solution.- 1.4. Spectral radius of a linear operator.- 1.5. Operators which commute with contraction operators.- 1.6. Case of a compact set.- 1.7. Estimating the Lipschitz constant.- 1.8. Equations with uniform contraction operators.- 1.9. Local implicit function theorem.- §2. Convergence of successive approximations.- 2.1. Successive approximations.- 2.2. Equations with a contraction operator.- 2.3. Linear equations.- 2.4. Fractional convergence.- 2.5. Nonlinear equations.- 2.6. Effect of the initial approximation on estimate of convergence rate.- 2.7. Acceleration of convergence.- 2.8. Distribution of errors.- 2.9. Effect of round-off errors.- §3. Equations with monotone operators.- 3.1. Statement of the problem.- 3.2. Cones in Banach spaces.- 3.3. Solvability of equations with monotone operators.- 3.4. Non-trivial positive solutions.- 3.5. Equations with concave operators.- 3.6. Use of powers of operators.- 3.7. The contracting mapping principle in metric spaces.- 3.8. On special metric.- 3.9. Equations with power nonlinearities.- 3.10. Equations with uniformly concave operators.- §4. Equations with nonexpansive operators.- 4.1. Nonexpansive operators.- 4.2. Example.- 4.3. Selfadjoint nonexpansive operators.- 2 Linear equations.- §5. Bounds for the spectral radius of a linear operator.- 5.1. Spectral radius.- 5.2. Positive linear operators.- 5.3. Indecomposable operators.- 5.4. Comparison of the spectral radii of different operators.- 5.5. Lower bounds for the spectral radius.- 5.6. Upper bounds for the spectral radius.- 5.7. Strict inequalities.- 5.8. Examples.- §6. The block method for estimating the spectral radius..- 6.1. Fundamental theorem.- 6.2. Examples.- 6.3. Conic norm.- 6.4. Generalized contracting mapping principle.- §7. Transformation of linear equations.- 7.1. General scheme.- 7.2. Chebyshev polynomials.- 7.3. Equations with selfadjoint operators in Hilbert spaces.- 7.4. Equations with compact operators.- 7.5. Seidel’s method.- 7.6. Remark on convergence.- §8. Method of minimal residuals.- 8.1. Statement of the problem.- 8.2. Convergence of the method of minimal residuals.- 8.3. The moment inequality.- 8.4. ?-processes.- 8.5. Computation scheme.- 8.6. Application to equations with nonselfadjoint operators.- §9. Approximate computation of the spectral radius.- 9.1. Use of powers of an operator.- 9.2. Positive operators.- 9.3. Computation of the greatest eigenvalue.- 9.4. Approximate computation of the eigenvalues of selfadjoint operators.- 9.5. The method of normal chords.- 9.6. The method of orthogonal chords.- 9.7. Simultaneous computation of several iterations.- §10. Monotone iterative processes.- 10.1. Statement of the problem.- 10.2. Choice of initial approximations.- 10.3. Acceleration of convergence.- 10.4. Equations with nonpositive operators.- 3 Equations with smooth operators.- §11. The Newton-Kantorovich method.- 11.1. Linearization of equations.- 11.2. Convergence.- 11.3. Further investigation of convergence rate.- 11.4. Global convergence condition.- 11.5. Simple zeros.- §12. Modified Newton-Kantorovich method.- 12.1. The modified method.- 12.2. Fundamental theorem.- 12.3. Uniqueness ball.- 12.4. Modified method with perturbations.- 12.5. Equations with compact operators.- 12.6. Equations with nondifferentiable operators.- 12.7. Remark on the Newton-Kantorovich method.- §13. Approximate solution of linearized equations.- 13.1. Statement of the problem.- 13.2. Convergence theorem.- 13.3. Application of the method of minimal residuals.- 13.4. Choice of initial approximations.- §14. A posteriori error estimates.- 14.1. Error estimates and existence theorems.- 14.2. Linearization of the equation.- 14.3. Rotation of a finite-dimensional vector field.- 14.4. Rotation of a compact vector field.- 14.5. Index of a fixed point.- 14.6. Index of a fixed point and a posteriori error estimates.- 14.7. Special case.- 14.8. Relaxing the compactness condition.- 4 Projection methods.- § 15. General theorems on convergence of projection methods..- 15.1. Projection methods.- 15.2. Fundamental convergence theorem.- 15.3. The aperture of subspaces of a Hilbert space.- 15.4. The Galerkin method for equations of the second kind.- 15.5. Supplements to Theorem 15.3..- 15.6. Derivation of an equation of the second kind.- 15.7. The collocation method.- 15.8. The factor method.- § 16. The Bubnov-Galerkin and Galerkin-Petrov methods.- 16.1. Convergence of the Bubnov-Galerkin method for equations of the second kind.- 16.2. Necessary and sufficient conditions for convergence of the Galerkin-Petrov method for equations of the second kind.- 16.3. Convergence of the Bubnov-Galerkin method for equations with a positive definite principal part.- 16.4. Lemma on similar operators.- 16.5. Convergence of the Bubnov-Galerkin method.- 16.6. Regular operators.- 16.7. Rate of convergence of the Galerkin-Petrov method.- 16.8. The method of moments.- §17. The Galerkin method with perturbations and the general theory of approximate methods.- 17.1. Statement of the problem.- 17.2. Lemma on invertibility of the operators I - ?T and I - ?Tn.- 17.3. Convergence theorem.- 17.4. Relation to Kantorovich’s general theory of approximate methods.- 17.5. Perturbation of the Galerkin approximation.- 17.6. Bases in subspaces of a Hilbert space.- 17.7. Lemma on Gram matrices.- 17.8. Stability of the Galerkin-Petrov method.- 17.9. Convergent quadrature processes.- 17.10. The method of mechanical quadratures for integral equations.- §18. Projection methods in the eigenvalue problem.- 18.1. The eigenvalue problem.- 18.2. Convergence of the perturbed Galerkin method.- 18.3. Rate of convergence.- 18.4. The Bubnov-Galerkin method; eigenvalue estimates.- 18.5. Estimates for eigenelements.- 18.6. Improved eigenvalue estimates.- 18.7. The Galerkin-Petrov method.- 18.8. Case of unbounded operators.- 18.9. Operators in real spaces.- §19. Projection methods for solution of nonlinear equations.- 19.1. Statement of the problem.- 19.2. Solvability of nonlinear equations.- 19.3. Convergence of the perturbed Galerkin method.- 19.4. A posteriori error estimate.- 19.5. Rotation of a compact vector field.- 19.6. Second convergence proof.- 19.7. The method of mechanical quadratures for nonlinear integral equations.- 19.8. The method of finite differences.- 19.9. The Galerkin method.- 19.10. The Galerkin-Petrov method. Stability.- 19.11. Projection methods assuming one-sided estimates.- 19.12. Minimum problems for functional.- 19.13. The factor method for nonlinear equations.- 5 Small solutions of operator equations.- §20. Approximation of implicit functions.- 20.1. Fundamental implicit function theorem.- 20.2. Multilinear operators and Taylor’s formula.- 20.3. Differentiation of an implicit function.- 20.4. Method of undetermined coefficients.- 20.5. Successive approximations.- 20.6. Asymptotic approximations to implicit functions.- 20.7. Formal series and formal implicit functions.- 20.8. Analyticity of an implicit function.- §21. Finite systems of equations.- 21.1. Review of ring theory.- 21.2. Rings of power series.- 21.3. The Newton diagram.- 21.4. Branching of solutions of scalar analytic equations.- 21.5. The method of elimination.- 21.6. Structurally stable case.- 21.7. Systems of analytic equations.- §22. Branching of solutions of operator equations.- 22.1. Statement of the problem.- 22.2. Decomposable operators.- 22.3. Matrix representations of decomposable operators.- 22.4. Branching equations.- 22.5. Asymptotic approximations of branching equations.- 22.6. Equations in formal power series.- 22.7. Equations in analytic operators.- § 23. Simple solutions and the method of undetermined coefficients.- 23.1. Simple solutions.- 23.2. Asymptotic approximations to simple solutions.- 23.3. Simple solutions and branching equations.- 23.4. Quasi-solutions of operator equations.- 23.5. The method of undetermined coefficients.- 23.6. Equations with analytic operators.- §24. The problem of bifurcation points.- 24.1. Statement of the problem.- 24.2. Equations with compact operators.- 24.3. Use of the branching equation.- 24.4. Amplitude curves.