Non-Self-Adjoint Boundary Eigenvalue Problems, 192권Gulf Professional Publishing, 2003. 6. 26. - 500페이지 This monograph provides a comprehensive treatment of expansion theorems for regular systems of first order differential equations and n-th order ordinary differential equations. In 10 chapters and one appendix, it provides a comprehensive treatment from abstract foundations to applications in physics and engineering. The focus is on non-self-adjoint problems. Bounded operators are associated to these problems, and Chapter 1 provides an in depth investigation of eigenfunctions and associated functions for bounded Fredholm valued operators in Banach spaces. Since every n-th order differential equation is equivalent to a first order system, the main techniques are developed for systems. Asymptotic fundamental systems are derived for a large class of systems of differential equations. Together with boundary conditions, which may depend polynomially on the eigenvalue parameter, this leads to the definition of Birkhoff and Stone regular eigenvalue problems. An effort is made to make the conditions relatively easy verifiable; this is illustrated with several applications in chapter 10. The contour integral method and estimates of the resolvent are used to prove expansion theorems. For Stone regular problems, not all functions are expandable, and again relatively easy verifiable conditions are given, in terms of auxiliary boundary conditions, for functions to be expandable. Chapter 10 deals exclusively with applications; in nine sections, various concrete problems such as the Orr-Sommerfeld equation, control of multiple beams, and an example from meteorology are investigated.
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목차
Operator functions in Banach spaces | 1 |
First order systems of ordinary differential equations | 53 |
Boundary eigenvalue problems for first order systems | 101 |
Birkhoff regular and Stone regular boundary eigenvalue | 129 |
Expansion theorems for regular boundary eigenvalue | 203 |
nth order differential equations | 249 |
Regular boundary eigenvalue problems for nth order | 279 |
tions | 297 |
nth order differential equations and nfold expansions | 389 |
Applications | 409 |
APPENDIX A Exponential sums A 1 The convex hull of sums of complex numbers | 441 |
Estimates of exponential sums | 450 |
Improved estimates for exponential sums | 473 |
| 475 | |
| 497 | |
| 499 | |
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A₁ associated functions associated vectors assume assumptions Banach space belongs Birkhoff regular boundary conditions boundary eigenvalue operator boundary eigenvalue problem c₁ C₂ canonical system coefficients consider converges Corollary CSRF d₁ defined definition depends holomorphically diagonal eigenfunctions and associated eigenvalue operator function eigenvalue parameter eigenvalue problem 5.1.1 eigenvectors eigenvectors and associated estimate expansion theorems exponential sum f₁ follows Fredholm operator fulfilled fundamental matrix fundamental matrix function fundamental system given Hence HÖLDER'S inequality holds implies infer integral invertible K₁ Lemma linear linearly independent Lp(a m₁ matrix polynomial multiplicity n-th order differential obtain order systems polynomial problem is Birkhoff Proposition 2.3.1 proves r₁ representation root functions s-regular satisfied shows Sobolev spaces Stone regular subset Suppose system of root V₁ vector function y₁ yields zero of order Σ Σ