Complete Second Order Linear Differential Equations in Hilbert Spaces

Incomplete second order linear differential equations in Banach spaces as well as first order equations have become a classical part of functional analysis. This monograph is an attempt to present a unified systematic theory of second order equations y" (t) + Ay' (t) + By (t) = 0 including well-posedness of the Cauchy problem as well as the Dirichlet and Neumann problems. Exhaustive yet clear answers to all posed questions are given. Special emphasis is placed on new surprising effects arising for complete second order equations which do not take place for first order and incomplete second order equations. For this purpose, some new results in the spectral theory of pairs of operators and the boundary behavior of integral transforms have been developed. The book serves as a self-contained introductory course and a reference book on this subject for undergraduate and post- graduate students and research mathematicians in analysis. Moreover, users will welcome having a comprehensive study of the equations at hand, and it gives insight into the theory of complete second order linear differential equations in a general context - a theory which is far from being fully understood.

I. Well-posedness of boundary-value problems.- to Part I.- 1. Joint spectrum of commuting normal operators and its position. Estimates for roots of second order polynomials. Definition of well-posedness of boundary-value problems.- 1. Introductory notes.- 2. Joint spectrum of operators.- 3. Position of the joint spectrum.- 4. Estimates for roots of characteristic polynomials.- 5. Definitions of well-posedness and weak well-posedness of boundary-value problems for equation (1).- 6. Spaces of boundary data.- 7. (Weak) well-posedness and uniform (weak) well-posedness.- 2. Well-posedness of boundary-value problems for equation (1) in the case of commuting self-adjoint A and B.- 1. The Cauchy problem.- 2. The Dirichlet problem.- 3. The Neumann problem.- 4. The inverse Cauchy problem.- 3. The Cauchy problem.- 1. Distinction of the general case of commuting normal operators A and B.- 2. A criterion for the weak well-posedness.- 3. Proof of Theorem 3.1.- 4. A criterion for the well-posedness.- 5. The (weak) well-posedness in particular cases.- 6. Spectrum of the associated operator pencil.- 7. Fattorini’s definitions of well-posedness of the Cauchy problem.- 4. Boundary-value problems on a finite segment.- 1. The (weak) well-posedness of the Dirichlet problem.- 2. The (weak) well-posedness of the Dirichlet problem in particular cases.- 3. Boundary conditions for the Dirichlet problem.- 4. The weak well-posedness and the well-posedness of the Neumann problem.- 5. Boundary conditions for the Neumann problem.- 6. The inverse Cauchy problem.- II. Initial data of solutions.- to Part II.- 5. Boundary behaviour of an integral transform R(t) as t ? 0 depending on the sub-integral measure.- 1. Analogy with Tauberian theorems.- 2. A model example.- 3. Proof of Lemma 5.1.- 4. Further results.- 5. Continuity of R(t) on R+. in extreme cases.- 6. Continuity of R(t) on R+.- 7. Continuity, boundedness, and integrability of R(t) on a finite segment [0,T].- 8. Equivalence of conditions on a sub-integral measure.- 6. Initial data of solutions.- 1. The set of initial data of solutions.- 2. When FC = D(B) x (D(A) ? D(|B|1/2))?.- 3. When FC = D(B) x D(A)?.- 4. E-sequences of vectors and the general expression for weak solutions.- 5. The set of initial data of weak solutions.- 6. When $${F_C}^\prime = H \times {H_{ - 1}}$$?.- 7. Fatou-Riesz property.- III. Extension, stability, and stabilization of weak solutions.- to Part III.- 7. The general form of weak solutions.- 1. Another general expression for weak solutions.- 2. Continuity, boundedness, and integrability of R(t) on [0.T] in a more general case.- 3. The general form of weak solutions where (2.2) holds.- 4. Initial data of weak solutions where (2.2) holds.- 5. Weak well-posedness of the Cauchy problem in a special space of initial data.- 8. Fatou-Riesz property.- 1. Fatou-Riesz property.- 2. Two-sided Fatou-Riesz property.- 3. First order equation and incomplete second order equations.- 4. The case of self-adjoint A and B.- 5. Spectrum of the associated operator pencil.- 9. Extension of weak solutions.- 1. Extension of weak solutions on a finite interval.- 2. Boundedness of weak solutions on a finite interval.- 3. Exponential growth of weak solutions.- 4. Two-sided extension of weak solutions.- 5. Spectrum of the associated operator pencil.- 6. Comparison of the results on extension of weak solutions and bounded weak solutions.- 7. Intermediate classes of weak solutions.- 8. Extension of weak solutions and weak well-posedness of boundary-value problems.- 10. Stability and stabilization of weak solutions.- 1. Stability and uniform stability of an equation.- 2. Stabilization of the Cesaro means for weak solutions.- 3. Stabilization of a weak solution.- 4. Stabilization of weak solutions and asymptotic stability of an equation.- 5. Exponential stability and uniform exponential stability of an equation.- 6. Stabilization of $$\frac{{y(t)}}{t}$$ for weak solutions of an equation.- 7. The case of self-adjoint A and B.- IV. Boundary-value problems on a half-line.- to Part IV.- 11. The Dirichlet problem on a half-line.- 1. Classes of (weak) uniqueness.- 2. Existence of (weak) solutions.- 3. A criterion for the (weak) well-posedness.- 4. Boundary data of solutions.- 12. The Neumann problem on a half-line.- 1. Classes of uniqueness (weak uniqueness).- 2. Existence of solutions and weak solutions.- 3. Criteria for the well-posedness and the weak well-posedness.- 4. Boundary data of solutions and weak solutions.- Commentaries on the literature.- List of symbols.