Some uniqueness theorems for first order hyperbolic systems (original) (raw)

Spectral theory for systems of ordinary differential equations with distributional coefficients

Journal of Differential Equations

We study the spectral theory for the first-order system Ju ′ + qu = wf of differential equations on the real interval (a, b) when J is a constant, invertible skew-Hermitian matrix and q and w are matrices whose entries are distributions of order zero with q Hermitian and w non-negative. Also, we do not pose the definiteness condition often required for the coefficients of the equation. Specifically, we construct minimal and maximal relations, and study self-adjoint restrictions of the maximal relation. For these we determine Green's function and prove the existence of a spectral (or generalized Fourier) transformation. We have a closer look at the special cases when the endpoints of the interval (a, b) are regular as well as the case of a 2 × 2 system. Two appendices provide necessary details on distributions of order zero and the abstract spectral theory for relations.

On an eigenvalue problem of Ahmad and Lazer for ordinary differential equations

Proceedings of the American Mathematical Society, 1987

Lazer, we show the existence of a class of nonselfadjoint eigenvalue problems related to the equation y(n) + Xp(x)y = 0 for which the general eigenvalues comparison is not true. We use a comparison principle for the zeros of the corresponding Cauchy problem. This paper provides a contribution to the understanding of a problem raised by S. Ahmad and A. C. Lazer [1] in connection with the comparison of the eigenvalues for some multi-point boundary value problems which are not selfadjoint. One is given the equation (1) Lny + Xp(x)y = 0, where p(x) is a continuous function of constant sign on an interval /, A is a parameter, and Lny is a linear differential disconjugate operator of order n, that is, the only solution of Lny = 0 with n zeros on I (counting multiplicity) is y = 0. Let us consider the eigenvalue problem given by equation (1) and the system of boundary conditions ,, Lzy(a)=0, iG{ii,...,ik}, L]V(b)=0, JGiJu.-.Jn-k}, where o, b G I, 1 < k < n-1, Liy, i = 0,..., n-1, are the quasi-derivatives of y(x) (see [7]), and {t'i,..., ¿fc}, {ji, ■ ■ ■ ,jn-k) are two arbitrary sets of indices from the set {0,... ,n-1}. Problems of this type have been studied extensively (cf. [2, 3, 5]). In particular, Elias [5] has shown that if (-l)n_fcp(x) < 0, then the eigenvalues of problems (1) and (2) are real and nonnegative and form a divergence sequence {Am}m£N-Ahmad and Lazer [1] have considered a particular type of boundary condition (2), that is (3) y(a)=y'(a) =-= yik-1\a) = 0, y(b)=y'(b) =-=y(n-k-i\b)=0, and showed that if we set p = Pi, where p¿, i-1,2, are two continuous functions, considering the corresponding sequence of eigenvalues (A¿,m)m6N, i = 1,2, ordered by magnitude, then the condition (4)_ (-l)n-kp2(x) < (-l)"-fepi(x) < 0

Some algebraic properties of the hyperbolic systems

Annali Dell'universita' Di Ferrara, 2006

The technique of quasi-symmetrizer has been applied to the well-posedness of the Cauchy problem for scalar operators [10], [13] and linear systems [5], [15], [4], and to the propagation of analitycity for solutions to semi-linear systems [6]. In all these works, it is assumed that the principal symbol depends only on the time variable. In this note we illustrate, in some special cases, a new property of the quasisymmetrizer which allows us to generalize the result in [6] to semi-linear systems with coefficients depending also on the space variables [21]. Such a property is closely connected with some interesting inequalities on the eigenvalues of a hyperbolic matrix. We expect that this technique applies also to other problems. Keywords: First order hyperbolic systems, Quasi-symmetrizer, Glaeser inequality