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In the first part inequalities for solutions of Riccati matrix difference equations are obtained which correspond to the linear Hamiltonian difference system ⌬ X s A X q B U , ⌬U s C X y A T U , k k k q 1 k k k k kq1 k k where A , B , C , X , U are n = n-matrices with symmetric B and C. If the k k k k k k k matrices X are invertible, then the matrices Q s U(More)
We consider symplectic difference systems involving a spectral parameter together with general separated boundary conditions. We establish the so-called oscillation theorem which relates the number of finite eigenvalues less than or equal to a given number to the number of focal points of a certain conjoined basis of the symplectic system. Then we prove(More)
In this paper we prove the differentiability properties of solutions of nonlinear dynamic equations on time scales with respect to parameters. This complements the previous work of the first and third authors regarding the existence and continuity of solutions with respect to parameters. In addition, we treat separately time scale dynamic equations which(More)
In this paper we consider linear Hamiltonian differential systems without the controllability (or normality) assumption. We prove the Rayleigh principle for these systems with Dirichlet boundary conditions, which provides a variational characterization of the finite eigenvalues of the associated self-adjoint eigenvalue problem. This result generalizes the(More)
In this paper, we consider linear Hamiltonian differential systems which depend in general nonlinearly on the spectral parameter and with Dirichlet boundary conditions. Our results generalize the known theory of linear Hamiltonian systems in two respects. Namely, we allow nonlinear dependence of the coefficients on the spectral parameter and at the same(More)
In this article we treat the algebraic eigenvalue problem for real, symmetric, and banded matrices of size N × N , say. For symmetric, tridi-agonal matrices, there is a well-known two-term recursion to evaluate the characteristic polynomials of its principal submatrices. This recursion is of complexity O(N) and it requires additions and multiplications(More)
In this paper we study a general even order symmetric Sturm–Liouville matrix differential equation, whose leading coefficient may be singular on the whole interval under consideration. Such an equation is new in the current literature, as it is equivalent with a system of Sturm–Liouville equations with different orders. We identify the so-called normal form(More)