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A nonlinear calculus of variations problem on time scales with variable endpoints is considered. The space of functions employed is that of piecewise rd-continuously ∆-differentiable functions (C 1 prd ). For this problem, the Euler-Lagrange equation, the transversality condition, and the accessory problem are derived as necessary conditions for weak local optimality. Assuming the coercivity of the second variation, a corresponding second order sufficiency criterion is established.

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We present a theory of the definiteness (nonnegativity and positivity) of a quadratic functional F over a bounded time scale. The results are given in terms of a time scale symplectic system (S), which is a time scale linear system that generalizes and unifies the linear Hamiltonian differential system and discrete symplectic system. The novelty of this paper resides in removing the assumption of normality in the characterization of the positivity of F , and in establishing equivalent conditions for the nonnegativity of F without any normality assumption. To reach this goal, a new notion of generalized focal points for conjoined bases (X, U ) of (S) is introduced, results on the piecewise-constant kernel of X(t) are obtained, and various Picone-type identities are derived under the piecewise-constant kernel condition. The results of this paper generalize and unify recent ones in each of the discrete and continuous time setting, and constitute a keystone for further development in this theory.

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Abstract. 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 traditional Rayleigh principle to possibly abnormal linear Hamiltonian systems. The main tools are the extended Picone formula, which is proven here for this general setting, results on piecewise constant kernels for conjoined bases of the Hamiltonian system, and the oscillation theorem relating the number of proper focal points of conjoined bases with the number of finite eigenvalues. As applications we obtain the expansion theorem in the space of admissible functions without controllability and a result on coercivity of the corresponding quadratic functional.

In this paper we study the existence and properties of the principal solutions at infinity of nonoscillatory linear Hamiltonian systems without any controllability assumption. As our main results we prove that the principal solutions can be classified according to the rank of their first component and that the principal solutions exist for any rank in the range between explicitly given minimal and maximal values. The minimal rank then corresponds to the minimal principal solution at infinity introduced by the authors in their previous paper, while the maximal rank corresponds to the principal solution at infinity developed by W. T. Reid, P. Hartman or W. A. Coppel. We also derive a classification of the principal solutions, which have eventually the same image. The proofs are based on a detailed analysis of conjoined bases with a given rank and their construction from the minimal conjoined bases. We illustrate our new theory by several examples.

Key words Linear Hamiltonian system, self-adjoint eigenvalue problem, proper focal point, conjoined basis, finite eigenvalue, oscillation, controllability, normality, quadratic functional MSC (2010) 34L05, 34C10, 49N10, 93B60, 34L10In 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 time we do not impose any controllability and strict normality assumptions. We introduce the notion of a finite eigenvalue and prove the oscillation theorem relating the number of finite eigenvalues which are less than or equal to a given value of the spectral parameter with the number of proper focal points of the principal solution of the system in the considered interval. We also define the corresponding geometric multiplicity of finite eigenvalues in terms of finite eigenfunctions and prove that the algebraic and geometric multiplicities coincide. The results are also new for Sturm-Liouville differential equations, being special linear Hamiltonian systems.The coefficients are n × n-matrix-valued functions such that the Hamiltoniandefined on [a, b] × R is symmetric, satisfies certain smoothness assumptions (see Section 2), and the monotonicity assumption