Publication details

Nonnegativity and positivity of quadratic functionals in the discrete calculus of variations: Survey

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Year of publication 2005
Type Article in Periodical
Magazine / Source Journal of Difference Equations and Applications
MU Faculty or unit

Faculty of Science

Field General mathematics
Keywords Second variation; Euler-Lagrange difference equation; Discrete quadratic functional; Nonnegativity; Positivity; Linear Hamiltonian difference system; Conjugate interval; Coupled interval; Conjoined basis; Riccati difference equation
Description In this paper we provide a survey of characterizations of the nonnegativity and positivity of discrete quadratic functionals which arise as the second variation for nonlinear discrete calculus of variations problems. These characterizations are in terms of (i) (strict) conjugate and (strict) coupled intervals, (ii) the conjoined bases of the associated Jacobi difference equation, and (iii) the solution of the corresponding Riccati difference equation. The results depend on the form of the boundary conditions of the quadratic functional and, basically, we distinguish three types: (a) separable endpoints with zero right endpoint (this of course includes the simplest case of both zero endpoints), (b) separable endpoints, and (c) jointly varying endpoints.
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