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@article{575741, author = {Hilscher, Roman and Zeidan, Vera}, article_number = {9}, 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}, language = {eng}, issn = {1023-6198}, journal = {Journal of Difference Equations and Applications}, title = {Nonnegativity and positivity of quadratic functionals in the discrete calculus of variations: Survey}, url = {http://journalsonline.tandf.co.uk/app/home/contribution.asp?wasp=567302076cf74cd5bc178e8900c08c9c&referrer=parent&backto=issue,6,7;journal,2,45;linkingpublicationresults,1:103361,1}, volume = {11}, year = {2005} }
TY - JOUR ID - 575741 AU - Hilscher, Roman - Zeidan, Vera PY - 2005 TI - Nonnegativity and positivity of quadratic functionals in the discrete calculus of variations: Survey JF - Journal of Difference Equations and Applications VL - 11 IS - 9 SP - 857-875 EP - 857-875 PB - Taylor and Francis SN - 10236198 KW - Second variation KW - Euler-Lagrange difference equation KW - Discrete quadratic functional KW - Nonnegativity KW - Positivity KW - Linear Hamiltonian difference system KW - Conjugate interval KW - Coupled interval KW - Conjoined basis KW - Riccati difference equation UR - http://journalsonline.tandf.co.uk/app/home/contribution.asp?wasp=567302076cf74cd5bc178e8900c08c9c&referrer=parent&backto=issue,6,7;journal,2,45;linkingpublicationresults,1:103361,1 N2 - 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. ER -
HILSCHER, Roman a Vera ZEIDAN. Nonnegativity and positivity of quadratic functionals in the discrete calculus of variations: Survey. \textit{Journal of Difference Equations and Applications}. Taylor and Francis, 2005, roč.~11, č.~9, s.~857-875. ISSN~1023-6198.
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