Nonnegativity of a discrete quadratic functional in terms of the (strengthened) Legendre and Jacobi conditions

Computers and Mathematics with Applications

Volumn 45, Issue 6-9, 2003, Pages 1369-1383

  Hilscher, R ^a   Zeidan, V ^a  

^a MICHIGAN STATE UNIVERSITY   (United States)

Author keywords

Conjugate interval; Discrete calculus of variations; Discrete quadratic functional; Jacobi difference equation; Legendre condition

Indexed keywords

DIFFERENCE EQUATIONS; DIFFERENTIATION (CALCULUS); MATRIX ALGEBRA; RICCATI EQUATIONS; VECTORS;

CONJUGATE INTERVAL; DISCRETE QUADRATIC FUNCTIONAL; JACOBI DIFFERENCE EQUATION; LEGENDRE CONDITION;

BOUNDARY CONDITIONS;

EID: 0037732716     PISSN: 08981221     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0898-1221(03)00109-3     Document Type: Article

References (28)

1. R.P. Agarwal, C.D. Ahlbrandt, M. Bohner and A. Peterson, Discrete linear Hamiltonian systems: A survey, Discrete and Continuous Hamiltonian Systems, Special Issue of Dynam. Systems Appl., (R.P. Agarwal and M. Bohner. Guest Editors) 8 (3/4), 307-333, (1999).
2. C.D. Ahlbrandt, Discrete variational inequalities, In General Inequalities 6, Proceedings of the 6th International Conference on General Inequalities, Oberwolfach, 1990, International Series of Numerical Mathematics, Volume 123, (Edited by W. Walter), pp. 93-107, Birkhäuser Verlag, Basel, (1992).
3. C.D. Ahlbrandt, Equivalence of discrete Euler equations and discrete Hamiltonian systems, J. Math. Anal. Appl. 180, 498-517, (1993).
4. C.D. Ahlbrandt, C. Chicone, S.L. Clark, W.T. Patula and D. Steiger, Approximate first integrals for discrete Hamiltonian systems, Dynam. Contin. Discrete Impuls. Systems 2, 237-264, (1996).
5. C.D. Ahlbrandt and J.W. Hooker, A variational view of nonoscillation theory for linear difference equations, In Differential and Integral Equations, Proceedings of the 12th and 13th Midwest Conferences, Iowa City, IA, 1983, Argonne, IL, 1984, (Edited by J.L. Henderson), pp. 1-21, Institute of Applied Mathematics, University of Missouri-Rolla, Rolla, MO, (1985).
6. C.D. Ahlbrandt and A.C. Peterson, Discrete Hamiltonian Systems: Difference Equations, Continued Fractions, and Riccati Equations, Kluwer Academic, Boston, MA, (1996).
7. M. Bohner, Linear Hamiltonian difference systems: Disconjugacy and Jacobi-type conditions, J. Math. Anal. Appl. 199, 804-826, (1996).
8. M. Bohner, Riccati matrix difference equations and linear Hamiltonian difference systems, Dynam. Contin. Discrete Impuls. Systems 2 (2), 147-159, (1996).
9. M. Bohner and O. Došlý, Disconjugacy and transformations for symplectic systems, Rocky Mountain J. Math. 27 (3), 707-743, (1997).
10. M. Bohner, O. Došlý and W. Kratz, Discrete Reid roundabout theorems, Discrete and Continuous Hamiltonian Systems, Special Issue of Dynam. Systems Appl., (R.P. Agarwal and M. Bohner, Guest Editors) 8 (3/4), 345-352, (1999).
11. R. Hilscher and V. Zeidan, Discrete optimal control: The accessory problem and necessary optimality conditions, J. Math. Anal. Appl. 243 (2), 429-452, (2000).
12. R. Hilscher and V. Zeidan, Second order sufficiency criteria for a discrete optimal control problem, J. Difference Equ. Appl. 8 (6), 573-602, (2002).
13. W.G. Kelley and A.C. Peterson, Difference Equations: An Introduction with Applications, Academic Press, San Diego, CA, (1991).
14. C.D. Ahlbrandt and J.W. Hooker, Riccati matrix difference equations and disconjugacy of discrete linear systems, SIAM J. Math. Anal. 19 (5), 1183-1197, (1988).
15. H. Kawasaki, A conjugate points theory for a nonlinear programming problem, SIAM J. Control Optim. 40 (1), 54-63, (2001).
16. M. Bohner, Calculus of variations on time scales, Author preprint, (July 2000).
17. R. Hilscher, Reid roundabout theorem for symplectic dynamic systems on time scales, Appl. Math. Optim. 43 (2), 129-146, (2001).
18. M. Bohner, Inhomogeneous discrete variational problems, In Advances in Difference Equations, Proceedings of the Second International Conference on Difference Equations, Veszprem, 1995, (Edited by S. Elaydi, I. Györi and G. Ladas), pp. 89-97, Gordon and Breach, Amsterdam, (1997).
19. M. Bohner and O. Došlý, Positivity of block tridiagonal matrices, SIAM J. Matrix Anal. 20 (1), 182-195, (1998).
20. O. Došlý and R. Hilscher, Linear Hamiltonian difference systems: Transformations, recessive solutions, generalized reciprocity, Discrete and Continuous Hamiltonian Systems, Special Issue of Dynam. Systems Appl., (R.P. Agarwal and M. Bohner, Guest Editors) 8 (3/4), 401-420, (1999).
21. L. Erbe and P. Yan, Disconjugacy for linear Hamiltonian difference systems, J. Math. Anal. Appl. 167, 355-367, (1992).
22. L. Erbe and P. Yan, Qualitative properties of Hamiltonian difference systems, J. Math. Anal. Appl. 171, 334-345, (1992).
23. R. Hilscher, Disconjugacy of symplectic systems and positivity of block tridiagonal matrices, Rocky Mountain J. Math. 29 (4), 1301-1319, (1999).
24. C.D. Ahlbrandt, Dominant and recessive solutions of symmetric three term recurrences, J. Differential Equations 107, 238-258, (1994).
25. V. Zeidan, Sufficient conditions for variational problems with variable endpoints: Coupled points, Appl. Math. Optim. 27 (2), 191-209, (1993).
26. P. Zezza, The Jacobi condition in optimal control, In Control Theory, Stochastic Analysis and Applications, (Edited by S. Chen and J. Yong), pp. 137-149, World Scientific, River Edge, NJ, (1991).
27. P. Zezza, Jacobi condition for elliptic forms in Hilbert spaces, J. Optim. Theory Appl. 76 (2), 357-380, (1993).
28. D.G. Luenberger, Linear and Nonlinear Programming, Second Edition, Addison-Wesley, Reading, MA, (1984).