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@article{488395,
author = {Hilscher, Roman and Zeidan, Vera},
article_location = {USA},
article_number = {1},
keywords = {discrete quadratic functional; coupled interval; Jacobi difference equation; conjugate interval; Legendre condition; discrete calculus of variations},
language = {eng},
issn = {0022-247X},
journal = {Journal of Mathematical Analysis and Applications},
title = {Coupled intervals in the discrete calculus of variations: necessity and sufficiency},
volume = {276},
year = {2002}
}
TY - JOUR
ID - 488395
AU - Hilscher, Roman - Zeidan, Vera
PY - 2002
TI - Coupled intervals in the discrete calculus of variations: necessity and sufficiency
JF - Journal of Mathematical Analysis and Applications
VL - 276
IS - 1
SP - 396-421
EP - 396-421
PB - Acad.Press
SN - 0022247X
KW - discrete quadratic functional
KW - coupled interval
KW - Jacobi difference equation
KW - conjugate interval
KW - Legendre condition
KW - discrete calculus of variations
N2 - In this work we study nonnegativity and positivity of a discrete quadratic functional with separately varying endpoints. We introduce a notion of an interval coupled with 0, and hence, extend the notion of conjugate interval to 0 from the case of fixed to variable endpoint(s). We show that the nonnegativity of the discrete quadratic functional is equivalent to each of the following conditions: the nonexistence of intervals coupled with 0, the existence of a solution to Riccati matrix equation and its boundary conditions. Natural strengthening of each of these conditions yields a characterization of the positivity of the discrete quadratic functional. Since the quadratic functional under consideration could be a second variation of a discrete calculus of variations problem with varying endpoints, we apply our results to obtain necessary and sufficient optimality conditions for such problems. This paper generalizes our recent work in [18], where the right endpoint is fixed.
ER -
HILSCHER, Roman a Vera ZEIDAN. Coupled intervals in the discrete calculus of variations: necessity and sufficiency. \textit{Journal of Mathematical Analysis and Applications}. USA: Acad.Press, 2002, roč.~276, č.~1, s.~396-421. ISSN~0022-247X.
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