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[IJ R . Agarwal, M. Bohner , and A. Pet erson. Inequalities on t ime scales: A survey. Mat h.In equal. A ppl., 4(4) :535- 557 , 2001.
[2J R. P. Agarwal. Difference Equa tions and Inequalit ies. Marcel Dekker , Inc. , New York, 1992.[3J R. P. Agarwal and M . Bohner . Quadratic fun ction als for seco nd orde r matrix equat ions on
time scales. Nonlin ear A nal. , 33(7) :675- 692, 1998.[4J R . P. Agarwal and M. Bohner. Basi c ca lculus on t ime scales and some of its applications.
Results Math., 35(1-2) :3- 22, 1999 .[5] R . P. Agarwal , M . Bohner , and D. O 'Regan. Time scale sys tems on infinite intervals. N on
lin ear Anal., 47:837- 848, 2001.[6J R . P. Agarwal , M. Bohner , and D. O 'Regan . Time scale boundary value problems on in
finite intervals . J. Comput. Appl. Math ., 141(1-2) :27- 34 , 2002 . Special Issu e on "Dynam icEquations on Time Scales" , ed ited by R . P. Agarwal , M . Bohner , and D. O'Regan .
[7J R. P. Agarwal, M. Bohner , D. O 'Regan, and A . Peterson. Dynamic equat ions on t imescales : A sur vey. J. Comput. Appl. Math., 141(1-2 ):1- 26, 2002. Special Issu e on "Dy nam icEquations on Time Scales" , edi ted by R. P . Agar wal , M. Bohner , and D. O 'Regan. Preprintin Ulme r Seminare 5.
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[10] R . P. Ag arwal, M. Boh ner , and P. J . Y. Wong. Sturm- Liouv ille eigenva lue problems on t imescales . Appl. Ma th. Comput., 99( 2-3) :153-166, 1999 .
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Index
tl.-in t egr able , 135, 141tl.-meas urable , 158tl.-pre-antiderivative, 141'V-in t egrable , 1258 , 348 , 10$, 10
Abel' s lemma, 143Abel' s t heorem
converse, 97h igh er order equation , 257seco nd order eq uation , 63, 64se lf-adjoint equat ion , 96
a bsolut ely co nvergentim prop er in t egr a l, 146
adjoint eq uation , 19, 58, 77adjoint operator , 59ad m issib le, 300a lmost every whe re , 161 , 162alpha d iffere ntiable, 12antiderivative, 8 , 117associated solution , 295Avery-Hender son fixed point theorem , 225,
229
backward graininess , 47back ward ju m p operator, 1Banach space
partiall y ordered, 190Bendixson's formula , 33Bernoulli equat ion, 34, 38bou ndary cond it ions
joint , 328se parated , 321, 323
boundary value problemco nj ugate, 210im pulsive , 233righ t focal, 193, 210 , 230
Caratheodo ry extension, 157Cauchy criter ion
improper int egr al , 146Cauchy function, 195 , 197 , 267
hi gh er order equation, 81Cauchy integral , 8 , 117change of variable , 141
345
characteristic polynomialE u ler eq uation , 24lin ea r eq uat ion , 19, 65 , 94
ci rcle d ot multiplication, 34cir cle minus subtract ion, 10
a lpha case, 13m atrix case, 75nabl a case, 48scala r case, 10
circle p lus add it ion , 10a lpha case, 13m atrix case, 75nabla case , 48scala r case, 10
ci rcle squarea lpha case, 14delta case , 40nabla case , 48
Clairaut eq uat ion, 43Cld ,73compar iso n test
im p ro p er int egral , 147, 155co m parison t heorem, 177, 193concave , 169co ndit ionally co nve rgent
im p ro per in t egral , 146co ne, 190
expans ion a nd compression, 225, 236reproducing , 190solid, 190
impulsive problem, 233indefin ite integral , 8infinite intervals , 285initial value problem
first order linear, 10, 19, 58- 60matrix case, 77, 78seco nd order linear, 61, 66
complex roots, 70distinct real roots, 68double root, 71
inner product , 97integr abl e
Cauchy criteri on , 120
delta , 118Riem ann, 121 , 122
int egr alCauchy, 8, 117co nsec utive points , 89Darbou x, 117im prop er , 145 , 155indefinit e, 8Lebesgue, 159nabla, 124Newton, 117pr op er ti es, 8, 9Riem ann, 121
integr ation by parts, 8, 137integrati on by substitution , 141interior, 12int erpolating families , 257intrinsic growt h fun ction, 32isolated ,2
J acobi 's cond it ion, 114streng t hened, 311
join t bou nd ary condit ions, 328jump operator
backward, 1forward, 1
Kiguradze inequali ti es , 273Krein-Rutman t heory, 272kth qu as i-A d eri vative, 263
L'H6pit al 's ruledelt a de rivatives, 86nabla derivat ives , 86
par t ial der ivati ve, 33par tition, 118PBVP, 166periodic boundary value problem , 166P er ron theor em , 272Picone identity, 112, 304, 305
extended, 307P 6lya factorization, 102P6lya mean value t heore m , 263population mod el , 11,32positive definite , IIIp ositively reg ress ive , 10, 18 , 53Priifer transformat ion , 331pre-an t ider ivative, 8, 117pre-differen ti able, 6princip al solution, 295principal sys tem of so lutions, 265product rule, 3, 13, 74
quadrat ic convergence, 172 , 176quadra ti c fun cti on al , 300
nonhom ogeneous, 318quasi-A deriva ti ve, 263quotien t rul e , 3, 13, 74