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Index
tl.-in t egr able , 135 , 141
tl.-meas urable , 158
tl.-pre- antiderivative, 14 1
'V-in t egr able , 125
8 , 34
8 , 10
$, 10
characte ristic p olynomial
E u ler eq uatio n , 24
lin ea r eq u at io n , 19 , 65 , 94
ci rcle d ot multiplication , 34
cir cle minus su btract ion, 10
a lph a case, 13
m a trix case, 75
n a bl a case, 48
scala r case, 10
circle p lus add it io n , 10
a lp ha case, 13
m a trix case, 75
n abla case , 48
scala r case, 10
ci rcle square
a lpha case, 14
delta case , 40
n abla case , 48
Clairaut eq uat io n, 43
Cld ,73
compar iso n test
im p ro p er int egral , 147, 155
co m parison t heorem, 177, 193
concave , 169
co ndit io n ally co nve rgent
im p ro p er in t egral , 146
co ne, 190
expans ion a nd compressio n, 225 , 236
reproducing , 190
solid, 190
co njoined
b ases , 295
solution, 295
s pecial normali zed b as es , 327
co nj ugate point , 297
co nj ugate problem, 210
co nj ug a te trans pose, 76
co nvex, 169
Cr d , 7
critical so lution , 32
Abel' s lemma, 143
Abel' s t heorem
converse, 97
h igh er o rder eq uatio n , 257
se co nd order eq uation , 63, 64
se lf-adjoint eq uat ion , 96
a bso lut ely co nvergent
im prop er in t egr a l, 146
adjoint eq uatio n , 19 , 58, 77
adjoint operator , 59
ad m issi b le, 300
a lmost eve ry whe re , 161 , 162
alpha d iffere ntiable, 12
antiderivative, 8 , 117
associated so lution , 295
Avery-Hender son fixed p oi nt theorem , 225 ,
229
b ackward gr aininess , 47
b ack ward j u m p o perator, 1
Banach s pace
p a rtiall y o rdered, 190
Bendixson's formula , 33
Bernoulli eq uat io n, 34, 38
bou ndary co nd it io ns
joint , 328
se parated , 321, 323
boundary value problem
co nj ugate, 210
im pulsive , 233
r igh t focal, 193, 210 , 230
Caratheodo ry extension, 157
Cauchy criter ion
improper int egr al , 146
Cauchy function , 195 , 197 , 267
hi gh er order equation, 81
Cauchy integral, 8 , 117
change of var iable , 141
D arb ou x
tl.- integr al , 118
'V-int egr a l, 125
int egr al , 117
d el t a
diffe rent iable , 2
345
346
integrable, 118
integral, 8
dense, 2
dense point , 296
derivative
d efinition, 2, 74
exchanging ~ and \7 , 88
polynomials , 3
properties, 2, 3, 74
Descartes system , 257, 258
Dirichlet-Abel test
improper integral , 148
disconjugate, 100 , 258
symplectic case, 296 , 298
dominant solution, 105
dynamic equation
Bernoulli, 38
Clairaut, 43
Euler, 24
first order linear, 19
higher order linear, 19
logistic, 30
Ricc ati,40
Verhulst, 30
environmental carrying capacity, 32
equilibrium solution, 32
Euler equat ion , 24
multiple root case , 26
Euler 's formula, 53
Euler -Lagrange equation, 300
existe nce theorem
antiderivatives, 8
pre-antiderivatives , 7
existence-un iq uenes s theorem
matrix cas e, 75
second order linear, 6 1
self- adjoint equation , 96
exp onent ial function , 10, 76
eq uiva lence of e a nd e, 90
harmonic numbers, 11, 55
properties , 10, 76
sign, 52
t able, 55, 56
extend ed Pi con e id entity, 307
Fekete syst em, 257, 258
first order lin ear eq ua t ion, 19
five fun ctionals fixed point theo rem, 241
fixed point theorem
Avery-Henderson, 225, 229
five fun ct ion als , 241
G atica-Smith, 218
Guo-Krasnosel'skii, 194
Leggett-Williams , 236 , 237
Sch auder , 176,220
t riple, 236
flow of sy mplect ic system s, 329
focal points, 296
INDEX
forward difference op er ator, 3
forw ard jump operator, 1
Frechet space, 275
Frobenius factorization, 258
fundamental system , 62
fundamental theorem of calculus, 137 , 138
Furi-Per a , 275
Furi-Pera theorem, 276
G atica-Smith fixed point theorem, 218
general solution, 62, 97
gener alized
exponential fun ction, 12
graininess, 12
polynomials, 79
quasilinearization, 165
square , 14, 40, 48
time scales, 12
zero, 100
higher order case, 254
sympletic case, 297
graininess, 2
Green 's formula, 97
Green 's function , 171 , 175, 184, 194, 198,
223, 225 , 237 , 241 , 267 , 268
symmetry condition, 198
Gronwall 's inequality, 290
Guo-Krasnosel'skii fixed point theorem, 194
GZ ,254
Hamiltonian sy stem, 294 , 332 , 334
harmonic numbers, 11, 55
Harnack ineq ua lit ies , 273
higher order Euler eq uat ion, 24
higher order linear equat ion, 19, 81
Hilger der ivative, 2
hyperbolic functions , 66 , 67
hyperbolic system , 330
hyperconcave, 166 , 169 , 174
hyperconvex , 166 , 169 , 174
improper integral
convergent, 146
diverg ent , 146
first kind , 145
second kind , 155
impulsive problem, 233
indefin ite integral, 8
infinit e intervals , 285
initial valu e problem
first order linear, 10, 19, 58- 60
matrix case, 77, 78
seco nd order linear, 61, 66
com plex roots, 70
distinct real roots, 68
double root, 71
inner product , 97
int egr abl e
Cauchy criteri on , 120
INDEX
d elt a , 118
Riem ann , 121 , 122
int egr al
Cauchy, 8, 117
co nsec utive point s, 89
Darbou x, 117
im prop er , 145 , 155
indefinit e, 8
Leb esgu e, 159
nabla, 124
Newt on, 117
pr op er ti es, 8, 9
Ri em ann , 121
int egr ation by p arts, 8, 137
integrati on by substitution , 141
interior, 12
int erpolating families, 257
intrinsic growt h fun ction, 32
isolated ,2
J acobi 's cond it ion, 114
streng t he ne d, 311
join t bou nd ary cond it ions, 328
jump ope rator
backward , 1
forward, 1
Kiguradze inequali ti es , 273
K rein-Rutman t heory, 272
kth qu as i-A d eri vative, 263
L'H6pit al 's rule
delt a de rivatives, 86
na bla derivat ives , 86
Lagr an ge br acket , 96
Lagr an ge ident ity, 58
self-adjoint equation, 96
ld- con tinuous, 47 , 73
Leb esgu e
A-int egral, 159
A-measure , 157 , 158
"V-integr al, 159
cr ite rion, 161
Leb esgu e dominated convergence theorem,
159, 161
left neighbo rho od , 85
left-dense, 2
left- scatter ed , 2
Legendre cond it ion , 297, 316
Legget t -Williams fixed point t heo re m, 236,
237
Ler ay- Sch auder nonlinear alternative , 207 ,
275
Lidst on e problem , 191, 194
linear eq uatio n
first or de r , 19
highe r or der, 19
Liou ville's formula , 78
Lip schi t z con d it ion, 130 , 179
347
Lip sch it z constant, 130
local right-maximum, 4
lo cal righ t-minimum, 4
logarithm, 35
logist ic equation , 30, 38, 42
lower
Darbou x A -int egr al , 118 , 125
Darboux A- sum, ll8
Darbou x "V-sum, 124
lower so lution , 167, 175, 271 , 283
PBVP, 177
Markov sys tem, 257, 258 , 262
matrix ex pone nt ia l, 76
mean value theor em , 5, 142 , 143 , 145
n abla
d erivative , 12
expone nt ia l fun ction, 49
matrix case, 76
sign , 53
hyperb olic fun ctions, 66
integr al , 124, 125, 162
Riccati eq uations , 73
t r igono me tric fun cti on s, 69 , 70
Wr onski an
scala r case, 62, 63
Newton int egr al , ll 7
nonoscillatory, 100
normali zed con joined bas es , 295
II-regr essive, 48 , 61
osc illatory, 100
par t ial der ivati ve, 33
par tition, 118
PBVP, 166
p eriodic boundary value problem , 166
P er ron theor em , 272
Picon e id entity, 112, 304, 305
extende d, 307
P 6lya factorization , 102
P6lya mean value t he ore m , 263
p opulation mod el , 11 ,32
positive d efinite, III
p ositively reg ress ive , 10, 18 , 53
Priifer transforma t ion , 331
pre-an t ider ivative, 8, 117
pre-differ en ti able, 6
prin cip al soluti o n, 295
principal sys tem of so lutions, 265
product rule, 3, 13 , 74
quadrat ic convergence, 172 , 176
quadra ti c fun cti on al , 300
nonhom ogen eous, 318
quasi -A d eriva ti ve, 263
quotien t rul e, 3, 13, 74
R , 10 , 75
348
INDEX
n+ ,18,53
rd-continuous, 7, 285
reachable boundary states, 318
recessive solution , 105
reduction of order, 70, 72
self-adjoint equation, 97
refinement, 118
regressive , 20 , 89 , 285
alpha case, 13
Euler equation, 24
first order equat ion, 59
group, 10, 75
matrix case, 75
matrix function , 252
scalar case, 10
second order eq uat ion , 63
symplectic cas e, 294
vector space, 18, 34
regulated, 7, 129
Reid roundabout theorem, 116, 328
Riccati equation, 40 , 42
symplectic case, 299
Riccati factoriz ation , 109
Riccati operator
mixed derivatives , 109
nabla, 73
symplectic case, 299
Riemann
~-integral, 118, 122
~-sum , 122
'V-integrable, 125
'V-sum , 125
integral , 121 , 127
right focal problem , 193, 210, 230
right neigborhood , 86
right-decr easing, 4
right-dense, 2
right-incr easing, 4
right-maximum
local, 4
right-minimum
local,4
right-scatter ed , 2
rising function , 55 , 80
Rolle's t heore m , 255 , 269
roundabout t heorem , 328
m atrix case, 76
sep arated boundary cond it ions, 321 , 323
se parated boundary value problem , 165
solution, 167
symplectic sys tem , 294
special normalized conjoine d bases , 296 , 327
Stirling's formula , 11
strengthened J acobi condition, 311
strongly (R~ : I)-normal, 322
strongly normal , 310
strongly oscillatory, 55
Sturm
comparison theorem , 115,316,317
sep aration theorem , 101, 316
Sturm-Liouville equation, 294 , 334
continuous cas e, 331
discrete cas e, 297
sublinear , 199, 201
sup erline ar, 199 , 200
superposition principle, 61
Sylvester 's id entity, 258
symplectic
matrix, 294
system, 293 , 294
saturation level , 32
SBVP, 165
Schauder fixed point theorem , 176 , 220
Schauder-Tychonoff theorem, 275
second order line ar eq ua t ions
const a nt coe fficie nts, 94
se lf-ad joint form , 92 , 93
sector, 167
self-ad joint equation
mix ed d erivatives, 92
self-r eciprocal , 329
semigroup property, 11, 49
Wallis product , 57
well-posed, 253
Wronskian, 96 , 97 , 256
identity, 295
time scal e, 1
topological transversality method, 194 , 207
trace, 75
Trench factorization , 103, 264
trigonometric
fun ctions , 69 , 70
sy st em , 329
triple fixed point t heore m , 236
upper
Darboux ~- integral, 118
Darboux ~-sum, 118
Darboux 'V-integral, 125
Darboux 'V-sum, 124
upper solution , 175, 271 , 283
PBVP,l77
variation of paramet er s
first order , 19, 59, 60
higher order , 81
m atrix cas e, 77, 78
Verhulst eq ua t ion, 30

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