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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

conjoinedb ases , 295solution, 295special normali zed b ases , 327

co njugate point , 297co nj ugate problem, 210co nj ug ate trans pose, 76co nvex, 169Cr d , 7critical so lution , 32

Darbou xtl.- integr al , 118'V-int egr a l, 125int egral , 117

d el t adiffe rent iable , 2

346

integrable, 118integral, 8

dense, 2dense point , 296derivative

definition, 2, 74exchanging ~ and \7 , 88polynomials , 3properties, 2, 3, 74

Descartes system , 257, 258Dirichlet-Abel test

improper integral , 148disconjugate, 100 , 258

symplectic case, 296 , 298dominant solution, 105dynamic equation

Bernoulli, 38Clairaut, 43Euler, 24first order linear, 19higher order linear, 19logistic , 30Ricc ati,40Verhulst , 30

environmental carrying capacity, 32equilibrium solution, 32Euler equat ion , 24

multiple root case , 26Euler 's formula, 53Euler-Lagrange equation, 300existe nce theorem

antiderivatives, 8pre-antiderivatives , 7

existence-uniqueness theoremmatrix cas e, 75second order linear, 61self- adjoint equation , 96

exp onent ial function , 10, 76equivalence of e a nd e, 90harmonic numbers, 11, 55properties , 10, 76sign, 52t able, 55, 56

extended Pi con e identity, 307

Fekete syst em, 257, 258first order lin ear equa t ion, 19five fun ctionals fixed point theo rem, 241fixed point theorem

Avery-Henderson, 225, 229five fun ct ion als , 241Gatica-Smith, 218Guo-Krasnosel'skii , 194Leggett-Williams, 236 , 237Schauder , 176,220t riple, 236

flow of symplect ic system s, 329focal points, 296

INDEX

forward difference op er ator, 3forw ard jump operator, 1Frechet space, 275Frobenius factorization, 258fundamental system , 62fundamental theorem of calculus, 137 , 138Furi-Pera , 275Furi-Pera theorem, 276

G atica-Smith fixed point theorem, 218general solution, 62, 97gener alized

exponential fun ction, 12graininess, 12polynomials, 79quasilinearization, 165square , 14, 40, 48time scales, 12zero, 100

higher order case, 254sympletic case, 297

graininess, 2Green 's formula, 97Green 's function , 171 , 175, 184, 194, 198,

223, 225 , 237 , 241 , 267, 268symmetry condition, 198

Gronwall 's inequality, 290Guo-Krasnosel 'skii fixed point theorem, 194GZ ,254

Hamiltonian system, 294 , 332 , 334harmonic numbers, 11, 55Harnack inequalit ies , 273higher order Euler equat ion, 24higher order linear equat ion, 19, 81Hilger der ivative, 2hyperbolic functions , 66 , 67hyperbolic system , 330hyperconcave, 166 , 169 , 174hyperconvex , 166 , 169 , 174

improper integralconvergent, 146diverg ent , 146first kind, 145second kind , 155

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

Lagrange bracket , 96Lagr an ge ident ity, 58

self-adjoint equation, 96ld- continuous, 47 , 73Leb esgue

A-int egral , 159A-measure , 157 , 158"V-integr al , 159cr ite rion, 161

Leb esgu e dominated convergence theorem,159, 161

left neighborhood , 85left-dense, 2left- scattered , 2Legendre cond it ion , 297, 316Legget t -Williams fixed point t heo rem, 236,

237Leray- Schauder nonlinear alternative , 207 ,

275Lidst one problem , 191, 194linear eq uatio n

first orde r , 19highe r order, 19

Liou ville's formula , 78Lipschi t z con dit ion, 130 , 179

INDEX

Lipsch itz constant, 130local right-maximum, 4local righ t-minimum, 4logarithm, 35logist ic equation, 30, 38, 42lower

Darbou x A-integr al , 118 , 125Darboux A- sum, ll8Darbou x "V-sum, 124

lower so lution , 167, 175, 271 , 283PBVP, 177

Markov sys tem, 257, 258 , 262matrix exponent ia l, 76mean value theor em , 5, 142 , 143 , 145

nabladerivative , 12exponent ia l fun ction, 49

matrix case, 76sign , 53

hyperb olic fun ctions, 66integr al , 124, 125, 162Riccati equations , 73t r igonome tric fun cti ons, 69 , 70Wronski an

scalar case, 62, 63Newton integr al , ll7nonoscillatory, 100normali zed conjoined bases , 295II-regressive, 48 , 61

osc illatory, 100

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

R , 10, 75

347

348

n+ ,18,53rd-continuous, 7, 285reachable boundary states, 318recessive solution , 105reduction of order, 70, 72

self-adjoint equation, 97refinement, 118regressive , 20 , 89 , 285

alpha case, 13Euler equation, 24first order equat ion, 59group, 10, 75matrix case, 75matrix function , 252scalar case, 10second order equat ion , 63symplectic case, 294vector space, 18, 34

regulated, 7, 129Reid roundabout theorem, 116, 328Riccati equation, 40 , 42

symplectic case, 299Riccati factorization, 109Riccati operator

mixed derivatives , 109nabla, 73symplectic case, 299

Riemann~-integral, 118, 122~-sum , 122'V-integrable, 125'V-sum , 125integral , 121 , 127

right focal problem , 193, 210, 230right neigborhood , 86right-decreasing, 4right-dense, 2right-incr easing, 4right-maximum

local, 4right-minimum

local,4right-scatter ed , 2rising function , 55, 80Rolle's t heore m, 255, 269roundabout t heorem, 328

saturation level , 32SBVP, 165Schauder fixed point theorem , 176 , 220Schauder-Tychonoff theorem, 275second order line ar equa t ions

const ant coe fficients, 94se lf-ad joint form , 92 , 93

sector, 167self-ad joint equation

mix ed derivatives, 92self-r eciprocal , 329semigroup property, 11, 49

INDEX

matrix case, 76sep arated boundary condit ions, 321, 323separated boundary value problem , 165solution, 167

symplectic sys tem, 294special normalized conjoined bases , 296 , 327Stirling's formula , 11strengthened J acobi condition, 311strongly (R~ : I)-normal, 322strongly normal , 310strongly oscillatory, 55Sturm

comparison theorem, 115,316,317sep aration theorem , 101, 316

Sturm-Liouville equation, 294 , 334continuous cas e, 331discrete cas e, 297

sublinear , 199, 201sup erlinear, 199, 200superposition principle , 61Sylvester 's identity, 258symplectic

matrix, 294system, 293 , 294

time scal e, 1topological transversality method, 194 , 207trace, 75Trench factorization , 103, 264trigonometric

fun ctions , 69 , 70system, 329

triple fixed point t heore m, 236

upperDarboux ~-integral, 118Darboux ~-sum, 118Darboux 'V-integral , 125Darboux 'V-sum, 124

upper solution , 175, 271 , 283PBVP,l77

variation of paramet ersfirst order , 19, 59, 60higher order , 81matrix case, 77, 78

Verhulst equa t ion, 30

Wallis product , 57well-posed, 253Wronskian, 96 , 97 , 256

identity, 295