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Hartman P. Ordinary Differential Equations
Hartman P.  Ordinary Differential Equations









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: Ordinary Differential Equations

: Hartman P.

:

This work was originally published in 1973, and appeared in a second edition in 1982 (Birkhauser), of which this is a reprint. Covering the basics of the theory of ordinary differential equations, discussion centers on the integration of differential inequalities but also describes techniques involving simple topological arguments, fixed point theorems, and functional analysis. The text assumes a familiarity with matrix theory and the ability to deal with functions of real variables. Hartman taught mathematics at Johns Hopkins University.


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: 09.04.2005

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(u, v)-subset      281 291 293
Adjoint, dth order equation      66
Adjoint, dth order equation, linear system      62
Adjoint, dth order equation, system of total differential equations      119 (see also Associate operator)
Admissible      438 462 481
Angular distance      451
Arzela theorems      4
Ascoli theorems      4
Associate operator      486
Asymptotic integration, boundary layer theory      534
Asymptotic integration, boundary layer theory, difference equation      241(Ex. 5.4) 289(Ex. 290(Ex.
Asymptotic integration, boundary layer theory, dth order equation      314 447(Ex.9.2(c))
Asymptotic integration, boundary layer theory, perturbed linear system      212 259 273 445 447(Ex.
Asymptotic integration, boundary layer theory, second order equation      319(Ex. 17.4 17.5) 320(Ex. 369 375 446 447(Ex.
Asymptotic lines      107(Ex. 6.3(c))
Asymptotic phase      254
Asymptotic stability      see Stability
Attractor      160 213
Attractor, domain of attraction      548
Autonomous system      38 144 202
Banach spaces, admissible      438 462 481
Banach spaces, admissible, $L^p$, $L^{\infty}$, $L^{\infty}_0$, M      436 453
Banach spaces, admissible, associate      484
Banach spaces, admissible, class $\mathcal{J}$, $\mathcal{J}#$      452 484
Banach spaces, admissible, lean at $\omega$      439
Banach spaces, admissible, open mapping theorem      405 437 439 464
Banach spaces, admissible, quasi-full      453 467 471 474 475
Banach spaces, admissible, Schauder fixed point theorem      405 414 425
Banach spaces, admissible, stronger than L      437 453
Bessel equation      87(Ex. 12.3(a))
Bessel equation, $J_{\mu}^2+K_{\mu}^2$      518(Ex. 4.2). 519(Ex. 4.3)
Bessel equation, asymptotic behavior of solutions      371 (Ex. 8.1)
Bessel equation, integral of $J_{\mu}$      513 (Ex. 3.6)
Bessel equation, zeros of solutions      336(Ex. 3.2(d)) 519(Ex.
Blasius differential equation      520
Bohl theorem      199
Boundary layer theory      519
Boundary value problems      337 407 418 519
Boundary value problems, adjoint      410
Boundary value problems, linear, first order      407
Boundary value problems, linear, second order      418
Boundary value problems, nonlinear, second order      422 433 434(Ex.
Boundary value problems, periodic, nonlinear      412 435(Ex.5.9(6))
Boundary value problems, singular, third order      519
Boundary value problems, Sturm Liouville      337
Bounds for solutions      30
Bounds for solutions, autonomous system      543
Bounds for solutions, derivatives      428
Bounds for solutions, equation of variation      110
Bounds for solutions, second order equation      373(Ex. 8.6) 374(Ex.
Brouwer fixed point theorem      278
Cantor selection theorem      3
Cauchy, characteristic strips      133 135
Cauchy, characteristic strips, initial value problem      131 137 140
Center      159 215 216(Ex.
Characteristic direction      209 220
Characteristic equation (polynomial)      65
Characteristic roots and exponents      61 252 253
Characteristic strips      133 135
Comparison theorem, principal solutions      358
Comparison theorem, principal solutions, second order equations      333
Comparison theorem, principal solutions, second order systems      391(Ex. 10.2)
Comparison theorem, principal solutions, Sturm      333
Complete integrability      118 123 128
Complete Riemann metric      541 546
Complete system of linear partial differential equations      119 120(Ex. 124
Conjugate, points (Jacobis theorem)      391(Ex. 10.1)
Conjugate, points (Jacobis theorem), solutions      385 386 399
Constant coefficients, dth order equation      65
Constant coefficients, dth order equation, linear system      57
Constant coefficients, dth order equation, second order equation      324 (Ex. 1.1)
Continuum      16
Curvature, quadratic form      106(Ex. 6.2 63(b))
Curvature, quadratic form, lines of      107(Ex. 63(d))
Curvature, quadratic form, Riemann tensor      106(Ex. 6.2)
Cycles      see Limit cycles
Derivate      26
Dichotomies, adjoint systems      484
Dichotomies, adjoint systems, definitions      453
Dichotomies, adjoint systems, first order systems      474
Dichotomies, adjoint systems, Greens functions      461 476 477
Dichotomies, adjoint systems, higher order systems      478
Dichotomies, adjoint systems, second order equations      483(Ex. 7.1) 496(Ex. 13.2)
Difference equations      241 (Ex. 5.4) 289 290(Ex.
Differential forms      101 120
Differential inequalities      24
Differential inequalities, equations of variation      110
Differential inequalities, linear systems      54
Differential inequalities, partial      140 141 142(Ex.
Disconjugate equation of second order      351 362
Disconjugate equation of second order, positive solutions      351 352(Ex.
Disconjugate equation of second order, variational principles      352 (see also Nonoscillatory)
Disconjugate system of second order      384
Disconjugate system of second order, criterion      388 390 391 10.2) 420 421(Ex.
Domain of attraction      548
Egress point      37 175 202 278 281 520
Eigenfunction      338 342
Eigenfunction, $L^2$-approximation (completeness)      338
Eigenfunction, interpolation      344(Ex. 4.4)
Equations of variation      96 110
Equations of variation, bounds for solutions      110
Equicontinuity      3
Equivalence, differential equations      258 271(Ex.
Equivalence, differential equations, maps      258
Ergodic      193 194 198 199
Euler differential equation      85
Existence in large      29
Existence in large for linear systems      31 45
Existence theorems, boundary value problems      337 407 418 519
Existence theorems, boundary value problems, Cauchy problem      137
Existence theorems, boundary value problems, invariant manifolds      234 242 296
Existence theorems, boundary value problems, linear system of partial differential equations      124
Existence theorems, boundary value problems, maximal solution      25
Existence theorems, boundary value problems, monotone solution      357 506 514
Existence theorems, boundary value problems, PD-solution      497
Existence theorems, boundary value problems, Peano      10
Existence theorems, boundary value problems, Picard Lindelof      8 (see also Periodic solution Solutions
Extension theorem      12
Exterior, derivatives      102
Exterior, derivatives, forms      101 120
Fixed point theorems      see Brouwer Schauder Tychonov
Floquet theory      60 66 71 302(Ex.
Focus      160 215 216(Ex.
Formal power series      78 79 261
Frechet space      405 436
Frobenius, factorization      67
Fuchs, theorem      85
Fuchs, theorem, type of differential equation      86(Ex. 12.2)
Fundamental matrix or solution      47
Fundamental matrix or solution, adjoint system      62
Fundamental matrix or solution, analytic system      70
Fundamental matrix or solution, characteristic exponents and roots      61
Fundamental matrix or solution, Floquet theory      60 71 302(Ex.
Fundamental matrix or solution, system with constant coefficients      57
Fundamental matrix or solution, unitary      62(Ex. 7.1)
Geodesies      106(Ex. 6.2)
Global asymptotic stability      537
Greens formula      62 67 327 385
Greens functions      328 328(Ex. 2.2) 338 409(Ex. 439 441 461 476 477
Gronwall inequality      24
Gronwall inequality, generalized      29
Gronwall inequality, systems      29 (Ex. 4.6)
Haars lemma      139
Half-trajectory      202
Hermitian part of a matrix      55 420
Homann differential equation      520
Hypergeometric equation      509 (Ex. 2.6(b))
Hypergeometric equation, Kummers confluent form      509 (Ex. 2.6(c))
Implicit function theorem      5 11
INDEX      see Jordan curve Stationary
Indicial equation      85
Inhomogeneous equations      see Variation of constants
Integrability conditions      118 119 123 128
Integral, first      114 124
Integral, first, (m+1)st      478
Invariant manifold      228 234 242 296
Invariant set      184 184(Ex. 11.3)
Jacobi, system of partial differential equations      120
Jacobi, theorem on conjugate points      391 (Ex. 10.1)
Jordan curve, flow on      190
Jordan curve, flow on, index and vector field      149 173
Jordan curve, flow on, theorem      146
Jordan normal form      58 68
Kamke uniqueness theorem      31
Kneser, H., theorem      15
Kronecker theorem      194
Lagrange identity      62 67 327
Legendre equation      87(Ex. 12.3(6))
Legendre equation, associated equation      87(Ex. 12.3(c)) 508(Ex.
Lettenmeyer theorem      87 91
Lienard equation      179
Limit cycle      145 151 152 156 178 181 10.5) 190 253
Limit points, $\omega-$ and $\alpha-$      145 184
Limit points, $\omega-$ and $\alpha-$, set of $\omega$-limit points      145 154 155 158 184(Ex. 11.4) 190 193
Linearizations, differential equations      244 257(Ex. 258
Linearizations, differential equations, maps      194 245 257
Linearly independent solutions      46 64 326
Liouville, formula      46
Liouville, formula, Sturm Liouville problems      337
Liouville, formula, substitution      331
Liouville, formula, volume preserving map      96(Ex. 3.1)
Lipschitz, continuity      3
Lipschitz, continuity, S- and L-Lipschitz continuity      107
Lyapunov, asymptotic stability      38 40 539
Lyapunov, asymptotic stability, function      38 40 539
Lyapunov, asymptotic stability, order number      56 294 301 303
Lyapunov, asymptotic stability, stability      38 40
Lyapunov, asymptotic stability, theorem on second order equation      346
Lyapunov, asymptotic stability, uniform stability      40
Manifold, definition of 2-manifold      182
Manifold, definition of 2-manifold, flow on      183
Manifold, definition of 2-manifold, invariant      228 234 242 296
Manifold, definition of 2-manifold, P(B, D)-manifold      484
Manifold, definition of 2-manifold, stable and unstable      238 244 255
Maps, associated with general solution      94 96 231
Maps, associated with general solution, associated with periodic solution      251
Maps, associated with general solution, equivalence      258
Maps, associated with general solution, invariant manifolds      234
Maps, associated with general solution, linearization      194 245 257
Maps, associated with general solution, volume preserving      96(Ex. 3.1)
Matrix, exponential of      57
Matrix, exponential of, factorization of analytic      75
Matrix, exponential of, Jordan normal form      58 68
Matrix, exponential of, logarithm of      61
Matrix, exponential of, norm      54
Matrix, exponential of, trace (tr)      46 (see also Fundamental matrix)
Maximal interval of existence      12
Maximal solution      25
Minimal set      184 184(Ex. 11.4) 185 190 193
Monotony      500
Monotony, functions of solutions      500 510 518 519(Ex.
Monotony, solutions      357 506
Nagumo theorems      32 428
Node      160 216 216(Ex. 3.3) 219
Nonoscillatory equations of second order      350 362
Nonoscillatory equations of second order, criteria      362
Nonoscillatory equations of second order, necessary conditions      367 368
Norm, matrix      54
Norm, matrix, on $R^d$      3
Open mapping theorem      405 437 439 464
Operator, associate      486
Operator, associate, closed      405
Operator, associate, graph      405
Operator, associate, null space      462
Order number      56 294 301 303
Oscillatory      333 351 369 510
Osgood uniqueness theorem      33
Peano, differentiation theorem      95
Peano, differentiation theorem, existence theorem      10
Periodic solution, associated map on transversal      251
Periodic solution, associated map on transversal, characteristic roots      61 252
Periodic solution, existence      151 178 179(Ex. 179 181 198 407 412 415 416 435(Ex.
Periodic solution, existence, Floquent theory      60 66 71 302(Ex. Stability)
Perron Frobenius theorem      507(Ex. 2.2)
Perron Frobenius theorem, total differential systems      117 120
Perron, singular differential equation      91(Ex. 13.1)
Perron, singular differential equation, theorem of Perron Frobenius      507 (Ex. 2.2)
Perturbed linear systems      212 259 273
Perturbed linear systems, invariant manifolds      242 296
Perturbed linear systems, linearization      244 257(Ex.
Picard Lindelof theorem      8
Poincare Bendixson theorem      151
Poincare Bendixson theorem, generalized      185
Polyas mean value theorem      67 (Ex. 8.3)
Principal solution of second order equation      350 355
Principal solution of second order equation, comparison      358
Principal solution of second order equation, continuity      360
Principal solution of second order equation, definition and existence      355 357
Principal solution of second order equation, monotone      357
Principal solution of second order system      392 398
Prufer transformation      332
Quasi-full Banach space      453 467 471 474 475
Reduction of order      49
Reduction of order, second order equation      64 327
Regular growth      514(Ex. 3.7)
retract      278
Retract, quasi-isotopic deformation      280(Ex. 2.1)
Riccati, equations      331 364
Riccati, equations, differential inequality      362 364 368
Riccati, equations, generalized equation      226(Ex. 4.6)
Riesz, F,. theorem      387
Rotation number      191 198
Rotation point      158 173(Ex.
Saddle-point      161 216 218
Sauvage theorem      73
Sauvage theorem, partial converse      74
Schauder fixed point theorem      405 414 425
Schwartz, A. J., theorem      185
Second order, linear equation      322
Second order, linear equation, linear system      384 418
Second order, linear equation, nonlinear      174 422 Boundary Bounds Comparison Dichotomies Disconjugate Liouville Monotony Nonoscillatory Principal Prufer Riccati Solutions Sturm Variation Zeros)
Sectors (elliptic, parabolic, hyperbolic)      161
Sectors (elliptic, parabolic, hyperbolic), index of stationary point      166
Sectors (elliptic, parabolic, hyperbolic), level curves      173 (Ex. 9.1)
Self-adjoint, operator      342
Self-adjoint, operator, dth order equation      398
Singular point, simple      73 78(Ex. 84 86(Ex.
Singular point, simple, regular      73 78(Ex. 85 86
Small at $\infty$      486
Solution, definition      1 46(Ex.
Solution, definition, continuity with respect to initial conditions or parameters      94
Solution, definition, D-solution      437
Solution, definition, differentiability with respect to initial conditions or parameters      95 100 104 115
Solution, definition, PD-solution      462
Solutions tending to 0, binary systems      161 208 209 211 220
Solutions tending to 0, binary systems, linear systems      500
Solutions tending to 0, binary systems, perturbed linear systems      259 294 300 304 445
Solutions tending to 0, binary systems, second order equations      510 514 (Ex.
spirals      151 159 190 211 216 220
Square root, differential operators      354 392
Square root, non-negative Hermitian matrix      503(Ex. 1.2)
Stability, asymptotic      38 40 537
Stability, asymptotic, global      537
Stability, asymptotic, Lyapunov      38 40
Stability, asymptotic, orbital      157 254
Stability, asymptotic, periodic solutions      158 178 179 253 302(Ex.
Stability, asymptotic, uniform      40
Stationary point      144 183 209 212 220
Stationary point, index      149 166 173 Focus Node Saddle-point)
Sturm, comparison theorems      333 362
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