Gripenberg G., Londen S.O., Staffans O. — Volterra integral and functional equations :: Электронная библиотека попечительского совета мехмата МГУ

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Gripenberg G., Londen S.O., Staffans O. — Volterra integral and functional equations

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Название: Volterra integral and functional equations

Авторы: Gripenberg G., Londen S.O., Staffans O.

Аннотация:

The rapid development of the theories of Volterra integral and functional equations has been strongly promoted by their applications in physics, engineering and biology. This text shows that the theory of Volterra equations exhibits a rich variety of features not present in the theory of ordinary differential equations. The book is divided into three parts. The first considers linear theory and the second deals with quasilinear equations and existence problems for nonlinear equations, giving some general asymptotic results. Part III is devoted to frequency domain methods in the study of nonlinear equations. The entire text analyzes n-dimensional rather than scalar equations, giving greater generality and wider applicability and facilitating generalizations to infinite-dimensional spaces.

Язык:

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

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Год издания: 1990

Количество страниц: 725

Добавлена в каталог: 24.08.2014

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Предметный указатель

$(a\ast b)(t,s)$       19[1.3.9] 226 228[9.2.3] 228[9.2.4] 243[9.5.3] 250[9.6.7]
$(a\ast\beta)(t,E)$       294[10.3.5]
$(a\ast\mu)(t)$       293[10.3.4]
$(f\ast\beta)(E)$       292[10.3.2]
$(\alpha\ast b)(t,s)$       294[10.3.5]
$(\alpha\ast f)(t)$       285[10.2.2] 287[10.2.3] 292[10.3.2]
$(\alpha\ast\beta)(t,E)$       20[1.3.10] 287 287[10.2.3] 294[10.3.5]
$(\mu\ast b)(t)$       293[10.3.4]
$(\mu\ast\beta)(E)$       285[10.2.2] 287[10.2.3]
$<<\varphi,\nu>>$       496
$adj[\nu]$       124 135
$BBV(J;\eta)$ , $BBV_{0}(J;\eta)$       116
$BC(J;\eta)$ , $BC_{0}(J;\eta)$       116 315[11.2.3] 316[11.2.4]
$BL^{P}(J)$ , $BL^{P}_{0}(J)$       xviii 73
$BL^{P}(J;\eta)$ , $BL^{P}_{0}(J;\eta)$       115 116
$BUC(J;\eta)$       116
$BV(J;\eta)$       116
$B^{\infty}(J)$ , $B^{\infty}_{0}(J)$ , $B^{\infty}_{\ell}(J)$ , $B^{\infty}_{loc}(J)$       xviii
$B^{\infty}(J;\eta)$ , $B^{\infty}_{0}(J;\eta)$       116
$B^{\infty}_{\varepsilon,K}$       361
$C_{\varepsilon, M}$       348
$det[\nu]$       124 135
$diam\{\Omega\}$       xxi 346
$K_{T}$       353 358[12.2.9]
$L^{p}$ -anti-accretive type      see "Kernel Lp-anti-accretive
$L^{p}(J;\eta)$       115
$L^{P}_{k}$       357 361
$PT_{aco}$       516[16.5.4] see anti-coercive
$Q(\phi,\eta,T)$       xxii
$T_{\infty}$       348
$W^{I,p}(J)$       xix
$Z(\mu)$       see "Kernel Fourier
$[\nu,\phi]$       xxi
$\alpha\{\Omega\}$       346 379
$\alpha_{\varphi}$       120[4.4.1]
$\ast$ -product      xxi see
$\Gamma(\varphi)$ , $\Gamma_{+}(\varphi)$ , $\Gamma_{-}(\varphi)$       453[15.2.1] see set"
$\hat{L}^{1}(\varphi)$ , locally in $\hat{L}^{1}(\varphi)$       199[7.3.1] 199[7.3.2]
$\hat{L}^{1}(\varphi)$ , locally in $\hat{L}^{1}(\varphi)$ at $\infty$       199[7.3.1]
$\hat{L}^{1}(\varphi)$ -smooth of order p      200[7.3.5] 200[7.3.7]
$\hat{L}^{1}(\varphi)$ -zero of order p      200[7.3.6] 200[7.3.7]
$\hat{M}(\varphi)$ , locally in $\hat{M}(\varphi)$       199[7.3.3] 200[7.3.4]
$\hat{M}(\varphi)$ , locally in $\hat{M}(\varphi)$ at $\infty$       199[7.3.3]
$\mathcal{D}$ , $\mathcal{D}'$       xix 527 527—530
$\mathcal{F}(L^{p};J)$       227[9.2.2]
$\mathcal{K}$       227[9.2.2] see "
$\mathcal{M}$       see "Kernel nonconvolution
$\mathcal{M}(B^{\infty};J)$       284[10.2.1]
$\mathcal{M}(B^{\infty};L^{1};J)$       292[10.3.1]
$\mathcal{M}(B^{\infty}_{loc};J)$       290[10.2.11]
$\mathcal{M}(B^{\infty}_{loc};L^{1}_{loc};J)$       295[10.3.6]
$\mathcal{T}_{s}$       xvii
$\mathcal{V}$       see "Kernel"
$\mathcal{V}(B^{\infty};J)$       242[9.5.1]
$\mathcal{V}(B^{\infty}_{loc};J)$       242[9.5.1]
$\mathcal{V}(C;J)$       242[9.5.2]
$\mathcal{V}(L^{p};J)$       227[9.2.2]
$\mathcal{V}(L^{p}_{loc};J)$       240[9.4.1]
$\mathcal{V}(M;B^{\infty};J)$       293[10.3.3]
$\mathcal{V}(M_{loc};B^{\infty}_{loc};J)$       295[10.3.6]
$\nu_{a}$ , $\nu_{d}$ , $\nu_{s}$       xxi 79 113 123
$\omega_{\varphi}$       120[4.4.1]
$\overline{R}$       176
$\psi$ -sequence      486
$\rho(\varphi)$       472[15.5.1] 473[15.5.3]
$\rho^{\infty}(\varphi)$       476[15.6.1]
$\rho^{\infty}_{\omega}(\phi)$       476[15.6.1]
$\rho_{\omega}(\varphi)$       472[15.5.1] 472[15.5.2] 473[15.5.3] 474[15.5.6] 475[15.5.7] 475[15.5.8] 485
$\sigma$ -algebra      285 294
$\sigma$ -finite measure space      90[3.4.1] 91
$\Sigma(\mu)$       461[15.4.1] see of
$\sigma(\phi)$       xx 462[15.4.4] 462 see of
$\sigma^{\infty}(\phi)$       476[15.6.1] 476[15.6.2] 548[17.3.7]
$\succeq$ , $\succ$       xxi
$\tau_{h}$       xx 63[2.7.1] 63[2.7.2] 63[2.7.3] 316 453[15.2.2] 492
$\tilde{L}^{1}$ , locally in $\tilde{L}^{1}$       176[6.3.1] 177[6.3.2] 177[6.3.3] 179[6.3.5] 179[6.3.6] 183 185[6.6.1] 185[6.6.2] 185[6.6.3]
$\tilde{L}^{1}$ , locally in $\tilde{L}^{1}$ at $\infty$       176[6.3.1] 177[6.3.4] 179[6.3.5] 179[6.3.6] 185[6.6.1] 185[6.6.2]
Abel's equation      158
Absolute continuity      427[14.2.2] 442[14.4.1] 442[14.4.3] 443[14.4.5] 443[14.4.6]
Accretive operator      624
Adjoint, Banach space adjoint      213
Adjoint, equation      274
Adjoint, matrix      see "Matrix adjoint"
Adjoint, semigroup      see "Semigroup adjoint"
Admissible      278 338
Admissible, $(L^{1};L^{1})$       272[9.10.4]
Admissible, $(L^{1};L^{p})$       277[9.11.3]
Admissible, $(L^{p};L^{q})$       278
Admissible, $(L^{p};L^{\infty})$       277[9.11.3]
Admissible, $(L^{\infty};L^{\infty})$       271 [9.10.2]
Aerofoil section      12[1.2.8]
Almost-periodic function      65 470[15.4.17] 471 472[15.4.19] 484-485
Almost-periodic function in the sense of Stephanov      65
Analytic      177[6.3.3] 177[6.3.4] see analytic"
Anti-coercive type      516[16.5.4] see anti-coercive
Anti-symmetric matrix      141[5.2.1] 493[16.2.3] 501[16.3.3]
Approximate, convolution identity      67[2.7.4] 528
Approximate, equation      404
Approximate, multiplicative identity      528
Arithmetic measure      511
Arzela — Ascoli theorem      68[2.7.5] 68(2.7.6]
Associative algebra      233[9.3.2] 233[9.3.4] 234[9.3.5] 239[9.3.17] 290[10.2.10]
Asymptotic, directional spectrum      476[15.6.1]
Asymptotic, spectral subspace      476[15.6.1]
Asymptotic, spectrum      476[15.6.1] 476[15.6.2] 548[17.3.7]
Asymptotic, stability      426
Asymptotically, almost-periodic function      39 65
Asymptotically, drifting periodic function      459[15.3.4] 459[15.3.5] 476[15.6.2] 483
Asymptotically, periodic function      39 65
Asymptotically, slowly varying function      458[15.3.2] 458[15.3.3] 476[15.6.2] 480[15.8.1] 483-484
Autonomous, equation      316
Autonomous, operator      454[15.2.5] 454[15.2.7] 456[15.2.9] 456[15.2.11] 480[15.8.1] 624 see invariant
Banach algebra      228
Banach algebra of $L^{1}$ -functions      275-276
Banach algebra of convolution measure kernels      113[4.1.3] 119[4.3.4] 136[4.6.2]
Banach algebra of nonconvolution kernels      19[1.3.9] 228[9.2.4] 243[9.5.3] 247[9.6.2] 249[9.6.5] 250[9.6.7] 251[9.6.9] 252[9.6.12] 256[9.7.5] 275[9.11.1] 275[9.11.2]
Banach algebra of nonconvolution measure kernels      20[1.3.10] 287[10.2.3]
Banach module      228 228[9.2.4] 247[9.6.2] 249[9.6.5] 251[9.6.9] 252[9.6.12] 287[10.2.3]
BBV(J), $BBV_{0}(J)$       xviii 73
Bernstein function      161
Bernstein's theorem      17[1.3.5] 143(5.2.5]
beta function      415[13.6.3]
Bochner's theorem      26[1.3.16] 498[16.2.7] 499 506 595—596
Boundary of the spectrum      467[15.4.13] 467[15.4.14]
Boundedness principle      427[14.2.2] 428[14.2.3] 429[14.2.4] 429[14.2.5] 430[14.2.6] 431[14.2.8]
Caratheodory conditions      355 356
Cauchy's theorem      153
Causal      345 348 348[12.2.1] 360[12.3.1] 362[12.4.1] 363[12.4.2] 533
Characteristic, equation      137 193[7.2.2] 219
Characteristic, exponent      192-196 193[7.2.2] 198 200-202 219
coercivity constant      27[1.3.16] 516[16.5.4] 516[16.5.5] 516[16.5.6] 517[16.5.7] 519 531
Compact      see "Equicontinuity"
Compact in $BC_{0}(J;\eta)$       315[11.2.3] 316[11.2.4]
Compact in $L^{1}$       357[12.2.8] 365[12.4.4] 365 373
Compact open topology      see "Uniform convergence on compact subsets"
Compact, mapping      40 243[9.5.3] 255 256[9.7.5] 314[11.2.2] 632
Compact, mapping from $L^\infty$ to $L^{1}$       367[12.4.5]
Compact, mapping from $L^{1}$ to $L^{1}$       253[9.7.1] 254[9.7.3]
Compact, mapping from $L^{p}$ to $L^{p}$       253 254[9.7.2] 255[9.7.4] 279 368[12.4.6]
Compact, mapping, convolution      40[2.2.5]
Compact, mapping, weakly compact      254
Compact, measure of noncompactness      346 379
Compact, set of solutions      404(13.4.4]
Compact, weak      156
Comparison, equation      257[9.8.2] 298[10.3.10] 359
Comparison, result      344 345[12.1.2] 353[12.2.4] 353 358 406[13.4.5] 406[13.4.7] 409[13.4.10] 410[13.5.2] 412[13.5.6]
Completely monotone function      142[5.2.3] 143[5.2.4] 143[5.2.5] 147[5.2.8] see completely
Completely monotone, kernel      17[1.3.5] see completely
Completely monotone, Laplace transform of a completely monotone function      144[5.2.6] 147[5.2.7]
Completely monotone, resolvent      see "Resolvent completely

Cone      403[13.4.1] 492[16.2.2] 625[20.4.7] 625[20.4.8] 626[20.4.10] 627[20.4.12]
Cone, solution in a cone      627[20.4.12] see maximal"
Continuation      343 see noncontinuable"
Continuous dependence      see "Differential resolvent continuous "Resolvent continuous "Resolvent measure continuous "Solution continuous
Contraction mapping principle      314[11.2.1]
control systems      8[1.2.4]
Convex      see "Function convex" "Kernel convex"
Convolution      see " $\ast$ -product"
Convolution, $a\ast b$ of functions      36 38[2.2.1]
Convolution, $\mu\ast a$ of a measure and a function      77 79[3.2.1]
Convolution, $\mu\ast\nu$ of measures      112[4.1.1]
Convolution, $\nu\ast\mu$ of distributions      529
Convolution, $\nu\ast\varphi$ of a distribution and a test function      496
Convolution, compact mapping in $L^{p}$       40[2.2.5]
Convolution, derivative of a convolution      99[3.7.1] 100[3.7.2] 100[3.7.3] 101
Convolution, operator mapping $L^{2}$ into $L^{2}$       332[11.6.1]
Convolution, properties      39[2.2.2] 40[2.2.3] 40[2.2.4] 63[2.7.2] 67[2.7.4] 96[3.6.1] 98[3.6.2] 113[4.1.2] 113[4.1.4] 119[4.3.4] 119[4.3.5] 133[4.5.1] 465[15.4.11]
Cosmic ray transport      4[1.2.1]
Critical line      18[1.3.7] 198 203
Cross section      353 358(12.2.9]
Delay equation      20[1.3.10] 283 345
Difference equation      16[1.3.4] 137
Differential resolvent      15[1.3.3] 77 81[3.3.2]
Differential resolvent in $L^{1}(J,\varphi)$       130[4.4.13] 130[4.4.14]
Differential resolvent in $L^{2}$       88[3.3.13] 334
Differential resolvent in a weighted space      130[4.4.13] 130[4.4.14] 130(4.4.15] 131[4.4.16] 131
Differential resolvent of a completely monotone kernel      17[1.3.5] 150[5.4.1]
Differential resolvent of a kernel of positive type      604[19.4.1] 605[19.4.2] 606[19.4.3] 609[19.4.6]
Differential resolvent, $L^{1}$ -remainder      192 [7.2.1] 200[7.3.7]
Differential resolvent, bounded      445 [14.5.2]
Differential resolvent, continuous dependence      82[3.3.4]
Differential resolvent, derivative in $L^{1}(J;\varphi)$       130[4.4.13] 130[4.4.14]
Differential resolvent, equation      15[1.3.3] 21[1.3.10] 77 81[3.3.1] 295
Differential resolvent, exponential decay      150[5.4.1]
Differential resolvent, exponentially growing      192[7.2.1]
Differential resolvent, integrable derivative      83[3.3.5]
Differential resolvent, integrable derivative, whole line      83[3.3.7]
Differential resolvent, integrate      83[3.3.5] 85[3.3.9] 89[3.3.16] 89[3.3.17] 150[5.4.1] 181[6.4.3] 181[6.4.4]
Differential resolvent, integrate, whole line      83[3.3.7] 86[3.3.10]
Differential resolvent, jump discontinuity      83[3.3.7]
Differential resolvent, maps $L^{2}$ into $L^{2}$       89[3.3.14]
Differential resolvent, nonconvolution      21[1.3.10] 295 296[10.3.7] 296[10.3.8] 299[10.4.1]
Differential resolvent, of a convex kernel      184 187
Differential resolvent, parameter-dependence      184 187
Differential resolvent, periodic      88(3.3.12]
Differential resolvent, polynomially growing      200[7.3.7]
Differential resolvent, real-valued      81
Differential resolvent, type $L^{1}$       300[10.4.3] 302[10.4.5] 446[14.5.6]
Differential resolvent, type BC      301[10.4.4] 303[10.4.6]
Differential resolvent, type BUC      299[10.4.2]
Differential resolvent, unbounded      192[7.2.1] 200[7.3.7]
Differential resolvent, unique      81[3.3.1] 83[3.3.7] 130[4.4.14] 296[10.3.7]
Differential resolvent, used in a nonlinear equation      603
Differential resolvent, used in a perturbed linear equation      313 324 334
Differential resolvent, variation of constants formula      299[10.4.1]
Differential resolvent, whole line      78 83[3.3.7] 84[3.3.8] 86[3.3.10] 130[4.4.14] 130[4.4.15] 131[4.4.17] 193[7.2.1]
Differential resolvent, zero at infinity      89[3.3.17]
Differential-delay equation      2 24[1.3.14] 76 105 283 427 434 447[14.5.9]
Directional spectrum      472[15.5.1] 473[15.5.3]
distribution      527
Distribution, bounded      531
Distribution, convolution of two distributions      529
Distribution, derivative of a distribution      99[3.7.1] 100 103 528
Distribution, Fourier transform of a convolution      528
Distribution, Fourier transform of a distribution      103 495 521[16.6.2] 524[16.6.5] 528 529[16.8.1]
Distribution, integrable      183
Distribution, Laplace transform of a distribution      496 529 529(16.8.1]
Distribution, Plancherel's theorem for distributions      496 522
Distribution, positive type      533
Distribution, support of a distribution      528
Distribution, tempered      26[1.3.16] 495[16.2.5] 495 497[16.2.6] 514[16.5.2] 521[16.6.2] 524[16.6.5] 527
Duality mapping      216[8.3.5]
Eigenfunction      219-220
Eigenvalue      219 see
Epidemiology      6[1.2.3] 449
equation      see "Fredholm" "Functional" "Solution "Solution "Volterra"
Equicontinuity      315[11.2.3] 349 353[12.2.3] 356[12.2.7] 392[13.2.2]
Existence of solution      see "solution of a nonlinear equation exists"
Exponential polynomial      192
Exponentially growing solution      see "Solution of a linear equation exponentially
Extended semigroup      see "Semigroup extended"
Feedback      2—3 8[1.2.4]
Fejer kernel      51
Finite delay      203 207
Finitely generated limit set      see "Limit set finitely
First kind      see "Volterra equation of the first kind"
Fixed point theorem      22[1.3.11] 22[1.3.12] 314[11.2.1] 314[11.2.2] 346[12.1.3] see
Forcing function      35 225
Forcing function semigroup      see "Semigroup forcing
Fourier coefficient      57
Fourier transform      41 50
Fourier transform in $L^{2}$       530[16.8.2]
Fourier transform of a convex function      170[6.2.2] 186
Fourier transform of a distribution      495 529[16.8.1] see Fourier
Fourier transform of a kernel of anti-coercive type      516[16.5.6]
Fourier transform of a kernel of positive type      494[16.2.4] 497[16.2.6]
Fourier transform of a kernel of strict positive type      510[16.4.5]
Fourier transform of a kernel of strong positive type      507-508
Fourier transform of a nonnegative, nonincreasing function      173[6.2.3]
Fourier transform of an $L^{1}$ -function      177[6.3.2] 464[15.4.8]
Fourier transform of bounded variation      467[15.4.15] 548[17.3.8]
Fourier transform, absolutely continuous      182[6.5.1]
Fourier transform, Hoelder-continuous      467[15.4.15] 548[17.3.8]
Fourier transform, inversion formula      183 499
Fourier zero-set      see "Kernel Fourier
Frechet-derivative      397
Fredholm, integral equation      3 48[2.4.6]
Fredholm, integrodifferential equation      15[1.3.3] 86[3.3.10]
Fredholm, kernel      14[1.3.2] 15[1.3.3] 16[1.3.4] 227[9.2.1]
Fredholm, resolvent      232[9.3.1] see whole "Resolvent whole
Fubini's theorem      90(3.4.1] 91(3.4.2] 91(3.4.3]
Function      see "Kernel"
Function, almost-periodic      65 470[15.4.17] 471 472[15.4.19] 484-485
Function, anti-coercive type      516[16.5.4]
Function, asymptotically almost-periodic      39 65
Function, asymptotically drifting periodic      459[15.3.4] 459[15.3.5] 476[15.6.2] 483
Function, asymptotically periodic      39 65
Function, asymptotically slowly varying      458[15.3.2] 458[15.3.3] 476[15.6.2] 480[15.8.1] 483—484
Function, completely monotone      142[5.2.3] 143[5.2.4] 143[5.2.5] 147[5.2.8]
Function, convex      500[16.3.1] 508[16.4.3]
Function, locally in $\hat{L}^{1}(\varphi)$       see $\hat{L}^{1}(\varphi)$
Function, locally in $\hat{M}(\varphi)$       see $\hat{M}(\varphi)$
Function, locally in $\tilde{L}^{1}$       see " $\tilde{L}^{1}$ "
Function, periodic      459(15.3.5] 460(15.3.7] 470(15.4.17]
Function, positive definite      426
Function, positive type      492(16.2.1]
Function, strict positive type      510(16.4.4]
Function, strong positive type      507(16.4.1]
Function, submultiplicative      118(4.3.2] 120[4.4.1]
Function, total variation      92(3.5.1]
Functional, differential equation      23[1.3.12] 77 107 203 345 360[12.3.1] 378 438 456—457 see
Functional, equation      22[1.3.12] 345 347 352[12.2.2] 353[12.2.3] 361 455
Fundamental solution      77 301
Gel'fand's theorem      121[4.4.2] 125[4.4.3] 125[4.4.4] 127[4.4.7] 127[4.4.8]
Gradient      539 see gradient"
Green's function      73 108 see whole "Resolvent whole
Gronwall's inequality      257[9.8.2] 291[10.2.15] 298[10.3.10] 306—307
Hammerstein operator      343
Hardy space      151 182 599
Hardy — Littlewood inequality      183
Hausdorff maximality principle      343
Heat equation      1
Heat flow      12[1.2.9]
Hoelder's inequality, matrix version      525 527
Homomorphism      121 126[4.4.6] 136[4.6.2]
Hyperstability      535
Ideal      126[4.4.6] 469 484[15.10.2]
identities      574[18.4.1] 576[18.4.4] 581[18.5.1] 583[18.5.4] 639[20.5.1] 643
Imaginary part      see "Matrix imaginary
Initial function correction      208 215

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