[Al]

[Ar]

[Bo]

[CL]

[Da]

[Di]

[Ed]

[En]

[ER]

[Faf]

[FF]

[Fi]

[Fvv1]

REFERENCES

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Artstein Z., The limiting equations of nonautonomuous differential equations, J. of Differential Equations, 25 (1977), 184-202.

Bokstein M.F., Theorems on the existence and uniqueness of solutions of systems of ordinary differential equations, Uchen. Zap. MGU, 15, N1 (1939), 3-72 (Russian).

Coddington E., Levinson N., The theory of Ordinary Differential Equations, McGraw-Hill, New York, 1955.

Davy J.L., Properties of solution set of a generalized differentional equation, Bull. Austral. Math. So., 6, N3 (1972), 379-398.

Dieudonne J., Foundations of Modern Analysis, New York - London, Academic Press, 1960.

Edwards R.E., Functional analysis, Holt, Rinehart and Winston, New York, 1965.

Engelking R., General topology, PWN, Warszawa, 1977.

Efimov N.Y., Rosendorn E.R., Linear Algebra and Multidimensional Geometry, Nauka, Moscow, 1970 (Russian).

Filippov A.F., Differential equations with discontinuous right hand sides, Math. Appl. (Soviet Ser.) 18, Kluwer, Dordrecht, 1988.

Fedorchuk V.V., Filippov V.V., General topology: Basic constructions, Moscow University Publishers, Moscow, 1988 (Russian).

Fichtengolz G. M., Course of differential and integral calculus, 3, Fizmatgiz, Moscow, 1963 (Russian).

Filippov V.V., Solution spaces of ordinary differential equations, Moscow Univer­sity Publishers, Moscow, 1993 (Russian).

508

[Fvv2]

[Fvv3]

[Fvv4]

[Fvv5]

[Fvv6]

[GOl]

[G02]

[Go]

[Ha]

[HW]

[HZ1]

[HZ2]

[Kl]

[Ku1]

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[Ku3]

[LR]

Filippov V.V., Ordinary differential equations: topological approach to the theory (to appear in Russian).

Filippov V.V., Topological structure of solution spaces of ordinary differential equations, Uspehi Mat. Nauk, 48, N1 (1993), 103-154 (Russian).

Filippov V.V., Basic topological structures of the theory of ordinary differential equations, Topology in nonlinear Analysis, Banach center publications, 35 (1996), 171-192.

Filippov V.V., On the acyclicity of solution set of ordinary differential equations, Doklady RAN, 352, N1 (1997) (Russian).

Filippov V.V., On Aronszajn theorem, Differenc. Uravnen., 33, N1 (1997), 11-15 (Russian).

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Gossez J.-P., Omari. P., Periodic solutions of a second order ordinary differen­tial equation: a necessary and sufficient condition for nonresonance, Journal of Differential Equations, 94, N1 (1991), 67-82.

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[LSJ

[Ma]

[Mar]

[Me]

[MS]

[RHL]

[Se1]

[Se2]

[Scl]

[Sc2]

[Yo]

[Za]

509

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INDEX

Trspace, 18 Z-neighborhood, 267 Z-retraction, 317 <I>-w-thin set, 232 <I>-thin set, 232 E-neighborhood, 15 E-neighborhood of a set 47 , cp-w-thin set, 232 cp-thin set, 232 e-lifting, 63 e-loop, 70 e-homotopic mappings, 63 e-homotopy, 63

absolutely continuous function, 118 adherent point, 13 admissible change of variables, 292 admissible control, 353 Alexander lemma, 92 anti-discrete topology, 9 approximate derivative, 139 arcwise connected space, 61 argument, 2 Arzela theorem, 88 asymptoticaly stable point, 458 at most countable set, 7 at most countable set with respect

to a space, 235 autonomous equation, 441 autonomous space, 441

Baire theorem, 26 ball, 15 Banach space, 72 base, 11 base of neighborhoods of a set, 29

510

bijective mapping, 5 boundary of a set, 13 bounded metric space, 4 7 Brouwer Fixed Point Theorem, 70

Cantor perfect set, 25 Cantor stairs, 113 Caratheodory conditions, 255 cardinality, 6 Cartesian product, 3 Cauchy problem for a differential

inclusion (equation), 199 Cauchy problem for a space, 202 Cauchy sequence, 25 characteristic polynomial, 335 characteristic vector, 501 closed ball, 15 closed convex hull, 51 closed cover, 23 closed curve, 61 closed path, 61, 62 closed set, 13 closure, 13 compact open topology, 73 compact space, 23 compactum, 23 complement, 3 complete space, 25 component, 59 conic space, 457 connected component, 59 connected space, 56 constant mapping, 5 contingent, 268 continuous mapping, 30

continuous mapping (at a point), 27

continuum, 58 contractive mapping, 31 convergence of a sequence of points

to a set, 399 convergence of series, 73 convergence of space sequences,

213 convergent generalized sequence,

22 convex combinations, 52 convex hull, 51 convex set, 50 countable set, 7 cover, 23 curve, 61

Davy conditions, 255 de Morgan formulae, 4 decreasing function, 108 definite integral, 128 degree of a mapping, 66 Denjoy integrable function, 140 dense subset, 26 density point, 138 derivative of a function with re­

spect to a set, 116 diameter of a set, 39 diameter of a set with respect to

a space, 442 difference of sets, 3 differential equation with multi-

valued right hand side, 193 differential inclusions, 192 directed set, 9 disconnected space, 56 discrete topology, 9 distance, 14 distance between subsets, 17 domain, 4 domain (of definition), 2

domain of definition, 4

Egorov theorem, 161 element of a set, 1 embedding, 43 empty set, 2 end of a path, 60 end of a simple arc, 62 endpoints, 50

511

endpoints of a path, 60 equi-absolutely continuous set, 358 equi-absolutely continuous sequence

of functions, 183 equicontinuous set of mappings,

87 equivalent sets, 6 Euclidean distance, 14 everywhere dense subset, 26 extendable mapping, 313

factor, 3, 38 family, 2 Fatou lemma, 137 filter, 9 finite intersection property, 24 finite set with respect to a space,

235 first approximation, 45 7 fixed point, 31, 69 function, 2 fundamental sequence, 25 fundamental system of solutions,

331 Fundamental theorem of algebra,

68

general solution, 264 generalized sequence, 17 graph, 5

Hahn-Banach Theorem, 166 Hausdorff distance, 80 Hausdorff metric, 80

512

Hausdorff space, 18 Hilbert cube, 45 homeomorphic spaces, 42 homogeneous linear equation, 328,

334

identity mapping, 5 image, 2 improper subset, 2 increasing function, 108 indefinite integral, 128, 141 index, 3 index of a point with respect to a

loop, 484 index of a point with respect to a

space, 485 index of a space with respect to a

loop, 484 index set, 2 induced topology, 14 initial point of a path, 60 initial point of a simple arc, 62 injective mapping, 5 integral, 128 interior, 12 interior point, 12 interval of a circle, 4 70 inverse mapping, 5 isolated point, 21

Jordan form, 333

Lebesgue integrable function, 128 Lebesgue lemma, 490 Lebesgue theorem, 136 left Z-neighborhood, 267 left contingent, 268 left increasing point, 109 Lienard transformation, 407 limit of a generalized sequence, 18 limit ordinal number, 36 limit point of a generalized se­

quence, 17

limit point of a set, 17 linear functional, 50 Liouville's formula, 332 Lipschitz constant, 30 Lipschitz condition, 30 locally arcwise connected space,

61 locally compact space, 49 locally integrable function, 129 loop, 61, 62 loop bounding a region, 62 Luzin theorem, 161

majorant, 271 mapping, 2 mapping onto ... , 5 measurable function, 104 measurable multi-valued mapping,

152 measurable set, 99 measure of a set, 99 measure of a set with respect to a

space, 232 metric, 14 Minkowski functional, 169 monotone function, 109

nearest point, 52 negative change of variables, 292 neighborhood, 12 Nemytski plane, 11 non-characteristic vector, 501 non-decreasing function, 108 non-homogeneous linear equation,

328, 334 non-increasing function, 108 non-isolated point, 21 nontrivial connected set, 58 norm, 50 normal space, 48 normed space, 50 nowhere dense subset, 26

one to one mapping, 5 open ball, 15 open cover, 23 open-closed set, 56 ordinal number, 35 ordinary differential equation, 191 outer measure, 96

partial order, 33 partial ordering, 33 path, 60 path connecting points, 60 Peano condition, 266 periodic space, 452 Picard condition, 267 Poincare-Bendixson theorem, 498 positive change of variables, 292 precedes, 33 preimage, 2 primitive, 128, 141 product of a family of sets, 3 projection, 38 proper subset, 2

region, 61 regular space, 28 relatively compact subset, 23 relatively weakly compact sequence,

180 restriction, 5 retraction, 317 right Z-neighborhood, 267 right contingent, 268 right increasing point, 109

Schauder Fixed Point Theorem, 70

Scorza-Dragoni theorem, 160 Scorza-Dragoni condition, 280 segment, 50 segment of a circle, 63 self-intersection, 61 sequence, 7

series, 73 set, 1

513

set free of stationary points, 483 set of measure zero, 103 simple arc, 62 simple closed arc, 62 simple closed path, 62 simple loop, 61, 62 simple path, 61 singleton, 8 slitting set, 61 solution of a control problem, 353 solution of a control problem with

restrictions, 353 solution of a differential inclusion

(equation ), 193 solution space, 193 space measurable sets, 179 space of compact subsets, 79 stationary point of a solution space,

443 strongly monotone function, 109 sub-base of neighborhoods of a

set, 29 subbase, 12 sub cover, 23 subproduct, 38 subsequence, 7 subset, 2 subset of full measure, 104 subspace of a topological space, 14 subspace of a metric space, 17 succeeds, 33 sufficient me-approximation, 27 4 sufficient mek-approximation, 27 4 sum of series, 73 surjective mapping, 5 system of neighborhoods, 10

topological limit, 22 total length, 95 transfinite, 35

514

triangle inequality, 14 trivial connected set, 58 Tychonoff neighborhood, 37 Tychonoff topology, 37 Tychonov theorem, 93

uniformly continuous family of func-tions, 88

uniformly convergent series, 73 upper semicontinuous mapping, 30 upper semicontinuous mapping (at

a point), 27

value of a function (mapping), 2 variation of constants, 334 Vietoris neighborhood, 77 Vietoris topology, 77

weak topology of a normed space, 168

weak topology on L •, 165 weakly convergent sequences, 171 Wronskian, 331

Zermelo theorem, 34 Zorgenfrei line, 10 Zorn lemma, 33

NOTATION

( + ), 292, 313 (- ), 292, 313 (X, p), 14 (v), 368 (c), 197 (e), 203 (!), 309 (k), 247 (l), 309 (n'), 198 (n), 196 (p ), 195 (q), 194 (r), 309 (s ), 195 (u), 203 (x, e)- inner (scalar) product, 50 A(V), 441 A 3 s, 2 A~ s, 2 A <l B, 2 A~ B, 2 A.(V), 441 B(V), 444 B(X, Y), 71 BC(X, Y), 71 B 72 A, 2 B ;2 A, 2 B*(V), 444 C(X, Y), 72 Cv(X, Y), 83 Cvc(X, Y), 83 D(F), 193 D(F,M), 193 DN- Cantor perfect set, 46 FY, 198

Ft, 198 L 1 , 173 L 1 ([a, b]), 173 Li([a, b]), 399 L 00 , 174 MY, 198 M1 (h, c), 233 M 2 (h, c), 234 Mt, 198 O(K,U), 73 O(x, c), 15 O(U1, ... , Uk), 77 0 1(M,c), 47 o,(x, c), 15 O,x, 15 pa(U), 452 P.a(U), 452 Q .. 0 (U), 271 Q;*0 (U), 271 Q**(U), 271 Q;*(U), 271 Q*0 (U), 271 Q*(U), 198 Qt(U), 211 Qd(<p)(U), 254 Qmek(U), 274 Qme(U), 271 R(M), 194 Ri(M), 194 R.(M), 194 R!(M), 194 xl n ... nxn, 3 X1 U ·· · UXn, 3 xl X ... X Xn, 3 z+, 304 z-, 304

516

z+, 204 z-+, 205, 306 z-, 204 zne(+), 308 zne(-), 308 zne, 308 ZM, 195 [M], 13 [M]x, 13 aM, 13 Gr(f), 5 Im(f), 5 n+(z), 446 n-(z), 446 Rex -- real part of x, 461 Stat(Z), 483 c(M), 51 n1 , 3 n{XI, ... 'Xn}, 3 n{ X a : o: E A}, 3 nr=lxi, 3 naEAXa, 3 cc(M), 51 cont(Z,x), 268 contr(Z, x), 268 cont1(Z, x), 268 u,, 3 U{X1, ... , Xn}, 3 U{Xa: o: E A}, 3 ur=lxi, 3 UaEAXa, 3 0, 2 expc X, 78 expc X, 79 (M), 12 (M) X' 12 (Z, Z 00 ), 444 lim top, 22 lim top inf, 19 lim top sup, 19 JL*(M), 95 JLifi(M), 232 wo, 36

w1 , 36 Wifi(M), 232 w"'(M), 232 1r(j) =A, 4 1rb, 84 7r_, 84 -<, 33 I11, 3 IJ{XI,···,Xn}, 3 IJ{Xa: 0: E A}, 3 rr~l xk, 3 rrz=l xk, 3 ITaEA Xa, 3 p(A, B), 17 ~' 33 rz, 267 (j;, 291 d(T), 95 f(M), 2 f- 1 (M), 2 j(Z, f), 484 j(Z, J, c), 483 ]z(a), 485 l1r, 84 8(U), 213 8 E A, 2 8 rf. A, 2 X = lim'P Xa, 18 Xa--> X, 18 y1 = <I>(t, y), 192 y 1 E F(t, y), 192 C - set of complex numbers,

complex plane, 68 C - set of complex numbers,

complex plane, 192 en, 192 N - set of positive integers, 7 Q - set of rational numbers, 8 ~-real line, 9 ~n - n-dimensional Euclidean

space, 4

Z - set of integers, 8 (n*), 267

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Y. Komatu: Distortion Theorems in Relation to Linear Integral Operators. 1996, 313 pp. ISBN 0-7923-4304-2

A.G. Chentsov: Asymptotic Attainability. 1997, 336 pp. ISBN 0-7923-4302-6 S.T. Zavalishchin and A.N. Sesekin: Dynamic Impulse Systems. Theory and Applications. 1997, 268 pp. ISBN 0-7923-4394-8

U. Elias: Oscillation Theory ofTwo-Term Differential Equations. 1997,226 pp. ISBN 0-7923-4447-2

D. O'Regan: Existence Theory for Nonlinear Ordinary Differential Equations. 1997, 204 pp. ISBN 0-7923-4511-8

Yu. Mitropolskii, G. Khoma and M. Gromyak: Asymptotic Methods for Investigat­ing Quasiwave Equations of Hyperbolic Type. 1997, 418 pp. ISBN 0-7923-4529-0

R.P. Agarwal and P.J.Y. Wong: Advanced Topics in Difference Equations. 1997, 518 pp. ISBN 0-7923-4521-5

N .N. Tarkhanov: The Analysis of Solutions of Elliptic Equations. 1997, 406 pp. ISBN 0-7923-4531-2

B. Rieean and T. Neubrunn: Integral, Measure, and Ordering. 1997, 376 pp. ISBN 0-7923-4566-5

N.L. Gol'dman: Inverse Stefan Problems. 1997, 258 pp. ISBN 0-7923-4588-6

S. Singh, B. Watson and P. Srivastava: Fixed Point Theory and Best Approxima­tion: The KKM-map Principle. 1997,230 pp. ISBN 0-7923-4758-7

A. Pankov: G-Convergence and Homogenization of Nonlinear Partial Differential Operators. 1997,263 pp. ISBN 0-7923-4720-X

Other Mathematics and Its Applications titles of interest:

S. Hu and N.S. Papageorgiou: Handbook of Multivalued Analysis. Volume I:

Theory. 1997,980 pp. ISBN 0-7923-4682-3 (Set of2 volumes: 0-7923-4683-1)

L.A. Sakhnovich: Interpolation Theory and Its Applications. 1997,216 pp. ISBN 0-7923-4830-0

G.V. Milovanovic: Recent Progress in Inequalities. 1998,531 pp. ISBN 0-7923-4845-1

V.V. Filippov: Basic Topological Structures of Ordinary Differential Equations.

1998,530 pp. ISBN 0-7293-4951-2

S. Gong: Convex and Starlike Mappings in Several Complex Variables. 1998,

208 pp. ISBN 0-7923-4964-4