almost periodic functions and differential equations

Almost periodic functions and differential equations Almost periodic functions and differential equations B.M.LEVITAN & V.V.ZHIKOV Translated by L W Longdon CAMBRIDGE UNIVERSITY PRESS Cambridge London New York New Rochelle Melbourne Sydney Published by the Press Syndicate of the University of Cambridge, The Pitt Building, Trumpington Street, Cambridge CB2 1RP 32 East 57th Street, New York, NY 10022, USA 296 Beaconsfield Parade, Middle Park, Melbourne 3206, Australia © Moscow University Publishing House 1978 English edition © Cambridge University Press 1982 Originally published in Russian as Pochti periodicheskie funktsii differentsial' nye uravneniya by the Moscow University Publishing House 1978 Assessed by E D Solomentsev and V A Sadovnichii First published in English, with permission of the Editorial Board of the Moscow University Publishing House, by Cambridge University Press 1982 - Printed in Great Britain at the University Press, Cambridge Library of Congress catalogue card number: 83 4352 British Library Cataloguing in Publication Data Levitan, B.M Almost periodic functions and differential equations Periodic functions I Title II Zhikov, V.V uravneniya English 515.8 QA331 ISBN 521 24407 III Pochtiperiodicheskie funktsii i differentsial'nye Contents Preface 1 2 Almost periodic functions in metric spaces Definition and elementary properties of almost periodic functions Bochner's criterion The connection with stable dynamical systems Recurrence A theorem of A A Markov 10 Some simple properties of trajectories 11 Comments and references to the literature 12 Harmonic analysis of almost periodic functions Prerequisites about Fourier—Stieltjes integrals Proof of the approximation theorem The mean-value theorem; the Bohr transformation; Fourier series; the uniqueness theorem Bochner—Fejer polynomials Almost periodic functions with values in a Hilbert space; Parseval's relation The almost periodic functions of Stepanov Comments and references to the literature Arithmetic properties of almost periods Kronecker's theorem The connection between the Fourier exponents of a ix function and its almost periods Limit-periodic functions 14 14 17 21 25 31 33 36 37 37 40 45 vi Contents Theorem of the argument for continuous numerical complex-valued almost periodic functions Comments and references to the literature 48 51 Generalisation of the uniqueness theorem (N-almost periodic functions) 53 Introductory remarks, definition and simplest properties of N-almost periodic functions Fourier series, the approximation theorem, and the uniqueness theorem Comments and references to the literature Weakly almost periodic functions Definition and elementary properties of weakly almost periodic functions Harmonic analysis of weakly almost periodic functions Criteria for almost periodicity Comments and references to the literature A theorem concerning the integral and certain questions of harmonic analysis The Bohl—Bohr—Amerio theorem Further theorems concerning the integral Information from harmonic analysis A spectral condition for almost periodicity Harmonic analysis of bounded solutions of linear 53 59 62 64 64 68 70 76 equations Comments and references to the literature Stability in the sense of Lyapunov and almost periodicity Notation The separation properties A lemma about separation Corollaries of the separation lemma Corollaries of the separation lemma (continued) A theorem about almost periodic trajectories Proof of the theorem about a zero-dimensional fibre Statement of the principle of the stationary point 77 77 81 87 91 92 96 98 98 98 101 105 107 109 113 116 Contents Realisation of the principle of the stationary point when the dimension m -._ Realisation of the principle of the stationary point under monotonicity conditions Comments and references to the literature Favard theory Introduction Weak almost periodicity (the case of a uniformly convex space) Certain auxiliary questions Weak almost periodicity (the general case) Problems of compactness and almost periodicity Weakening of the stability conditions On solvability in the Besicovitch class Comments and references to the literature 10 11 The method of monotonic operators General properties of monotonic operators Solvability of the Cauchy problem for an evolution equation The evolution equation on the entire line: questions of the boundedness and the compactness of solutions Almost periodic solutions of the evolution equation Comments and references to the literature Linear equations in a Banach space (questions of admissibility and dichotomy) Notation Preliminary results The connection between regularity and the exponential dichotomy on the whole line Theorems on regularity Examples Comments and references to the literature vii 117 121 123 124 124 127 130 134 135 140 142 147 149 149 153 157 161 165 166 166 166 170 172 176 181 The averaging principle on the whole line for parabolic equations Bogolyubov's lemma Some properties of parabolic operators 182 182 183 viii Contents The linear problem about averaging A non-linear equation The Navier—Stokes equation The problem on the whole space Comments and references to the literature 186 189 193 195 199 Bibliography 200 Index 208 Preface The theory of almost periodic functions was mainly created and published during 1924-1926 by the Danish mathematician Harald Bohr Bohr's work was preceded by the important investigations of P Bohl and E Esclangon Subsequently, during the 1920s and 1930s, Bohr's theory was substantially developed by S Bochner, H Weyl, A Besicovitch,, J Favard, J von Neumann, V V Stepanov, N N Bogolyubov, and others In particular, the theory of almost periodic functions gave a strong impetus to the development of harmonic analysis on groups (almost periodic functions, Fourier series and integrals on groups) In 1933 Bochner published an important article devoted to the extension of the theory of almost periodic functions to vector-valued (abstract) functions with values in a Banach space In recent years the theory of almost periodic equations has been developed in connection with problems of differential equations, stability theory, dynamical systems, and so on The circle of applications of the theory has been appreciably extended, and includes not only ordinary differential equations and classical dynamical systems, but wide classes of partial differential equations and equations in Banach spaces In this process an important role has been played by the investigations of L Amerio and his school, which are directed at extending certain classical results of Favard, Bochner, von Neumann and S L Sobolev to differential equations in Banach spaces We survey briefly the contents of our book In the first three chapters we present the general properties of almost periodic functions, including the fundamental approximation theorem From the The problem on the whole space 199 Now we briefly outline a proof of Theorem First we must show that the L„, (co coo) are uniformly correct Assuming otherwise we have a sequence w a > co and sequences lual, {fn } c C such that Lu n = fn, Ilunlic = 1, Ilfnlic * We take a point za such that lua (zn )1 = ta ) - 1- and put va (z ) = tia(z +z) It is important that {v a } is locally compact in C (a Nash type estimate); we may assume that v a > io Now we must realise a passage to the limit similar to that in Theorem A small feature is that as well as the main averaging procedure, in this case we have a passage to the limit with respect to x Therefore, /5 is a solution not necessarily of Lu = but of some equation rhu = • But this contradicts the regularity of L The uniform correctness of L.* (co> coo) is proved in exactly the same way But then the regularity of L a, follows directly from Proposition The proof of the remaining assertions of the theorem should not present difficulty In conclusion we mention a most commonly used sufficient condition for the regularity of L: ão(x)0 and every doh(x) * O This condition is fulfilled if, for example, 'do - 0, ao € 6(R m) and * Comments and references to the literature Bogolyubov's lemma is proved in his book [8] A very extensive bibliography on the averaging method is given in the monograph of Mitropol'skii [90] The contents of Chapter 11 are mainly from Zhikov's article [56] The results in § were obtained by Zhikov jointly with L Tsend and M Otel'baev (unpublished) A technically different approach to averaging in parabolic problems which has been developed by Simonenko [101] must be mentioned At this point it is important that the mean values of the coefficients are uniform with respect to x e R Bibliography Amerio, L (1955) Soluzioni quasi-periodiche, o limitate, di sistemi differenzali non lineari quasi-periodiche, o limitati, Annali di Matematica Pura ed Applicata (4), 39,97 119 Amerio, L 8z Prouse, G (1971) Almost periodic functions and functional equations, New York and London, van Nostrand Reinhold Arnol'd, V I (1975) Matematicheskie metody 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mathematics, Pacific Journal of Mathematics, 70, 67 81 Shubin, M A (1978) Almost periodic functions and partial differential operators, Uspekhi Matematicheskykh Nauk, 32 (2), 47 (Russian - - Mathematical Surveys, 33 (2), 52) Kozlov, S M (1978) Homogenization of differential operators with almost periodic rapidly oscillating coefficients, Matematicheskii Sbornik, 107, 199 217 (Mathematics of the USSR Sbomik, 35,481 98) Zhikov, V V (1979) A pointwise stabilization criterion for second order parabolic equations with almost periodic coefficients, Matematicheskii Sbornik, 38, Sbornik, 110, 309 18 ((1980) Mathematics of the USSR 279-92) - - - - - - Bibliography 207 Zhikov, V V., Kozlov, S M & Oleinik, O A (1982) Homogenization of parabolic operators with almost periodic coefficients, Matematicheskii Sbornik, 117, 69-85 Kozlov, S M., Oleinik, O A & Zhikov, V V (1981) Sur l'homogénéisation d'opérateurs différentiels paraboliques a coefficients presque périodiques, Comptes Rendus des Séances de l'Académie de Sciences, Paris, 293, Series 1, no 4,245-8 Zhikov, V V & Sirazhudinov, M M (1981) Homogenization of nondivergent second order elliptic and parabolic operators, and the stabilization of the solution of the Cauchy problem, Matematicheskii Sbomik, 116, 166-86 Index admissible pair, 167 almost automorphic function, 63 almost period E-almost period, E,N-almost period, 53 almost periodic function, 1, 60, 71, 73, 75, 79, 84, 99, 100, 111, 115, 124, 144 almost periodic solution, 109, 113, 115, 117, 121, 126,.138, 146, 147, 148, 163, 164, 165 Amerio, L., 36, 76, 96, 147 Amerio, L & Prouse, G., 76, 96, 147 approximation theorem, 17, 36 for N-almost periodic functions, 60 for Stepanov almost periodic functions, 36 basic semigroup, 100 basis finite, 26 integer, 26 rational, 26, 69 Baskakov, A G., 63, 97 Besicovitch almost periodic function, 142 Besicovitch almost periodic solution, 143, 163 Birkhoff minimal set, 9, 131 Bochner, S., 13, 33, 36, 52, 62, 96, 148 Bochner, S & von Neumann, J., 148 Bochner's criterion, 4, 6, 42, 70, 73 Bochner-Fejer composite kernel, 28 Bochner-Fejer kernel of order n, 28 Bochner-Fejer polynomial, 29, 36, 69, 70 Bochner-Fejer sum, 25 Bochner-Fejer summation method for weakly almost periodic functions, 69 Bogolyubov, N N., 15, 36, 199 Bogolyubov, N N & Krylov, N M., 36 Bogolyubov's lemma, 182, 189, 199 Bohl, P., 12, 96 Bohr, H., 12, 52, 96 Bohr, H & Neugebauer, 0., 97 Bohr's example, 100, 144 Bohr transformation, 23 of weakly almost periodic functions, 69 Boles Basit, R., 63, 97 Boles Basit, R & Tsend, L., 97 bounded almost periodic function, 70, 75 bounded solutions, 77, 92, 93, 94, 118, 120, 157, 160, 174, 177, 178 Brodskii, M S & Mil'man, D P., 13 Bronshtein, I U., 97, 123 Bronshtein, I U slir Chernyi, B F 0., 123 Cheresiz, V M., 123 coercive estimate, 156 coercive inequality, 184 coercive property, 151 compactness condition, 160, 1.73 compactness lemma, 137 compactness of a set of almost periodic functions, compact solution, 107, 140 condition for almost periodicity, 70, 91, 92, 97 condition of right solvability, 166 condition for uniform positive stability, 138 condition of weak continuity, 124 conditional exponential stability, 191 conditionally periodic function, 12 contraction mapping principle, 153, 192 Index convolution, 16 criterion for almost periodicity, 70, 91 criterion for N-almost periodicity, 58 criterion for a point spectrum, 139 Daletskii, Yu, L & Krein, M G., 179, 181 demicontinuity, 150 derivative of an almost periodic function, deviation from an almost periodic function, 31 duality, 154 Dunford, N & Schwartz, J T., 13 dynamical system (flow), Ellis, R., 123 Ellis semigroup, 101, 104 equi-almost periodic set, equitransitive system, 114, 140 Esclangon, E., 12 Esclangon's lemma, 97 exponential dichotomy, 167, 168, 169 extension completely non-distal, 132 distal, 98 of a minimal flow, 98 positively stable, 106 extreme point, 131 evolution equation, 99, 157, 160, 161, 164 Favard, J., 52, 97, 147, 148 Favard minimal set, 130, 131, 134, 135 Fejer sum, 25 Fejer kernel, 28 fibre, 99 distal, 99, 104, 105 minimal, 116 positively stable, 132 semidistal, 99 Flor, P., 123 flow, compact, 98 distal, 98, 105 equicontinuous, semidistal, 98 strictly ergodic, 101 two-sidely stable, Fourier coefficients of almost periodic function, 24 of N-almost periodic function, 60 of Stepanov almost periodic function, 35 minimal property of, 32 Fourier exponent, 24, 32, 40, 41, 50 Fourier-Bohr exponent, 42, 43 209 Fourier series of almost periodic function, 24 of N-almost periodic function, 59, 60 Fourier-Stieltjes integral, 16, 36, 56 Fourier-Stieltjes transform, 14 Fourier transform, 87 Furstenberg, H., 123 Gel'fand, I M., 148 Gorin, E A., 52 Gottschalk, W A & Hedlund, G A., 114, 123 harmonic analysis of weakly almost periodic functions, 68 homogeneous space, 118 hyperbolic equation, 140, 148, 165 identical invariance, 107 indefinite integral, 77, 81, 138 invariant section, 142, 143 Jessen, B., 52 Jessen, B (Sr Tornehave, H., 52 Kadets, M I., 74, 96, 133, 148 Krasnosel'skii, M A., Burd, B Sh & Kolesov, Yu, S., 181 Ladyzhenskaya, O A., 194 Ladyzhenskaya, O A., Solonnikov, V A Sr Uranseva, N N., 196 Lax, P D & Phillips, R S., 148 Levin, B Ya., 62 Levin, B Ya, & Levitan, B M., 62 Levitan, B M., 36, 52, 62, 76, 97, 144 limit periodic function, 46 linearly independent set, 26 Lions, J.-L., 165 local convergence, 87 Loomis, L H., 97 Lyapunov stable semigroup, 10 Lyapunov solution, 100 Lyapunov trajectory, 13 Maizel', A D., 181 Marchenko, V A., 62 Massera, L L & Schaffer, I I., 178, 181 mean value, 22 mean-value theorem, 22 measure of oscillation of an operatorfunction, 179 Millionshchikov, V M., 123 minimax method, 127, 135, 147 Mitropolskii, Yu A., 199 210 Index module, 59 rational hull ?rat, 115 Montgomery, D & Samelson, H., 118 Muckenhoupt, C F., 13 Mukhamadiev, E., 181 Sell, G R., 123 semigroup continuous, 11 distal, 98 Lyapunov stable, 10 semidistal, 98 semiseparated solution, 100 semiseparation condition, 127, 134, 147 semitrajectory, separated solution, 100 separation lemma, 102, 105 N-almost periodic function, 54, 57, 58, 63, 84, 111, 112, 146 N-almost periodic solution, 127 Navier-Stokes equation, 193 Nemytskii, V V & Stepanov, V V., 13, 123 sequence operator averaged, 182 coercive, 151, 152, 184 correct, 167, 169 evolution, 154 set monodromy, 178 monotonic, 149 non-expansive, 152 parabolic, 183, 189, 190, 196 regular, 167 semicontinuous, 149 strongly correct, 167 strongly elliptic, 184 uniformly regular, 172 weakly regular, 167 orbit, 116 Parsevars relation, 31, 32, 36 Pelczynski, A., 96 Perov, A I & Ta Kuang Khai, 148 Pérron, 0., 181 point of almost periodicity, 91 of non-almost periodicity, 91 regular, 87 stationary, 117 point spectrum of homogeneous problems, 143 P-property, 173 Poisson stable function, 80, 86, 97, 173, 185 Pontryagin, L S., 118 positive definite function, 17, 19, 55 principle of the stationary point, 116, 117, 121 quasiperiodic function, 12 f-increasing, 42 f-normal, 42 f-returning, 80 returning, 57, 78, 80, 81 weakly convergent, 64 equi-almost periodic, A-invariant, 116 invariant, minimal, 9, 130 relatively dense, 1, 86 Shcherbakov, B A., 96, 97, 123 Simonenko, N B., 199 space co, 81 C(X), 6(X), H8 (0), 193 Y, 1, 87 2'' (X), 33 21) (-00, co; B), 166 lte(X), 34 Ye(X), 34 et, 157 Xcomp, 157 komp 190 X, 190 not containing c o, 82 , reflexive embedded, 183 weakly complete, 64 spectrum, 24, 87, 143 spectral condition for almost periodicity, 91 stationary equation, 161 stationary subgroup, 118 Stepanov, V V., 33, 36 Stepanov almost periodic function, 33, 34, 36, 80 strictly convex norm, 121 rarified subset, 92 rational hull Erat, 115 recurrent function, 12, 80, 84 Reich, A., 63 resolvent, 93 theorem Amerio, 109, 111, 112 Birkhoff, 9, 102, 125, 139 Bochner-Khinchin, 17, 19, 36 Index theorem-continued Bogolyubov, 13, 41, 55, 62, 97 Bohl-Bohr-Amerio, 77, 80 Ellis, 104, 116 Favard, 62, 109 Kronecker, 37, 39, 46, 63, 66, 71, 115 Kronecker-Weyl, 51, 52, 144 Levitan, 109 Lyusternik, 7, 13 Markov, 10, 13, 136, 141 Zhikov, 110 theorem of the argument, 49, 101 trajectory absolutely recurrent, 10, 106 compact, 8, 20, 106, 137 compatibly recurrent, of a group, minimal closure of, recurrent, 9, 102, 115 of a semigroup, semiseparated, 100, 103, 106 separated, 100 weakly recurrent, 137 unconditionally bounded series, 82 uniform exponential dichotomy, 167 uniform positive stability, 108, 135 uniformly convex space, 74 uniformly exponentially stable to the right, 170 uniformly stable to the right, 170 211 uniqueness theorem for almost periodic functions, 24 for N-almost periodic functions, 62 two sided, 108 Valikov, K V., 148 Veech, W A., 63 V-monotonic equation, 107, 121, 123 Wallace, A D., 123 weak convergence, 64 weak limit, 64 weakly almost periodic function, 65, 66, 68, 69, 70, 71, 72, 75, 81, 126 weakly almost periodic solution, 127, 129, 131, 134, 140 weakly compact solution, 125, 130, 143 weakly N-almost periodic solution, 126, 127, 128, 134, 135 weakly recurrent solution, 139, 140, 141 Wiener, N., 36 Wintner, A., 52 Wolff, F., 97 Zhikov, V V , 62, 76, 97, 123, 148, 165, 181, 199 Zhikov, V V & Levitan, B M., 165 Zhikov, V V & Tyurin, V M., 181 Zhikov, V V & Valikov, K V., 181 [...]... for f 2 Now we are going to deduce further properties of almost periodic functions that are obtained more simply from Bochner's criterion than from our definition Property 6 The sum f(t)+g(t) of two almost periodic functions is almost periodic The product of an almost periodic function f(t) and a numerical almost periodic function 0 (t) is almost periodic Proof Let hn } be an arbitrary sequence of real... E -almost period for f(t) The position of conditionally periodic functions in the class of continuous almost periodic func- Comments and references 13 tions is discussed in Chapter 3, § 3, and the role of the system of inequalities (8) in the theory of almost periodic functions is considered in Chapter 3, § 2, and Chapter 4, § 1 (Bogolyubov's theorem) The extension of the theory of almost periodic functions. .. teJ Comments and references to the literature 1 The definition of an almost periodic function and its simplest properties for numerical functions is due to Bohr [17] and [22] Long before the publication of Bohr's work, Bohl [15] and Esclangon [120], [121] had discussed a special case of almost periodic functions § which are now known as conditionally periodic (or sometimes, quasiperiodic) functions In... spectral theory and in the theory of homogenisation of partial differential equations with almost periodic coe ffi cients The additional references are, in the main, concerned with this theme L W Longdon 1 Almost periodic functions in metric spaces 1 Definition and elementary properties of almost periodic functions Throughout the book J denotes the real line, X a complete metric space, and p = p(xl,... numbers nT is relatively dense It is easy to produce examples of almost periodic functions that are not periodic, for instance, f(t)= cos t + cos t.s12 2 Almost periodic functions in metric spaces We prove some of the simplest properties of almost periodic functions; these are straight-forward consequences of the definition Property 1 An almost periodic function f :J *X is compact in the is compact sense... arbitrarily, and take an e -almost period 7 = re such that to + T 1/2, that is, to+ //2 Then P(f(t0+ 7 ), fit0))= 8 Because to -FT E [-1/2, 1/2], the set Rfa is an e-net for Rf, as we required to prove Remark For numerical almost periodic functions (that is, when X = R 1 ) and for almost periodic functions with values in a finitedimensional Banach space, Property 1 reduces to the following: if f is an almost periodic. .. = (5, and so p i(g(f(t + r)), g(f(t))) -5 E Corollary Let f be a continuous almost periodic function with values in a Banach space X Then l[f(t)li k is a continuous numerical almost periodic function for all k > 0 Property 5 Suppose that fis an almost periodic function with values in a Banach space X If the (strong) derivative f' exists and it is uniformly continuous on J, then f' is an almost periodic. .. s' = t'+ re, and s"= t" + rE, From (1), (2) and the triangle inequality we have - PU (t" ), fit'))=P(Pt"), fis"))+P(Ps"),Ps1)) +P(fis'), f(e))< E Definition and elementary properties 3 Property 3 Let fn : J X, n = 0, 1, 2, , be a sequence of continuous almost periodic functions that converges uniformly on J to a function f Then f is almost periodic Proof We take an arbitrary e > 0 and let n = ne... x2, , Xn) and the norm 11x11 = kil l lixkil It follows easily from Bochner's criterion that if fi (t), f2(t), , fn (t) are almost periodic functions from J into X 1 , X2, , Xn, then the function f(t)= (Mt), f2(t), , fn (t)) is an almost periodic function from J into X The next property is easily deduced from this remark Property 7 Let fi(t), f2(t), , fn(t) be almost periodic functions from... Chapter 4 is devoted to the theory of N -almost periodic functions In comparison with the corresponding chapter of the book AlmostPeriodic Functions by B M Levitan (Gostekhizdat, Moscow (1953)), we have added a proof of the fundamental lemma of Bogolyubov about the structure of a relatively dense set Chapter 5 is concerned with the theory of weakly almost periodic functions developed mainly by Amerio Chapter .. .Almost periodic functions and differential equations Almost periodic functions and differential equations B.M.LEVITAN & V.V .ZHIKOV Translated by L W Longdon CAMBRIDGE... sum f(t)+g(t) of two almost periodic functions is almost periodic The product of an almost periodic function f(t) and a numerical almost periodic function (t) is almost periodic Proof Let hn... weakly almost periodic functions Harmonic analysis of weakly almost periodic functions Criteria for almost periodicity Comments and references to the literature A theorem concerning the integral and