FOURIER SERIES AND PERIODIC SOLUTIONS OF DIFFERENTIAL EQUATIONS

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1 FOURIER SERIES AND PERIODIC SOLUTIONS OF DIFFERENTIAL EQUATIONS Nguyen Thanh Lan Department of Mathematic Wetern Kentucky Univerity ABSTRACT: We ue Fourier erie to find a neceary and ufficient condition for exitence and uniquene of periodic olution of differential equation on Hilbert pace. AMS (MOS) Subject Claification: Primary 34 G 1, 34 K 6, Secondary 47 D INTRODUCTION Let E be a Banach pace and f(t) be a continuou function from [,T]toE. The Fourier coefficient of f(t) i defined a f k = 1 T Z T f()e 2kπi/T d, k ZZ. Then f(t) can be repreented by Fourier erie f(t) k= e 2kπit/T f k. Uing Fourier erie i a well known method for invetigating olution of differential equation, in particular for periodic and almot periodic olution (ee e. g. [1], [4], [7], [9], [1]). In thi paper we ytematically ue Fourier erie to find the condition for exitence and uniquene of periodic olution of functional differential equation x (t) = Z of the firt order differential equation [de()]x(t )+f(t), x (t) =Ax(t)+f(t), 1

2 and of the econd order differential equation x (t) =Ax(t)+f(t) on Hilbert pace. The paper i organized a the follow: Firt we find, for any function f C([, 1], E), a neceary and ufficient condition uch that thee equation have periodic olution. At the ame time, the tructure of the periodic olution i alo preented. Secondly, we characterize the exitence and uniquene of the periodic olution in term of the reolvent of the correponding operator. A a matter of fact, our proof i quite elementary and natural, and give a clear relationhip between periodic olution and the inhomogeneou term f(t). In Section 2, we begin with a functional differential equation on Hilbert pace. A a reult of Theorem 2.1, Theorem 2.2 extend the reult in [4] and [6] to abtract pace with improved proof. In Section 3 and 4, we deal with firt and econd order differential equation on Hilbert pace. Here we find imilar condition for the exitence of periodic olution. In Theorem 3.1 we extend Pru reult [9]. Section 4 preent a neceary and ufficient condition for the exitence and uniquene of 1-periodic olution of econd order differential equation (ee alo [1] for related reult). Let u fix ome notation. Firt, for the ake of implicity, we et the period T = 1. In a Hilbert pace E, letl 2 ([, 1],E)bethepaceofallmeaurablefunction f :[, 1] E uch that kfk 2 =( R 1 f(t) 2 dt) 1/2 i finite. L 2 ([, 1],E)ia Hilbert pace with w.r.t. the norm k k 2. The pace of bounded continuou function f :[, 1] E i denoted by C([, 1],E)anditnormikfk = up f(t). t [,1] We recall ome baic propertie of Fourier coefficient of function on Banach and Hilbert pace. Theorem 1.1 Let E beabanachpaceandletf(t) and g(t) be function on C([, 1],E). Thenf = g if and only if f n = g n for all n ZZ. Theorem 1.2 Let E beabanachpaceandf C 1 ([, 1],E), i.e. f be continuouly differentiable. Then f N (t) = P N k= f k.e 2kπit converge uniformly to f(t). Theorem 1.3 Let E be a Hilbert pace and f C([, 1],E). Then the fol- 2

3 lowing hold (Pareval equality). kfk 2 2 = f() 2 d = X f k 2. k Z For more detail of Fourier erie, we refer the reader to [2]. 2. PERIODIC SOLUTIONS OF FUNCTIONAL DIFFERENTIAL EQUATIONS On a Banach pace E, conider the following functional differential equation x (t) = Z [de()]x(t )+f(t), (2.1) where E() i continuou from the left and i of bounded total variation on [, ), i.e. γ = Z de() <. (2.2) Equation (1.1) ha been tudied by many author on C n (ee [4], [6] and reference therein). In thi ection we tart with a new approach. For each function f C([, 1],E) we give a neceary and ufficient condition for the exitence of periodic olution of (2.1) which i completely baed on the Fourier erie. The proof i natural and give a new framework for abtract pace. The idea come from the obervation that x(t) =ae 2nπit i a olution of (2.1) correponding to f(t) =be 2nπit if and only if (2nπi A n )a = b, where A n i defined on E by Thi lead to the following A n x = Z de()e 2nπi x. (2.3) Theorem 2.1 Let E be a Banach pace and {E(t)} t be a family of bounded operator in E atifying (2.2) and f C(E) be 1-periodic. Then the following are equivalent. (a) Equation (2.1) ha 1-periodic olution. 3

4 (b) For each n ZZ, f n Range(2πni A n ),wherea n i defined by (2.3), and there exit a equence {x n },where(2πin A n )x n = f n, uch that P kx n k 2 P < and N = N x n converge in E. Proof. (a) (b): Let x(t) be a 1-periodic olution correponding to f. Then we have the following relation: (2nπi A n )x n = f n, where f n and x n are Fourier coefficient of f and x, repectively. It implie f n Range(A n ) for n ZZ. Moreover, by Theorem 1.2, the function P x N (t) = N x n e 2πint P converge to x(t) inc([, 1],E). Hence, kx n k 2 = n= R 1 P kx(t)k2 dt < and N = N x n = x N () converge to x() in E. (b) (a): Let {x n } be a equence atifying (2nπi A n )x n = f n and N P = N x n converge in E. Define f N (t) = P N n= f n e 2πint and x N (t) = P Nn= x n e 2πint,thenx N (t) C 1 ([, 1],E) i 1-periodic olution of (2.1) correponding to f N (t). Integrating (2.1) from to t we have Z x N (t) =x N () + ( de()x N (τ ))dτ + f N (τ)dτ. (2.4) for t [, 1]. By Theorem 1.3, f N (t) f(t) and x N (t) x(t) = P n= x n e 2πint in L 2 ([, 1],E)aN. Hence, for N>Mwe have Z kx N (t) x M (t)k kx N () x M ()k + de() kx N (τ) x M (τ)kdτ k x i + () X x i k + + Z + kf N (τ f M (τ)kdτ de() kx N (τ) x M (τ)k 2 dτ kf N (τ f M (τ)k 2 dτ uniformly for t [, 1]. Hence x N (t) x(t) inc([, 1],E). In particular, x(t) i a 1-periodic function, and, by (2.4), i a olution of (2.1). 2 4

5 From the above proof, it follow that all 1-periodic olution of (2.1) are of P the form x(t) = x n e 2nπit,wherex n atify (2nπi A n )x n = f n and the erie converge uniformly. It i eay to ee that if (2nπi A n ) i injective for all n Zhen there i only one olution. The converity i alo true, ince if (2nπi A n ) i not injective for ome n, i.e, if there exit a x n 6= x n uch that (2nπi A n ) x n = f n,then x(t) =x(t)+ x n.e 2nπit i another 1-periodic olution. Uing thi obervation, we are now in a poition to characterize the exitence and uniquene of 1-periodic olution of (2.1) for each f(t) C([, 1],E). Theorem 2.2 Let E be a Hilbert pace. Then the following are equivalent (a) For each continuou 1-periodic function f equation (2.1) ha one and only one continuou 1-periodic olution (b) 2nπi ρ(a n ) for all n ZZ. Proof (a) (b): Let (a) be atified. By the above obervation, we only have to how that that (2nπi A n ) i urjective. Let y be an arbitrary point on E and x(t) be the 1-periodic olution of x (t) = R [de()]x(t )+e2nπit y. If x k i the k th Fourier coefficient of x(t), then x k = for all k 6= n and (2nπi A n )x n = y. HenceA n i urjective. (b) (a): By aumption, 2nπi ρ(a n ). Since ka n kγ, there exit acontantc uch that k(2nπi A n ) 1 kc/n for all n 6=. Letf be any function in C([, 1], E). To how the exitence of 1-periodic olution, P byvirtueoftheorem2.1ituffice to how that N = N (2kπi A k ) 1 f k converge in E. But for N>M> we have k N M k ( k(2kπi A k ) 1 f k k + () X +( k(2kπi A k ) 1 k 2 ) 1/2 ( () X k(2kπi A k ) 1 f k k kf k k 2 ) 1/2 () k(2kπi A k ) 1 k 2 ) 1/2 X ( kf k k 2 ) 1/2 5

6 a N,M. Thu n converge in E. The uniquene of the olution follow from the injectivity of (2nπi A n ), and thi complete the proof PERIODIC SOLUTIONS OF FIRST ORDER DIFFERENTIAL EQUATIONS On a Banach pace E, we conider the firt order differential equation x (t) =Ax(t)+f(t), (3.1) where A i the generator of a C -emigroup (T (t)) in E. A function x C(IR,E) i called a mild olution of (3.1) if x(t) =T (t )x()+ T (t τ)f(τ)d (3.2) for all t>in IR (ee more detail in [8]). If x(t) C 1, then it i called a trict olution. If x(t) atifie (3.2) on [,1], uch that x() = x(1), then it i clear that x(t) can be continuouly extended to a 1-periodic mild olution of (3.1), provided f ha been extended 1-periodically, too. Therefore, we call a mild olution of (3.1) 1-periodic if it atifie (3.2) for <t 1and x() = x(1). We have the following. Theorem 3.1 Let A be the generator of a C -emigroup on a Hilbert pace E and f be a continuou function on [, 1]. Then the following are equivalent. (a) Equation (3.1) ha 1-periodic mild olution. (b) f n Range(2nπi A) for each n ZZ and inf{ n= kx n k 2 : (2πin A)x n = f n } = M<. (3.3) Proof. (a) (b): Let x(t) be the 1-periodic olution correponding to f. Then the Fourier coefficient of f and x atify the following equation (ee [5]). (2nπi A)x n = f n. (1) 6

7 It implie that f n Range(2nπi A). Moreover, condition (3.3) i alo atified ince kx n k 2 = kx(t)k 2 dt <. n= (b) (a): We borrow the technique obtained in [9]. Let {x n } be a equence atifying (2nπi A)x n = f n and P n= kx n k 2 <. (Condition (3.3) P guarantee the exitence of thi equence). Define f N (t) = N f n e 2πint, x N (t) = N P n= x n e 2πint. n= Then f N (t),and by aumption, x N (t), converge to f(t) and a function x(t) inl 2 ([, 1],E) repectively. Moreover, x N (t) C 1 [, 1],E) and i a 1-periodic olution of (3.1). Hence it atifie x N (t) =T (t)x N () + T (t )f N ()d. (3.4) Taking t = 1 in (3.4), we get (I T (1))x N () = R 1 T (1 )f N()d R 1 T (1 )f()d. On the other hand, multiplying (3.4) by T (1 t) and integrating over [, 1], we obtain T (1)x N () = T (1 t)x N (t)dt ( T (1 τ)f N (τ)dτ)dt. The right-hand ide of thi equation converge a N. Hence x N () = T (1)x N () + (1 T (1))x N () converge in E. Therefore from (3.4) it follow that x N (t) x(t) inc([, 1],E). In particular, x(t) i a mild 1-periodic olution of (3.1). 2 P Again, all 1-periodic olution of (3.1) have the tructure x(t) = (2nπi A) 1 f n e 2nπit. It i eay to ee that thee 1-periodic olution are unique if and only if (2nπi A) i injective for each n ZZ. In particular, if for every f C[, 1], there exit a unique periodic olution of (3.1), then we get 2kπi ρ(a) for k ZZ. Moreover, n= k(2nπi A) 1 f n k 2 = n= kxk 2 n < 7

8 for each f C([, 1],E). Uing the imilar argument a in Theorem 4.2, we obtain up k ZZ k(2kπi A) 1 k <. Thu, the following Theorem, which wa obtained in [9], i followed from the above obervation. Theorem Let E be a Hilbert pace and A be the generator of a C -emigroup on E. Then the following are equivalent. (a) For each continuou 1-periodic function f equation (3.1) ha one and only one 1-periodic mild olution (b) 2kπi ρ(a), (k ZZ), and up k ZZ k(2kπi A) 1 k <. 4. PERIODIC SOLUTIONS OF SECOND ORDER DIFFERENTIAL EQUATIONS We are now conider the econd order differential equation u (t) =Au(t)+f(t), (4.1) where A i the generator of a coine family (C(t)) on a Banach pace E. For a given function f C(E), a function u(t) C 1 (E) i called a mild olution of (4.1) if u(t) =C(t )u()+s(t )u ()+ S(t τ)f(τ)d, (4.2) where S(t) i the aociated ine family (ee [3] for more detail). We call the mild olution u(t) a trict olution if u(t) C 2 (IR,E). If u(t) i a function atifying (4.2) on [,1] uch that u() = u(1) and u () = u (1), then it i clear that u(t) can be continuouly extended by periodicity to a 1-periodic mild olution of (4.1), provided f(t) ha been extended 1-periodically, too. Therefore, we call a mild olution of (4.1) 1-periodic if it atifie (4.2) for <t 1, u() = u(1) and u () = u (1). We have the following. Theorem 4.1 Let E be a Hilbert pace and f(t) be a function on C([, 1],E). Then the following are equivalent. (a) Equation (4.1) ha 1-periodic mild olution. 8

9 (b) f n Range(4π 2 n 2 + A) for n ZZ, and inf{ n 2 kx n k 2 :(4π 2 n 2 + A)x n = f n } <. (4.3) Proof. (a) (b): If u(t) i a 1-periodic mild olution correponding to f(t), then the coefficient of f and u atify the following equation (ee [7]) (4π 2 n 2 + A)u n = f n. It implie f n Range(4π 2 n 2 + A). Moreover, ince u(t) iinc 1 cla, we have u (t) C([, 1],E) and the Fourier coefficient of u (t) areu n =2πniu n. Hence, 4n 2 π 2 ku n k 2 = ku (t)k 2 dt <, n= from which (4.3) follow. (b) (a): Let {x n } be a equence atifying (4n 2 π 2 + A)x n = f n and P n= 4n 2 π 2 ku n k 2 <. (Condition (4.3) guarantee the exitence of uch a equence). Define f N (t) = N P n= f n e 2πint, x N (t) = N P n= x n e 2πint,wheref n i the Fourier coefficient of f and u n atifie (4n 2 π 2 A)u n = f n. Then f N (t), and by aumption, u N (t) andu N(t), converge to f(t) and to certain function u(t) andv(t) inl 2 ([, 1],E), repectively. Moreover, for N>M we have ku N (t) u M (t)k ( ku n k + 1 n 2 )1/2 ( () X ku n k n 2 ku n k 2 ) 1/2 () () X 1 X +( n 2 )1/2 ( n 2 ku n k 2 ) 1/2 a N,M. Thiimpliethatu N (t) converge to function u(t) uniformly. In particular, u i a continuou and 1-periodic function. To prove that u i in 9

10 C 1 cla, u () = u (1), and atifie (4.2), it uffice to how the convergence of u N (t) inc([, 1],E)aN. Todothat,wefirt oberve that u N(t) i a 1-periodic mild olution of (4.1) correponding to f N (t), i.e., From (4.4) it follow x N (t) =C(t)x N () + S(t)u N () + S(t )f N ()d. (4.4) u N (t) =C (t )u N ()+C(t )u N ()+ C(t τ)f N (τ)dτ. (4.5) Multiplying (4.4) by C (t ), (4.5) by C(t ), and ubtracting the reult, we have C(t )u N (t) C (t )u N (t) =u N ()+ C(τ )f N (τ)dτ. (4.6) Integrating (4.6) over [, +1] and uing the fact that R +1 C (t )u N (t)dt = (C(1) I)u N () R +1 C(t )u N(t)dt, we conclude that Z +1 2 C(t )u N(t)dt (C(1) I)u N () =. u N()+ By aumption, P Z +1 ³ C(τ )f N (τ)dτ dt for all [, 1]. (4.7) 4π 2 n 2 ku n k 2 <. It implie that u N(t) converge in L 2 ([, 1],E). Hence, from (4.7) it follow the convergence of u N() in C([, 1],E) and it complete the proof. 2 It i intereting that the periodic olution of (4.1) are alo of the form u(t) = P u ne 2nπit, where (4n 2 π 2 + A)u n = f n. Moreover, if condition(4.3) i atified, then u(t) automatically belong to C 1 cla. We now characterize the exitence and uniquene of periodic olution of econd order in term of the pectrum of A. Theorem 4.2 Let E be a Hilbert pace. Then the following are equivalent. (a) Equation (4.1) ha a unique 1-periodic mild olution for each function f C([, 1],E). 1

11 (b) { 4π 2 n 2 : n ZZ} ρ(a) and there i a contant C uch that for all n 6=, k(4π 2 n 2 + A) 1 k <C/n. Proof. (a) (b) follow directly from Theorem 4.1. To prove the invere, let f(t) be any function in C([, 1],E)andx(t) be the correponding 1- periodic olution. Uing the equation (4π 2 n 2 + A)u n = f n for every n ZZ, and the argument a in the proof of Theorem 2.2, we obtain that (4π 2 n 2 +A) i urjective and injective, i. e., 4π 2 n 2 ρ(a) andu n = (4π 2 n 2 + A) 1 f n. Now let u n be the Fourier coefficient of u,thenu n =2πniu n. Hence, 4π 2 n 2 k(4π 2 n 2 + A) 1 f n k 2 = ku n k2 = ku (t)k 2 dt < (4.8) for every f C([, 1],E). Aume contrarily that (b) i not true, i.e. up n kn(4π 2 n 2 + A) 1 k =. Then we can find a equence (n j ) j IN with n j and kn j (4π 2 n 2 j A) 1 k. Hence, from equence (n j ) j we can find a ubequence, and without lo of generality, denoted again by (n j ) j, uch that for each n j there i a point f nj with kf nj k = 1 and kn j (4π 2 n 2 j A) 1 f nj k >j 2. Let g nj = 1 f j 2 nj,thenkg nj k = 1 and kn j 2 j ((4π 2 n 2 j A) 1 g nj k > 1. It follow that P j= g nj e 2πjit uniformly converge in C([, 1],E). Hence, g = P j= g kj e 2πjit i a function in C([, 1],E), where j= kn j (4π 2 n 2 j A) 1 g nj k 2 =, which i contradictory to (4.8), and thi complete the proof. 2. REFERENCES [1] Dalleckii J. and Krein M. G.: Stability of olution of differential equation in Banach Space. Tranl. Math. Mono., vol 43, Amer. Math. Soc., Providence, R. I.,

12 [2] Dym,H.andMcKeanH.P.:Fourier erie and integral. Academic Pre, New York and London, [3] H.O.Fattorini: Second order linear differential equation in Banach pace. North Holland Mathematic Studie (18), [4] Hatvani, L. and Kriztin T.: On the exitence of periodic olution for linear inhomogeneou and quai-linear functional differential equation. J. Diff. Equ. 97 (1992), p [5] Haraux A.: Nonlinear evolution equation Lecture Note in Math., vol. 841 Springer Verlag, Heidelberg (1992), p [6] Langenhop Carl E.: Periodic and almot periodic olution of Volterra integral differential equation with infinite memory. J. Diff. Eq. 58, p [7] Lizama, C.: Mild almot periodic olution of abtract differential equation. J. Math. Anal. Appl. 143 (1989), p [8] Pazy A.: Semigroup of linear operator and application to partial differential equation. Springer Verlag, Berlin, [9] Pru, J.: On the pectrum of C -emigroup. Tran. Amer. Math. Soc. 284, no. 2, p [1] Schuler, E.: On the pectrum of coine function. J. Math. Anal. Appl. 229 (1999), no. 2, p

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