DSpace at VNU: On the existence of quasi periodic and almost periodic solutions of neutral functional differential equations

COMMUNICATIONS ON PURE AND APPLIED ANALYSIS Volume 3, Number 2, June 2004 Website: http://AIMsciences.org pp 291–300 ON THE EXISTENCE OF QUASI PERIODIC AND ALMOST PERIODIC SOLUTIONS OF NEUTRAL FUNCTIONAL DIFFERENTIAL EQUATIONS Nguyen Minh Man and Nguyen Van Minh Department of Mathematics Hanoi University of Science, 334 Nguyen Trai, Hanoi, Vietnam Abstract This paper is concerned with the existence of almost periodic d Dxt = solutions of neutral functional differential equations of the form dt Lxt + f (t), where D, L are bounded linear operators from C := C([−r, 0], Cn ) to Cn , f is an almost (quasi) periodic function We prove that if the set of imaginary solutions of the characteristic equations is bounded and the equation has a bounded, uniformly continuous solution, then it has an almost (quasi) periodic solution with the same set of Fourier exponents as f Introduction The problem we consider in this paper is to find such conditions that linear neutral functional differential equations of the form d Dxt = Lxt + f (t), (1.1) dt where D, L are bounded linear operators from C := C([−r, 0], Cn ) to Rn , f is an almost (quasi) periodic function in the sense of Bohr, have an almost (quasi) periodic solution with the same Fourier exponents as f This problem has a long history and is one of the main concerns in the qualitative theory of differential equations In the simplest case of ordinary differential equat tions, when one deals with the τ -periodicity of the integral F (t) = f (ξ)dξ, where f is τ -periodic and continuous, an easy computations shows that F is τ -periodic if and only if it is bounded This simple conclusion turns out to be a motivation for numerous works In this direction, for more details on results on the existence of periodic solutions we refer the reader to [19, 4, 16, 25, 22] When f is periodic the method of study in these works is to prove the existence of fixed points of the period maps However, this method does not work in the more general case where f is almost periodic A new method of study was introduced in [23, 7, 13, 12, 20] to overcome this obstacle which makes use of the so-called evolution semigroups associated with the evolutionary processes generated by equations In our equation (1.1), with general assumptions on D and L, the Cauchy problem may have no solutions, so one has no evolutionary processes (or in this case, solution semigroups) In our recent paper [21], we have begun studying conditions for the existence of almost periodic solutions to Eq (1.1) We have showed that if (1.1) has a bounded, uniformly continuous solution, ∆i \Sp(f ) is closed, ∆i is bounded, and Sp(f ) is 2000 Mathematics Subject Classification 34K14, 34K06 Key words and phrases Neutral functional differential equation, almost periodic solution, quasi periodic solution 291 292 NGUYEN MINH MAN AND NGUYEN VAN MINH countable, where Sp(f ) is the Beurling spectrum of f , that is equal to the closure of the set of the Fourier exponents of f in the case of almost periodic functions, and ∆i ∆(λ) := {ξ ∈ R : det ∆(iλ) = 0} := λDeλ· − Leλ· , λ ∈ C, then (1.1) has an almost periodic solution w such that Sp(w) ⊂ Sp(f ) In this paper, by showing that every bounded, uniformly continuous solution of (1.1) is almost periodic, we will prove (see Theorem 3.6) that if the set ∆i is bounded, then the existence of a bounded, uniformly continuous solution of (1.1) yields readily the existence of an almost periodic solution w with the same set of Fourier exponents as f without any additional conditions This implies in particular (see Corollary 3.7) the existence of quasi periodic solutions if the forcing term f is quasi periodic Spectral Theory of Functions and Almost Periodicity 2.1 Notations In the remaining part of this paper we will use the following notations: C, and z stand for the set of complex numbers and the real part of a complex number z, respectively BC(R, Cn ) and BU C(R, Cn ) denote the space of Cn -valued bounded continuous functions and the space of Cn -valued bounded uniformly continuous functions on R, respectively 2.2 Spectrum of a function We denote by F the Fourier transform, i.e +∞ e−ist f (t)dt (Ff )(s) := (2.1) −∞ (s ∈ R, f ∈ L1 (R)) Then the Beurling spectrum of u ∈ BU C(R, Cn ) is defined to be the following set Sp(u) := {ξ ∈ R : ∀ > ∃f ∈ L1 (R), suppFf ⊂ (ξ − , ξ + ), f ∗ u = 0} where +∞ f ∗ u(s) := f (s − t)u(t)dt −∞ Theorem 2.1 Under the notation as above, Sp(u) coincides with the set consisting of ξ ∈ R such that the Fourier- Carleman transform of u u ˆ(λ) = ∞ −λt e u(t)dt, ( λ ∞ λt − e u(−t)dt, ( > 0); λ < 0) (2.2) has no holomorphic extension to any neighborhood of iξ Proof For the proof we refer the reader to [24, Proposition 0.5, p.22] We collect some main properties of the spectrum of a function, which we will need in the sequel, for the reader’s convenience Theorem 2.2 Let f, gn ∈ BU C(R, Cn ), n ∈ N such that gn → f as n → ∞ Then Sp(f ) is closed, Sp(f (· + h)) = Sp(f ) ∀h ∈ R, If α ∈ C\{0}, Sp(αf ) = Sp(f ), If Sp(gn ) ⊂ Λ for all n ∈ N then Sp(f ) ⊂ Λ, QUASI PERIODIC AND ALMOST PERIODIC SOLUTIONS 293 Sp(ψ ∗ f ) ⊂ Sp(f ) ∩ suppFψ, ∀ψ ∈ L1 (R) Proof For the proof we refer the reader to [15] or [24, pp 20-21] As an immediate consequence of Theorem 2.2 we have that if Λ is a closed subset of R, then the function space Λ(Cn ) := {f ∈ BU C(R, Cn ) : Sp(f ) ⊂ Λ} is a closed subspace of BU C(R, Cn ) Definition 2.3 Let g ∈ BU C(R, Cn ) Then the almost periodic spectrum of g is defined to be the set SpAP (g) := {ξ ∈ R : ∀ > 0, ∃φ ∈ L1 (R) s.t suppFφ ⊂ (ξ − , ξ + ), φ ∗ g ∈ / AP (Cn )} Obviously, for any g ∈ BU C(R, Cn ), we have SpAP (g) ⊂ Sp(g) 2.3 Almost periodic functions We recall that a subset E ⊂ R is said to be relatively dense if there exists a number l > (inclusion length) such that every interval [a, a + l] contains at least one point of E Let f be a continuous function on R taking values in Cn f is said to be almost periodic in the sense of Bohr if to every > there corresponds a relatively dense set T ( , f ) (of -periods ) such that sup f (t + τ ) − f (t) ≤ , ∀τ ∈ T ( , f ) t∈R For all reals λ such that the following integrals (Fourier coefficients) T →∞ 2T T f (t)e−iλt dt a(λ, f ) := lim −T exist As is known (see e.g [6]), there are at most countably reals λ (Fourier exponents) such that a(λ, f ) = 0, the set of which will be denoted by σb (f ) and called Bohr spectrum of f Throughout the paper we will use the relation Sp(f ) = σb (f ) and denote by AP (X) the space of all almost periodic functions taking valued in X, here X is either Rn or Cn with sup norm We summarize several main properties of almost periodic functions, whose proofs can be found in [6], in the following theorem: Theorem 2.4 The following assertions hold: f ∈ BC(R, X) is almost periodic if and only if for every sequence {τn }∞ n=1 ⊂ R the sequence of functions {fτn = f (τn + ·)}∞ n=1 contains at least a convergent subsequence; If f is X-valued almost periodic, then f ∈ BU C(R, X); AP (X) is a closed subspace of BU C(R, X); If f ∈ AP (X) and f exists as an element of BU C(R, X), then f ∈ AP (X) Example 2.5 The following examples illustrate close relations between spectrum of a function with its behavior: If u(t) = aeiλt , = a ∈ Cn , λ ∈ R, ∀t ∈ R, then Sp(u) = {λ} If u(·) is a bounded Cn -valued continuous function, then u is τ -periodic (for some τ > if and only if Sp(u) ⊂ 2πZ τ For u ∈ L1 (R) we have Sp(u) = suppFu The most important criterion for the almost periodicity of a bounded continuous function which we will use in this paper is the following: 294 NGUYEN MINH MAN AND NGUYEN VAN MINH Theorem 2.6 Let u ∈ BU C(R, Cn ) Then u is almost periodic provided that SpAP (u) is countable Proof For the proof (of more general results) we refer the reader to [15], [14, Chap 4] Next, we gather some further properties of the almost periodic spectrum SpAP (f ) by quoting here another approach to this kind of spectrum via the notion of quotient groups (see [1]) Consider the shift group (S(t))t∈R on BU C(R, X) with generator ¯ B Since (S(t))t∈R leaves AP (X) invariant there is an induced C0 -group (S(t)) t∈R on Y := BU C(R, X)/AP (X) given by ¯ S(t)π(u) = π(S(t)u), ∀t ∈ R, u ∈ BU C(R, X), where π : BU C(R, X) → BU C(R, X)/AP (X) denotes the quotient mapping Its ˜ Let u ∈ BU C(R, X) Define generator is denoted by B u ˜(λ) = ∞ −λt ¯ e S(t)π(u)dt, ( ∞ ¯ − eλt S(−t)π(u)dt, λ > 0) ( λ < 0) Then we have Proposition 2.7 SpAP (u) coincides with the set of ξ ∈ R such that u ˜(λ) has no holomorphic extension to any neighborhood of iξ from the right half plane Proof For the proof see [1, §2, §3] Main Results Consider now the equation Eq (1.1) Using the Riesz representation of bounded linear functionals we can re-write D and L in the form Dφ = dµ(θ)φ(θ), ∀φ ∈ C (3.1) dη(θ)φ(θ), ∀φ ∈ C, (3.2) −r Lφ = −r where µ(θ) and η(θ) are n × n-matrices whose entries are functions of θ of bounded variation In the remaining part of this paper we denote ∆(λ) := λDeλ· − Leλ· , for every λ ∈ C, where Deλ· := eλθ dµ(θ); Leλ· := −r eλθ dη(θ) −r ρ(D, L) := {λ ∈ C : ∃ ∆−1 (λ)} Hence, for every λ ∈ C, ∆(λ) is a matrix depending analytically on λ Throughout this paper we make the standing assumption that det ∆(λ) ≡ (3.3) Lemma 3.1 ([21]) Under the above notations, the following assertions hold true ρ(D, L) is an open subset of C and σ(D, L) := C\ρ(D, L) is discrete; ∆−1 (λ) is an analytic function on ρ(D, L) Below we let ∆i := {ξ ∈ R : ∆(iξ) = 0} Then we have Lemma 3.2 Let x be a bounded, uniformly continuous solution of Eq (1.1) Then SpAP (x) ⊂ ∆i QUASI PERIODIC AND ALMOST PERIODIC SOLUTIONS 295 Proof Taking Carleman transforms of both sides of (1.1) and applying Fubini Theorem, for λ > we have +∞ +∞ d Dxt e−λt dt = dt Lxt e−λt dt + fˆ(λ) Thus we have +∞ d [ dt +∞ dµ(θ)xt (θ)]e−λt dt = dη(θ)xt (θ)]e−λt dt + fˆ(λ) [ −r −r Consequently, +∞ [ −r +∞ = dµ(θ)xt (θ)]e−λt |+∞ +λ 0 dµ(θ)xt (θ)]e−λt dt [ −r 0 dη(θ)xt (θ)]e−λt dt + fˆ(λ) [ −r Therefore, − dµ(θ)x0 (θ) + λ −r +∞ xt (θ)e−λt dt dµ(θ)[ −r 0 +∞ = xt (θ)e−λt dt + fˆ(λ) dη(θ)[ −r Re-writing the above expression we have +∞ −Dx0 + λ −r +∞ = x(t + θ)e−λ(t+θ) eλθ dt] + fˆ(λ) dη(θ)[ −r 0 +∞ x(ξ)e−λξ dξ] + fˆ(λ) eλθ dη(θ)[ = −r θ +∞ −r θ x(ξ)eλξ dξ] + fˆ(λ) x(ξ)e−λξ dξ − eλθ dη(θ)[ = x(t + θ)e−λ(t+θ) eλθ dt] dµ(θ)[ 0 So, we have 0 eλθ dµ(θ).ˆ x(λ) − λ −Dx0 + λ −r θ x(ξ)e−λξ dξ] eλθ dµ(θ).[ −r θ x(ξ)e−λξ dξ] + fˆ(λ) = Leλ· x ˆ(λ) − Leλ· [ Therefore, [λDeλ· − Leλ· ]ˆ x(λ) = φ(λ) + fˆ(λ), where θ −r x(ξ)e−λξ dξ] − dµ(θ)eλθ [ φ(λ) = Dx0 + λ θ x(ξ)e−λξ dξ] dη(θ)eλθ [ −r Finally, ∆(λ)ˆ x(λ) = φ(λ) + fˆ(λ) (3.4) 296 NGUYEN MINH MAN AND NGUYEN VAN MINH Now for s ∈ R let us denote S(s)u by us By a similar computation we have ∆(λ)xs (λ) = f s (λ) + ϕs (λ), (3.5) where xs (ξ)e−λξ dξ] − dµ(θ)eλθ [ −r θ θ ϕs (λ) := Dxs0 + λ −r xs (ξ)e−λξ dξ] dη(θ)eλθ [ which is analytic in λ ∈ C Let λ0 ∈ ∆i Then det ∆(iλ0 ) = As shown in [21], ∆−1 (λ) is analytic in λ in a neighborhood of iλ0 of the complex plane Hence, xs (λ) = ∆−1 (λ)f s (λ) + ∆−1 (λ)φs (λ), for λ in a neighborhood of iλ0 Obviously, for ∞ xs (λ) e−λt x(ξ + s)dξ ∞ ∞ e−λt [S(ξ)x](s)dξ = = λ>0 ∞ e−λt xs (ξ)dξ = = (3.6) e−λt S(ξ)xdξ (s) = (R(λ, B)x)(s) Similarly, f s (λ) = (R(λ, B)f )(s), ∀ λ > Substituting these expresions into (3.6) we have R(λ, B)x = ∆−1 (λ)R(λ, B)f + ∆−1 (λ)ϕ(λ), ˜ where ϕ(λ) ˜ :R (3.7) s s → ϕ (λ) Hence, in a neighborhood of iλ0 , we have π(R(λ, B)x) = π(∆−1 (λ)R(λ, B)f ) + π(∆−1 (λ)ϕ(λ)) (3.8) Since f ∈ AP (X) we have π(∆−1 (λ)R(λ, B)f ) = ∈ Y ˜ Therefore, using the indentity x ˜(λ) = R(λ, B)π(x) for ˜ x ˜(λ) = R(λ, B)π(x) λ > 0, we have = π(R(λ, B)x) = ∆−1 (λ)π(ϕ(λ)) = ∆−1 (λ)ϕ(λ) ˜ λ > (3.9) On the other hand, ϕ(λ) ˜ is analytic in λ ∈ C Finally, around iλ0 the function x ˜(λ) has a holomorphic extension from the right half plane That is λ0 ∈ SpAP (x) This proves the lemma Corollary 3.3 Under the standing assumption, any bounded and uniformly continuous solution of Eq (1.1) is almost periodic Proof Since the zero set of ∆(λ) in this case consists of isolated points, it is countable So is ∆i By the above lemma, the almost periodic spectrum of any bounded and uniformly continuous solution should be countable, so the solution is almost periodic Remark 3.4 The almost periodicity of bounded solutions of ordinary differential equations has been proved in [6] For abstract ordinary differential equations in Banach spaces this result has been considered in [14, Chap 6] with additional conditions (see also [1] for a more complete proof) QUASI PERIODIC AND ALMOST PERIODIC SOLUTIONS 297 Lemma 3.5 Let x be a bounded and uniformly continuous solution of Eq (1.1) Then σb (f ) ⊂ σb (x) ⊂ ∆i ∪ σb (f ) (3.10) Proof By Corollary 3.3, this solution x is almost periodic Thus, the Bohr transforms of both sides of Eq (1.1) exist Therefore, T T →∞ 2T −T T T →∞ 2T d dt e−iλt lim dµ(ξ)x(t + ξ) dt −r e−iλt = lim dη(ξ)x(t + ξ) dt + a(λ, f ) −r −T Therefore, integrating the above left side by parts we have −iλt e T →∞ 2T dµ(ξ)x(t + ξ)|T−T lim − lim T →∞ 2T T T →∞ 2T −r T (−iλ)e−iλt dµ(ξ)x(t + ξ)dt −T −r dη(ξ)x(t + ξ)e−iλt dt + a(λ, f ) = lim −T −r The first term of the left side vanishes as x and e−iλt are bounded Hence, iλ T T →∞ 2T e−iλt x(t + ξ)dt dµ(ξ) lim −r −T T = dη(ξ) lim T →∞ 2T −r e−iλt x(t + ξ)dt + a(λ, f ) −T Thus, iλ T →∞ 2T T e−iλθ x(θ)dθ dµ(ξ)[ lim −r T →∞ 2T T +ξ T →∞ 2T T T →∞ 2T T →∞ 2T T +ξ x(θ)e−iλθ ]eiλξ −T T x(θ)eiλθ dθ dη(ξ)[ lim −r −T +ξ x(θ)e−iλθ − lim + lim = −T −T T →∞ 2T −T +ξ e−iλθ dθ − lim + lim T x(θ)e−iλθ ]eiλξ −T Consequently, 0 dµ(ξ)eiλξ = a(λ, x) iλa(λ, x) −r dη(ξ)eiλξ + a(λ, f ) −r Hence, iλDeiλ· − Leiλ· a(λ, x) = a(λ, f ) If λ ∈ R, λ ∈ ∆i and λ ∈ σb (f ), then a(λ, x) = 0, i.e., λ ∈ σb (x) This shows that σb (x) ⊂ ∆i ∪ σb (f ) Next, if λ ∈ R and λ ∈ σb (x), then, by definition, a(λ, x) = Thus, a(λ, f ) = By definition, λ ∈ σb (f ), i.e., σb (f ) ⊂ σb (x) We are now in a position to prove the main result of this note 298 NGUYEN MINH MAN AND NGUYEN VAN MINH Theorem 3.6 Let ∆i be bounded Moreover, assume that Eq (1.1) has a bounded, uniformly continuous solution Then, there exists an almost periodic solution w to Eq (1.1) such that σb (w) = σb (f ) (3.11) Proof If ∆i is bounded, since it is part of the zero set of det ∆(λ) = it consists of finitely many points By the Corollary 3.3 the bounded solution, denoted by x, of ∞ Eq (1.1) is almost periodic Suppose that x ∼ k=0 eiλk t xk is the Fourier series of x Set P (t) = eiλk t xk λk ∈∆i ∩σb (x) Obviously, P (t) is a trigonometric polynomial Moreover, it is a solution to the homogeneous equation In fact, it suffices to show that every term P k (t) := eiλk t xk is a solution of the corresponding homogeneous equation By definition, ∆(iλk ) = Hence, d DP k (t) − LP k (t) dt = ∆(iλk )xk = Define w(t) = x(t) − P (t) Obviously, w is almost periodic Moreover, σb (w) = σb (x)\∆i Hence, σb (w) ⊂ σb (f ) It remains to show that w is an almost periodic solution to Eq (1.1) But this is obvious as P (t) is a solution to the corresponding homogeneous equation Combined with the inclusion σb (f ) ⊂ σb (w) of the Lemma 3.5, this proves the theorem A set of reals S is said to have an integer and finite basis if there is a finite subset T ⊂ S such that any element s ∈ S can be represented in the form s = n1 b1 + · · · + nm bm , where nj ∈ Z, j = 1, · · · , m, bj ∈ T, j = 1, · · · , m An almost periodic function f is said to be quasi periodic if it is of the form f (t) = F (t, t, , t), t ∈ R, where F (t1 , t2 , , ) is a Cn -valued continuous function of p variables which is periodic in each variable The function f is quasi periodic if and only if the set of its Fourier-Bohr exponents has an integer and finite basis (see [14, p.48]) Corollary 3.7 Let all assumptions of the above theorem be satisfied Moreover, let f be quasi-periodic Then if Eq (1.1) has a bounded, uniformly continuous solution, then it has a quasi-periodic solution w such that σb (w) ⊂ σb (f ) Proof This corollary is an immediate consequence of Theorem 3.6 Example 3.8 Consider the equation x(t) ˙ − x(t ˙ − 2) = Bx(t − 1) + f (t), x(t) ∈ Cn , (3.12) where B is a n × n-matrix and f is an almost periodic function ∆i is the set of λ ∈ R such that the matrix ∆(iλ) = (iλ − iλe−2iλ )I − e−iλ B is not invertible QUASI PERIODIC AND ALMOST PERIODIC SOLUTIONS 299 It is easy to see that if |λ| > B , then 1 |iλ − iλe−2iλ | = |λ||1 − e−2iλ | > B × = B 3 Therefore, the operator ∆(iλ) is invertible for large λ ∈ R Hence, ∆i is bounded Applying the above theorem we can conclude that if (3.12) has a bounded uniformly continuous solution, then it has an almost periodic solution w such that σb (w) = σb (f ) Example 3.9 Consider scalar equations of the form N M Ak x(t ˙ − τk ) = x(t) ˙ + Bj x(t − µj ) + f (t), x(t) ∈ R (3.13) j=1 k=1 where N, M ∈ N, Ak , Bj are reals, τk , µj are positive reals Then, the characteristic operator is of the form N M e−µj λ Bj e−τk λ Ak − ∆(λ) = λ + λ j=1 k=1 We now show that the function 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