A NOTE ON ALMOST PERIODIC VARIATIONAL EQUATIONS

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1 A NOTE ON ALMOST PERIODIC VARIATIONAL EQUATIONS PETER GIESL AND MARTIN RASMUSSEN Abstract. The variational equation of a nonautonomous differential equation ẋ = F t, x) along a solution µ is given by ẋ = D xf t, µt))x. We consider the question whether the variational equation is almost periodic provided that the original equation is almost periodic by a discussion of the following problem: Is the derivative D xf almost periodic whenever F is almost periodic? We give a negative answer in this paper, and the counterexample relies on an explicit construction of a scalar almost periodic function whose derivative is not almost periodic. Moreover, we provide a necessary and sufficient condition for the derivative D xf to be almost periodic. In addition, we also discuss this problem in the discrete case. We consider the variational equation x n+ = D xf n, µ n) of the almost periodic difference equation x n+ = F n, x n) along a solution µ n.. Introduction. When studying the dynamical behavior of a nonautonomous differential equation ẋ = F t, x) in the vicinity of a solution µ, one usually first analyzes the linearization along this solution, which is given by the variational equation ẋ = D x F t, µt))x. It is not clear a priori if certain structures of the right-hand side carry over to the variational equation. For instance, if the righthand side F is periodic with respect to t, then obviously D x F is periodic, and thus, the variational equation along a periodic solution is also periodic. Does the same hold for almost periodic differential equations? This question arose in [3], where almost periodicity of the variational equation was an extra assumption cf. [3, Theorem 4., 4.4]). Before we discuss the case of almost periodic differential equations ẋ = F t, x), let us consider scalar functions f : R R. The derivative of a periodic function f is obviously periodic, but is the derivative of an almost periodic function f also almost periodic? One of the first theorems in many books on almost periodic functions states that the derivative is almost periodic if and only if it is uniformly continuous see, e.g., [, Theorem.6] or [, Theorem.3,.8]). The theorem suggests that the derivative of an almost periodic function is not almost periodic in general, but to our best knowledge, a counterexample has not been provided in the literature yet. We will present the explicit construction of such a counterexample in Section.. This counterexample is then used in Section. to construct the right-hand side of an almost periodic differential equation ẋ = F t, x) such that D x F is not almost periodic. In Section.3, we provide a necessary and sufficient condition for the derivative D x F to be almost periodic which is similar to the scalar case. Namely, Date: July, 9. Mathematics Subject Classification. primary 34C7; secondary 6A4,4A75,6B5. Key words and phrases. Almost periodic differential equation, almost periodic difference equation, almost periodic function, variational equation.

2 PETER GIESL AND MARTIN RASMUSSEN D x F is almost periodic if and only if it is uniformly continuous on sets of the form R K, where K R N is compact. Moreover, we show that the variational equation ẋ = D x F t, µt))x along an almost periodic solution µ is almost periodic in this case. The discrete case of almost periodic sequences is then treated in the last part of this paper, and we prove that the counterexample of Section. can be used to construct an example of a function F : Z R N R N such that D x F is not discrete almost periodic. The discrete case turns out to be more complicated, since the application of the continuous function requires the treatment of a mixture of continuous and discrete variables. Notation. We denote by R N N the set of all real N N matrices. The Euclidean space R N is equipped with the Euclidean norm, which is induced by the scalar product,, where x, y := N i= x iy i. The N )-sphere of the R N is defined by S N := { x R N : x = }.. The continuous case. Throughout this section, we distinguish between almost periodic functions f : R R N and functions F : R R N R N which are almost periodic uniformly in the second argument. The definitions are given as follows. A function f : R R N is called Bohr) almost periodic if the set T f, ε) := { τ R : ft) fτ + t) < ε for all t R } is relatively dense in R for all ε >. Note that a set L R is relatively dense if and only there exists a T > such that [t, t+t ] L for all t R. An almost periodic function is necessarily uniformly continuous on R see, e.g., [, Theorem.3] or [, Corollary.5]). Next we consider almost periodic functions depending on a parameter x R N. Our studies are motivated by almost periodic differential equations ẋ = F t, x). We call a function F : R R N R N Bohr) almost periodic uniformly in x if for all compact sets K R N and ε >, the set T F, ε, K) := { τ R : F t, x) F τ + t, x) < ε for all t R and x K } is relatively dense in R. A function F which is almost periodic uniformly in x is necessarily uniformly continuous on sets of the form R K, where K R N is compact see, e.g., [4, Lemma 3]). If F is a C -function, the derivative of F with respect to x R N will be denoted by D x F : R R N R N N... A first counterexample. This subsection is devoted to the explicit construction of a continuously differentiable almost periodic function whose derivative is not almost periodic. The main idea is to construct a function g, which is not almost periodic, whereas its integral ft) := t gs) ds is almost periodic. g is constructed as a series of functions i N g it) consisting of peaks, cf. Figure, which integrate to zero. The functions g i t) are periodic functions with these peaks, where for increasing i, the peaks become thinner and the periods larger. In the following, we will describe the construction in detail.

3 A NOTE ON ALMOST PERIODIC VARIATIONAL EQUATIONS 3 Figure. The function h. Figure. The functions g, g, and g 3. The peak moves to the right and is contracted in t-direction. Definition of h. First of all, we define a continuous auxiliary function h : [, ] [, ] see also Figure ) by 4t : t [, 4], ht) := 4t : t [ 4, 4] 3, 4t 4 : t [ 3 4, ]. Note that ht) dt =. Definition of g i. Moreover, we define the continuous functions g i : R [, ], i N, by h i t + i) [ : t i, ) ) ] i, g i t) := [ : t / i, ) ) ] i. g i is a contracted version of h in t-direction, cf. Figure for i =,, 3. Note that the support of g i is a subset of [, ]. For later use, we show a property of the functions g i, i N, in the following. By the change of variables s = i t + i, we get g i t) dt = )i g i t) dt = i i θ+ i hs) ds

4 4 PETER GIESL AND MARTIN RASMUSSEN for all θ Figure 3. The function g: The wider peaks of g occur after, 6, and 4, the medium peaks of g occur after 4 and, and the thin peak of g 3 occurs after 8. [ max s R s ht) dt = 4 i, ), we obtain max θ R ) i ] ; for all other θ, the integral is zero. Hence, using Definition of g i. We now define g i : R [, ] by g i t) dt = i+)..) g i t) := k Z g i t + k ) i ) for all t R. This function is periodic with period i+. The t-values with g i t) belong to the set { t R : i t and i+ t }. Note that N+ g N i t) dt = for all N Z. Using.), the function g i t) satisfies max g θ R i t) dt = i+)..) Definition of g. Finally, we define g : R [, ] by gt) := i N g i t) for all t R.3) see Figure 3). Since the intersection of the support g i and the support of g j is empty for i j, at most one value g i t), i N, is non-zero for each t, and thus, the convergence of the sum in.3) follows. Note that N+ gt) dt = for all N Z. N The next theorem states that the integral of g is almost periodic with a derivative which is not almost periodic. Theorem.. Define the integral of g, ft) := t gs) ds for all t R, with g : R [, ] as defined above. Then the following statements hold:

5 A NOTE ON ALMOST PERIODIC VARIATIONAL EQUATIONS 5 i) f is continuously differentiable, ii) f = g is not almost periodic, iii) f is almost periodic. Proof. Since g is continuous, its integral f is continuously differentiable, and assertion i) follows immediately. However, f = g is not uniformly continuous, since the peaks of g i become thinner for i, and due to [, Theorem.3], this means that g is not almost periodic. It remains to show assertion iii), the almost periodicity of f. Thereto, note that ft) = i N f it), where f i t) = t g is) ds. We will show that the series i N f it) converges uniformly. Indeed, t f i t) = g i s) ds i+) by.). Moreover, the functions f i t) are periodic, since the functions g i t) are periodic and the integral over each peak is zero. Since the limit of a uniformly convergent sequence of almost periodic functions is an almost periodic function, cf., e.g., [, Theorem.6], the function f is almost periodic. This finishes the proof of this theorem... A second counterexample. In this subsection, we use the example from Section. to construct an example of a function F : R R N R N which is almost periodic uniformly in the second argument, but the derivative D x F does not fulfill this property. Theorem.. We define the C -function F : R R N R N by F j t, x) := f t N i= x ) i for all j {,..., N}, where f is the function of Theorem.. Then the following statements are fulfilled: i) F is almost periodic uniformly in x, ii) D x F is not almost periodic uniformly in x. Proof. To show that F is almost periodic uniformly in x, we first note that if τ R fulfills ft) fτ + t) < ε for all t R, then τ also fulfills f t N i= x i) f τ + t N i= x i) < ε for all t R and x K, where K is a compact set. This becomes clear by making the transformation t t N i= x i. Hence, for a compact set K R N and ε >, we have { } τ R : ft) fτ + t) < ε/ N for all t R { τ R : F t, x) F τ + t, x) < ε for all t R and x K }. Since the first set is relatively dense by Theorem., also the latter one is relatively dense, and this implies i). Before proving ii), note that we have D x F t, x) = g t N i= x i) MI for all t, x) R R N, where M I = ) i,j=,...,n R N N and g is the function from Section..

6 6 PETER GIESL AND MARTIN RASMUSSEN Let us now assume that D x F is almost periodic uniformly in x. Then each entry of the matrix-valued function D x F is almost periodic uniformly in x, and that means in particular for the compact set K = {} and any ε >, that the set T g, ε) = { τ R : gt) gt + τ) < ε for all t R } is relatively dense. This implies that g is almost periodic, which is a contradiction to Theorem. and finishes the proof..3. A necessary and sufficient condition for almost periodicity. This subsection is devoted to a necessary and sufficient condition for the almost periodicity of the derivative D x F of an almost periodic function F : R R N R N. More precisely, we prove that D x F t, x) is almost periodic uniformly in x if and only if it is uniformly continuous on sets of the form R K, where K R N is compact. We also show that the variational equation ẋ = D x F t, µt))x to an almost periodic solution µ is almost periodic in this case. Theorem.3. Let the C -function F : R R N R N be almost periodic uniformly in the second argument. We suppose that D x F : R R N R N N is uniformly continuous on sets of the form R K, where K R N is compact. Then the function D x F is also almost periodic uniformly in the second argument. Proof. We fix i {,..., N} and note that it is sufficient to show that the i-th row of the matrix D x F, denoted by grad F i, is an almost periodic function. For n N, we define the function ϕ i n : R R N S N R by ϕ i nt, x, ξ) := n F i t, x + ξ/n) F i t, x) ). Due to the mean value theorem, we obtain the representation ϕ i nt, x, ξ) = grad F i t, x + θn t, x, ξ)ξ/n ), ξ where the function θ n : R R N S N R fulfills θ n t, x, ξ). Now choose a compact set K R N and ε >. The uniform continuity of D x F implies that there exists an N > with grad F i t, x + θn t, x, ξ)ξ/n ) grad F i t, x) < ε 4 for all t R, x K, ξ S N and n N note that θ n t, x, ξ) ). Due to the Cauchy-Schwarz inequality, we obtain ϕ i n t, x, ξ) grad F i t, x), ξ < ε 4 for all t R, x K, ξ S N and n N. We define the compact set and let τ T F i, 4N, K ), where T F i, ε 4N, K ) = ε K := { x + ξ/n : x K, ξ }, { τ R : F i t, x) F i t + τ, x) <.4) ε 4N for t R and x K }.

7 A NOTE ON ALMOST PERIODIC VARIATIONAL EQUATIONS 7 Note that the set T ε F i, 4N, K ) is relatively dense, since F i is almost periodic. Then we have for all x K and ξ S N ϕ i N t, x, ξ) ϕ i N t + τ, x, ξ) ) N Fi t, x + ξ N Fi t + τ, x + ξ ) N + Fi t, x) F i t + τ, x) ) < ε.5) Hence, we obtain for all x K that grad F i t, x) grad F i t + τ, x) = sup ξ S N grad F i t, x) grad F i t + τ, x), ξ.4),.5) < sup grad F i t, x), ξ ϕ i N t, x, ξ) ξ S N + sup ξ S N ϕ i N t, x, ξ) ϕ i N t + τ, x, ξ) + sup ξ S N ϕ i N t + τ, x, ξ) grad F i t + τ, x), ξ ε 4 + ε + ε 4 = ε, and this implies that T ε F i, 4N, K ) T grad F i, ε, K ). Hence, the set on the right-hand side is relatively dense. Thus, grad F i is almost periodic, and this also means that D x F is almost periodic, since i has been chosen arbitrarily. This theorem, together with [4, Lemma 3], implies the following sufficient and necessary condition for the almost periodicity of D x F. Corollary.. Let the C -function F : R R N R N be almost periodic uniformly in x. Then D x F : R R N R N N is uniformly continuous on sets of the form R K, where K R N is compact, if and only if the function D x F is almost periodic uniformly in x. Finally, we apply the results to variational equations ẋ = D x F t, µt))x, where the solution µ is supposed to be almost periodic. Corollary.. Let the C -function F : R R N R N be almost periodic uniformly in x and D x F : R R N R N N be uniformly continuous on sets of the form R K, where K R N is compact. Then, given an almost periodic solution µ : R R N of the differential equation ẋ = F t, x), the variational equation along the solution µ, given by is almost periodic uniformly in x). ẋ = D x F t, µt))x, Proof. This follows from Theorem.3 and [, Theorem.].

8 8 PETER GIESL AND MARTIN RASMUSSEN 3. The discrete case. Similarly to the continuous case, we distinguish between almost periodic functions f : Z R N and functions F : Z R N R N which are almost periodic uniformly in the second argument. The definitions are given as follows. A function f : Z R N is called Bohr) discrete almost periodic if the set T Z f, ε) := { κ Z : fk) fκ + k) < ε for all k Z } is relatively dense for all ε >. Next we consider almost periodic functions depending on a parameter x R N. Our studies are motivated by almost periodic difference equations x n+ = F n, x n ). We call a function F : Z R N R N Bohr) discrete almost periodic uniformly in x if for all compact sets K R N and ε >, the set T Z F, ε, K) := { κ Z : F k, x) F κ + k, x) < ε for all k Z and x K } is relatively dense in R. A function F which is almost periodic uniformly in x is necessarily uniformly continuous on sets of the form Z K, where K R N is compact. To define a function F which is discrete almost periodic uniformly in x such that D x F is not discrete almost periodic uniformly in x, we need some preparations. Proposition 3.. We define ḡ : Z R by ḡk) := gπk) for all k Z, where g : R R is the function from Section. Then ḡ is not discrete almost periodic. Proof. We show that for ε = 4, we have T Zḡ, ε) = {}, which implies the assertion. Let κ. Denote κπ mod =: K, ). We can choose j N such that both i) K > j and ii) j+ > K; this is obviously possible in the limit j+ j. For some fixed j, we can now choose the minimal i N such that i > j + + κπ mod. 3.) j+ We show that i < j. Indeed, for i = j, we can satisfy 3.) since we have with ii) > K + j j+ > ; the last inequality holds due to i). Hence, j K + j+ = j + j + κπ ) mod, and 3.) is satisfied. Note that j+ 3.) means that j + j+ + κπ mod supp g i. 3.) We distinguish two cases. Case. g i + j + κπ mod ) < j+. We set x := + j. Since the set {kπ j mod j+ : k Z} is j+ dense in [, j+ ], there exists a sequence {k n } n N in Z such that lim n k n π j mod j+ = x. Due to g i x+κπ mod ) < and g jx) =, there exists an n > such that for all n n, we have because of 3.) and g i kn + κ)π mod ) 5 8, k n + κ)π mod supp g i g j k n π + j mod j+ ) 7 8

9 A NOTE ON ALMOST PERIODIC VARIATIONAL EQUATIONS 9 It follows that and ḡ k n + κ)π ) 5 8 ḡk n π) 7 8 for all n n, which means that ḡkn + κ)π) ḡ k n π ) 4 = ε for all n n. This implies that κ / T Z ḡ, ε). Case. g i j + j+ + κπ mod ). Note that because of i < j, either g i + j + κπ + j+ mod ) > j+ or g i + j + κπ j+ mod ) > j+ holds. We concentrate on the first case, so we assume that g i + j + κπ + j+ mod ) > j+ the second case can be treated analogously). We set x := + j + j+ = j+ + j. Since j+ the set {kπ i mod i+ : k Z} is dense in [, i+ ], there exists a sequence {k n } n N in Z such that lim n k n π i mod i+ = x + κπ mod. Due to g i x + κπ mod ) and g jx) =, there exists an n > such that for all n n, we have and It follows that and g i kn + κ)π + i mod i+) 3 8 g j k n π mod ) 8, k nπ mod supp g j. ḡ k n + κ)π ) 3 8 ḡk n π) 8 for all n n, which means that ḡk n + κ)π) ḡ k n π ) 4 = ε for all n n. This implies that κ / T Z ḡ, ε) and finishes the proof of this proposition. Lemma 3.. For any j N and any < τ < j, the set { k Z : kπ mod j [, τ] } is relatively dense. Proof. The set { kπ mod j : k Z } is dense in [, j ). Hence, the set {k Z : kπ mod j [, τ]} is unbounded. It remains to show that the distance between two of these k is bounded. There exists a minimal number N N such that N π mod j [, τ/], and similarly, there is a minimal N N such that N π mod j [ j τ/, j ). This proof is finished by the fact that the maximal distance between two k of the above set { k Z : kπ mod j [, τ] } is N = max { N, N }. Indeed, let k Z such that kπ mod j [, τ]. If kπ mod j [, τ/], then k + N )π [, τ], and if kπ mod j [τ/, τ], then k + N )π [, τ). Proposition 3.3. For any ε >, the set { k Z : fπt) fπt + k)) < ε for all t R } is relatively dense, where f : R R is the function from Theorem..

10 PETER GIESL AND MARTIN RASMUSSEN Proof. Let ε >, and choose j N such that ε > j+). Choose τ > such that τ < minε, ). By Lemma 3., the set { k Z : kπ mod j+ [, τ] } is relatively dense. We show that these k satisfy fπt) fπt + k)) < ε for all t R, which then proves this lemma. Thereto, let t R, and choose k Z such that kπ mod j+ [, τ]. Hence, there is an l Z such that kπ j+ l =: v with v τ <. We write tπ = t + θ, where t := tπ Z and θ [, ). Note that θ + v <. We obtain with kπ = j+ l + v fkπ + tπ) ftπ) = = j+ l+v+t +θ t +θ t+ t +θ gs) ds j+ l+t gs) ds + t+θ = gs) ds + t gs) ds t } + {{ } = j+ l+t +θ+v j+ l+t gs) ds, j+ l+t +θ+v + gs) ds j+ l+t since N+ gs) ds = for all N Z. N We now choose i N such that i t and i+ t and consider the two cases i j and i > j. Case. i j. In this case, we have i t + j+ l) and i+ t + j+ l). Indeed, in contradiction to the second statement, assume that i+ t + j+ l). Then, since i+ j+ l, this implies i+ t, which is a contradiction. Hence, there are m, m Z which allow the representations t = m ) i and t + j+ l = m ) i, and we obtain j+ l+t +θ+v j+ l+t gs) ds = = j+ l+t +θ t+θ t j+ l+t gs) ds + g i s) ds + gs) ds j+ l+t +θ+v j+ l+t +θ j+ l+t +θ+v j+ l+t +θ Since gs), we have fkπ + tπ) ftπ) gs) ds j+ l+t +θ+v j+ l+t +θ gs) ds t+θ t g i s) ds. gs) ds gs) ds v τ < ε. Case. i > j. We choose p N with p t + j+ l) and p+ t + j+ l). We have i j +, thus j+ i t and hence j+ t + j+ l), and this implies p j +. We show that j+ l+t +θ+v j+ l+t gs) ds = +v g p s) ds. This is clear for θ + v <. For θ + v <, the number j+ l + t + θ + v is odd, and we thus have j+ l+t +θ+v j+ l+t gs) ds = = +v g p s) ds.

11 A NOTE ON ALMOST PERIODIC VARIATIONAL EQUATIONS We arrive at j+ l+t +θ+v gs) ds j+ l+t t { +v max g p s) ds, t+θ gs) ds = } g i s) ds. +v g p s) ds g i s) ds By.), we have g is) ds i for all θ R. This implies j+ l+t +θ+v t+θ gs) ds gs) ds j+ l+t max { i, p} j+) < ε, t since i, p j +. Thus, we have fkπ + tπ) ftπ) < ε. This finishes the proof of this proposition. The following theorem is the discrete analogue to Theorem.. Theorem 3.4. We define the function F : Z R N R N by F j k, x) = f π k N i= x i) ) for all j {,..., N}, where f is the function of Theorem.. Then the following statements are fulfilled: i) F is discrete almost periodic uniformly in x, ii) D x F is not discrete almost periodic uniformly in x. Proof. To show that F is discrete almost periodic uniformly in x, we first note that if κ Z fulfills fπt) fπκ + t)) < ε for all t R, then κ also fulfills f π t N i= x i) ) f π κ + t N i= x i) ) < ε for all t R and x K, where K is a compact set. This becomes clear by making the transformation t t N i= x i. Hence, for a compact set K R N and ε >, we have { } κ Z : fπt) fπκ + t)) < ε/ N for all t R { κ Z : F t, x) F κ + t, x) < ε for all t R and x K }. Since the first set is relatively dense by Proposition 3.3, also the latter one is relatively dense, and this implies i). Before proving ii), note that we have D x F t, x) = πg π t N i= x i) ) M I for all t, x) R R N, where M I = ) i,j=,...,n R N N and g is the function from Section.. Let us now assume that D x F is discrete almost periodic uniformly in x. Then each entry of the matrix-valued function D x F is discrete almost periodic uniformly in x, and that means in particular for the compact set K = {} and any ε >, that the set T Z g, ε) = { κ Z : gπt) gπt + κ)) < ε for all t R } is relatively dense. This implies that ḡt) = gπt) is almost periodic, which is a contradiction to Proposition 3. and finishes the proof.

12 PETER GIESL AND MARTIN RASMUSSEN REFERENCES [] C. Corduneanu, Almost Periodic Functions, Interscience Tracts in Pure and Applied Mathematics, no., Interscience Publishers, New York, 968. [] A. M. Fink, Almost Periodic Differential Equations, Springer Lecture Notes in Mathematics, vol. 377, Springer, Berlin, Heidelberg, 974. [3] P. Giesl and M. Rasmussen, Borg s criterion for almost periodic differential equations, Nonlinear Analysis. Theory, Methods & Applications 69 8), no., [4] G. R. Sell, Nonautonomous differential equations and dynamical systems I. The basic theory, Transactions of the American Mathematical Society 7 967), 4 6. Peter Giesl, Department of Mathematics, Mantell Building, University of Sussex, Falmer, BN 9RF, UK address: p.a.giesl@sussex.ac.uk Martin Rasmussen, Department of Mathematics, Imperial College, London SW7 AZ, United Kingdom address: m.rasmussen@imperial.ac.uk Research supported by the Bayerisches Eliteförderungsgesetz of the State of Bavaria, Germany, and by a Marie Curie Intra European Fellowship of the European Community Grant Agree Number: 638).