LASALLE TYPE STATIONARY OSCILLATION THEOREMS for AFFINE-PERIODIC SYSTEMS Hongren Wang Xue Yang Yong Li Xiaoyue Li

Home , Wang (surname)

DISCRETE AND CONTINUOUS doi:10.3934/dcdsb.2017156 DYNAMICAL SYSTEMS SERIES B Volume 22, Number 7, September 2017 pp. 2907–2921

LASALLE TYPE STATIONARY OSCILLATION THEOREMS FOR AFFINE-PERIODIC SYSTEMS

Hongren Wang

College of mathematics, Jilin Normal University Jilin 136000, China College of Mathematics, Jilin University Changchun 130012, China

Xue Yang

School of Mathematics and Statistics, Northeast Normal University Changchun 130024, China Center for Mathematics and Interdisciplinary Sciences, Northeast Normal University Changchun 130024, China College of Mathematics, Jilin University Changchun 130012, China Institute of Mathematics, Academy of Mathematics and Systems Science Chinese Academy of Sciences, Beijing 100190, China

Yong Li∗

Xiaoyue Li

School of Mathematics and Statistics, Northeast Normal University Changchun 130024, China

(Communicated by Yuan Lou)

Abstract. The paper concerns the existence of affine-periodic solutions for affine-periodic (functional) differential systems, which is a new type of quasi- periodic solutions if they are bounded. Some more general criteria than LaSalle’s one on the existence of periodic solutions are established. Some applications are also given.

2010 Mathematics Subject Classiﬁcation. Primary: 34C27; Secondary: 34C25. Key words and phrases. LaSalle type stationary oscillation principle, Q-aﬃne T -periodic solutions, asymptotic stability. The second author is supported by NSFC (grant No. 11201173). The third author is supported by National Basic Research Program of China (grant No. 2013CB834100), NSFC (grant No. 11571065) and NSFC (grant No. 11171132). ∗ Corresponding author: Yong Li.

2907 2908 HONGREN WANG, XUE YANG, YONG LI AND XIAOYUE LI

1. Introduction. As is well known, regular solutions of differential equations are generally split into the following types: stationary, periodic, quasi-periodic, almost periodic, almost automorphic or recurrent ones. It is an important topic what conditions, for example, stability, can guarantee the existence of such solutions. Some classical results of this aspect can be found in books [27] and [1]. Here we would like to mention LaSalle’s stationary oscillation theorem [13], which is a simple but flexible criterion in applications. More precisely, consider the equation d x˙ = f(t, x)(· ≡ ), (1) dt where f : R1 ×Rn → Rn is continuous, T -periodic in t, and satisfies some regularity conditions.

1 1 Theorem 1.1. (LaSalle stationary oscillation principle, [13]) Let a : R+ → R+ \{0} be continuous and satisfy lim a(t) = 0. Assume f(t, x) is Lipschitz in x, T -periodic t→∞ 1 in t, and all solutions are bounded on R+; moreover

|x(t, x0) − x(t, y0)| ≤ a(t)|x0 − y0| ∀t ≥ 0, where x(t, x0) denotes the solution of system (1) with the initial value x(0) = x0. Then system (1) has a unique asymptotically stable T -periodic solution. Some systems exhibit the following affine periodicity: −1 1 n f(t + T, x) = Qf(t, Q x) ∀t ∈ R , x ∈ R , where Q is a nonsingular n × n matrix. When Q = I (the identity matrix), system (1) is just a usual T -periodic one; when Q is an orthogonal matrix, system (1) is a quasi-periodic one. From physics’ views it is natural to ask whether system (1) admits a solution x(t) with the following property x(t + T ) = Qx(t) ∀t, where f(t, x) possesses the affine periodicity. We call such a solution x(t) a Q-affine T -periodic solution of (1). As mentioned above, a Q-affine T -periodic solution x(t) might be T -periodic if Q = I, quasi- periodic if Q is orthogonal, even unbounded like eλty(t) with quasi-periodic y(t) if Q is only nonsingular. For periodic solutions and anti-periodic solutions, some works can be found in [2, 4, 5, 7, 8, 10, 11, 12, 15, 16, 17, 18, 19, 21, 22, 25, 26]. Recently, there have been some literature in studying the existence of affine-periodic solutions, see, for example, [3, 14, 20, 23, 24, 28]. In this paper, we will prove similar stationary oscillation theorems for Q-affine periodic systems with orthogonal matrix Q, make some discusses about more general stationary oscillation conditions, and give more flexible versions. The paper is organized as follows. In Section 2, we state a stationary oscillation theorem on Q-affine T periodic solutions. Section 3 provides an analogous result for functional differential equations. In Section 4, we illustrate some applications. 2. Stationary oscillation theorem for ODE (1). From now on we always assume Q is an orthogonal matrix. In this section, we will prove the following LaSalle type theorem on Q-affine T periodic solutions for system (1).

Theorem 2.1. Consider the Q-aﬃne T -periodic system (1), and assume f : R1 × n n 1 1 R → R is continuous. Let a : R+ → R+ \{0} be continuous and satisfy r = LASALLE TYPE STATIONARY OSCILLATION THEOREMS 2909

lim a(kT ) < 1. Assume the following hold: k→∞ 1) There exists a solution z(t) of system (1) defined on [0,T ]; 2) Any two solutions x(t) and y(t) satisfy |x(t) − y(t)| ≤ a(t)|x(0) − y(0)| whenever they exist. Then system (1) has a unique asymptotically stable Q-affine T periodic solution. Lemma 2.2. Assume the Q-affine T -periodic system (1) admits the uniqueness to initial value problems. Let x(t, x0) be a solution of (1) with the initial value 1 condition x(0) = x0 defined on R+. Then −1 −1 Q x(t + T, x0) ≡ x(t, Q x(T, x0)).

1 Proof. Note that for t ∈ R+, Z t+T x(t + T, x0) = x0 + f(s, x(s, x0))ds 0 Z T Z t+T = x0 + f(s, x(s, x0))ds + f(s, x(s, x0))ds 0 T Z t = x(T, x0) + f(s + T, x(s + T, x0))ds 0 Z t −1 = x(T, x0) + Q f(s, Q x(s + T, x0))ds, 0 which together with the uniqueness of solutions to the initial value problem implies that

−1 −1 Q x(t + T, x0) ≡ x(t, Q x(T, x0)).

We now give the proof of Theorem 2.1.

Proof. We take the following three steps: 1 Step 1. Prove that z(t) is extensible on R+. Once done, by 2), it follows that

|x(t, x0) − z(t)| ≤ a(t)|x0 − z(0)| ∀t ≥ 0. 1 Hence all the solutions x(t, x0) of system (1) are deﬁned on R+. By 2), x(t, x0) is uniquely determined by the initial value x0, i.e., the equation admits the continuous dependence of solutions with respect to initial values. By Peano’s theorem and 2), (1) has a unique solution w(t) with the initial value condition w(0) = Q−1z(T ), and it exists on [0,T ]. Note that dQw(t − T ) dQw(t − T ) = dt d(t − T ) = Qf(t − T,Q−1Qw(t − T )) = f(t, Qw(t − T )) ∀t ∈ [T, 2T ]. Therefore, Qw(t − T ) is the solution of (1) with the initial value condition z(T ) on [T, 2T ]. 2910 HONGREN WANG, XUE YANG, YONG LI AND XIAOYUE LI

Define   z(t), t ∈ [0,T ], z(t) =  Qw(t − T ), t ∈ [T, 2T ]. Hence z(t) exists on [T, 2T ]. Repeating this procedure, we obtain that z(t) exists 1 on R+. Step 2. Prove that the equation has Q-affine T periodic solutions. We define the Poincarémap P : Rn → Rn by −1 n P (x0) = Q x(T, x0) ∀x0 ∈ R , and P is well-defined. By the definition of r, there exist positive integer K and r ∈ (r, 1) such that a(kT ) ≤ r ∀k ≥ K. By induction and the uniqueness of solutions to initial value problem, we have j j−1 −j P (x0) = P ◦ P (x0) = Q x(jT, x0) ∀j ≥ 1. In fact, j j−1 −1 j−1 P (x0) = P ◦ P (x0) = Q x(T,P x0) ∀j ≥ 1. Note that Z t+T x(t + T, x0) = x0 + f(s, x(s, x0))ds 0 Z t+T = x(T, x0) + f(s, x(s, x0))ds. T Therefore Z t −1 −1 −1 Q x(t + T, x0) = Q x(T, x0) + f(s, Q x(s + T, x0))ds. 0 Further we have Z t −1 −1 −1 x(t, Q x(T, x0)) = Q x(T, x0) + f(s, x(s, Q x(T, x0)))ds. 0 From the above, we can see −2 −1 −1 Q x(t + T, x0) = Q x(t, Q x(T, x0)). In the same manner, it follows that j −j P (x0) = Q x(jT, x0) ∀j ≥ 1. Thus K K −K −K |P (x0) − P (y0)| = |Q x(KT, x0) − Q x(KT, y0)|

= |x(KT, x0) − x(KT, y0)| n ≤ r|x0 − y0| ∀x0, y0 ∈ R . Therefore P is a r-contraction mapping of order K. Now, by Banach’s contraction n mapping ﬁxed point theorem, P has a unique ﬁxed point x∗ ∈ R , i.e., −1 Q x(T, x∗) = x∗. Again by the uniqueness of solutions to the initial value problem, we have

x(t + T, x∗) ≡ Qx(t, x∗) ∀t, LASALLE TYPE STATIONARY OSCILLATION THEOREMS 2911 which shows that x(t, x∗) is a unique Q-aﬃne T periodic solution of the system. Step 3. Prove the stability and the asymptotic stability. Let us prove the following estimate inductively:

m |x(t + mKT, x∗) − x(t + mKT, x0)| ≤ a(t)|x∗ − x0|r ∀t ≥ 0, m ≥ 1. Firstly, for m = 1, by Lemma 1 and assumption 2), we have

|x(t + KT, x∗) − x(t + KT, x0)| −1 −1 =|Q x(t + KT, x∗) − Q x(t + KT, x0)| −1 −1 =|x(t + (K − 1)T,Q x(T, x∗)) − x(t + (K − 1)T,Q x(T, x0))| −1 −1 −1 −1 =|Q x(t + (K − 1)T,Q x(T, x∗)) − Q x(t + (K − 1)T,Q x(T, x0))| −1 −1 =|x(t + (K − 2)T,Q x(T,Q x(T, x∗))) −1 −1 − x(t + (K − 2)T,Q x(T,Q x(T, x0)))| −2 −2 =|x(t + (K − 2)T,Q x(2T, x∗)) − x(t + (K − 2)T,Q x(2T, x0))| = ...... −K −K =|x(t, Q x(KT, x∗)) − x(t, Q x(KT, x0))|

≤a(t)|x(KT, x∗) − x(KT, x0)|

≤a(t)a(KT )|x∗ − x0|

≤a(t)|x∗ − x0|r. Secondly, suppose that for k = m − 1, the estimate holds. Then for k = m, by Lemma 2.2 and 2), we have

We now take t ∈ [0,T ]. By the arbitrariness of m and r ∈ (0, 1), we can get the desired stability and asymptotic stability by the above inequality. This completes the proof. Remark 1. In Theorem 2.1, we do not assume that f(t, x) satisﬁes Lipschitz conditions. Remark 2. In addition, we do not assume that all the solutions of the system are bounded in forward. Remark 3. We do not require that lim a(t) = 0. t→∞ Remark 4. When f(t, x) = f(x), i.e., the system is autonomous one, the assumption condition 1) can be removed. In fact, we assume that [0, b) is the right maximum existence interval of the solution x(t, x0), and b < ∞. For any h, by autonomy of the system, x(t + h, x0) also is a solution of the equation. So, from assumption, for t, t + h ∈ [0, b), we have

|x(t + h, x0) − x(t, x0)| = |x(t, x(h, x0)) − x(t, x0)|

≤ a(t)|x(h, x0) − x0| → 0 (h → 0). By Cauchy’s principle, ξ = lim x(t) exists. Consider the initial value problem t→b−0 x˙ = f(x), x(b) = ξ.

By Peano’s theorem, there exists h0 > 0 such that the above initial value problem has a solution on [b, b + h0]. Hence, x(t, x0) can be extended to [0, b + h0], which contradicts the assumption of [0, b). Then b = ∞. Remark 5. We do not know whether the assumption condition 1) can be removed for non-autonomous systems.

3. A version to functional differential equations. We have considered above the stationary oscillation theorem on Q-affine T periodic solutions in case of ordinary differential equations. Below we will prove an analogous result for functional differential equations. Consider the functional differential equation (FDE, for short)

x˙ = F (t, xt), (2) where xt(s) = x(t + s), s ∈ [−r0, 0], r0 > 0 is a constant, F (t, ϕ) is a completely 1 n n continuous function from R × C([−r0, 0], R ) to R , with Q-affine-periodicity F (t + T, ϕ) = QF (t, Q−1ϕ). Then (2) is called a Q-affine T -periodic functional differential equation.

Deﬁnition 3.1. A solution x(t) of (2) on R1 is said to be Q-aﬃne T -periodic if x(t + T ) = Qx(t) ∀t. We have the following.

Theorem 3.2. Consider the Q-aﬃne T -periodic system (2), and assume F : R1 × n n 1 1 C([−r0, 0], R ) → R is completely continuous. Let a : R+ → R+\{0} be continuous and satisfy r = lim kakT k < 1, where akT (s) = a(kT + s), s ∈ [−r0, 0] and || · || = k→∞ sup | · |. Assume the following hold: [−r0,0] LASALLE TYPE STATIONARY OSCILLATION THEOREMS 2913

1) There exists a solution z(t) of (2) with the initial value condition z0 = ϕ0 ∈ n C([−r0, 0], R ) deﬁned on [0,T ]; 2) Any two solutions x(t) and y(t) satisfy

||xt(·) − yt(·)|| ≤ a(t)||x0 − y0|| whenever they exist, where x0 and y0 denote the initial values of x(t) and y(t), respectively. Then FDE (2) has a unique asymptotically stable Q-aﬃne T periodic solution.

n Proof. Put C = C([−r0, 0], R ). Similar to the proof in Theorem 2.1, by Peano type theorem on FDEs and 2), we have that for any ϕ ∈ C, all the solutions x(t, ϕ) 1 of (2) exist on R+, where x(t, ϕ) denotes the solution of (2) with the initial value condition x0 = ϕ. Define a Poincarémap P : C → C by P (ϕ) = Q−1x(T, ϕ) ∀ϕ ∈ C, and P is well-defined. By the definition of r, there exist positive integer K and r ∈ (r, 1) such that

||akT || ≤ r ∀k ≥ K. It is not hard to prove j −j P (ϕ) = Q xjT (·, ϕ) ∀j ≥ 1. Thus K K −K −K kP (ϕ0) − P (ψ0)k = kQ xKT (·, ϕ0) − Q xKT (·, ψ0)k

= kxKT (·, ϕ0) − xKT (·, ψ0)k

≤ rkϕ0 − ψ0k ∀ϕ0, ψ0 ∈ C. Therefore P is a r contraction mapping of order K. Now, by Banach’s contraction mapping ﬁxed point theorem, P has a unique ﬁxed point ϕ∗ ∈ C, i.e., −1 Q xT (·, ϕ∗) = ϕ∗. Again by the uniqueness of solutions to the initial value problem, we have

x(t + T, ϕ∗) ≡ Qx(t, ϕ∗) ∀t, which shows that x(t, ϕ∗) is a unique Q-aﬃne T periodic solution of (2). The stability is similar to the proof of Theorem 2.1. The proof is complete.

4. Applications. In this section we give some applications of Theorems 2.1-3.2. The ﬁrst is to Theorem 2.1:

Theorem 4.1. Consider the Q-affine T -periodic system (1), and assume f : R1 × n n 1 n×n 1 R → R is continuous. Let D : R+ → R be a C real positive definite matrix function with D = D> (transpose), and let f be C1 in x. Assume ∂f >(t, x) ∂f(t, x) D˙ (t) + D(t) + D(t) ≤ α(t)D(t), (3) ∂x ∂x D(kT ) = D(0) ∀k ∈ N (the set of all natural numbers), 1 1 where α : R+ → R is a locally integrable function with α(t) ≤ α0 (positive con- R kT stant), and satisfies 0 α(s)ds ≤ −σ(σ > 0) for any integer k ≥ k0 (k0 > 0). Then affine-periodic system (1) admits a unique asymptotically stable Q-affine T-periodic solution. 2914 HONGREN WANG, XUE YANG, YONG LI AND XIAOYUE LI

1 Proof. We firstly prove that all the solutions x(t) of (1) are extensible on R+, which verifies 1) in Theorem 2.1. By Picard’s theorem, x(t) exists locally and is unique. Let [0,L) be any existence interval with L < ∞. Put V (t, x) = x>D(t)x. Then the Lyapunov function V (t, x) is positive definite on Rn. Hence along the solution x(t), dV | =x(t)>D˙ (t)x(t) + (f(t, x(t)) − f(t, 0))>D(t)x(t) dx x=x(t) + x(t)>D(t)(f(t, x(t)) − f(t, 0)) + f(t, 0)>D(t)x(t) + x(t)>D(t)f(t, 0) > > =x(t) D˙ (t)x(t) + x(t) fx(t, θtx(t))D(t)x(t) > > + x(t) D(t)fx(t, θtx(t))x(t) + 2f(t, 0) D(t)x(t) ≤α(t)x(t)>D(t)x(t) + |Df(t, 0)|2 + |x(t)|2 (by (3)) ≤|D(t)f(t, 0)|2 + cV (t, x(t)) for some positive constant c. Consequently, for t ∈ [0,L), Z t Z t V (t, x(t)) ≤ V (0, x(0)) + |D(s)f(s, 0)|2ds + c V (s, x(s))ds. 0 0 Gronwall’s inequality implies that Z t V (t, x(t)) ≤ (V (0, x(0)) + |D(s)f(s, 0)|2ds)ect ∀t ∈ [0,L), 0 which leads to that x(L−) exists. By Picard’s theorem, x(t) exists on [L, L + δ), for 1 some δ > 0. By the arbitrariness of L, x(t) exists on R+, as desired. Now let us continue to showing condition 2). For any two solutions u(t) and v(t) of (1), put x = u(t) − v(t). Then along x(t), we have dV =x(t)>D˙ (t)x(t) dx ∂f + x(t)>( (t, v(t) + θ x(t))>D(t)x(t) ∂x t ∂f + x(t)>D(t) (t, v(t) + θ x(t))x(t) ∂x t ≤α(t)x(t)>D(t)x(t) (by (3)) =α(t)V (t, x(t)) ∀t ≥ 0. Integrating on [0, t], we have

R t α(s)ds V (t, x(t)) ≤ V (0, x(0))e 0 . In particular, x(kT )>D(0)x(kT ) =V (0, x(kT )) =x(kT )>D(kT )x(kT ) =V (kT, x(kT )) (4)

R kT α(s)ds ≤V (0, x(0))e 0 LASALLE TYPE STATIONARY OSCILLATION THEOREMS 2915

n for all integer k ≥ k0. Now we choose the following norm in R , q n |x| = x>D(0)x ∀x ∈ R .

R t α(s)ds Put a(t) = e 0 ∀t ≥ 0. It follows from (4) that condition 2) in Theorem 2.1 holds. By virtue of Theorem 2.1, system (1) admits a unique asymptotically stable Q-aﬃne T -periodic solution, and the proof is complete.

Corollary 1. Consider the Q-affine T -periodic system (1), and assume f : R1 × Rn → Rn is continuous. Let D be an n×n real positive definite matrix with D = D> (transpose), and let f be C1 in x. Assume ∂f > ∂f D + D ≤ αD, (5) ∂x ∂x 1 1 where α : R+ → R is a locally integrable function with α(t) ≤ α0 (a positive R kT constant), and satisfies 0 α(s)ds ≤ −σ(σ > 0) for any integer k ≥ k0 (k0 > 0). Then affine-periodic system (1) admits a unique asymptotically stable Q-affine T- periodic solution. ∂f Remark 6. If we only assume that ∂x is bounded and every eigenvalues λ(t, x) satisfies Reλ(t, x) ≤ −σ ∀(t, x), where σ is a positive constant, then we can conclude that (1) has a unique Q-affine T -periodic solution by using dichotomy theory [6]; however it is not enough to imply stability. For example, consider a linear Q-affine T -periodic system n x˙ = A(t)x + e(t), x ∈ R , (6) where A : R1 → Rn×n is continuous and A(t + T ) ≡ QA(t)Q−1 ∀t; e(t + T ) ≡ Qe(t) ∀t. Let x(t) be a Q-affine T -periodic solution of the system. Then x(t) is asymptotically stable if and only if the trivial solution x ≡ 0 of thex ˙ = A(t)x is asymptotically stable. In general, even if every eigenvalue λ(t) of A(t) satisfies that Reλ(t) ≤ −σ < 0 ∀t, we can not obtain the stability of the zero solution. Nevertheless, it follows from (5) the following. Corollary 2. If A>(t) + A(t) is negative definite, then 1) System (6) admits a unique asymptotically stable Q-affine T -periodic solution. 2) The zero solution of x˙ = A(t)x is asymptotically stable. Furthermore, we have Corollary 3. Assume A>(t) + A(t) ≤ α(t)I ∀t, 1 1 R kT where α : R → R is locally integrable and satisfies 0 α(s)ds ≤ −σ < 0 for any integer k ≥ k0 > 0. Then 1) The zero solution of x˙ = A(t)x is asymptotically stable; 2) System (6) admits a unique asymptotically stable Q-affine T -periodic solution. 2916 HONGREN WANG, XUE YANG, YONG LI AND XIAOYUE LI

Via Lyapunov’s method, we have generally the following abstract version.

Theorem 4.2. Consider the Q-aﬃne T -periodic system (1), and assume f : R1 × n n 1 n n 1 1 R → R is continuous. Let V : R+ × R × R → R+ be a C function with 1 V (t, x, y) = V (t + kT, x, y) ∀k ∈ N , V (t, x, y) = V (t, Q±1x, Q±1y), a(|x − y|) ≤ V (t, x, y) ≤ b(|x − y|) ∀(t, x, y), 1 1 where a, b : R+ → R+ are wedges, namely, they are strictly increasing continuous functions with a(0) = 0 = b(0), a(∞) = ∞ = b(∞), and satisﬁes that, for some β ∈ (0, 1), a(βr) inf > 0. (7) r>0 b(r) Moreover, assume along any two solutions x(t) and y(t) of (1), ˙ V (t, x, y)|(x,y)=(x(t),y(t)) ≤ α(t)V (t, x, y)|(x,y)=(x(t),y(t)) ∀t ≥ 0, (8)

1 1 R kT where α : R+ → R is locally integrable and satisﬁes 0 α(s)ds ≤ −σ (σ > 0) for any integer k ≥ k0 > 0. Assume there exists a solution z(t) of (1) on [0,T ]. Then (1) admits a unique asymptotically stable Q-aﬃne T -periodic solution.

1 −1 Proof. We ﬁrst prove that z(t) is extensible on R+. Let x = z(t), y = Q z(t + T ). If x(t) ≡ y(t), i.e. z(t) ≡ Q−1z(t + T ), then z(t) is a Q-aﬃne T -periodic solution of (1), and hence z(t) exists on R1, as desired. Thus we let x(t) 6= y(t). By (8) and other assumptions, we obtain a(|x(t) − y(t)|) ≤ V (t, x(t), y(t)) R t α(s)ds ≤ V (0, x(0), y(0))e 0 R t α(s)ds ≤ b(|x(0) − y(0)|)e 0 ∀t ∈ [0,T ], which implies that −1 R t α(s)ds |x(t) − y(t)| ≤ a (b(|x(0) − y(0)|)e 0 ) ∀t ∈ [0,T ]. In the same way as step 1 in the proof of Theorem 2.1, it follows that z(t) exists on 1 1 R+. Hence all the solutions of (1) exist on R+ by repeating above arguments. Inductively, by Lemma 2.2, the properties of V , similar to the proof of Theorem n 2.1, we have that for any m ≥ 1, t ≥ 0, and x0, y0 ∈ R ,

V (t + mKT, x(t + mKT, y0), x(t + mKT, x0)) (9) −1 −1 =V (t + mKT, Q x(t + mKT, y0),Q x(t + mKT, x0)) −1 −1 =V (t + mKT, x(t + (mK − 1)T,Q x(T, y0)), x(t + (mK − 1)T,Q x(T, x0))) = ...... −mK −mK =V (t + mKT, x(t, Q x(mKT, y0)), x(t, Q x(mKT, x0))) R t α(s)ds ≤e 0 V (0, x(mKT, y0), x(mKT, x0)) R t α(s)ds =e 0 V (KT, x(mKT, y0), x(mKT, x0)) R t α(s)ds −1 ≤e 0 V (KT,Q x((mK − 1)T, −1 −1 −1 Q x(T, y0)),Q x((mK − 1)T,Q x(T, x0))) LASALLE TYPE STATIONARY OSCILLATION THEOREMS 2917

R t α(s)ds −(m−1)K −(m−1)K =e 0 V (KT,Q x(KT,Q x((m − 1)KT, y0)), −(m−1)K Q x(KT, x((m − 1)KT, x0))) R t α(s)ds R KT α(s)ds ≤e 0 e 0 V (0, x((m − 1)KT, y0), x((m − 1)KT, x0)) ≤ ......

R t α(s)ds R KT α(s)ds m ≤e 0 (e 0 ) V (0, y0, x0) R t α(s)ds R KT α(s)ds m ≤e 0 (e 0 ) b(|y0 − x0|).

Fix integers K ≥ k0 and m = N > 0 such that a(βr) e−σN < inf . r>0 b(r) We derive from assumption and (9) that −(NK) −(NK) |Q x(NKT, y0) − Q x(NKT, x0)| ≤ β|y0 − x0| ∀y0, x0, which shows that P NK is a contract map. Thanks to Banach’s fixed Point theorem, P has a unique fixed point x∗, that is, x(t, x∗) is a Q-affine T -periodic solution of (1). Stability and asymptotic stability follows from (9). This completes the proof. Remark 7. The proof of Theorem 4.2 shows that if we assume Z (k+1)T α(s)ds ≤ −σ < 0 ∀k ≥ k0, kT then the condition V (t, Q±1x, Q±1y) = V (t, x, y) can be removed. Example 1. Consider the system   sin ω1t  cos ω1t    2  .  x˙ = −a(t)x − b(t)|x| x +  .  ≡ f(t, x), 2m = n,    sin ωmt  cos ωmt where a, b : R1 → R are continuous, and satisfy a(t + T ) = a(t), b(t + T ) = b(t), Z T a(t)dt = σ > 0, b(t) ≥ 0, T > 0. 0 Then f(t + T, x) = Qf(t, Q−1x). Put cos ω T sin ω T cos ω T sin ω T Q = diag 1 1 ··· m m . − sin ω1T cos ω1T − sin ωmT cos ωmT Choose a Lyapunov function 1 V (x) = |x|2. 2 2918 HONGREN WANG, XUE YANG, YONG LI AND XIAOYUE LI

Put g(x) = |x|2x. Then       x1 1 ··· 1 x1 ∂g = |x|2I + 2  ..   . .   ..  ∂x  .   . .   .  xn 1 ··· 1 xn ≡ |x|2I + h(x).

∂g Note that h(x) is positive semi-definite. Thereby, ∂x is positive definite. Thus by the mean value theorem, along any two solutions x(t) and y(t) of the system dV (x(t) − y(t)) =(x(t) − y(t))>(−a(t)I dt ∂g − b(t) (y(t) + θ (x(t) − y(t)))(x(t) − y(t)) ∂x t ≤ − a(t)V (x(t) − y(t)). Note that Z mT a(t)dt = mσ → ∞ (m → ∞). 0 By virtue of Theorem 4.2, the system admits a unique asymptotically stable Q-affine T -periodic solution, which is also a quasi-periodic solution with frequency (ω1, ··· , ωm). The next is to Theorem 3.2. Let FDE(2) have the following form:

x˙ = L(t, xt) + g(t, xt) ≡ F (t, xt), (10) where L, g : R1 × C → Rn satisfy besides the continuity and Q-aﬃne periodicity that there exist locally L1 integrable functions l, q : R1 → R1 such that |L(t, ϕ)| ≤ q(t)||ϕ||, 1 |g(t, ϕ) − g(t, ψ)| ≤ l(t)||ϕ − ψ|| ∀t ∈ R , ϕ, ψ ∈ C. (11) Furthermore, L(t, ϕ) is homogeneous linear in ϕ. Let T (t, s) denote the solution operator of the homogenous linear equation

x˙ = L(t, xt), (12) that is,

yt(·, s, ϕ) = T (t, s)ϕ, where y(t, s, ϕ) denotes the solution of (12) with the initial value ys = ϕ. By virtue of Theorem 1.1 in Chapter 6 of [9], y(t, s, ϕ) exists on [s, ∞). We have Theorem 4.3. Under the above assumptions, assume there exist a constant c and a locally L1 integrable function β : R1 → R1 such that 1) the norm R tβ(s)ds ||T (t, s)|| ≤ ce s ∀t ≥ s;

2) the limit Z kT (cl(s) + β(s))ds → −∞ as k → ∞, 0 LASALLE TYPE STATIONARY OSCILLATION THEOREMS 2919 where {k} ⊂ N. Then FDE(10) admits a unique asymptotically stable Q-aﬃne T -periodic solution. Proof. We have from the variation-of-constants formula (Theorem 2.1 in Chapter 6 of [9]) that Z t xt(0, ϕ) = T (t, 0)ϕ + T (t, s)X0(·)g(s, xs(0, ϕ))ds ∀t ≥ 0, ϕ ∈ C, 0 where xt(0, ϕ) = xt(·, 0, ϕ), and   0, −r ≤ θ < 0, X0(θ) =  I, θ = 0. Thus, it follows from (11), 1) and the variation-of-constants formula that

||xt(0, ϕ) − xt(0, ψ)|| Z t R t β(s)ds R tβ(τ)dτ ≤ce 0 ||ϕ − ψ|| + c e 0 l(s)||xs(0, ϕ) − xs(0, ψ)||ds, 0 and hence − R t β(s)ds e 0 ||xt(0, ϕ) − xt(0, ψ)|| Z t −R sβ(τ)dτ ≤c||ϕ − ψ|| + cl(s)e 0 ||xs(0, ϕ) − xs(0, ψ)||ds. 0 Applying Gronwall’s inequality yields R t(cl(s)+β(s))ds ||xt(0, ϕ) − xt(0, ψ)|| ≤ c||ϕ − ψ||e 0 ∀t ≥ 0. (13) R t(cl(s)+β(s))ds Put a(t) = ce 0 . Then assumption 2) implies that a(kT ) → 0 as k → ∞. This shows together with (13) that all the conditions of Theorem 3.2 hold. By virtue of Theorem 3.2, equation (10) admits a unique asymptotically stable Q-affine T - periodic solution. This completes the proof. Let us illustrate Theorem 4.3 as follows: Example 2. Consider a delay differential equation (DDE for short) x˙(t) = Ax(t) + Bx(t − τ) + f(t), (14) where τ is a positive constant, −2 −1 0.2 0.4 A = ,B = , −2 −3 0.5 −0.2 and f(·): R1 → R2 is T -periodic. Define L(t, ϕ) = Aϕ(0) + Bϕ(−τ), g(t, ϕ) = f(t). Then L, g : R1 × C → R2 satisfy 1 |L(t, ϕ)| ≤ (|A| + |B|)||ϕ||, |g(t, ϕ) − g(t, ψ)| = 0 ∀t ∈ R , ϕ, ψ ∈ C. The corresponding characteristic equation for the homogenous linear equation x˙(t) = Ax(t) + Bx(t − τ) (15) 2920 HONGREN WANG, XUE YANG, YONG LI AND XIAOYUE LI is h(λ) = det |λI − A − Be−τλ| = 0.

It can be proved easily that α0 := max{Reλ : h(λ) = 0} < 0. Then from Theorem 6.2 in Chapter 1 of [9] there is a constant K(α0) such that the solution yt(·, s, ϕ) = T (t, s)ϕ of (15) with the initial value ys = ϕ satisﬁes ||T (t, s)|| ≤ Ke−0.5α0(t−s) ∀t ≥ s. Therefore DDE (14) admits a unique asymptotically stable T -periodic solution. Example 3. Consider the system with delay   sin ω1t  cos ω1t     .  x˙ = −ax(t) + x(t − r) +  .  ≡ F (t, xt),    sin ωmt  cos ωmt where r, a > 0 are constant. Put Q is the same as in Example 1, then F (t + T, ϕ) = QF (t, Q−1ϕ). Let λ ∈ R1 such that λ + a − e−λr = 0. It is well known that Reλ < 0. It follows from Corollary 4.1 in Chapter 7 of [9] that there exist constants c, β > 0 such that ||T (t, s)|| ≤ ce−β(t−s) ∀t ≥ s. By Theorem 4.3, the system admits a unique asymptotically stable Q-aﬃne T - periodic solution.

Acknowledgments. The authors express sincere thanks to the anonymous referee for his/her comments and suggestions.

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