Periodic solutions for a class of non-autonomous differential delay equations

Transkript

1 Nonlinear Differ. Equ. Appl. 16 (2009), c 2009 Birkhäuser Verlag Basel/Switzerland /09/ published online July 4, 2009 DOI /s Nonlinear Differential Equations and Applications NoDEA Periodic solutions for a class of non-autonomous differential delay equations Rong Cheng, Junxiang Xu Abstract. By applying symplectic transformation, Floquet theory and some results in critical point theory, we establish the existence of periodic solutions for a class of non-autonomous differential delay equations, which can be changed to Hamiltonian systems. Mathematics Subject Classification (2000). 34C25, 37J45. Keywords. Hamiltonian system, Floquet theory, symplectic transformation, periodic solution, delay equation. 1. Introduction and main result In 1974, Kaplan and Yorke [1] studied a class of differential delay equations x (t) = f(x(t 1)) (1.1) and x (t) = [f(x(t 1)) + f(x(t 2))], (1.2) where f(x) C(R, R) is odd, xf(x) > 0forx 0,andf satisfies some asymptotically linear conditions both at origin and at infinity. They proved that the Eqs. 1.1 and 1.2 have periodic solutions with period 4 and 6, respectively. They also conjectured that similar results should be true for (1.1) or(1.2) with more than two delays under similar conditions on f. This is called the Kaplan Yorke s conjecture. Since then the two equations and their general cases have been studied by many authors through various methods. In 1978, Nussbaum [3] obtained the existence of a 2n-periodic solution for a class of differential delay equations which include the Eqs. 1.1 and 1.2 as two special cases, where n is a positive integer. This work is supported by the National Natural Science Foundation of China ( ).

2 794 R. Cheng and J. Xu NoDEA Thus, the Kaplan Yorke s conjecture was solved. But the method used in [3] is quite different from that in [1]. Recently many authors [4 9] made use of Kaplan and Yorke s original idea to study some general differential delay equations by changing them into Hamiltonian systems. Li and He [4, 5] studied the existence of multiple periodic solutions of the following equation x (t) = [f(x(t r 1 )) + f(x(t r 2 )) + + f(x(t r n 1 ))] (1.3) where r i (i =1, 2,...,n 1) are positive constants. By variational methods they proved that the system (1.3) has multiple periodic solutions. According to the variational methods used in [4, 5], Cima and his collaborators [6] applied qualitative theory of ordinary differential equations to study the following equation x (t) = [f(x(t r 1 ))g(x(t r 2 )) + f(x(t r 2 ))g(x(t r 1 ))]. They found a suitable condition such that the above equation has periodic solutions. Then Llibre and Tarta [7] generalized the work in [6]. They considered the following system x (t) =af(x(t r 1 ))g(x(t r 2 )) + bf(x(t r 1 ))g(x(t r 3 )) af(x(t r 2 ))g(x(t r 1 )) + cf(x(t r 2 ))g(x(t r 3 )) bf(x(t r 3 ))g(x(t r 1 )) cf(x(t r 1 ))g(x(t r 2 )). They got a sufficient condition for the existence of periodic solutions of the system, where a, b, c take the values 1 or 1. When the delays r i (i = 1, 2,...,n 1) take positive integers, Jekel and Johnston [8] studied the following system x (t) = [f(x(t 1)) + f(x(t 2)) + + f(x(t (n 1)))], (1.4) and by using some known results on the existence of closed orbits for Hamiltonian vector fields they showed that the Eq. 1.4 has a periodic solution. Almost at the same time, Fei [9] took a different approach from those employed in previous papers to study the existence of multiple periodic solutions for the Eq He applied the pseudo index theory established in [10] to obtain the multiple periodic solutions of (1.4) when f(x) is asymptotically linear both at zero and at infinity with respect to x. The previous work is mainly concerned with autonomous differential delay equations. In this paper, we study the existence of periodic solutions for a class of non-autonomous differential delay equations with the following form x (t) = [f(t, x(t τ)) + f(t, x(t 2τ)) + + f(t, x(t (n 1)τ))], (1.5) where τ = π/n and f(t, x) is odd and asymptotically linear both at zero and at infinity with respect to x. We shall apply symplectic transformation and some well known results in critical point theory to obtain the existence of periodic solutions for the Eq. 1.5.

3 Vol. 16 (2009) Periodic solutions for non-autonomous 795 Before introducing our main results, we make the following assumptions. (H 1 ) f(t, x) C(R R, R) is odd with respect to x and periodic in t with period τ. (H 1) f(t, x) C 1 (R R, R) is odd with respect to x and periodic in t with period τ. (H 2 ) For all t [0,τ], f(t, x) f(t, x) lim = β 0 (t), lim = β (t), x 0 x x x i.e., the two limits are uniformly in t. It is obvious that β 0 (t) andβ (t) are periodic functions with period τ. Since there is much difference between n =2N(N Z + )andn =2N +1, we consider the two cases, respectively. In this paper, we always assume that n =2N is even. Let I 2N be the identity matrix in R 2N and α R. We denote the matrix αi 2N by α for convenience. Let A 2N be a 2N 2N skew symmetric matrix defined by A 2N = Let σ(ia 1 2N ) denotes the spectrum of the matrix ia 1 2N. Then from Lemma 4 of [5] and a direct computation, we have σ(ia 1 2N )= ctan 2l 1 } π : l =1, 2,...,2N. 4N Write α 0 = 1 τ τ 0 β 0(t)dt and α = 1 τ τ 0 β (t)dt. Then our main results read as follows. Theorem 1.1. Let (H 1 ) and (H 2 ) hold. Suppose that (H 3 ) α 0 kctan 2l 1 4N π, α kctan 2l 1 4N π, for all k Z+ and 1 l 2N, and (H 4 ) There exist at least an integer l 0 with 1 l 0 2N and some k 0 Z + such that minα 0,α } <k 0 ctan 2l 0 1 4N π<maxα 0,α }. Then the equation 1.5 has at least one nontrivial periodic solution.

4 796 R. Cheng and J. Xu NoDEA Theorem 1.2. Let (H 1) and (H 2 ) hold. Suppose that (H 5 ) α kctan 2l 1 4N π, for all k Z+ and 1 l 2N. Thereexistk i Z + and l i Z + with either 1 l 1 <l 2 < <l n0 N or N +1 l 1 <l 2 < < l n0 2N such that α 0 = k 1 ctan 2l 1 1 4N π = k 2 ctan 2l 2 1 4N π = = k n 0 ctan 2l n 0 1 4N π, where 1 i n 0 and 1 n 0 N. and (H 6 ) There are at least n 0 +1 integers m 1,m 2,...,m n0+1 with 1 m 1 <m 2 < <m n0+1 2N and n 0 +1 integers k 1,k 2,...,k n0+1 such that maxα 0,α } k 1 ctan 2m 1 1 4N π> >k n 0+1 ctan 2m n π>minα 0,α }. 4N Then the equation 1.5 has at least one nontrivial periodic solution. Remark 1.3. In the autonomous case as considered in [4, 5, 9], multiple periodic solutions are obtained. In non-autonomous case, we can only obtain single periodic solution since the functional is not S 1 -invariant. Remark 1.4. As we pointed out above, Theorems 1.1 and 1.2 are concerned with the existence of periodic solutions for non-autonomous differential delay Eq Therefore, our results generalize the results obtained in the references. 2. Proof of the main results We will prove our main results by several steps. We first transform the system (1.5) into an equivalent Hamiltonian system. Then we construct a symplectic transformation which reduces the linear parts of the Hamiltonian system to constant coefficients. Subsequently, some lemmas will be given. Finally, we use two critical point results and Galerkin approximation method to prove Theorems 1.1 and Theorem An equivalent Hamiltonian system We first change the Eq. 1.5 to a form of Hamiltonian system. Suppose that x(t) is a solution for the Eq. 1.5 satisfying x(t) = x(t nτ). Let x 1 (t) =x(t), x 2 (t) =x(t τ),...,x n (t) =x(t (n 1)τ). (2.1) Then X(t) =(x 1 (t),x 2 (t),...,x n (t)) is a solution of the following system d dt X(t) =A 2N F (t, X(t)), (2.2) where F (t, X) =(f(t, x 1 ),f(t, x 2 ),...,f(t, x n )) and A 2N is defined in Sect. 1. Conversely, if a solution X(t) of the system (2.2) has the following symmetric structure x 1 (t) = x n (t τ),x 2 (t) =x 1 (t τ),...,x n (t) =x n 1 (t τ), (2.3)

5 Vol. 16 (2009) Periodic solutions for non-autonomous 797 then x(t) =x 1 (t) is a solution of the Eq. 1.5 satisfying x(t nτ) = x(t). Thus, solutions of the system (2.2) with the symmetric structure (2.3) give solutions of the differential delay equation (1.5). Hence, solving the Eq. 1.5 is equivalent to finding solutions of the Eq. 2.2 with the symmetric structure (2.3). Since A 2N is a nonsingular skew symmetric ( Hamiltonian ) matrix, i.e., it satisfies JA 2N + A 2N J = 0, where J = I2N 0 is the standard symplectic I 2N 0 matrix, the system (2.2) can be written as the following Hamiltonian system y (t) =A 2N y H(t, y), (2.4) where H(t, y) = y 1 0 f(t, x)dx + + y 2N 0 f(t, x)dx, y =(y 1,y 2,...,y 2N ) R 2N,and y H(t, y) denotes the gradient of H(t, y) with respect to y. The matrix A 2N is called the symplectic structure associated to the Hamiltonian system (2.4). Below, we turn to study the Hamiltonian system (2.4) Reducibility of the linear parts The problem now is equivalent to finding periodic solutions of the Hamiltonian system (2.4) with the symmetric structure (2.3). By the assumption (H 1 )and (H 2 ), the system (2.4) is asymptotically linear with respect to y both at origin and at infinity with coefficients of the linear parts depending on t. If we can find a global symplectic transformation in the sense of the symplectic structure A 2N which transforms (2.4) into an equivalent Hamiltonian system with constant coefficients of the linear parts both at zero and at infinity, then we can apply some well known results in [12,13] to prove our theorems. Since the coefficients of the linear parts at origin and infinity may be different, the Floquet theory cannot be applied directly. We want to use Hamiltonian flow mapping to construct such global symplectic transformation. For this purpose, we only need to construct the corresponding Hamiltonian function. By the assumption (H 2 ), we have β0 (t)x + o( x ), as x 0, f(t, x) = β (t)x + o( x ), as x. Then H(t, y) is even with respect to y and satisfies β0 (t)y + o( y ), as y 0, y H(t, y) = β (t)y + o( y ), as y. And the corresponding Hamiltonian system (2.4) satisfies y (t) =A 2N β 0 (t)y + o( y ) as y 0, (2.5) y (t) =A 2N β (t)y + o( y ) as y. (2.6) For the system (2.5), let y = P 1 (t, z), where P 1 (t, z) =e Q1(t) z, Q 1 (t) =A 2N t 0 γ 0 (ξ)dξ and γ 0 (t) =β 0 (t) α 0. Then the transformation y = P 1 (t, z) is symplectic.

6 798 R. Cheng and J. Xu NoDEA Then with the transformation y = P 1 (t, z) the system (2.5) is transformed into the following Hamiltonian system z (t) =α 0 A 2N z + o( z ) as z 0. (2.7) Similarly, let y = P 2 (t, z), where P 2 (t, z) =e Q2(t) z, Q 2 (t) =A 2N t 0 γ (ξ)dξ and γ (t) =β (t) α. In the same way we have that the transformation y = P 2 (t, z) is also symplectic. Then the differential equation (2.6) can be transformed to the following Hamiltonian system z (t) =α A 2N z + o( z ) as z. (2.8) Noting that Q i (t)(i = 1, 2) are τ-periodic, the functions e Qi(t) (i = 1, 2) are bounded. Thus, there exist two positive constants r and R with r < R such that r y e Qi(t) y R y (i =1, 2). (2.9) We now manage to construct a global symplectic transformation Ψ(t, z) such that P1 (t, z), z < r/ y =Ψ(t, z) = R P 2 (t, z), z > R/ r. (2.10) Let Γ 0 (t) = t 0 γ 0(ξ)dξ and Γ (t) = t 0 γ (ξ)dξ. Then Γ 0 (t) and Γ (t) are two τ-periodic functions. Let ρ 1 (w) andρ 2 (w) be two smooth functions satisfying ρ 1 ( w 2 )= 1, as w < r 0, as w > R, ρ 2 ( w 2 )= 0, as w < r 1, as w > R. Let H t (w) = 1 2 Γ 0(t) w 2 ρ 1 ( w 2 )+ 1 2 Γ (t) w 2 ρ 2 ( w 2 ), where t is regarded as a parameter. It is obvious that H t (w) isaτ-periodic function with respect to the parameter t. Consider the following Hamiltonian system dw ds = A 2N w Ht (w). (2.11) Note that the above Hamiltonian system (2.11) is autonomous with respect to s andt is regarded as a parameter. Let Ψ t (s, w) be the general solution of (2.11). By the uniqueness of solutions of ordinary differential equations, Ψ t (s, w) is τ-periodic with respect to the parameter. Now we prove that the flow mapping Ψ t (s, w) is symplectic with respect to the symplectic form A 2N, that is, ( ) Ψ t (s, w) Ψt (s, w) A 2N = A 2N. w w Since A 2N is skew symmetric and non-degenerate, there is a non-degenerate matrix S such that SA 2N S = J,

7 Vol. 16 (2009) Periodic solutions for non-autonomous 799 where J is the standard symplectic matrix. Let w = S 1 z. Then the system (2.11) is transformed to the following standard Hamiltonian system d z ds = J z Ht ( z), (2.12) where Ht ( z) = H t (S 1 z). Let Ψ t (s, z) = SΨ t (s, S 1 z). Then Ψ t (s, z) isthe general solution of the Hamiltonian system (2.12). Thus, we have that Ψ t ( Ψ t z ) = J, i.e., S Ψ ( t w S 1 J S Ψ ) t w S 1 = J S Ψ t w (S 1 J(S 1 ) ) Ψ ( ) t w A Ψt 2N = A 2N. w ( ) Ψt S = J w z J Let y =Ψ(t, z) =Ψ t (s, w) s=1,w=z =Ψ t (1,z). Then we have ( ) Ψ Ψ z A 2N = A 2N. z For w < r, the system (2.11) becomes dw/ds=γ 0 (t)a 2N w.so,w(s)=e Γ0(t)A2N s z. If s < 1, z < r/ R, then w(s) = e Γ0(t)A2N s z R z < r. Thusfor s < 1, z < r/ R, wehavew(s) =e Γ0(t)A2N s z. Therefore, Ψ(t, z) =P 1 (t, z) as z < r/ R. In the same way we can prove that Ψ(t, z) =P 2 (t, z) as z > R/ r. Let z =Φ(t, y) be the inverse mapping of the transformation y =Ψ(t, z). Then the system (2.4) is changed to the following system dz dt = A 2N z Ĥ(t, z)+ Φ t, (2.13) where Ĥ(t, z) =H(t, Ψ(t, z)). By a direct computation, A 1 2N 2 Φ t y is symmetric. Then according to [15], there is a smooth function R such that A 2N z R = Φ t. Then H (t, z) =Ĥ(t, z)+r(t, z) is the Hamiltonian function of the system (2.13) and the Hamiltonian system (2.13) can be rewritten as dz dt = A 2N z H (t, z). (2.14) By (2.7) and (2.8), H (t, z) satisfies the following properties. z H (t, z) =α 0 z + o( z ) as z 0 uniformly in t, (2.15) z H (t, z) =α z + o( z ) as z uniformly in t. (2.16)

8 800 R. Cheng and J. Xu NoDEA Now in order to prove the main results of this paper, we only need to consider the Hamiltonian system (2.14). In the following of this paper, we prove the following two theorems. Theorem 2.1. Under the assumptions of Theorem 1.1, the system 2.14 has at least one nontrivial periodic solution. Theorem 2.2. Under the assumptions of Theorem 1.2, the system 2.14 has at least one nontrivial periodic solution Some lemmas We now have constructed the global symplectic transformation y = Ψ(t, z) which transforms the system (2.4) into the Hamiltonian system (2.14). Note that the system (2.14) satisfies the two properties (2.15) and (2.16). We shall use two well known critical point results given in [12, 13] to prove the two theorems. We work in the Hilbert space E = W 1 2,2 (S 1, R 2N ) with the inner product, and the norm. The space E consists of all z(t) inl 2 (S 1, R 2N )whose Fourier series + z(t) =a 0 + (a k coskt + b k sinkt) satisfies a The inner product on E is defined by z 1,z 2 =(a 1 0,a 2 0)+ 1 2 k( a k 2 + b k 2 ) < +. k[(a 1 k,a 2 k)+(b 1 k,b 2 k)], where z i = a i (ai k coskt + bi ksinkt)(i =1, 2) and (, ) denotes the inner product in R 2N. Recall that A 1 2N is also a nonsingular skew symmetric matrix. For z,y E, we define an operator A on E by extending the bilinear form Az, y = 2π 0 (A 1 2N z (t),y(t))dt. (2.17) It is not difficult to compute that A is a bounded self-adjoint linear operator on E and Az = + (A 1 2N b kcoskt A 1 2N a ksinkt). (2.18)

9 Vol. 16 (2009) Periodic solutions for non-autonomous 801 For each z E, we define a functional ψ on E by ψ(z) = 1 2π 2 Az(t),z(t) H (t, z(t))dt. It is well known that critical points of ψ are solutions of the system (2.14). Hence, to seek periodic solutions of the system (2.14) is equivalent to seeking critical points of ψ. For any α R and z,y E, we define another operator B α on E by 0 2π B α z,y = (αz(t),y(t))dt. (2.19) 0 Then B α is also a bounded self-adjoint linear operator on E. Moreover,B α is compact and for any z E, + 1 B α z = αa 0 + k ( αa kcoskt αb k sinkt). (2.20) For any z(t) E and by (2.19) and (2.20), we have (A + B α)z(t) = αa (( A 1 2N b k 1 ) ( k αa k coskt + A 1 2N a k 1 ) ) k αb k sinkt. (2.21) Lemma 2.3. Suppose that (H 1 ) and (H 2 ) hold. Then the functional ψ satisfies ψ (z) (A + B α0 )z = o( z ) as z 0, (2.22) ψ (z) (A + B α )z = o( z ) as z. (2.23) Proof. From (2.15), for any r >0, there exists a constant C( r) (here and in the following C denotes various constants) such that x H (t, x) α 0 x r x + C( r) x 2, for each x R 2N. Notice that ψ(z) = 1 2π ( 2 (A + B α 0 )z(t),z(t) H (t, z(t)) 1 ) 2 (α 0z(t),z(t)) dt. Thus, we have 0 ψ (z) (A + B α0 )z C z H (t, z) α 0 z C( r z + C( r) z 2 ), which means (2.22). By (2.16), we have x H (t, x) α x r x + C( r), for each x R 2N. Then (2.23) holds similarly.

10 802 R. Cheng and J. Xu NoDEA In order to obtain solutions of (2.14) with the symmetric structure (2.3), we define a 2N 2N matrix T 2N with the following form T 2N = Then by T 2N, for any z(t) E, define an action on z by δz(t) =T 2N z(t τ). (2.24) Then by a direct computation we have that δ 2N z(t) = z(t 2Nτ),δ 4N z(t) =z(t) and G = δ, δ 2,...,δ 4N } is a compact group action over E. Ifδz(t) =z(t) holds, then through a direct check, we have that z(t) has the symmetric structure (2.3). Write SE = z E : δz(t) =z(t)}. Then from z(t) = z(t 2Nτ) and a direct computation, we have } SE = z E : z(t) = (a k cos(2k 1)t + b k sin(2k 1)t). Let ψ SE be the restriction of ψ on SE. We have the following lemma. Lemma 2.4. Critical points of ψ SE over SE are critical points of ψ on E. Proof. We first check that T 2N Ψ t (s, w) =Ψ t (s, T 2N w), where Ψ t (s, w) is the flow mapping generated by the system (2.11). In fact, by the definition of H, weget w (s) =A 2N Γ 0 (t)wρ 1 ( w 2 )+Γ 0 (t) w 2 ρ 1( w 2 )w +Γ (t)wρ 2 ( w 2 )+Γ (t) w 2 ρ 2( w 2 )w}. Notice that T 2N T2N = I and T 2N commutes with A 2N and J. Set z(s) = Ψ t (s, T 2N w), ỹ(s) =Ψ t (s, w), then d z(s) = A 2N Γ 0 (t)t 2N wρ 1 ( w 2 )+Γ 0 (t) w 2 ρ ds 1( w 2 )T 2N w +Γ (t)t 2N wρ 2 ( w 2 )+Γ (t) w 2 ρ 2( w 2 )T 2N w} = T 2N A 2N Γ 0 (t)wρ 1 ( w 2 )+Γ 0 (t) w 2 ρ 1( w 2 )w +Γ (t)wρ 2 ( w 2 )+Γ (t) w 2 ρ 2( w 2 )w} dỹ(s) = T 2N ds, this, jointly with Ψ t (0,T 2N w)=t 2N w = T 2N Ψ t (0,w) shows that T 2N Ψ t (s, w) = Ψ t (s, T 2N w). The definition of H, together with a direct computation implies that H (t, T 2N z)=h (t, z), z H (t, T 2N z)=t 2N z H (t, z).

11 Vol. 16 (2009) Periodic solutions for non-autonomous 803 Combining these with (2.24) and the fact that any z(t) E is 2π-periodic, we can verify that ψ(δz) =ψ(z), ψ (δz) =δψ (z) =ψ (z). That means ψ is G-invariant and ψ is G-equivariant. Moreover, ψ (z) SE. Therefore, if ψ (z) =0onSE, then ψ (z) =0onE. Hence, the conclusion of Lemma 2.4 holds. Remark 2.5. It follows from Lemma 2.4 that we only need to find critical points of ψ SE on SE. In the following, we use Galerkin approximation method and two critical point results to obtain critical points of ψ SE. We first analyze the spectrum of the operator A + B α. Observing (2.21), we only need to analyze the spectrum of the following matrix ( αi2n ka 1 ) T k (α) = 2N. αi 2N ka 1 2N Let λ 0 be an eigenvalue of A 2N.ByLemma4of[5], Proposition 3.6 of [16] and a direct computation, we have det(λ 0 I 2N A 2N )= 1 2 ((1 λ 0) 2N +(1+λ 0 ) 2N )=0. (2.25) Then (2.25) gives σ(a 2N )= ±itan 2l 1 π : l =1, 2,...,N 4N }, (2.26) i.e., A 2N has N pairs of pure imaginary eigenvalues. Let λ be an eigenvalue of the matrix T k (α). Then det(λi 4N T k (α)) = det((λ + α) 2 I 2N +(ka 1 2N )2 ) = det((λ + α) 2 I 2N (ika 1 2N )2 ) = det((λ + α)i 2N ika 1 2N )det((λ + α)i 2N + ika 1 2N ) = det(λi 2N (ika 1 2N αi 2N ))det(λi 2N ( ika 1 2N αi 2N)). Notice that ika 1 2N αi 2N and ika 1 2N αi 2N have the same spectrum. It is easy to see that if λ takes the eigenvalue of ika 1 2N αi 2N, then det(λi 4N T k (α)) = 0. By (2.26), one has σ(ia 1 2N )= ±ctan 2l 1 4N π : l =1, 2,...,N }, (2.27) i.e. ia 1 2N has N pairs of real eigenvalues. In order to prove our main results we need to use the following index. By M ( ) andm 0 ( ) we denote the negative

12 804 R. Cheng and J. Xu NoDEA and the zero Morse indices of the symmetric matrix, respectively. For a constant diagonal matrix αi 2N, we define our index as i (α) = (M (T k (α)) 2N) and i 0 (α) = M 0 (T k (α)). Since for k large enough, one has M (T k (α)) = 2N and M 0 (T k (α)) = 0. Hence i (α) andi 0 (α) are well defined. The following lemma gives the relation between α R and the indices i 0 (α),i (α). Lemma 2.6. (1) Suppose that (H 3 ) hold. Then i 0 (α 0 )=i 0 (α )=0. (2) Suppose that (H 3 ) and (H 4 ) hold. Then i (α 0 ) i (α ). (3) Suppose that (H 5 ) and (H 6 ) hold. Then i (α ) / [i (α 0 ),i (α 0 )+i 0 (α 0 )]. Proof. Case (1). From (2.27), if α 0 kctan 2l 1 4N π, for all k Z+ and 1 l 2N, then λ 0, where λ is the eigenvalue of T k (α 0 ). That means M 0 (T k (α 0 )) = 0 for k 1. Thus i 0 (α 0 )= M 0 (T k (α 0 )) = 0. Similarly, we have i 0 (α )=0. Case (2). Without loss of generality, we suppose that α 0 <α. Then by the conditions (H 3 ) and (H 4 ), α 0 <k 0 ctan 2l 0 1 4N π<α. Then by a direct check, we have M (T k0 (α 0 )) + 2 M (T k0 (α )). For each k k 0, one can check easily that M (T k (α 0 )) M (T k (α )). Hence, one has (M (T k (α 0 )) 2N) < (M (T k (α )) 2N), since M (T k (α)) = 2N for k large enough. This yields that i (α 0 ) <i (α ). Then case (2) holds. Case (3). By the assumption (H 5 ), i 0 (α 0 )=2n 0. Without loss of generality, let α 0 <α. Then from (H 6 ) and with a similar argument to case (2), we get M (T kj (α )) 2+M (T kj (α 0 )), for j =1, 2,...,n Moreover, M (T k (α )) M (T k (α 0 )) for k k j (j =1, 2,...,n 0 + 1). Thus, i (α ) (i (α 0 )+i 0 (α 0 )) 2. That means i (α ) / [i (α 0 ),i (α 0 )+i 0 (α 0 )]. For k 1, set SE(k) =z E : z(t) =a k cos(2k 1)t + b k sin(2k 1)t} and SE m = SE(1) SE(2) SE(m). Let Ω = P m : m =1, 2,...} be a sequence of orthogonal projections. Ω is called a Galerkin approximation scheme with respect to the operator A + B α,if it satisfies the following properties:

13 Vol. 16 (2009) Periodic solutions for non-autonomous 805 (1) The image of P m, as a subspace of SE, has finite dimension; (2) P m z z as m for any z SE; (3) P m commutes with A + B α. Let P m : SE SE m be the orthogonal projection. One may check easily that P m satisfies the above properties (1 3). Thus, Ω is a Galerkin approximation scheme with respect to the operator A + B α. Let A m and ψ SEm be the restrictions of A and ψ SE on SE m, respectively. We have the following important lemma. Lemma 2.7. Let (H 1 ) and (H 2 ) hold. If α kctan 2l 1 4N π, for all k Z+ and 1 l 2N, then (1) ψ SE satisfies (PS) condition over SE, i.e., every sequence z j } SE with z j SE j, ψ SEj (z j ) 0 and ψ SE (z j ) being bounded, possesses a convergent subsequence. (2) ψ SEj satisfies (PS) condition, i.e., every sequence z i } SE j, ψ SEj (z i ) 0 and ψ SEj (z i ) being bounded, possesses a convergent subsequence. Proof. From Lemma 2.6, α kctan 2l 1 4N π, for all k Z+ and 1 l 2N means that i 0 (α ) = 0 which yields that A + B α has a bounded inverse. Then the proof is standard. Observe that our functional ψ SEm acts on the finite dimensional space SE m. The following two lemmas are well known results in critical point theory. Please consult [2, 12, 13]. Lemma 2.8. Let ψ SEm be a C 1 function satisfying (2.22) and (2.23). IfM 0 (A m + P m B α0 P m )=M 0 (A m +P m B α P m )=0and M (A m +P m B α0 P m ) M (A m + P m B α P m ), then ψ SEm has at least one nontrivial critical point. Lemma 2.9. Let ψ SEm be a C 2 function satisfying (2.22) and (2.23). IfM 0 (A m + P m B α P m )=0and M (A m +P m B α P m ) / [M (A m +P m B α0 P m ),M (A m + P m B α0 P m )+M 0 (A m +P m B α0 P m )], then ψ SEm has at least one nontrivial critical point. To apply Lemmas 2.8 and 2.9 to get critical points of ψ SE, we introduce the following lemma. Lemma If α kctan 2l 1 4N π, for all k Z+ and 1 l 2N, then M (A m + P m B α P m ) M (A m )=M (A k + P k B α P k ) M (A k ) for m, k large enough. Note that α kctan 2l 1 4N π, for all k Z+ and 1 l 2N, means that i 0 (α) = 0. Then the proof is the same as in [13]. For the linear compact operator B α, we define I (B α )=k = M (A m + P m B α P m ) M (A m ), for infinitely many m}, I 0 (B α )=k = M 0 (A m + P m B α P m ), for infinitely many m}.

14 806 R. Cheng and J. Xu NoDEA Let I (ψ, ) =I (B α ), I (ψ, 0) = I (B α0 ); I 0 (ψ, ) =I 0 (B α ), I 0 (ψ, 0) = I 0 (B α0 ). By Lemma 2.10, the index above is singleton and well defined. Now we prove the following lemma. Lemma With respect to the approximation scheme Ω, one has I (ψ SE, 0) = (M (T k (α 0 )) 2N) =i (α 0 ), I (ψ SE, ) = I 0 (ψ SE, 0) = (M (T k (α )) 2N) =i (α ), M 0 (T k (α 0 )) = i 0 (α 0 ). Proof. It follows from the definition of the negative Morse index of T k (α 0 )and (2.21) that the dimension of the negative eigenspace of the operator A m +P m B α0 P m on SE m = m j=1 SE(j) is equal to m (M (T k (α 0 )). For α 0 =0,M (T k (α 0 )= 2N. Therefore, the formula I (ψ SE, 0) = (M (T k (α 0 )) 2N) =i (α 0 ) holds. Similarly, we have the other formulas Proof of Theorem 2.1 We are now ready to prove the main results. We first give the proof of Theorem 2.1. ProofofTheorem2.1 Under the assumptions of Theorem 2.1 and by (2.23) and Lemma 2.6, forɛ = 1 2 (A + B α ) 1 1, there is a large R>0 such that That yields ψ SE (z) (A + B α )(z) <ɛ z for z >R. ψ SE (z) (A + B α ) 1 1 z ψ SE (z) (A + B α )(z) >ɛ z for z >R. This means ψ SE has no critical points outside the ball B R. Since A + B α0 has bounded inverse and (I P m )B α0 0asm 0, there exists a constant C such that (A m + P m B α0 P m ) 1 <C.By(2.22), we can take r small enough with r<rsuch that ψ SE (z) (A + B α0 )(z) < 1 z for z <r. 2C

15 Vol. 16 (2009) Periodic solutions for non-autonomous 807 Thus, for m large enough and for z SE m with z <r, one has ψ SE (z) (A m + P m B α0 P m )(z) ψ SE (z) (A + B α0 )(z) < 1 2C z From the above inequality, we get < 1 2 (A m + P m B α0 P m ) 1 1 z. ψ SE (z) > (A m + P m B α0 P m )(z) 1 2 (A m + P m B α0 P m ) 1 1 z > (A m + P m B α0 P m ) 1 1 z 1 2 (A m + P m B α0 P m ) 1 1 z = 1 2 (A m + P m B α0 P m ) 1 1 z. Therefore, 0 is the unique critical point of ψ SE inside B r. By Lemmas 2.6 and 2.11, M (A m + P m B α0 P m ) M (A m + P m B α P m ). Then Lemmas 2.3 and 2.8 yield ψ SEm has a nontrivial critical point z m with r< z m <R. By Lemma 2.7, ψ SE satisfies the (PS) condition. So, z m } has a subsequence convergent to a point z, which is just a critical point of ψ SE with r< z <R Proof of Theorem 2.2 In this section, we prove Theorem 2.2. ProofofTheorem2.2 Since M 0 (T m (α 0 )) = 0 for m large enough, the null space of (A + B α0 ) SE is included in SE m, where (A + B α0 ) SE is the restriction of A + B α0 on SE. Thus, there exists a constant C such that (A m + P m B α0 P m ) <C for m large enough, where (A m + P m B α0 P m ) denotes the inverse of A m + P m B α0 P m restricted in the range of A m + P m B α0 P m.by(2.22) we can take a small r such that ψ SEm (z) (A + B α0 ) 1 as z <r. 2C As z < 2r, one has ψ SEm (z) (A m + P m B α0 P m ) ψ SEm (z) (A + B α0 ) 1 2C < 1 2 (A m + P m B α0 P m ) 1. Then by a similar argument with [13, Theorem 1.3], we have the fact that if ψ SEm has critical points, then at least one of them is outside B r. By Lemmas 2.6 and

16 808 R. Cheng and J. Xu NoDEA 2.11, we have for m large enough M (A m + P m B α P m ) / [ M (A m + P m B α0 P m ),M (A m + P m B α0 P m ) +M 0 (A m + P m B α0 P m ) ]. Hence, Lemma 2.9, jointly with Lemma 2.3 and the proof of Theorem 2.1 yields that ψ SEm has a nontrivial critical point z m with r< z m <R. By Lemma 2.7, ψ SE satisfies the (PS) condition. So, z m } has a subsequence convergent to a point z, which is just a critical point of ψ SE with r< z <R. References [1] Kaplan, J., Yorke, J.: Ordinary differential equations which yield periodic solutions of differential delay equations. J. Math. Anal. Appl. 48, (1974) [2] Amann, H., Zehnder, E.: Nontrivial solutions for a class of nonresonance problems and applications to nonlinear differential equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci (4) 8, (1980) [3] Nussbaum, R.: Periodic solutions of special differential delay equations: an example in non-linear functional analysis. Proc. R. Soc. Edinb. 81A, (1978) [4] Li, J., He, X., Liu, Z.: Hamiltonian symmetric groups and multiple periodic solutions of differential delay equations. Nonlinear Anal. T.M.A. 35, (1999) [5] Li, J., He, X.: Multiple periodic solutions of differential delay equations created by asymptotically linear Hamiltonian systems. Nonlinear Anal. T.M.A. 31, (1998) [6] Cima, A., Jibin, L., Llibre, J.: Periodic solutions of delay differential equations with two delays via bi-hamiltonian systems. Ann. Diff. Equ. 17, (2001) [7] Llibre, J., Tarta, A.-A.: Periodic solutions of delay equations with three delays via bi-hamiltonian systems. Nonlinear Anal. T.M.A. 64, (2006) [8] Jekel, S., Johnston, C.: A Hamiltonian with periodic orbits having several delays. J. Diff. Equ. 222, (2006) [9] Fei, G.: Multiple periodic solutions of differential delay equations via Hamiltonian systems (I). Nonlinear Anal. T.M.A. 65, (2006) [10] Fannio, L.O.: Multiple periodic solutions of Hamiltonian systems with strong resonance at infinity. Discrete Cont. Dyn. Syst. 3, (1997) [11] Long, Y., Zehnder, E.: Morse theory for forced oscillations of asymptotically linear Hamiltonian systems. In: Albeverio, S. et al. (eds.) Stochatic processes, Physics and Geometry. Proceedings of the Conference in Asconal/Locarno, Switzerland, pp , World Scientific, Singapore, 1990

17 Vol. 16 (2009) Periodic solutions for non-autonomous 809 [12] Chang, K.C.: Solutions of asymptotically linear operator equations via Morse theory. Comm. Pure Appl. Math. 34, (1981) [13] Li, S., Liu, J.: Morse theory and asymptotically linear Hamiltonian systems. J. Diff. Equ. 78, (1989) [14] Mawhin, J., Willem, M.: Critical point theory and Hamiltonian systems, vol. 74. Appl. Math. Sci. Springer, Heidelberg (1989) [15] Meyer, K.R., Hall, G.R.: Introduction to Hamiltonian dynamical systems and the n-body problem. Springer, New York (1992) [16] Liu, Z., Li, J.: Hamiltonian systems and periodic solutions for differential delay equations. Science Publishhouse, Beijing (2000) R. Cheng, J. Xu Department of Mathematics, Southeast University, Nanjing, People s Republic of China mathchr@163.com J. Xu xujun@seu.edu.cn R. Cheng College of Mathematics and Physics, Nanjing University of Information Science and Technology, Nanjing, People s Republic of China Received: 24 December Accepted: 11 June 2009.