Abstract
We study the multiplicity of periodic solutions of a class of nonautonomous delay differential equations. By making full use of the Clark dual, the dual variational functional is considered. Some sufficient conditions are obtained to guarantee the existence of multiple periodic solutions.
2000 Mathematics Subject Classification: 34K13; 34K18.
Keywords:
periodic solution; Clark dual; MorseEkeland index; delay differential equation; asymptotic linearity1 Introduction
The existence and multiplicity of periodic solutions of delay differential equations have been investigated since 1962. Various methods have been used to study such a problem [110]. Among those methods, critical point theory is a very important tool. By combining with KaplanYorke method, it can be used indirectly to study the existence of periodic solutions of delay differential equation [1116]. By building the variational frame for some special systems, it can be used directly to study the existence of delay differential systems [17,18]. However, the variational functionals in the above two cases are strongly indefinite. They are hard to be dealed with.
In this article, we study multiple periodic solutions of the following nonautonomous delay differential equations
where x(t) ∈ ℝ^{n}, f ∈ C (ℝ × ℝ^{n}, ℝ^{n}). We assume that
(f1) f(t, x) is odd with respect to x and π/2periodic with respect to t, i.e.,
(f2) There exists a continuous differentiable function F(t, x), which is strictly convex with respect to x uniformly in t, such that f(t, x) is the gradient of F(t, x) with respect to x;
(f3)
where A_{0}, A_{∞ }are positive definite constant matrices.
By making use of the Clarke dual, we study the dual variational functional associated with (1), which is an indefinite functional. Since the dimension of its negative space is finite, we define it as the Morse index of the dual variational functional. This Morse index is significant. Then Z_{2}index theory can be used and some sufficient conditions are obtained to guarantee the existence of multiple periodic solutions of (1).
The rest of this article is organized as follows: in Section 2, some preliminary results will be stated; in Section 3, linear system is discussed and the Morse index of the variational functional associated with linear system is defined; in Section 4, our main results will be stated and proved.
2 Preliminaries
Denote by ℕ, ℕ*, ℤ, ℝ the sets of all positive integers, nonnegative integers, integers, real numbers, respectively. We define S^{1 }: = ℝ/(2πℤ).
For a matrix A, denote by σ(A) the set of eigenvalue of A. The identity matrix of order n is denoted by I_{n }and for simplicity by I.
Let x, y ∈ L^{2 }(S^{1}, ℝ^{n}). For every z ∈ C^{∞ }(S^{1}, ℝ^{n}), if
then y is called a weak derivative of x, denoted by .
The space H^{1 }= W^{1,2}(S^{1}, ℝ^{n}) consists of 2πperiodic vectorvalued functions with dimension n, which possess square integrable derivative of order 1. We can choose the usual norm and inner product in H^{1 }as follows:
where  · , (·,·) denote the usual norm and inner product in ℝ^{n}, respectively. Then H^{1 }is a Hilbert space.
Define the shift operator K : H^{1 }→ H^{1 }by Kx(·) = x(· + π/2), for all x ∈ H^{1}. Clearly, K is a bounded linear operator from H^{1 }to H^{1}. Set
Then E is a closed subspace of H^{1}. If x ∈ E, its Fourier expansion is
where a_{k}, b_{k }∈ ℝ^{n}. In particular, E does not contain ℝ^{n }as its subspace. In addition, for all x ∈ E.
The dual variational functional corresponding to (1) defined on H^{1 }is
By Hypothesis (f3), F(t, x)/x → +∞ as x → ∞ uniformly in t. Since F(t, ·) is strictly convex, Proposition 2.4 of [19] implies that F*(t, ·) ∈ C^{1}(ℝ^{n}, ℝ). Since f satisfies (f3), it follows the discussion in Chapter 7 of [19] that
Lemma 2.1. If y ∈ E is a critical point of J, then the function x defined by is a 2πperiodic solution of (1).
Proof. Since f(·, x) is π/2periodic, it follows that F(·, x) and then are π/2periodic. So is . (4) implies that there exist positive constants a_{1}, a_{2 }such that F*(t, y) ≤ a_{1 }+ a_{2}y^{s }for some s > 2 and all t ∈ ℝ, y ∈ H^{1}. We define φ by the formulas
It follows Proposition B. 37 of [20] that φ ∈ C^{1}(H^{1}, ℝ) and
We claim: φ'(y) ∈ E if y ∈ E.
To prove the above claim, let z ∈ H^{1 }and y ∈ E,
The arbitrary of z implies that φ'(y) ∈ E.
Since φ ∈ C^{1}(H^{1}, ℝ), it is easy to check that J ∈ C^{1}(H^{1}, ℝ) and
Assume that y ∈ E is a critical point of J. For any h ∈ H^{1}, h = h_{1 }+ h_{2}, where h_{1 }∈ E, h_{2 }∈ E^{⊥}. Then
Since φ'(y) ∈ E, it is easy to check that J'(y) ∈ E and < J'(y), h_{2 }> = 0. Since y is a critical point of J on E, then < J'(y), h_{1 }> = 0. Thus y is a critical point of J on H^{1}. Applying the fundamental Lemma (cf. [19]), there exists c_{1 }such that
Setting
we obtain x ∈ H^{1}, and by duality
Thus,
i.e.,
Moreover, x(0) = x(2π) since x ∈ H^{1}. It follows a regular discussion that is a periodic solution of (1). □
Let X be a Hilbert space, Φ ∈ C^{1}(X, ℝ), i.e., Φ is a continuously Fréchet differentiable functional defined on X. If X_{1 }is a closed subspace of X, denote by the orthogonal complement of X_{1 }in X. Fix a prime integer p > 1. Define a map μ : X → X such that μx = x for any x ∈ X and μ^{p }= id_{X}, where id_{X }is the identity map on X. Then μ is a linear isometric action of Z_{p }on X, where Z_{p }is the cyclic group with order p.
A subset A ⊂ X is called μinvariant if μ(A) ⊂ A. A continuous map h : A → E is called μequivariant if h(μx) = μh(x) for any x ∈ A. A continuous functional H : X → ℝ is called μinvariant if H(μx) = H(x) for any x ∈ X.
Φ is said to be satisfying (PS)condition if any sequence {x_{j}} ⊂ X for which {Φ(x_{j})} is bounded and Φ'(x_{j}) → 0 as j → ∞, possesses a convergent subsequence. A sequence {x_{j}} is called (PS)sequence if {Φ(x_{j})} is bounded and Φ'(x_{j}) → 0 as j → ∞.
Lemma 2.2 [21]. Let Φ ∈ C^{1}(X, ℝ) be a μinvariant functional satisfying the (PS)condition. Let Y and Z be closed μinvariant subspaces of X with codimY and dimZ finite and
Assume that the following conditions are satisfied:
(F1) Fix_{μ }⊂ Y, Z ∩ Fix_{μ }= {0};
(F2) inf_{x∈Y }Φ(x) > ∞;
(F3) there exist constants r > 0 and c < 0 such that Φ(x) ≤ c whenever x ∈ Z and x = r;
(F4) if x ∈ Fix_{μ }and Φ'(x) = 0, then Φ(x) ≥ 0.
Then there exists at least dimZ  codimY distinct Z_{p}orbits of critical points of Φ outside of Fix_{μ }with critical value less than or equal to c.
3 Morse index
Let A be a positive definite constant matrix. We consider the periodic boundary value problem
Since F(t, x) = 1/2(Ax, x), it is easy to verify that its Legendre transform F*(t, y) is of the form F*(t, y) = 1/2(By, y), where B = A^{1}. The dual action of (6) is defined on H^{1 }by
A is positively definite, so is B. Thus there exists δ_{1 }> 0 such that (By, y) ≥ δ_{1}y^{2 }for all y ∈ ℝ^{n}. Wirtinger's inequality implies that the symmetric bilinear form given by
define an inner product on E. The corresponding norm  · _{1 }is such that
Proposition 3.1. The norm  · _{1 }is equivariant to the stand norm  ·  of E.
Proof. Since B is positive, Wirtinger's inequality and (9) imply that there exists a positive constant δ_{2 }such that
On the other hand, since B is a positive definite matrix, there exists a constant M > 0 such that
Thus those two norms are equivariant to each other, which completes our proof. □
Let us define the linear operator L on E by setting
One can easily check that L is a compact selfadjoint operator. (7) can be rewritten as
It follows from the spectral theory that E can be decomposed as the orthogonal sum of Ker(I  L), E^{+ }and E^{ }with I  L positive definite (resp. negative definite) on E^{+ }(resp. E^{}). Since L is compact, it has at most finite many eigenvalues (counting the multiplicity) greater than one. Thus the index dimE^{ }< ∞.
Definition 3.1. The index i(A) is the Morse index of χ_{A}, i.e. the supermum of the dimension of the subspace of E on which χ_{A }is negative definite.
On the other hand, there exists a positive constant δ_{3 }such that
Setting δ = δ_{1}δ_{3 }> 0, we reduce from (9) and (11) the estimates
Proposition 3.2. The dimension of Ker(I  L) is equal to the number of linearly independent solutions of (6).
Proof. If y ∈ E, by a Fourier argument, y ∈ Ker(I  L) if and only if
for some c_{2 }∈ ℝ^{n }and a.e. t ∈ [0, 2π]. Since B = A^{1 }is invertible, y is continuous differentiable. Thus (13) hold for all t ∈ [0, 2π], and . Set
In view of (13), (14) and the Clark dual, we have
Thus x is a solution of (6). □
Conversely, assume now that x ∈ E ⊕ ℝ^{n }is a solution of (6). Then
and hence there exists an unique y ∈ E such that x = φy, where φ is defined by (14).
Thus
for a.e. t ∈ [0, 2π]. Integrating the above equality, we have x(t) = y(t + π/2) + c_{3 }for some c_{3 }∈ ℝ^{n }and all t ∈ [0, 2π]. Consequently,
for a.e. t ∈ [0, 2π], which is equivariant to (13) and shows that y ∈ Ker(I  L). Thus φ is an isomorphism between Ker(I  L) and the space of solutions of (6).
Consider the following eigenvalue problem
For any z ∈ E, we have (((I  L)y, z))_{1 }= >λ((y, z))_{1}. Computing directly, we have
Denote by e_{i}, i = 1, 2, ..., n the basic of ℝ^{n}. Choosing
it follows that
Since B = A^{1}, then
Thus
Proposition 3.3. If σ(A) ∩ {4l + 1, l ∈ ℕ*} = ∅, then I  L is invertible and codimE^{+ }= i(A).
Proof. The above analysis implies that Ker (I  L) = {0} if σ(A) ∩ {4l + 1, l ∈ ℕ*} = ∅. Thus I  L is invertible. If we decompose E as E = E^{+ }⊕ E^{}, then codimE^{+ }= i(A). □
4 Main results and their proofs
Now we consider the multiplicity of periodic solutions of (1). The main result reads as follows:
Theorem 4.1. Assume that (f1)(f3) are satisfied. Moreover,
(A1) σ (A_{∞}) ∩ {4l + 1, l ∈ ℕ*} = ∅;
(A2) i(A_{0}) > i(A_{∞}).
Then (1) has at least i(A_{0})  i(A_{∞}) pairs of nontrivial 2πperiodic solutions.
By (f3), f (t, 0) = 0 uniformly in t. Since F(t, ·) is strictly convex, it follows that 0 is the unique equilibrium point of (1). Without loss of generality, we can assume that F (t, 0) = 0 uniformly in t. Since f (t, 0) = 0, then F*(t, 0) = 0 uniformly in t.
Lemma 4.1. The functional J satisfies (PS)condition.
Proof. Denote by ((·, ·))_{1,∞ }the inner product, defined by (8) replacing B by B_{∞}. Let L_{∞ }be linear selfadjoint operator, defined by (10), under the inner product ((·, ·))_{1,∞}. We define the operator N over E by setting
Let be a (PS)sequence. Define f_{j }: = y_{j } L_{∞ }y_{j }+ Ny_{j}, j ∈ ℕ. Since
then f_{j }→ 0 in E as j → ∞. Then there exists R > 0 such that for all j ∈ ℕ,  f_{j } ≤ R. (A1) implies that M = I  L_{∞ }is invertible. Thus it follows from (4) that there exists a positive constant c_{4 }such that, for all y ∈ E
where  · _{1,∞ }denotes by the norm corresponding to ((·, ·))_{1,∞}. Since  f_{j } ≤ R, we have
The above inequality implies that {y_{j}} is bounded. A standard argument as Lemma 4.5 in [19] shows that {y_{j}} has a convergent sequence. □
Lemma 4.2. The functional J is bounded from below on a closed invariant subspace Y of E of codimension i(A_{∞}).
Proof. Consider the linear delay differential system
Its dual variational functional is
where ((·, ·))_{1,∞}, L_{∞ }are defined as Lemma 4.1.
Because of (A1) and Proposition 3.3, the operator I  L_{∞ }is invertible. We decompose E as , where is the positive (resp. negative) definite eigenspace of I  L_{∞}. Then . Let . Inequality (12) implies that there exists a positive constant δ_{4 }such that, for each y ∈ Y,
It follows (4) that there exists a positive constant c_{5 }such that
for each y ∈ ℝ^{n}. Hence, by the mean value theorem,
Consequently, we have, for y ∈ Y,
and J is bounded from below on Y.
Lemma 4.3. There exists an invariant subspace Z of E with dimension i(A_{0}) and some r > 0 such that J(y) < 0 whenever y ∈ Z and y_{1 }= r.
Proof. Consider the linear delay differential system
Its dual variational functional is
where ((·, ·))_{1,0}, L_{0 }are defined by (8) and (10) with B replaced by B_{0}, respectively.
We decompose E as , where , is the positive (resp. negative) definite eigenspace of I  L_{0}. Let . Then Z is a finite dimensional space. Inequality (12) implies that there exists δ_{5 }> 0 such that, for each y ∈ Z,
whenever y ∈ Z. By (5), there exists ρ > 0 such that
for y ∈ ℝ^{n }with y ≤ ρ. Hence, by the mean value theorem, we have
whenever y ≤ ρ. Consequently, if y ∈ Z and 0 < y < ρ, we get
If J(y) = 0, then , which implies that for a.e. t ∈ [0, 2π]. Thus y ∈ ℝ^{n }∩ Z = {0}, which contradicts with 0 < y < ρ. Thus J(y) < 0.
Proof of Theorem 4.1. Now, we apply Lemma 2.2 to prove Theorem 4.1.
Define the action μ : E → E by μx = x. Then μ is a generator of group Z_{2 }and Fix_{μ }= {0}. Obviously, (F1) and (F4) hold. It is easy to check that J is μinvariant. Lemma 4.1 implies that J satisfies (PS)condition.
Let Y, Z define as in Lemma 4.2 and 4.3. Then Y, Z are μinvariant subspaces and (A2) implies that
It follows from Lemmas 4.2 and 4.3 that (F2) and (F3) hold. Applying Lemma 2.2, J has at least i(A_{0})  i(A_{∞}) pairs of distinct critical points with critical values less than or equal to c. Since J(0) = 0 and E ∩ ℝ^{n }= {0}, if follows that all those i(A_{0})  i(A_{∞}) pairs of distinct critical points are nonconstant.
If y is a critical point of J, then is a 2πperiodic solution of (1). Clearly, y ≠ 0 implies that x ≠ 0. Let x_{1}, x_{2 }are two periodic solutions satisfying x_{1 }= x_{2}. Setting and , then
Integrating he above equality, we have y_{1 }=  y_{2 }+ c_{6 }for some c_{6 }∈ ℝ^{n}. Since y_{1}, y_{2 }∈ E, it follows that c_{6 }= 0. Then y_{1}, y_{2 }belongs the same Z_{2}orbit. Thus (1) has i(A_{0})  i(A_{∞}) pairs of nontrivial periodic solutions.
Corollary 4.1. Under the hypotheses of Theorem 4.1, (1) has at least 2[i(A_{0})  i(A_{∞})] nontrivial 2πperiodic solutions.
Since every Z_{2}orbit has two elements and those two elements are different from each other, this corollary is obvious.
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
The manuscript was drafted by HX and it is based on his PhD thesis. JY and ZG were the supervisors of the thesis and gave detailed comments on the manuscript. All authors read and approved the final manuscript.
Acknowledgements
This project was supported by the National Natural Science Foundation of China (Nos. 11031002 and 10871053). The authors are very grateful to the anonymous referee whose careful reading of the manuscript and valuable comments enhanced presentation of the manuscript.
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