Periodic Solutions of Semilinear Equations

at Resonance with a 2n-Dimensional Kernel

Ma Shiwang^{1}, Wang Zhicheng^{2}

Abstract. In this paper, we obtain some sufficient conditions for the existence of 2π-periodic solutions of some semilinear equations at resonance where the kernel of the linear part has dimension 2n(n ≥ 1). Our technique is essentially based on the Brouwer degree theory and Mawhin’s coincidence degree theory.

Key words: Semilinear equation, resonance, periodic solution, kernel, dimension, Brouwer degree, coincidence degree.

1991 AMS Classification: 34K15

1. Introduction

For a long time, many authors have payed much attention to the existence problem of periodic solutions for the perturbed systems of ordinary as well as functional differential equations. In recent years, we see an increasing interest in the more difficult problem “at resonance” in the sense that the associated linear homogenous system has a nontrivial periodic solution. In this side, some useful techniques, say the averaging method, have been developed and many significant results have been obtained for the existence of periodic solutions to some nonlinear systems of first order differential equations at resonance that involve a small parameter (see [1,2] and references therein).

Much research has also been devoted to the study of existence results for some nonlinear systems whose nonlinearities satisfy so-called Landesman-Lazer conditions. Several of these results are mentioned in [3]. However, less is known when the linear part has a two-dimensional kernel. Some work has been done by Lazer & Leach ^{[4]}, Cesari^{[5]}, Iannacci & Nkashama^{[6]}, Nagle & Sinkala^{[7,8]} and Ma, Wang & Yu^{[9]}. To the best of our knowledge, few authors have considered the case when the linear part has dimension greater than two. In this direction, an example with

Typeset by AMS-TEX

This project was supported by NNSF of China

1). Department of Automatic Control, Huazhong University of Science and Technology

2). Dept. of Appl. Math., Hunan University, Changsha 410082, P. R. China

a three-dimensional kernel and a fourth order ordinary differential equation are considered in [8] and [10] respectively. In a recent paper^{[12]}, the results in [8] have been improved and unified by Ma, Wang & Yu.

This paper is concerned with the existence of 2π-periodic solutions for the nonlinear system of first order functional differential equations of mixed type

(1.1)  ẋ_{j}(t) = B_{j}x_{j}(t) + F_{j}(t, x(t + ·)) + p_{j}(t), j = 1, 2, ⋯, n

where x_{j}(t) ∈ R^{2}, x(t + ·) ∈ BC(R, R^{2n}) is defined by x(t + s) = (x_{1}(t + s), x_{2}(t + s), ⋯, x_{n}(t + s)), p_{j} ∈ C(R, R^{2}) is 2π-periodic, and F_{j} : R × BC(R, R^{2n}) → R^{2} is continuous, bounded and 2π-periodic in its first variable t. The constant matrix B_{j} has a pair of purely imaginary eigenvalues ±im_{j} with m_{j} some positive integer. Without loss of generality, we assume

B_{j} = ( \begin{matrix}0 & m_{j} \\ −m_{j} & 0 \end{matrix} ) , j = 1, 2, ⋯, n.

In this paper, we also need the following hypothesis

(F) There exists a permutation k_{1}, k_{2}, ⋯, k_{n} consisting of 1, 2, ⋯, n and for any positive integer j with 1 ≤ j ≤ n, there exist τ_{j} ∈ R, H_{j} ∈ BC(R^{2}, R^{2}) with the asymptotic limits H_{j}(±, ±) = lim_{r,s→±∞} H_{j}(r, s) and G_{j} : R × BC(R, R^{2n}) → R^{2}, which is continuous, bounded and 2π-periodic with respect to its first variable t, such that for any t ∈ R and φ ∈ BC(R, R^{2n}),

F_{j}(t, φ) = H_{j}(φ_{2k_{j}−1}(−τ_{j}), φ_{2k_{j}}(−τ_{j})) + G_{j}(t, φ).

2. Main Results

In order to state our main results, we need some notations. For any positive integer N, we will denote by | · | the Euclidean norm in R^{N}. We always denote by A the matrix

A = ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) .

Let m and l be some positive integers. If p ∈ C(R, R^{2}) is 2π-periodic, we set

(2.1)  p(m) := \frac{1}{2π} ∫_{0}^{2π} e^{A^{T} (ms)}p(s)ds.

where “T ” denotes the transpose and e^{·} denotes the exponential of an operater. For H ∈ C(R^{2}, R^{2}), whenever the asymptotic limits

H(±, ±) = lim_{r,s→±∞} H(r, s)

exist, we set

(2.2)  \begin{aligned}W^{H} := & \frac{1}{2π} [( \begin{matrix}1 & −1 \\ 1 & 1 \end{matrix} ) H(+, +) + ( \begin{matrix}−1 & −1 \\ 1 & −1 \end{matrix} ) H(+, −) \\ & + ( \begin{matrix}−1 & 1 \\ −1 & −1 \end{matrix} ) H(−, −) + ( \begin{matrix}1 & 1 \\ −1 & 1 \end{matrix} ) H(−, +)]; \end{aligned}

(2.3) \begin{aligned}W^{H}(m, l) := & \frac{1}{2π} [∫_{0}^{π/2} L(m, l)(s)dsH(+, +) + ∫_{π/2}^{π} L(m, l)(s)dsH(+, −) \\ & + ∫_{π}^{3π/2} L(m, l)(s)dsH(−, −) + ∫_{3π/2}^{2π} L(m, l)(s)dsH(−, +)] , \end{aligned}

where the matrix value mapping L(m, l) : R → R^{2×2} is defined by

(2.4)  L(m, l)(s) = \frac{1}{l} ∑_{k=0}^{l−1} e^{A^{T} ( \frac{m}{l} (s+2kπ))}.

It is easy to verify that if m = l, then W^{H}(m, l) = W^{H}. Finally, let X be a normed space, if G : X → R^{N} is continuous and bounded, we denote by M_{G} the supremum of G, i.e.,

(2.5)  M_{G} := sup_{x∈X} |G(x)|.

Theorem 2.1. If, in addition to (F), we assume that for any 1 ≤ j ≤ n,

(2.6)  |m_{j} − m_{k_{j}}| < \frac{1}{2}m_{k_{j}}

and

(2.7) |W^{H_{j}}(m_{j}, m_{k_{j}})| > \frac{1}{2}(M_{G_{j}} + |p_{j}(m_{j})|) + \frac{1}{2} (∑_{i=1}^{n}(M_{G_{i}} + |p_{i}(m_{i})|)^{2})^{1/2}

hold, then Eq.(1.1) has at least one 2π-periodic solution.

The following is a direct corollary of Theorem 2.1.

Corollary 2.1. If, in addition to (F), we assume that for any 1 ≤ j ≤ n, (2.6) and

(2.8)  |W^{H_{j}}(m_{j}, m_{k_{j}})| > \frac{1}{2}M_{G_{j}} + \frac{1}{2} (∑_{i=1}^{n} M_{G_{i}}^{2})^{1/2} + (∑_{j=1}^{n} |p_{j}(m_{j})|^{2})^{1/2}

hold, then Eq.(1.1) has at least one 2π-periodic solution.

3. Proof of Main Results

Let X and Z be real normed spaces with respective norms ∥ · ∥_{X} and ∥ · ∥_{Z}, and L : domL ⊂ X → Z be a linear Fredholm mapping of index zero. Let P be a continuous projection in X onto ker L, I − Q be a continuous projection in Z onto ImL, and K_{P} : ImL → domL ∩ ker P be the (unique) pseudo-inverse of L associated to P in the sense that LK_{P} z = z for all z ∈ ImL and P K_{P} = 0. Let J : ImQ → ker L be an isomorphism. In addition, we assume that N : X → Z is L-completely continuous and that ⟨·, ·⟩ is an inner product on ker L.

The folowing useful lemma is proved in [11].

Lemma 3.1 [11]. Assume that dim ker L ≥ 2 and there exist M > 0, a bounded open subset Ω_{0} ⊂ ker L with 0 ∈ Ω_{0} and ∂Ω_{0} a connected subset in ker L, such that the following conditions hold:

(I) ∥K_{P} (I − Q)Nx∥_{X} ≤ M, for all x ∈ X;

(II) For any x ∈ X with P x ∈ ∂Ω_{0} and ∥(I − P)x∥_{X} < M,

(3.1)  ⟨JQNx, JQNx⟩ > 0;

(III) There exist a continuous mapping η : Ω̄_{0} → ker L and a family of continuous mappings η_{i} : ker L → ker L (i = 1, 2, ⋯, N) satisfying

(3.2) ⟨η_{i}(u), η_{i}(u)⟩ > 0, ⟨u, η_{i}(u)⟩ ≠ 0 for i = 1, 2, ⋯, N and u ≠ 0,

such that for any u ∈ ∂Ω_{0},

(3.3)  ⟨JQNu − η(u), η_{1}η_{2} ⋯ η_{N}(u)⟩ ≠ 0;

(3.4)  ⟨JQNu, JQNu − η(u)⟩ ≠ 0.

Then Lx = Nx has at least one solution x satisfying

P x ∈ Ω̄_{0}  and  ∥(I − P)x∥_{X} ≤ M.

Let N be a positive integer. Set P_{2π}^{(N)} = {x ∈ C(R, R^{N}) : x(t + 2π) = x(t), ∀t ∈ R}, ∥x∥ = sup_{t∈R} |x(t)| = sup_{t∈[0,2π]} |x(t)|. Then P_{2π}^{(N)} ⊂ BC(R, R^{N}) is a Banach space.

Let D = diag(B_{1}, B_{2}, ⋯, B_{n}), where

B_{j} = ( \begin{matrix}0 & m_{j} \\ −m_{j} & 0 \end{matrix} ) .

Define the operator L : P_{2π}^{(2n)} → P_{2π}^{(2n)} by Lx(t) = ẋ(t) − Dx(t),

domL = {x ∈ P_{2π}^{(2n)} : ẋ(t) exists and is continuous}.

It is not hard to check that L is a Fredholm mapping of index zero. Let J : ker L → ker L be the identical operator and let P = Q : P_{2π}^{(2n)} → P_{2π}^{(2n)} be the projections defined by

(3.5)  P x(t) = \frac{1}{2π} e^{Dt} ∫_{0}^{2π} e^{D^{T} s}x(s)ds.

Then the (unique) pseudo-inverse of L associated to P, denoted by K : ImL → domL ∩ ker P, is a compact operator with ∥K∥ ≤ 2π (see [11] for details).

Define the operator N : P_{2π}^{(2n)} → P_{2π}^{(2n)} by

Nx(t) = (N_{1}x(t), N_{2}x(t), ⋯, N_{n}x(t)),
N_{j}x(t) = F_{j}(t, x(t + ·)) + p_{j}(t), j = 1, 2, ⋯, n.

Then N is continuous and takes bounded sets into bounded sets, and hence is Lcompletely continuous. Moreover, Eq.(1.1) is equivalent to the operater equation Lx = Nx.

It is easy to see that H : R^{2n} → ker L defined by

H(a) = e^{Dt}a, for a ∈ R^{2n}

is an isometric isomorphism. In this paper, we identify a ∈ R^{2n} with its image H(a) ∈ ker L, i.e., H(a) = a, a ∈ R^{2n}.

For the sake of convenience, we also introduce the following notations. Let m, l be some positive integers and H ∈ C(R^{2}, R^{2}). For any real number ρ ≥ 0, we set

(3.6)  M^{H}(ρ) := \frac{1}{2π} ∫_{0}^{2π} e^{A^{T} s}H((ρ sin s, ρ cos s)^{T})ds;

(3.7)  M^{H}(ρ, m, l) := \frac{1}{2lπ} ∫_{0}^{2lπ} e^{A^{T} ( \frac{ms}{l})}H((ρ sin s, ρ cos s)^{T})ds.

It is easy to know that for any positive integer m,

M^{H}(ρ, m, m) = M^{H}(ρ)

In what follows, the following lemmas are needed.

Lemma 3.2 [11]. Let m, l be some positive integers and 0 < r_{0} < 1. If H ∈ C(R^{2}, R^{2}) is bounded and the asymptotic limits H(±, ±) = lim_{r,s→±∞} H(r, s) exist, then

(3.8)  lim_{ρ→∞} M^{H}(rρ, m, l) = W^{H}(m, l)

and

(3.9)  lim_{ρ→∞} M^{H}(rρ) = W^{H}

uniformly for r in [r_{0}, 1].

Lemma 3.3. For any permutation k_{1}, k_{2}, ⋯, k_{n} consisting of 1, 2, ⋯, n there exists a family of continuous mappings η_{i} : R^{2n} → R^{2n} (i = 1, 2, ⋯, N_{1}) with

(3.10)  ⟨η_{i}(u), η_{i}(u)⟩ > 0, ⟨u, η_{i}(u)⟩ > 0, for u ∈ R^{2n}\{0},

such that for any a_{j} ∈ R^{2} (j = 1, 2, ⋯, n),

(3.11)  η_{1}η_{2} ⋯ η_{N_{1}}(a_{1}, a_{2}, ⋯, a_{n}) = (a_{k_{1}}, a_{k_{2}}, ⋯, a_{k_{n}})

holds.

Proof. It suffices to show that there exists a family of continuous mappings ζ_{i} : R^{2n} → R^{2n} (i = 1, 2, ⋯, n_{1}) with

(3.12)  ⟨ζ_{i}(u), ζ_{i}(u)⟩ > 0, ⟨u, ζ_{i}(u)⟩ > 0, for u ∈ R^{2n}\{0},

such that for any a_{j} ∈ R^{2} (j = 1, 2, ⋯, n) and 1 ≤ j_{1} < j_{2} ≤ n,

(3.13) ζ_{1}ζ_{2} ⋯ ζ_{n_{1}}(a_{1}, ⋯, a_{j_{1}}, ⋯, a_{j_{2}}, ⋯, a_{n}) = (a_{1}, ⋯, a_{j_{2}}, ⋯, a_{j_{1}}, ⋯, a_{n}),

Define ζ_{i} : R^{2n} → R^{2n} (i = 1, 2, ⋯, 6) by

ζ_{i}(u) = (ζ_{i}^{(1)}, ζ_{i}^{(2)}, ⋯, ζ_{i}^{(n)}),

ζ_{i}^{(k)} = \{ \begin{matrix}u_{k}, k ≠ j_{1}, j_{2} \\ \frac{\sqrt{2}}{2}u_{j_{1}} + \frac{\sqrt{2}}{2}u_{j_{2}}, k = j_{1} \\ − \frac{\sqrt{2}}{2}u_{j_{1}} + \frac{\sqrt{2}}{2}u_{j_{2}}, k = j_{2} \end{matrix} , i = 1, 2

ζ_{i}^{(k)} = \{ \begin{matrix}u_{k}, k ≠ j_{1} \\ ((\frac{\sqrt{2}}{2}, \frac{−\sqrt{2}}{2})u_{j_{1}}^{T}, (\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})u_{j_{1}}^{T}), k = j_{1} \end{matrix} , i = 3, 4, 5, 6

where u = (u_{1}, u_{2}, ⋯, u_{n}) ∈ R^{2n}, u_{k} ∈ R^{2}, k = 1, 2, ⋯, n.

It is easy to see that ζ_{i} (i = 1, 2, ⋯, 6) is continuous, moreover, (3.12) and (3.13) hold with n_{1} = 6. This completes the proof.

We are now in a position to prove our main result.

Proof of Theorem 2.1. Let M = 4π[(∑_{j=1}^{n} M_{F_{j}}^{2})^{1/2} + (∑_{j=1}^{n} ∥p_{j}∥^{2})^{1/2}], then

for any x ∈ P_{2π}^{(2n)}, ∥K(I − Q)Nx∥ ≤ M, and hence the condition (I) of Lemma 3.1 holds.

Let ρ > 0, take

Ω_{0} = {u ∈ ker L : u = (r_{1}ρa_{1}, r_{2}ρa_{2}, ⋯, r_{n}ρa_{n}), a_{s} ∈ ∂B_{1}(0) ⊂ R^{2},
0 ≤ r_{s} < 1, s = 1, 2, ⋯, n}.

Then Ω_{0} is a bounded open set in ker L and

∂Ω_{0} = ⋃_{j=1}^{n}{u ∈ ker L : u = (r_{1}ρa_{1}, r_{2}ρa_{2}, ⋯, r_{n}ρa_{n}), a_{s} ∈ ∂B_{1}(0) ⊂ R^{2},
0 ≤ r_{s} ≤ 1, s = 1, 2, ⋯, n, r_{j} = 1}

For x ∈ P_{2π}^{(2n)} with x_{j}(t) = r_{j}ρe^{B_{j}t}a_{j} + x̄_{j}(t), a_{j} ∈ ∂B_{1}(0) = {a ∈ R^{2} : |a| = 1} ⊂ R^{2}; x̄ = (x̄_{1}, x̄_{2}, ⋯, x̄_{n}) ∈ ImL, x̄_{j}(t) ∈ R^{2}, ∥x̄_{j}∥ ≤ M, j = 1, 2, ⋯, n, it is not hard to verify that

(3.14)  JQNx = ((JQNx)_{1}, (JQNx)_{2}, ⋯, (JQNx)_{n}),

(3.15)  (JQNx)_{j} = e^{B_{j}^{T} τ_{j}}Y_{j}(ρ, a_{k_{j}}, r_{k_{j}}) + X_{j}(x) + p_{j}(m_{j}),

where

(3.16)  X_{j}(x) = \frac{1}{2π} ∫_{0}^{2π} e^{B_{j}^{T} s}G_{j}(s, x(s + ·))ds,

(3.17)  Y_{j}(ρ, a_{k_{j}}, r_{k_{j}}) = \frac{1}{2π} ∫_{0}^{2π} e^{B_{j}^{T} s}H_{j}(r_{k_{j}}ρe^{B_{k_{j}} s}a_{k_{j}} + x̄_{k_{j}}(s + τ))ds.

By using the fact that ∥x̄_{k_{j}}∥ ≤ M and a similar argument used in the proof of Lemma 3.2, it is not hard to show that

(3.18)  lim_{ρ→∞} |Y_{j}(ρ, a_{k_{j}}, 1)| = |W^{H_{j}}(m_{j}, m_{k_{j}})|

uniformly for a_{k_{j}} in ∂B_{1}(0) ⊂ R_{2} and ∥x̄_{k_{j}}∥ ≤ M.

It follows from (2.7) and (3.16) that

(3.19)  |W^{H_{j}}(m_{j}, m_{k_{j}})| > M_{G_{j}} + |p_{j}(m_{j})| ≥ |X_{j}(x)| + |p_{j}(m_{j})|

If r_{j_{0}} = 1 for some j_{0}, then (3.14), (3.15), (3.18) and (3.19) imply that for ρ sufficiently large,

JQN(x) ≠ 0.

Thus, we have proved that for ρ sufficiently large,

JQN(x) ≠ 0

for any x ∈ P_{2π}^{(2n)} with P x ∈ ∂Ω_{0} and ∥(I − P)x∥ ≤ M, that is, the condition (II) of Lemma 3.1 also holds.

Define the mapping η : Ω̄_{0} → ker L by

η(u) = (η^{(1)}(u), η^{(2)}(u), ⋯, η^{(n)}(u)),

η^{(j)}(u) = \frac{1}{2π} ∫_{0}^{2π} e^{B_{j}^{T} s}G_{j}(s, r_{1}ρe^{B_{1}(s+·)}a_{1}, ⋯, r_{n}ρe^{B_{n}(s+·)}a_{n})ds + p_{j}(m_{j}),

u = (r_{1}ρa_{1}, ⋯, r_{n}ρa_{n}), a_{j} ∈ ∂B_{1}(0) ⊂ R^{2}, 0 ≤ r_{j} ≤ 1, j = 1, 2, ⋯, n.

Then it is easy to see that η is continuous.

Let β_{j} (−π < β_{j} ≤ π) be defined by

sin β_{j} = \frac{W_{1}^{H_{j}}(m_{j}, m_{k_{j}})}{|W^{H_{j}}(m_{j}, m_{k_{j}})|}, cos β_{j} = \frac{W_{2}^{H_{j}}(m_{j}, m_{k_{j}})}{|W^{H_{j}}(m_{j}, m_{k_{j}})|}

here

W^{H_{j}}(m_{j}, m_{k_{j}}) = (W_{1}^{H_{j}}(m_{j}, m_{k_{j}}), W_{2}^{H_{j}}(m_{j}, m_{k_{j}})).

Let N_{2} be a positive integer satisfying

|\frac{τ_{j} − β_{j}/m_{j}}{N_{2}} | < \frac{π}{2m_{j}} , j = 1, 2, ⋯, n.

Define η_{i} : ker L → ker L (i = 1, 2, ⋯, N_{2}) by

η_{i}(u) = (η_{i}^{(1)}(u), η_{i}^{(2)}(u), ⋯, η_{i}^{(n)}(u)),

η_{i}^{(j)}(u) = e^{B_{j}^{T} γ_{j}}u_{j}, γ_{j} = \frac{τ_{j} − β_{j}/m_{j}}{N_{2}} , j = 1, 2, ⋯, n,

where u = (u_{1}, u_{2}, ⋯, u_{n}), u_{j} ∈ R^{2} (j = 1, 2, ⋯, n). Then it is clear that η_{i} (i = 1, 2, ⋯, N_{2}) is continuous and (3.2) holds.

By Lemma 3.3, we can also define η_{i} : ker L → ker L (i = N_{2} + 1, ⋯, N_{2} + N_{1}), which are continuous and satisfy (3.2), such that

η_{N_{2}+1}η_{N_{2}+2} ⋯ η_{N_{2}+N_{1}}(a_{1}, a_{2}, ⋯, a_{n}) = (a_{k_{1}}, a_{k_{2}}, ⋯, a_{k_{n}})

holds for any a_{j} ∈ R^{2} (j = 1, 2, ⋯, n).

In the sequel, we assume that u = (r_{1}ρa_{1}, r_{2}ρa_{2}, ⋯, r_{n}ρa_{n}) ∈ ∂Ω_{0}, a_{j} ∈ ∂B_{1}(0) ⊂ R^{2}, 0 ≤ r_{j} ≤ 1 j = 1, 2, ⋯, n. Clearly, we may assume, without loss of generality, that r_{k_{j_{0}}} = 1 for some j_{0}.

Let α_{j} = α_{j}(a_{j}) (−π < α_{j} ≤ π) defined by sin α_{j} = a_{j}^{(1)}, cos α_{j} = a_{j}^{(2)}, where a_{j} = (a_{j}^{(1)}, a_{j}^{(2)}).

Therefore, we have

(3.20)  η̄(u) := η_{1}η_{2} ⋯ η_{N_{2}+N_{1}}(u) = (η̄_{1}(u), η̄_{2}(u), ⋯, η̄_{n}(u)),

(3.21) η̄_{j}(u) = r_{k_{j}}ρe^{B_{j}^{T} (τ_{j}−β_{j}/m_{j})}a_{k_{j}} = ρr_{k_{j}}e^{B_{j}^{T} τ_{j}}e^{Aα_{k_{j}}} W^{H_{j}}(m_{j}, m_{k_{j}})/|W^{H_{j}}(m_{j}, m_{k_{j}})|

(3.22)  ξ̄(u) := JQNu − η(u) = (ξ̄_{1}(u), ξ̄_{2}(u), ⋯, ξ̄_{n}(u)),

(3.23)  \begin{aligned}ξ̄_{j}(u) & = \frac{1}{2π} e^{B_{j}^{T} τ_{j}} ∫_{0}^{2π} e^{B_{j}^{T} s}H_{j}(r_{k_{j}}ρe^{B_{k_{j}} s}a_{k_{j}})ds \\ & = e^{B_{j}^{T} τ_{j}}e^{A(m_{j}α_{k_{j}} /m_{k_{j}})}M^{H_{j}}(r_{k_{j}}ρ, m_{j}, m_{k_{j}}). \end{aligned}

It follows from (3.20)-(3.23) that

(3.24) \begin{matrix}⟨ξ̄(u), η̄(u)⟩ \\ = ρ ∑_{j=1}^{n} \frac{r_{k_{j}}}{|W^{H_{j}}(m_{j}, m_{k_{j}})|}⟨W^{H_{j}}(m_{j}, m_{k_{j}}), e^{Aϖ(j)α_{k_{j}}} M^{H_{j}}(r_{k_{j}}ρ, m_{j}, m_{k_{j}})⟩. \end{matrix}

where ϖ(j) = \frac{m_{j}−m_{k_{j}}}{m_{k_{j}}}.

Since |α_{k_{j}}| ≤ π and |m_{j} − m_{k_{j}}| < \frac{1}{2}m_{k_{j}}, we have

|ϖ(j)α_{k_{j}} | ≤ |ϖ(j)π| < \frac{π}{2} , j = 1, 2, ⋯, n.

Hence,

⟨W^{H_{j}}(m_{j}, m_{k_{j}}), e^{Aϖ(j)α_{k_{j}}} W^{H_{j}}(m_{j}, m_{k_{j}})⟩ = |W^{H_{j}}(m_{j}, m_{k_{j}})|^{2} cos (ϖ(j)α_{k_{j}})

(3.25)  ≥ |W^{H_{j}}(m_{j}, m_{k_{j}})|^{2} cos (ϖ(j)π) > 0,

For j ≠ j_{0}, we set

I_{j}^{0} = [0, |W^{H_{j_{0}}} (m_{j_{0}}, m_{k_{j_{0}}} )| cos (ϖ(j_{0})π)/(4nM_{H_{j}})) ,
I_{j}^{1} = [|W^{H_{j_{0}}} (m_{j_{0}}, m_{k_{j_{0}}} )| cos (ϖ(j_{0})π)/(4nM_{H_{j}}), 1] ,

then by (3.24), and noting that r_{k_{j_{0}}} = 1, we have

(3.26)  ⟨ξ̄(u), η̄(u)⟩ = ρ [Z_{0} + Z_{1} + Z_{2}]

where, (3.27)

Z_{0} = \frac{1}{|W^{H_{j_{0}}} (m_{j_{0}}, m_{k_{j_{0}}} )|}⟨W^{H_{j_{0}}} (m_{j_{0}}, m_{k_{j_{0}}} ), e^{A(ϖ(j_{0})α_{k_{j_{0}}} )}M^{H_{j_{0}}} (ρ, m_{j_{0}}, m_{k_{j_{0}}} )⟩

(3.28)

Z_{1} = ∑_{j≠j_{0},r_{k_{j}} ∈I_{j}^{0}} \frac{r_{k_{j}}}{|W^{H_{j}}(m_{j}, m_{k_{j}})|}⟨W^{H_{j}}(m_{j}, m_{k_{j}}), e^{A(ϖ(j)α_{k_{j}})}M^{H_{j}}(r_{k_{j}}ρ, m_{j}, m_{k_{j}})⟩

(3.29)

Z_{2} = ∑_{j≠j_{0},r_{k_{j}} ∈I_{j}^{1}} \frac{r_{k_{j}}}{|W^{H_{j}}(m_{j}, m_{k_{j}})|}⟨W^{H_{j}}(m_{j}, m_{k_{j}}), e^{A(ϖ(j)α_{k_{j}})}M^{H_{j}}(r_{k_{j}}ρ, m_{j}, m_{k_{j}})⟩

Since M^{H_{j}}(r_{k_{j}}ρ, m_{j}, m_{k_{j}}) → W^{H_{j}}(m_{j}, m_{k_{j}}) (ρ → ∞) uniformly for r_{k_{j}} in I_{j}^{1} by Lemma 3.2, we have

(3.30)  Z_{2} > 0,

for large ρ.

By the Schwartz inequility, we also find

(3.31)  |Z_{1}| ≤ ∑_{j≠j_{0},r_{k_{j}} ∈I_{j}^{0}} r_{k_{j}}M_{H_{j}} ≤ |W^{H_{j_{0}}} (m_{j_{0}}, m_{k_{j_{0}}} )| cos (ϖ(j_{0})π)/4.

Therefore, it follows from (3.26)-(3.31) that for ρ suffciently large,

⟨ξ̄(u), η̄(u)⟩ > 0.

Thus, (3.3) holds for any u ∈ ∂Ω_{0}.

On the other hand, it is not hard to show that

⟨JQNu, JQNu − η(u)⟩

= ∑_{j=1}^{n} |M^{H_{j}}(r_{k_{j}}ρ, m_{j}, m_{k_{j}})|^{2}

+ ∑_{j=1}^{n}⟨M^{H_{j}}(r_{k_{j}}ρ, m_{j}, m_{k_{j}}), e^{A^{T} (m_{j}α_{k_{j}} /m_{k_{j}})}e^{B_{j}τ_{j}}X_{j}(u)⟩

+ ∑_{j=1}^{n}⟨M^{H_{j}}(r_{k_{j}}ρ, m_{j}, m_{k_{j}}), e^{A^{T} (m_{j}α_{k_{j}} /m_{k_{j}})}e^{B_{j}τ_{j}}p_{j}(m_{j})⟩,

where

X_{j}(u) = \frac{1}{2π} ∫_{0}^{2π} e^{B_{j}^{T} s}G_{j}(s, r_{1}ρe^{B_{1}(s+·)}a_{1}, ⋯, r_{n}ρe^{B_{n}(s+·)}a_{n})ds.

By using the Schwartz inequality, it follows that

⟨JQNu, JQNu − η(u)⟩ ≥ ∑_{j=1}^{n} |M^{H_{j}}(r_{k_{j}}ρ, m_{j}, m_{k_{j}})|^{2}

− ∑_{j=1}^{n} |M^{H_{j}}(r_{k_{j}}ρ, m_{j}, m_{k_{j}})|[M_{G_{j}} + |p_{j}(m_{j})|]

= ∑_{j=1}^{n}[|M^{H_{j}}(r_{k_{j}}ρ, m_{j}, m_{k_{j}})| − \frac{1}{2}(M_{G_{j}} + |p_{j}(m_{j})|)]^{2}

− \frac{1}{4} ∑_{j=1}^{n}(M_{G_{j}} + |p_{j}(m_{j})|)^{2}.

Since r_{k_{j_{0}}} = 1 for some j_{0}, and M^{H_{j_{0}}} (ρ, m_{j_{0}}, m_{k_{j_{0}}} ) → W^{H_{j_{0}}} (m_{j_{0}}, m_{k_{j_{0}}} ) (ρ → ∞), it follows from (2.7) that for ρ sufficiently large,

⟨JQNu, JQNu − η(u)⟩ > 0.

Thus, (3.4) also holds for any u ∈ ∂Ω_{0}.

By virtue of Lemma 3.1, Eq.(2.1) has at least one 2π-periodic solution and the proof is complete.

Finally, we give an example to illustrate our main results. Example Consider the system

(3.32)  \{ \begin{matrix}x′_{1} & = x_{2} + x_{3}/(1 + x_{3}^{2}) + arctan x_{1} + p_{1}(t) \\ x′_{2} & = −x_{1} + 3 arctan x_{4} + \frac{1}{2} arctan x_{2} + p_{2}(t) \\ x′_{3} & = x_{4} + x_{5}e^{−x_{5}^{2}} + \sqrt{2} sin x_{3} + p_{3}(t) \\ x′_{4} & = −x_{3} − \sqrt{6} arctan x_{6} + \sqrt{2} cos x_{3} + p_{4}(t) \\ x′_{5} & = x_{6} + arctan x_{5} − 2 arctan x_{1} + p_{5}(t) \\ x′_{6} & = −x_{5} + \frac{1}{2} arctan x_{6} − 2 arctan x_{2} + p_{6}(t) \end{matrix}

where p_{j}(j = 1, 2, ⋯, 6) are continuous, 2π-periodic functions. By Corollary 2.1, it is easy to check that Eq.(3.32) has at least one 2π−periodic solution provided

\sqrt{|c_{1}|^{2} + |c_{2}|^{2} + |c_{3}|^{2}} < \sqrt{6} − \frac{\sqrt{2}}{2} − \frac{1}{4\sqrt{2}}\sqrt{5π^{2} + 32},

where

c_{k} = \frac{1}{2π} ∫_{0}^{2π} ( \begin{matrix}cos s & − sin s \\ sin s & cos s \end{matrix} ) ( \begin{matrix}p_{2k−1}(s) \\ p_{2k}(s) \end{matrix} ) ds, k = 1, 2, 3.

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