吳 越, 安天慶

(河海大學(xué) 理學(xué)院, 南京 210098)

0 引 言

考慮非自治二階哈密頓系統(tǒng)

(1)

其中:T>0;F: [0,T]×RN→R滿足如下假設(shè)：

(H) 對每個x∈RN,F(t,x)關(guān)于t可測, 對幾乎所有的t∈[0,T],F(t,x)關(guān)于x連續(xù)可微, 且存在a∈C(R+,R+),b∈L1(0,T;R+), 使得

F(t,x)≤a(x)b(t),F(t,x)≤a(x)b(t),

對所有的x∈RN和a.e.t∈[0,T]成立.

則φ連續(xù)可微且弱下半連續(xù)[1], 其中

是Hilbert空間, 其上的范數(shù)定義為

直接計算可得

泛函φ的臨界點即為問題(1)的解[1].

目前, 利用臨界點理論研究問題(1)周期解的存在性與多重性已有許多結(jié)果[1-12]. Mawhin等[1]在F是凸函數(shù)的條件下, 利用極小極大化方法證明了問題(1)至少有一個周期解. 對于非凸情形, 文獻[2-12]給出了各種合理的限制條件, 例如次可加條件和次二次增長條件等. 除文獻[5]外, 目前已有文獻多是在條件

(2)

1 主要結(jié)果

定理1設(shè)F(t,x)=F1(t,x)+F2(x)滿足假設(shè)(H)及以下條件：

F1(t,x)≤f(t)x+g(t);

(3)

2) 存在常數(shù)α∈[0,2)和r>0, 使得對一切x,y∈RN, 有

(F2(x)-F2(y),x-y)≥-r;

(4)

3)

(5)

注1令

定理2設(shè)F(t,x)=F1(t,x)+F2(x)滿足假設(shè)(H)、 定理1中條件1)及以下條件：

(F2(x)-F2(y),x-y)≥-r;

(6)

2)

(7)

注2令

定理3設(shè)F(t,x)=F1(t,x)+F2(x)滿足假設(shè)(H)、 定理1中條件1),2)及以下條件：

1) 存在β∈[0,2)和C>0, 使得對一切x,y∈RN, 有

(F2(x)-F2(y),x-y)≤C;

(8)

2)

(9)

注3令

其中:h∈L1(0,T;RN);r>0. 此時定理3條件1)中的α=5/3, 定理3成立, 但不滿足文獻[1-12].

2 定理的證明

2.1 定理1的證明

(10)

由式(11),(12)可知

2.2 定理2的證明

(14)

(15)

由式(15),(16)可知

2.3 定理3的證明

則易知

(18)

與定理1的證法類似, 有

(19)

由式(19)及式(12), 有

其中

(22)

進而

(23)

其中0

由定理3的條件1), 有

由φ(un)的有界性及式(22),(23), 有

φ(u)→∞,u→∞.

(25)

由定理1中條件2)及Sobolev不等式, 有

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