The method of lower and upper solutions for third-order periodic boundary value problems
The method of lower and upper solutions for third-order periodic boundary value problems
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DOI：
10.1006/jmaa.1995.1375
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发表时间：
1995-10
影响因子：
1.3
通讯作者：
A. Cabada
中科院分区：
文献类型：
--
作者：
A. Cabada
本文研究了问题L n u（t）= μ i（t，u（t））; u（i）（0）-u（i）（2π）= μ i ∈ R ; i = 0，.. n-1，其中Ln是n阶线性算子，λ a是Caratheodory函数，我们得到了保证单调方法有效的充分条件.为此，我们研究了算子T− 1 n的绿色函数的符号，其中Tnu = Lnu + Mu，M > 0。进一步得到了存在性结果。我们表明，得到的结果是最佳的。给定一个下解α和一个上解β，当α ≤ β或α ≥ β时，得到了α和β之间解的存在性.这一一般性研究应用于三阶问题u ∈（t）+ Pu″（t）+ Qu′（t）= μ i（t，u（t））; u（i）（0）− u（i）（2π）= μ i ∈ R ; i = 0，1，2，其中P，Q ∈ R .对于这个问题，我们得到常数M的最佳值，以保证算子T-13的绿色函数为正。
Abstract In this paper we develop the monotone method in the presence of lower and upper solutions for the problem L n u(t) = ƒ(t, u(t)); u (i) (0) − u (i) (2π) = μ i ∈ R ; i = 0,..., n − 1, with Ln an nth-order linear operator and ƒ a Caratheodory function, We obtain sufficient conditions to guarantee the validity of the monotone method for this problem. For this, we study the sign of the Green function of the operator T−1n, with Tnu = Lnu + Mu, M > 0. Furthermore existence results are obtained. We show that the results obtained are optimal. Given a lower solution α and an upper solution β, we obtain the existence of solution between α and β when either α ≤ β or α ≥ β. This general study is applied to the third-order problem u‴(t) + Pu″(t) + Qu′(t) = ƒ(t,u(t)); u (i) (0) − u (i) (2π) = μ i ∈ R ; i = 0,1,2, with P, Q ∈ R . For this problem we obtain the best value on the constant M to guarantee that the Green function of the operator T−13 is positive.