Unbounded perturbations of forced second order ordinary differential equations at resonance
Unbounded perturbations of forced second order ordinary differential equations at resonance
复制标题
DOI：
10.1016/0022-0396(87)90121-5
复制
发表时间：
1987-09
影响因子：
2.4
通讯作者：
R. Iannacci;M. N. Nkashama
中科院分区：
文献类型：
--
作者：
R. Iannacci;M. N. Nkashama
我们证明了微分方程 x ″(t)+ m 2 x (t)+ g (t, x (t))= e (t) 的至少一个解的存在性，其周期性条件为 x (0)− x (2Π)= x′(0)− x′(2Π)= 0，其中 m⩾ 0 是整数，e 可积，g 满足 Caratheodory 条件。我们的结果是在线性二阶微分方程 λ ″(t)+ λx (t)= 0, λϵ R 且 x (0)− x (2π)= x′(0)− x′(2π)= 0 的特征值 m 2 处存在共振的情况下获得的。函数 g 可能是无界的，并且“触及”特征值 (m+ 1) 2 (resp.(m− 1) 2 if m> 0) 在正测量的子集上是允许的。当周期性条件被狄利克雷或诺依曼边界条件取代时，我们的方法也适用。证明基于拓扑度、Mawhin 连续定理和 Leray-Schauder 技术。
We prove the existence of at least one solution for the differential equation x ″(t)+ m 2 x (t)+ g (t, x (t))= e (t) with periodicity conditions x (0)− x (2Π)= x′(0)− x′(2Π)= 0, where m⩾ 0 is an integer, e is integrable and g satisfies Caratheodory conditions. Our results are obtained for the case when there is resonance at the eigenvalue m 2 of the linear second order differential equation λ ″(t)+ λx (t)= 0, λϵ R with x (0)− x (2Π)= x′(0)− x′(2Π)= 0. The function g may be unbounded and “touching” of the eigenvalue (m+ 1) 2 (resp.(m− 1) 2 if m> 0) on a subset of positive measure is allowed. Our approach also works when periodicity conditions are replaced by Dirichlet or Neumann boundary conditions. The proofs are based on Topological degree, Mawhin's continuation theorem and Leray-Schauder techniques.