Positive periodic solution for Brillouin electron beam focusing system
The authors consider the problem of existence of positive solutions of the equations of the form \[ x^{''}+a(1+\cos {2t})x=\frac{1}{x}, \tag{1} \] where \(a>0\) is a constant. The approach is first to consider the following singular Liénard equation \[ x''(t)+f(x(t))x^{'}(t)+g(t,x(t))=0, \tag{2} \] where \(f : \mathbb R\rightarrow \mathbb R\) is continuous, \(g : [0,T]\times (0, \infty)\rightarrow \mathbb R\) is an \(L^2\)-Carathéodory function and \(g(t,x)\) is unbounded as \(x\rightarrow 0^{+}\). The following theorem (Theorem 3.1) on equation (2) is proved. Assume that (\(H_1\)) There exist constants \(0<D_1<D_2\) such that if \(x\) is a positive continuous \(T\)-periodic function satisfying \[ \int_{0}^{T}g(t,x(t)) dt=0, \] then \[ D_1\leq x(\tau)\leq D_2 \] for some \(\tau \in [0,T]\); (\(H_2\)) \(\bar {g}(x)<0\) for all \(x \in (0,D_1)\), and \(\bar {g}(x)>0\) for all \(x>D_2\), where \(\bar {g}(x):= \frac{1}{T} \int_{0}^{T}g(t,x)dt\); (\(H_3\)) \(g(t,x)=g_0(x)+g_1(t,x)\), where \(g_0 \in C((0,\infty);\mathbb R)\) and \(g_1\) is an \(L^2\)-Carathéodory function above; (\(H_4\)) \[ \int_{0}^{1} g_0(x) dt=-\infty; \] then equation (2) has at least one positive \(T\)-periodic solution if \[ a(1+\cos {2t})<\frac{2\pi^2}{T^2}. \] Mawhin's topological degree theorem and some inequalities are used to prove the theorem (Theorem 3.1). Theorem 3.1 is later applied to the specialized case of Brillouin electron beam focusing equation given by the equation (1), i.e., if in equation (1), \(0<a<1\), then it has at least one positive \(\pi\)-periodic solution.
scientific article; zbMATH DE number 5933123