Applications of global bifurcation to existence theorems for Sturm–Liouville problems
Volume 83 / 2004
Abstract
We prove an existence theorem for Sturm–Liouville problems $$ \cases{ u''(t) + \varphi(t,u(t),u'(t)) = 0 & \hbox{for a.e. }t\in(a,b), \cr l(u) = 0, } \tag*{$(*)$} $$ where $\varphi:[a,b]\times\mathbb R^k\times\mathbb R^k\to\mathbb R^k$ is a Carathéodory map. We assume that $\varphi(t,x,y) = m_1 \varphi_0(t,x,y) + o(|x|+|y|)$ as $|x|+|y|\to 0$ and $\varphi(t,x,y) = m_2 \varphi_0(t,x,y) + o(|x|+|y|)$ as $|x|+|y|\to \infty$, where $m_1,m_2$ are positive constants and $\varphi_0$ belongs to a class of nonlinear maps. The proof bases on global bifurcation results. We define a map $f:(0,\infty)\times C^1([a,b],\mathbb R^k)\to C^1([a,b],\mathbb R^k)$ such that if $f(1,u)=0$, then $u$ is a solution of $(*)$. Then we show that there exists a connected set ${\cal C}$ of nontrivial zeroes of $f$ such that there exist $(\lambda_1,u_1),(\lambda_2,u_2)\in{\cal C}$ with $\lambda_1<1<\lambda_2$. In the last section we give examples of maps $\varphi_0$ leading to specific existence theorems.
