Applications of global bifurcation to existence theorems for Sturm–Liouville problems

Jacek Gulgowski¹

¹ Institute of Mathematics University of Gdańsk Wita Stwosza 57 80-952 Gdań}sk, Poland

Annales Polonici Mathematici, Tome 83 (2004) no. 3, pp. 221-229

Résumé

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 $\lambda_11\lambda_2$. In the last section we give examples of maps $\varphi_0$ leading to specific existence theorems.

Détail

DOI : 10.4064/ap83-3-4

Keywords: prove existence theorem sturm liouville problems cases varphi hbox tag* * where varphi times mathbb times mathbb mathbb carath odory map assume varphi varphi varphi varphi infty where positive constants varphi belongs class nonlinear maps proof bases global bifurcation results define map infty times mathbb mathbb solution * there exists connected set cal nontrivial zeroes there exist lambda lambda cal lambda lambda section examples maps varphi leading specific existence theorems