Multiple solutions for a super-linear three-point boundary value problem
The authors study the three-point BVP \[ x''(t)+g(x(t))=p(t,x(t),x'(t)),\quad x(0)=0, x(\eta)=\beta x(1), \tag{1} \] where \(\eta\in (0,1)\), \(\beta>0, \beta\not=1\), \(g: \mathbb{R}\rightarrow \mathbb{R}\) is a continuous function with \(g(0)=0\), \(g(x)x>0\) for \(x\not= 0\). Moreover \(g(x)/x\) is supposed to be increasing and \(\lim_{| x| \rightarrow\infty}(g(x)/x)= \infty\). Further, \(p:[0,1]\times \mathbb{R}^2 \rightarrow \mathbb{R}\) is supposed to satisfy the Carathéodory conditions and \[ | p(t,x,y)| \leq M_1(t,\max(| x| ,| y| )),\quad (t,x,y)\in [0,1]\times \mathbb{R}^2, \] where \(M_1(t,s)\) is integrable in \(t\) on \([0,1]\) and increasing in \(s\) on \([0,\infty)\) with \[ \lim_{s\rightarrow \infty} {1\over s} \int_0^1 M_1(t,s)\,dt =0. \] This problem with \(\beta =1\) was studied earlier by \textit{M. Henrard} [Adv. Differ. Equ. 2, 753--778 (1997; Zbl 1023.34501)] where a variant of the Leray-Schauder continuation theorem was used. Here, the authors use some of the ideas and results of Hernard and prove the existence of at least two solutions to (1).
super-linear three-point boundary value problem
scientific article; zbMATH DE number 1787998
scientific article