Multiple nonnegative solutions to fourth-order ordinary differential equations (Q2717975)

The author considers the existence of multiple nonnegative solutions to the equation NEWLINE\[NEWLINE\frac{d^4y}{dx^4}+\lambda h(x)f(y(x))=0,\quad \lambda >0,\tag{1}NEWLINE\]NEWLINE with the boundary condition NEWLINE\[NEWLINEy(0)=y(1)=y''(0)=y''(1)= 0 \tag{2}NEWLINE\]NEWLINE which, for \(f(y)=y\), describes the deformations of an elastic beam both of whose ends simply supported at 0 and 1. NEWLINENEWLINENEWLINEBy means of the fixed-point index, the main result obtained is given by the following theorem: Assume that \( f \in C([0,\infty),[0,\infty))\), \(f(0)=0\), \(h\in C([0,1],[0,\infty))\) and \(h(x) \not\equiv 0\) on any subinterval of \([0,1]\), respectively that \( \lim_{y \to 0^+} \frac{f(y)}{y}=0\), \(\lim_{y \to \infty} \frac{f(y)}{y}=0\) and that \(\varliminf_{y \to \infty} f(y)=\infty\). NEWLINENEWLINENEWLINEThen, for an arbitrarily given real number \(M>0\), there exists a \(\sigma>0\), such that the BVP (1), (2) has at least two nonnegative solutions \(\varphi_\lambda^{(1)}(x)\), \(\varphi_\lambda^{(2)}(x)\) for any \(\lambda \geq \sigma\), with \(\varphi_\lambda^{(i)} \not\equiv 0\), \(i=1,2\), and \( \max \varphi_\lambda^{(1)}(x)>M x\in [0,1] \).