A characterization related to a two-point boundary value problem

In this short note, we establish the following result: Let $f:[0,+\infty[\to [0,+\infty[$,$\alpha:[0,1]\to ]0,+\infty[$betwocontinuousfunctions,with$f(0)=0$\mathrm{.}Assumethat,forsome$a>0$,thefunction$\xi\to {{\int_0^{\xi}f(t)dt}\over {\xi^2}}$isnon-increasingin$]0,a]$. Then, the following assertions are equivalent: $(i)$ for each $b>0$, the function $\xi\to {{\int_0^{\xi}f(t)dt}\over {\xi^2}}$isnotconstantin$]0,b]$;$(ii)$ for each $r>0$, there exists an open interval $I\subseteq ]0,+\mathrm{\infty }[$ such that, for every $\lambda \in I$, the problem \cases {-u''=\lambda\alpha(t)f(u) & in $[0,1]$\cr & \cr u>0 & in $]0,1[$\cr & \cr u(0)=u(1)=0\cr} has a solution $u$ satisfying $${\int }_0^1\mathrm{\mid }{u}^{\mathrm{\prime }}(t){\mathrm{\mid }}^{2}dt<r\mathrm{.}$$