A characterization related to a two-point boundary value problem

In this short note, we prove the following result: let $f:[0,+infty[ o [0,+infty[$, $alpha:[0,1] o ]0,+infty[$ be two continuous functions, with $f(0)=0$. Assume that, for some $a>0$, the function $\xi o {{int_0^{\xi}f(t)dt}over {\xi^2}}$ is non-increasing in $]0,a]$.par Then, the following assertions are equivalent:par oindent $(i)$hskip 5pt for each $b>0$, the function $\xi o {{int_0^{\xi}f(t)dt}over {\xi^2}}$ is not constant in $]0,b]$ ;par oindent $(ii)$hskip 5pt for each $r>0$, there exists an open interval $Isubseteq ]0,+infty[$ such that, for every $lambdain I$, the problem $$cases {-u''=lambdaalpha(t)f(u) & in $[0,1]$cr & cr u>0 & in $]0,1[$cr & cr u(0)=u(1)=0cr}$$ has a solution $u$ satisfying $$int_0^1|u'(t)|^2dt