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Existence of three solutions for a nonautonomous two point boundary value problem. (English) Zbl 0980.34015

Here, the author considers the two-point boundary value problem

${u}^{\text{'}\text{'}}+\lambda f\left(t,u\right)=0,\phantom{\rule{1.em}{0ex}}u\left(a\right)=u\left(b\right)=0,\phantom{\rule{2.em}{0ex}}\left(1\right)$

where $f:\left[a,b\right]×ℝ\to ℝ$ is a continuous function. Under some assumptions with respect to $f\left(t,u\right)$ there exist an open interval ${\Lambda }\subseteq \right]0,+\infty \left[$ and a positive real number $q$ such that, for each $\lambda \in {\Lambda }$, the problem (1) admits at least three solutions belonging to ${C}^{2}\left(\left[a,b\right]\right)$, whose norms in ${W}_{0}^{1,2}\left(\left[a,b\right]\right)$ are less than $q$.

##### MSC:
 34B15 Nonlinear boundary value problems for ODE 34B24 Sturm-Liouville theory