## Existence and multiplicity results for nonlinear differential equations depending on a parameter in semipositone case.(English)Zbl 1273.34023

Consider the boundary value problem $\begin{gathered} -(a(t)x')'+ b(t) x=\lambda f(t,x),\quad t\in I,\\ x(0)= x(2\pi),\;a(0)x'(0)= a(2\pi) x'(2\pi),\end{gathered}\tag{$$*$$}$ where $$I:= [0,2\pi]$$, $$\lambda$$ is a positive parameter, $$f\in \text{Car}(I\times \mathbb{R}^+,\mathbb{R})$$, $$a,b: I\to\mathbb{R}^+$$ are measurable functions satisfying $a(t)>0, \quad b(t)\geq0, \quad b(t)\not\equiv 0,\quad \int^{2\pi}_0 {dt\over a(t)}<\infty,\quad \int^{2\pi}_0 b(t)\,dt<\infty.$ The authors derive conditions on $$f(t,x)/x$$ such that there exist $$0<\underline\lambda<\overline\lambda$$ with the property that $$(*)$$ has for $$\lambda<\underline\lambda$$ no positive solution, $$(*)$$ has for $$\lambda>\overline\lambda$$ at least two positive solutions.
The proofs are based on the nonlinear alternative principle of Leray-Schauder and on Krasnosel’skii’s fixed point theorem in cones.

### MSC:

 34B09 Boundary eigenvalue problems for ordinary differential equations 34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations 47N20 Applications of operator theory to differential and integral equations
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### References:

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