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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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