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Semipositone boundary value problems with nonlocal, nonlinear boundary conditions. (English) Zbl 1318.34034

Author’s abstract: We demonstrate the existence of at least one positive solution to \[ \begin{aligned} -&y''(t)=\lambda f(t,y(t)), t\in (0,1),\\ & y(0)=H(\varphi (y)), y(1)=0,\end{aligned} \] where \(H:\mathbb R\to \mathbb R\) is a continuous function and \(\varphi :C([0,1])\to \mathbb R\) is a linear functional so that the boundary condition at \(t=0\) may be both nonlocal and nonlinear. Since the continuous function \(f:[0,1] \times \mathbb R\to \mathbb R\) may assume negative values, our results apply to semipositone problems. The classical Leray-Schauder degree is utilized to derive the existence result, which we obtain in the case wheN \(\lambda\) is small and which permits \(f\) to be negative on its entire domain. The result is illustrated by an example.

MSC:

34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34B09 Boundary eigenvalue problems for ordinary differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
47H07 Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces
47H30 Particular nonlinear operators (superposition, Hammerstein, Nemytskiń≠, Uryson, etc.)
47B40 Spectral operators, decomposable operators, well-bounded operators, etc.
47G10 Integral operators
47H11 Degree theory for nonlinear operators
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Full Text: Euclid