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On nonlinear boundary conditions involving decomposable linear functionals. (English) Zbl 1322.34038

The paper investigates the solvability of the boundary value problem \[ -y''(t)=\lambda f(t,y(t)),\quad t\in (0,1), \eqno(1) \]
\[ y(0)=H(\varphi(y)),\quad y(1)=0, \eqno (2) \] where \(\lambda>0\) is a parameter, \(f\in C([0,1]\times \mathbb{R})\), \(H\in C(\mathbb{R})\) and \[ \varphi(y)=\int_{[0,1]}y(t)d\alpha(t), \quad \alpha\in BV[0,1]. \] The author assumes that \(\varphi\) can be decomposed in a special way, namely, that there are linear functionals \(\varphi_1\) and \(\varphi_2\) such that \(\varphi(y)=\varphi_1(y)+\varphi_2(y)\) and \(\varphi_2\) sastisfies \(\varphi_2(y)\geq C_0 \|y\|\) for some constant \(C_0>0\) and all \(y\) in a suitable cone, and \(\varphi_1\) essentially captures the signed part of \(\varphi\). Under further assumptions imposed on data functions he proves that problem (1), (2) has at least one positive solution for each \(\lambda\in (0,\lambda_0)\) with \(\lambda_0\) explicitely computed. He also provides conditions giving the existence result for all \(\lambda>0\). The proofs are based on two different approaches, one involving classical degree theory and the second involving the concept of Fréchet dfferentiability at \(+\infty\).

MSC:

34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
47B40 Spectral operators, decomposable operators, well-bounded operators, etc.
47G10 Integral operators
47H11 Degree theory for nonlinear operators
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34B09 Boundary eigenvalue problems for ordinary differential equations
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