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Existence of small solutions to a resonant boundary value problem with large nonlinearity. (English) Zbl 0699.34020
If the problem \((L(y)\equiv)y''+p(x)y'+q(x)y=0\), \(M_ 1(y)=0\), \(M_ 2(y)=0\) with general two-point linear boundary conditions has a one- dimensional space of solutions spanned by \(\phi,\eta =\pm 1\) and \(f\in L^ 1\), then the problem \(L(y)+\eta y^ 3=f,M_ 1(y)=0,M_ 2(y)=0\) has at least one solution if there exists a function w such that \(\int^{b}_{a}w(t)\phi (t)dt=0\), \(L(w)=\phi^ 3\), and the problem \(L(y)=w\phi^ 2,M_ 1(y)=0\), \(M_ 2(y)=0\) has no solution.
Reviewer: W.Seda

MSC:
34B15 Nonlinear boundary value problems for ordinary differential equations
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