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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
##### Keywords:
two-point linear boundary conditions
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##### References:
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