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Differential equations at resonance. (English) Zbl 0843.34029
For \(p \in C [0,1] \cap C^1 (0,1)\), \(p(t) > 0\) on \((0,1)\), \({1 \over p} \in {\mathcal L}^1 [0,1]\), \(q,r \in {\mathcal L}^1_p [0,1]\), \(q(t) > 0\) a.e. on \([0,1]\) let \(A\) be a selfadjoint operator in \({\mathcal L}^2_{pq} [0,1]\), generated by the expression \(- {1 \over pq} [(py')' + pry]\) and usual selfadjoint boundary conditions. The spectrum of \(A\) is considered as consisting of simple eigenvalues \(\lambda_0 < \lambda_1 < \lambda_2 < \cdots\). If some conditions are satisfied, then the nonlinear equation \(- pqAy + pq \lambda_m y = p(t) f(t,y,py')\) has at least one solution. Here \(pf : [0,1] \times \mathbb{R}^2 \to \mathbb{R}\) is a so-called \({\mathcal L}^1\)-Carathéodory function.
Reviewer’s note. For periodic boundary conditions the simplicity of the spectrum of \(A\) is not ensured.

MSC:
34B15 Nonlinear boundary value problems for ordinary differential equations
34B24 Sturm-Liouville theory
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