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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
##### Keywords:
boundary value problems; eigenvalue; resonance; existence
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