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On the eigenfunction expansions associated with semilinear Sturm-Liouville-type problems. (English) Zbl 1162.34363
Summary: We consider semilinear second-order ordinary differential equations, mainly autonomous, in the form $$-u{^{\prime\prime}}=f(u)+\lambda u$$, supplied with different sets of standard boundary conditions. Here $$\lambda$$ is a real constant or it plays the role of a spectral parameter. Mainly, we study problems in the interval (0,1). It is shown that in this case each problem that we deal with has an infinite sequence of solutions or eigenfunctions. Our aim in the present article is to review recent results on basis properties of sequences of these solutions or eigenfunctions. In a number of cases, it is proved that such a system is a basis in $$L_{2}$$ (in addition, a Riesz or Bari basis). In addition, we briefly consider a problem for the half-line $$(0,\infty )$$. In this case, the spectrum of the problem fills a half-line and an analog of the expansions into the Fourier integral is obtained. The proofs are mainly based on the Bari theorem and, in addition, on our general result on sufficient conditions for a sequence of functions to be a Riesz basis in $$L_{2}$$.

##### MSC:
 34L10 Eigenfunctions, eigenfunction expansions, completeness of eigenfunctions of ordinary differential operators 34B15 Nonlinear boundary value problems for ordinary differential equations 34B40 Boundary value problems on infinite intervals for ordinary differential equations 34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations 34L30 Nonlinear ordinary differential operators 34L99 Ordinary differential operators 42C15 General harmonic expansions, frames
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