On the eigenfunction expansions associated with semilinear Sturm–Liouville-type problems

We consider semilinear second-order ordinary differential equations, mainly autonomous, in the form −u^″=f(u)+λu$-u^{″}=f\left(u\right)+\lambda u$, supplied with different sets of standard boundary conditions. Here λ$\lambda$ is a real constant or it plays the role of a spectral parameter. Mainly, we study problems in the interval (0,1)$(0,1)$. 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 _L2$L_2$ (in addition, a Riesz or Bari basis). In addition, we briefly consider a problem for the half-line (0,∞)$(0,\infty )$. 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 _L2$L_2$.