Sufficient conditions for functions to form Riesz bases in \(L_2\) and applications to nonlinear boundary-value problems (Q2761581)

The aim of this paper is to improve a result due to \textit{P. E. Zhidkov} [Nonlinear Anal., Theory Methods and Appl. 43A, No. 4, 471-483 (2001; Zbl 0972.34016)], where the following nonlinear problem is considered NEWLINE\[NEWLINEu''= f(u^2)u,\quad u= u(x),\quad x\in (0,1),\quad u(0)= u(1)= 0,NEWLINE\]NEWLINE where \(f(u^2)u\) is continuously differentiable for \(u\in \mathbb{R}\), \(f(0)\geq 0\) and \(f(+\infty)= -\infty\). New properties of the standard system \(\{u_n\}\) in \(L_2\) are presented.