Widths of weighted Sobolev classes with restrictions $f(a)=\cdots =f^{(k-1)}(a)=f^{(k)}(b)=\cdots =f^{(r-1)}(b)=0$ and spectra of non-linear differential equations

In this paper we prove that Kolmogorov widths of weighted Sobolev classes with restrictions $f(a)=\cdots =f^{(k-1)}(a)=f^{(k)}(b)=\cdots =f^{(r-1)}(b)=0$ in a weighted Lebesgue space and spectral numbers of some non-linear differential equation coincide. Here we suppose that the embedding is compact.