On a Quadratic Eigenvalue Problem and its Applications

Solving an initial boundary value problem on thesemiinfinite interval

“…In [2], we proved that under conditions (9) and (11) (ii) If b 0 = 0 , then Eq. (4) has, besides the zero eigenvalue, precisely 2N − 2 distinct nonzero real eigenvalues so that in this case the zero eigenvalue is defective (there is a generalized eigenvector associated with the eigenvector corresponding to the zero eigenvalue).…”

“…We know that there exist one zero eigenvalue and 2N − 1 nonzero distinct real eigenvalues provided that r 0 = 0 , r j > 0 for j = 1, · · · , N − 1 and b 0 ̸ = 0 ; see [2]. We denote the nonzero eigenvalues by Proof If…”

“…When ϵ = 0 (b 0 ̸ = 0), we know from [2] that zero is an eigenvalue, and besides there are 2N − 1 nonzero eigenvalues λ 1 < · · · < λ 2N −1 . When we include the parameter ϵ > 0 into the problem these eigenvalues λ j become functions of ϵ: λ j = λ j (ϵ) and the zero eigenvalue is deformed to a nonzero eigenvalue.…”

“…In [2], we proved that under conditions ( 9) and (11) the eigenvalues are all real and λ = 0 is an eigenvalue of Eq. (4).…”

“…We also consider conditions of the form lim t→∞ u(t) = v and lim t→∞ [u(t) − (v + tw)] = 0 instead of the second condition in (2), where v and w are constant vectors.…”