Abstract
We study an inverse problem on the half-linear Dirichlet eigenvalue problem , where with and r is a positive function defined on . Using eigenvalues and nodal data (the lengths of two consecutive zeros of solutions), we reconstruct and its derivatives. Our method is based on (Law and Yang in Inverse Probl. 14:299-312, 779-780, 1998; Shen and Tsai in Inverse Probl. 11:1113-1123, 1995), and our result extends the result in (Shen and Tsai in Inverse Probl. 11:1113-1123, 1995) for the linear case to the half-linear case. MSC:34A55, 34B24, 47A75.
Highlights
The subject under investigation is the half-linear eigenvalue problem consisting of ⎧⎨–(|y (x)|p– y (x)) = (p – )λr(x)|y(x)|p– y(x),⎩y( ) = y( ) =, ( )where p > with p =, and r is a positive function defined on [, ]
By [ – ], it is well known that the problem ( ) has countably many eigenpairs {(λn, yn(x)) : n ∈ N}, and the eigenfunction yn(x) has exactly n – nodal points in (, ), say = x( n) < x( n) < · · · < x(nn–) < x(nn) =
We intend to give the representation of the function r(x) and its derivatives in ( ) by using eigenvalues and nodal points
Summary
The subject under investigation is the half-linear eigenvalue problem consisting of. where p > with p = , and r is a positive function defined on [ , ]. We intend to give the representation of the function r(x) and its derivatives in ( ) by using eigenvalues and nodal points. This formation is treated as the reconstruction formula. The authors in [ ] studied ( ) with Dirichlet boundary conditions y( ) = y( ) = , while the authors in [ ] studied ( ) with eigenparameter dependent boundary conditions y( ) = , αy ( ) + λy( ) = for α = Both of them first used the modified Prüfer substitution to derive the asymptotic expansion of eigenvalues and nodal points and gave the reconstruction formula of q(x) by using the nodal data.
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