An inverse problem for a differential pencil using nodal points as data

Abstract

An inverse nodal problem is studied for a differential pencil with nonseparated boundary conditions. We prove that a dense subset of nodal points uniquely determines the boundary data and potential functions. We also provide a constructive procedure for the solution of the inverse nodal problem.

Energy dependent potential problems for the one dimensionalp-Laplacian operator

“…where studied in several papers, see [1,2,3,10,22,28,35,36], among other works. The existence of a discrete set of eigenvalues in the linear case is a difficult problem, and it was solved first by Friedman and Shinbrot [11] using functional analytic techniques for linear compact symmetric operators which can be applied to the inverses of the differential operators.…”

“…The spectral inverse problems for energy dependent potentials started with the works of Jaulent and Jean [19,20,21], and was studied later in [3,18,27,37,36].…”

“…both h and q are terms of the potential, and were determined in the linear case p = 2 by Buterin and Shieh, see [3], and by Yang in [36] in terms of a dense subset of nodes and the integrals of h and q. Their proofs are based on precise estimates on the location of zeros of eigenfunctions obtained from the regularized traces of the pencil of operators and the corresponding Weyl function.…”

“…So, we introduce a different approach here, of more elemental character. Compared with previous works in the linear case (i.e., [3,36]), we are using more information about the nodes, since they assume only that a dense subset of zeros is given, and also it is enough to know them on more than half the interval. On the other hand, we are not assuming that the integral of q and h are given, and only the case α = 1/2 was studied in these works.…”

On a lipschitz stability problem for p-Laplacian Bessel equation

“…Nodal length of this problem is denoted by l n j = x n j+1 x n j for j = 1; 2; :::; n 1: Using these type nodal datas, some uniqueness and reconstruction results for di¤erent type of operators have been expressed by several authors (see [21], [22], [23], [24] ). …”