The uniqueness of the solution of dual equations of an inverse indefinite Sturm–Liouville problem

Abstract

In this paper we consider a linear second-order equation of Sturm-Liouville type:with Dirichlet boundary conditions y(−1) = y(1) = 0, where q is a positive sufficiently smooth function on [−1, 1] and λ is a real parameter. We investigate the uniqueness of the solution for the dual equations of the indefinite inverse spectral problem (I). This result is necessary for expressing inverse problem of indefinite Sturm-Liouville equation.  2005 Elsevier Inc. All rights reserved.

Inverse problem for differential systems having a singularity and turning point of even or odd order

“…Asymptotic approximation of the solution of second-order differential equations with two turning points was investigated in [11,14,23]. Note that, the canonical solution of the equation with one turning point of odd order was studied in [12], and the existence and the uniqueness of the solution for corresponding dual equations were investigated. For boundary value problems with singular points, see the works [2,3,6,8,13,19,20] and the references therein.…”

On the uniqueness of the solution of dual equation of a singular Sturm–Liouville problem

“…In [5], the infinite product representation of solution of the Sturm-Liouville equation with one turning point was studied and they proved the existence and uniqueness of the solution for dual equations. In [3], the authors solved an inverse problem related to (2) in the case where there is a turning point inside the (prescribed) interval of definition.…”

On the canonical solution of indefinite problem with m turning points of even order

“…The importance of asymptotic analysis in obtaining information on the solution of a SturmLiouville equation with multiple turning points was realized by Leung [20], Olver [26][27][28], Heading [12], and Eberhard, Freiling and Schneider in [6]. The results of Dorodnicyn [4], Kazarinoff [15], McKelvey [23], Langer [18], Olver [28], Wazwaz [32], Dyachenko [5], Tumanov [30], Kheiri, Jodayree and Mingarelli in [16] and Neamaty [25] bring important innovations to the asymptotic approximation of solutions of Sturm-Liouville equations with two turning points.…”