Document Type : Research Paper
Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Kashan, Kashan 87317-53153, Iran.
In this paper, linear second-order differential equations of Sturm-Liouville type having a finite number of singularities and turning points in a finite interval are investigated. First, we obtain the dual equations associated with the Sturm-Liouville equation. Then, we prove the uniqueness theorem for the solutions of dual initial value problems.
 P.J. Browne and B.D. Sleeman, Inverse nodal problems for Sturm-Liouville equations with eigenparameter dependent boundary conditions, Inv. Prob., 12 (1996), pp. 377-381.
 A. Dabbaghian and Sh. Akbarpoor, The nodal points for uniqueness of inverse problem in boundary value problem with aftereffect, World Appl. Sci. J., 12 (2011), pp. 932-934.
 G. Freiling and V.A. Yurko, Inverse problems for Sturm-Liouville equations with boundary conditions polynomially dependent on the spectral parameter, Inv. Prob., 26 (2010), pp. 1-17.
 G. Freiling and V.A. Yurko, Inverse Sturm-Liouville problems and their applications, NOVA Science Publishers, New York, 2001.
 I.M. Gelfand and B.M. Levitan, On the determination of a differential equation from its spectral function, Amer. Math. Soc. Transl. Ser. 2., 1 (1955), pp. 253-304.
 O.H. Hald and J.R. Mclaughlin, Solution of inverse nodal problems, Inv. Prob., 5 (1989), pp. 307-347.
 H. Kheiri, A. Jodayree Akbarfam, and A.B. Mingarelli, The uniqueness of the solution of dual equations of an inverse indefinite Sturm-Liouville problem, J. Math. Anal. Appl., 306 (2005), pp. 269-281.
 H. Koyunbakan, The inverse nodal problem for a differential operator with an eigenvalue in the boundary condition, Appl. Math. Lett., 18 (2010), pp.173-180.
 V.A. Marchenko, Some problems in the theory of a second-order differential operator, Dokl. Akad. Nauk. SSSR., 72 (1950), pp. 457-460.
 S. Mosazadeh, A new approach to uniqueness for inverse Sturm-Liouville problems on finite intervals, Turk. J. Math., 41 (2017), pp. 1224-1234.
 S. Mosazadeh, Infinite product representation of solution of indefinite Sturm-Liouville problem, Iranian J. Math. Chem., 4 (2013), pp. 27-40.
 A.S. Ozkan and B. Keskin, Spectral problems for Sturm-Liouville operators with boundary and jump conditions linearly dependent on the eigenparameter, Inv. Prob. Sci. Eng., 20 (2012), pp. 799-808.
 J. Poschel and E. Trubowitz, Inverse Spectral Theory, Academic Press, London, 1987.
 W.A. Pranger, A formula for the mass density of a vibrating string in terms of the trace, J. Math. Anal. Appl., 141 (1989), pp. 399-404.
 C.T. Shieh and V.A. Yurko, Inverse nodal and inverse spectral problems for discontinuous boundary value problems, J. Math. Anal. Appl., 347 (2008), pp. 266-272.
 N. Topsakal, Inverse problem for Sturm-Liouville operators with Coulomb potential which have discontinuity conditions inside an interval, Math. Phys. Anal. Geom., 13 (2010), pp. 29-46.
 Y.P. Wang, A uniqueness theorem for Sturm-Liouville operators with eigenparameter dependent boundary conditions, Tamkang J. Math., 43 (2012), pp. 145-152.
 Y.P. Wang, Inverse problems for discontinuous Sturm-Liouville operators with mixed spectral data, Inv. Prob. Sci. Eng., 23 (2015), pp. 1180-1198.
 Y.P. Wang and V.A. Yurko, On the inverse nodal problems for discontinuous Sturm-Liouville operators, J. Diff. Equ., 260 (2016), pp. 4086-4109.
 C.F. Yang, Inverse nodal problems of discontinuous Sturm-Liouville operators, J. Diff. Equ., 254 (2013), pp. 1992-2014.
 V.A. Yurko, Method of Spectral Mappings in the Inverse Problem Theory, Inverse and Ill-posed Problems Series, Utrecht: VSP, 2002.
 V.A. Yurko, Inverse spectral problems for differential pencils on the half-line with turning points, J. Math. Anal. Appl., 320 (2006), pp. 439-463.