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# Journal of Inverse and Ill-posed Problems

Editor-in-Chief: Kabanikhin, Sergey I.

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# Inverse nodal problems for the Sturm–Liouville operator with nonlocal integral conditions

Yi-Teng Hu
• Corresponding author
• Department of Applied Mathematics, School of Science, Nanjing University of Science and Technology, Nanjing, 210094, Jiangsu, P. R. China
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• Other articles by this author:
/ Chuan-Fu Yang
• Department of Applied Mathematics, School of Science, Nanjing University of Science and Technology, Nanjing, 210094, Jiangsu, P. R. China
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• Other articles by this author:
/ Xiao-Chuan Xu
• Department of Applied Mathematics, School of Science, Nanjing University of Science and Technology, Nanjing, 210094, Jiangsu, P. R. China
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• Other articles by this author:
Published Online: 2017-05-05 | DOI: https://doi.org/10.1515/jiip-2017-0017

## Abstract

In this work, we consider inverse nodal problems of the Sturm–Liouville equation with nonlocal integral conditions at two end-points. We prove that a dense subset of nodal points uniquely determine the potential function of the Sturm–Liouville equation up to a constant.

MSC 2010: 34A55; 34B24; 47E05

## References

• [1]

S. Albeverio, R. O. Hryniv and L. P. Nizhnik, Inverse spectral problems for non-local Sturm–Liouville operators, Inverse Problems 23 (2007), 523–535.

• [2]

A. V. Bitsadze and A. A. Samarskii, Some elementary generalizations of linear elliptic boundary value problems, Dokl. Akad. Nauk SSSR 185 (1969), 739–740. Google Scholar

• [3]

P. J. Browne and B. D. Sleeman, Inverse nodal problem for Sturm–Liouville equation with eigenparameter dependent boundary conditions, Inverse Problems 12 (1996), 377–381.

• [4]

V. Bryuns, Generalized boundary-value problem for an ordinary linear differential operator, Dokl. Akad. Nauk SSSR 198 (1971), 1255–1258. Google Scholar

• [5]

S. A. Buterin and C. T. Shieh, Inverse nodal problem for differential pencils, Appl. Math. Lett. 22 (2009), 1240–1247.

• [6]

B. Chanane, Computing the eigenvalues of a class of nonlocal Sturm–Liouville problems, Math. Comput. Modelling 50 (2009), 225–232.

• [7]

Y. T. Chen, Y. H. Cheng, C. K. Law and J. Tsay, ${L}^{1}$ convergence of the reconstruction formula for the ponential function, Proc. Amer. Math. Soc. 130 (2002), 2319–2324. Google Scholar

• [8]

Y. H. Cheng and C. K. Law, On the quasi-nodal map for the Sturm–Liouville problem, Proc. Roy. Soc. Edinburgh Sect. A 136 (2006), 71–86.

• [9]

Y. H. Cheng, C. K. Law and J. Tsay, Remarks on a new inverse nodal problem, J. Math. Anal. Appl. 248 (2000), 145–155.

• [10]

R. Čiupaila, Ž. Jesevičuite and M. Sapagovas, Eigenvalue problem for ordinary differential operator subject to integer condition, Nonlinear Anal. 9 (2004), 109–116. Google Scholar

• [11]

S. Currie and B. A. Waston, Inverse nodal problems for Sturm–Liouville equations on graphs, Inverse Problems 23 (2007), 2029–2040.

• [12]

L. Greenberg and M. Marletta, Numerical solution of non-self-adjoint Sturm–Liouville problems and related systems, SIAM J. Numer. Anal. 38 (2001), 1800–1845.

• [13]

O. H. Hald and J. R. McLaughlin, Solutions of inverse nodal problems, Inverse Problems 5 (1989), 307–347.

• [14]

G. Infante, Eigenvalues of some non-local boundary-value problems, Proc. Edinb. Math. Soc. (2) 46 (2003), 75–86. Google Scholar

• [15]

N. I. Ionkin and E. A. Valikova, On the eigenvalues and eigenfunctions of a nonclasscial boundary value problem (in Russian), Math. Model. Russia 1 (1996), 53–63. Google Scholar

• [16]

C. K. Law, C. L. Shen and C. F. Yang, The inverse nodal problem on the smoothness of the potential function, Inverse Problems 15 (1999), 252–263. Google Scholar

• [17]

C. K. Law and J. Tsay, On the well-posedness of the inverse nodal problem, Inverse Problems 17 (2001), 1493–1512.

• [18]

C. K. Law and C. F. Yang, Reconstructing the potential function and its derivatives using nodal data, Inverse Problems 14 (1998), 299–312.

• [19]

B. M. Levitan, Inverse Sturm–Liouville Problems, VNU Science Press, Utrecht, 1987. Google Scholar

• [20]

B. M. Levitan and I. S. Sargsjan, Sturm–Liouville and Dirac Operators (in Russian), Nauka, Mosocow, 1988. Google Scholar

• [21]

V. A. Marchenko, Sturm–Liouville Operators and Their Applications (in Russian), Naukova Dumka, Kiev, 1977. Google Scholar

• [22]

C. M. McCarthy and W. Rundell, Eigenparameter dependent inverse Sturm–Liouville problems, Numer. Funct. Anal. Optim. 24 (2003), 85–105.

• [23]

J. R. McLaughlin, Inverse spectral theory using nodal points as data—A uniqueness result, J. Differential Equations 73 (1988), 354–362.

• [24]

S. Pečiulytė, O. Štikonienė and A. Štikonas, Sturm–Liouville problem for stationary differential operator with nonlocal integral boundary conditions, Math. Model. Anal. 10 (2005), 377–392. Google Scholar

• [25]

J. Pöschel and E. Trubowitz, Inverse Spectral Theory, Academic Press, Orlando, 1987. Google Scholar

• [26]

W. Rundell and P. Sacks, Numerical technique for the inverse resonance problem, J. Comput. Appl. Math. 170 (2004), 337–347.

• [27]

M. P. Sapagovas, The eigenvalues of some problem with a nonlocal condition (in Russian), Differ. Equ. 7 (2002), 1020–1026. Google Scholar

• [28]

M. P. Sapagovas and A. D. Štikonas, On the structure of the spectrum of a differential operator with a nonlocal condition, Differ. Equ. 41 (2005), 1010–1018.

• [29]

C. L. Shen, On the nodal sets of the eigenfunctions of the string equations, SIAM J. Math. Anal. 19 (1988), 1419–1424.

• [30]

C. L. Shen and C. T. Shieh, An inverse nodal problem for vectorial Sturm–Liouville equation, Inverse Problems 16 (2000), 349–356.

• [31]

C. T. Shieh and V. A. Yurko, Inverse nodal and inverse spectral problems for discontinuous boundary value problem, J. Math. Anal. Appl. 347 (2008), 266–272.

• [32]

A. Štikonas and O. Štikonienė, Characteristic functions for Sturm–Liouville problems with nonlocal boundary conditions, Math. Model. Anal. 14 (2009), 229–246.

• [33]

C. F. Yang, Inverse nodal problem for a class of nonlocal Sturm–Liouville operator, Math. Model. Anal. 15 (2010), 383–392.

• [34]

C. F. Yang, Stability in the inverse nodal solution for the interior transmission problem, J. Differential Equations 260 (2016), 2490–2506.

• [35]

C. F. Yang and V. A. Yurko, Recovering Dirac operator with nonlocal boundary conditions, J. Math. Anal. Appl. 440 (2016), 155–166.

• [36]

X. F. Yang, A solution of the inverse nodal problem, Inverse Problems 13 (1997), 203–213.

• [37]

V. A. Yurko, Integral transforms connected with discontinious boundary value problems, Integral Transforms Spec. Funct. 10 (2000), 141–164.

• [38]

V. A. Yurko, Inverse Spectral Problems for Differential Operators and Their Applications, Gordon and Breach, Amsterdam, 2000. Google Scholar

• [39]

V. A. Yurko, Inverse nodal problems for Sturm–Liouville operators on star-type graphs, J. Inverse Ill-Posed Probl. 16 (2008), 715–722. Google Scholar

• [40]

V. A. Yurko and C. F. Yang, Recovering differential operators with nonlocal boundary conditions, Anal. Math. Phys. (2015), 10.1007/s13324-015-0120-6. Google Scholar

Revised: 2017-02-22

Accepted: 2017-02-27

Published Online: 2017-05-05

Published in Print: 2017-12-01

Funding Source: National Natural Science Foundation of China

Award identifier / Grant number: 11171152

Award identifier / Grant number: 11611530682

Award identifier / Grant number: 91538108

Funding Source: Natural Science Foundation of Jiangsu Province

Award identifier / Grant number: BK 20141392

This work was supported in part by the National Natural Science Foundation of China (11171152, 11611530682 and 91538108) and Natural Science Foundation of Jiangsu Province of China (BK 20141392).

Citation Information: Journal of Inverse and Ill-posed Problems, Volume 25, Issue 6, Pages 799–806, ISSN (Online) 1569-3945, ISSN (Print) 0928-0219,

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