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Journal articles on the topic 'Sturm-Liouville type'

Author: Grafiati

Published: 6 September 2023

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1

Goktas, Sertac. "A New Type of Sturm-Liouville Equation in the Non-Newtonian Calculus." Journal of Function Spaces 2021 (October 31, 2021): 1–8. http://dx.doi.org/10.1155/2021/5203939.

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In mathematical physics (such as the one-dimensional time-independent Schrödinger equation), Sturm-Liouville problems occur very frequently. We construct, with a different perspective, a Sturm-Liouville problem in multiplicative calculus by some algebraic structures. Then, some asymptotic estimates for eigenfunctions of the multiplicative Sturm-Liouville problem are obtained by some techniques. Finally, some basic spectral properties of this multiplicative problem are examined in detail.

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2

Cernea, Aurelian. "Variational inclusions for a Sturm-Liouville type differential inclusion." Mathematica Bohemica 135, no. 2 (2010): 171–78. http://dx.doi.org/10.21136/mb.2010.140694.

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3

Bas, Erdal, Ramazan Ozarslan, and Dumitru Baleanu. "Sturm-Liouville difference equations having Bessel and hydrogen atom potential type." Open Physics 16, no. 1 (December 26, 2018): 801–9. http://dx.doi.org/10.1515/phys-2018-0100.

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Abstract In this work, we bring a different approach for Sturm-Liouville problems having Bessel and hydrogen atom type and we provide a basis for direct and inverse problems. From this point of view, we find representations of solutions and asymptotic expansions for eigenfunctions. Furthermore, some numerical estimations are given to illustrate the necessity of the Sturm-Liouville difference equations with the potential function for the convenience to the spectral theory. The behavior of eigenfunctions for the Sturm-Liouville problem having Bessel and hydrogen atom potential type is analyzed and compared to each other. And then, comparisons are showed by tables and figures.

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4

Li, Shuang, Jinming Cai, and Kun Li. "Matrix Representations for a Class of Eigenparameter Dependent Sturm–Liouville Problems with Discontinuity." Axioms 12, no. 5 (May 15, 2023): 479. http://dx.doi.org/10.3390/axioms12050479.

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Matrix representations for a class of Sturm–Liouville problems with eigenparameters contained in the boundary and interface conditions were studied. Given any matrix eigenvalue problem of a certain type and an eigenparameter-dependent condition, a class of Sturm–Liouville problems with this specified condition was constructed. It has been proven that each Sturm–Liouville problem is equivalent to the given matrix eigenvalue problem.

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5

Buterin, Sergey, and G. Freiling. "Inverse spectral-scattering problem for the Sturm-Liouville operator on a noncompact star-type graph." Tamkang Journal of Mathematics 44, no. 3 (September 30, 2013): 327–49. http://dx.doi.org/10.5556/j.tkjm.44.2013.1422.

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We study the Sturm-Liouville operator on a noncompact star-type graph consisting of a finite number of compact and noncompact edges under standard matching conditions in the internal vertex. We introduce and investigate the so-called spectral-scat\-tering data, which generalize the classical spectral data for the Sturm-Liouville operator on the half-line and the scattering data on the line. Developing the idea of the method of spectral mappings we prove that the specification of the spectral-scattering data uniquely determines the Sturm-Liouville operator on the graph.

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6

Karahan, D., and K. R. Mamedov. "ON A q-BOUNDARY VALUE PROBLEM WITH DISCONTINUITY CONDITIONS." Bulletin of the South Ural State University series "Mathematics. Mechanics. Physics" 13, no. 4 (2021): 5–12. http://dx.doi.org/10.14529/mmph210401.

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In this paper, we studied q-analogue of Sturm–Liouville boundary value problem on a finite interval having a discontinuity in an interior point. We proved that the q-Sturm–Liouville problem is self-adjoint in a modified Hilbert space. We investigated spectral properties of the eigenvalues and the eigenfunctions of q-Sturm–Liouville boundary value problem. We shown that eigenfunctions of q-Sturm–Liouville boundary value problem are in the form of a complete system. Finally, we proved a sampling theorem for integral transforms whose kernels are basic functions and the integral is of Jackson’s type.

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7

Rynne, Bryan P. "The asymptotic distribution of the eigenvalues of right definite multiparameter Sturm-Liouville systems." Proceedings of the Edinburgh Mathematical Society 36, no. 1 (February 1993): 35–47. http://dx.doi.org/10.1017/s0013091500005873.

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This paper studies the asymptotic distribution of the multiparameter eigenvalues of a right definite multiparameter Sturm–Liouville eigenvalue problem. A uniform asymptotic analysis of the oscillation number of solutions of a single Sturm–Liouville type equation with potential depending on a general parameter is given; these results are then applied to the system of multiparameter Sturm–Liouville equations to give the asymptotic eigenvalue distribution for the system as a function of a “multi-index” oscillation number.

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8

Porter, D., and D. S. G. Stirling. "Integral operators of Sturm-Liouville type." Integral Equations and Operator Theory 38, no. 1 (March 2000): 51–65. http://dx.doi.org/10.1007/bf01192301.

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9

JOHNSON, RUSSELL, and LUCA ZAMPOGNI. "SOME REMARKS CONCERNING REFLECTIONLESS STURM–LIOUVILLE POTENTIALS." Stochastics and Dynamics 08, no. 03 (September 2008): 413–49. http://dx.doi.org/10.1142/s0219493708002391.

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We consider the class of reflectionless Sturm–Liouville potentials and the subclass consisting of "Sato–Segal–Wilson" potentials. We construct illustrative examples lying in these classes, in particular we adapt the constructions of Craig and Levitan to the Sturm–Liouville case. We also discuss an inverse problem of Kotani type in the context of Sato–Segal–Wilson potentials.

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10

Buterin, Sergey. "An inverse spectral problem for Sturm-Liouville-type integro-differential operators with robin boundary conditions." Tamkang Journal of Mathematics 50, no. 3 (September 2, 2019): 207–21. http://dx.doi.org/10.5556/j.tkjm.50.2019.3347.

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The perturbation of the Sturm--Liouville differential operator on a finite interval with Robin boundary conditions by a convolution operator is considered. The inverse problem of recovering the convolution term along with one boundary condition from the spectrum is studied, provided that the Sturm--Liouville potential as well as the other boundary condition are known a priori. The uniqueness of solution for this inverse problem is established along with necessary and sufficient conditions for its solvability. The proof is constructive and gives an algorithm for solving the inverse problem.

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11

Mukhtarov, Oktay Sh, and Merve Yücel. "A Study of the Eigenfunctions of the Singular Sturm–Liouville Problem Using the Analytical Method and the Decomposition Technique." Mathematics 8, no. 3 (March 13, 2020): 415. http://dx.doi.org/10.3390/math8030415.

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The history of boundary value problems for differential equations starts with the well-known studies of D. Bernoulli, J. D’Alambert, C. Sturm, J. Liouville, L. Euler, G. Birkhoff and V. Steklov. The greatest success in spectral theory of ordinary differential operators has been achieved for Sturm–Liouville problems. The Sturm–Liouville-type boundary value problem appears in solving the many important problems of natural science. For the classical Sturm–Liouville problem, it is guaranteed that all the eigenvalues are real and simple, and the corresponding eigenfunctions forms a basis in a suitable Hilbert space. This work is aimed at computing the eigenvalues and eigenfunctions of singular two-interval Sturm–Liouville problems. The problem studied here differs from the standard Sturm–Liouville problems in that it contains additional transmission conditions at the interior point of interaction, and the eigenparameter λ appears not only in the differential equation, but also in the boundary conditions. Such boundary value transmission problems (BVTPs) are much more complicated to solve than one-interval boundary value problems ones. The major difficulty lies in the existence of eigenvalues and the corresponding eigenfunctions. It is not clear how to apply the known analytical and approximate techniques to such BVTPs. Based on the Adomian decomposition method (ADM), we present a new analytical and numerical algorithm for computing the eigenvalues and corresponding eigenfunctions. Some graphical illustrations of the eigenvalues and eigenfunctions are also presented. The obtained results demonstrate that the ADM can be adapted to find the eigenvalues and eigenfunctions not only of the classical one-interval boundary value problems (BVPs) but also of a singular two-interval BVTPs.

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12

Bas, Erdal, and Ramazan Ozarslan. "Sturm-Liouville problem via coulomb type in difference equations." Filomat 31, no. 4 (2017): 989–98. http://dx.doi.org/10.2298/fil1704989b.

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We present Sturm-Liouville problem via Coulomb type in difference equations. The representation of solutions is found. We proved that these solutions satisfy the equation. Asymptotic formulas of eigenfunctions are set.

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13

Allahverdiev, B. P., and H. Tuna. "Теория Титчмарша - Вейля сингулярного уравнения Хана - Штурма - Лиувилля." Владикавказский математический журнал, no. 3 (September 23, 2021): 16–26. http://dx.doi.org/10.46698/y9113-7002-9720-u.

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In this work, we will consider the singular Hahn--Sturm--Liouville difference equation defined by $-q^{-1}D_{-\omega q^{-1},q^{-1}}D_{\omega ,q}y( x) +v(x) y( x) =\lambda y(x)$, $x\in (\omega _{0},\infty),$ where $\lambda$ is a complex parameter, $v$ is a real-valued continuous function at $\omega _{0}$ defined on $[\omega _{0},\infty)$. These type equations are obtained when the ordinary derivative in the classical Sturm--Liouville problem is replaced by the $\omega,q$-Hahn difference operator $D_{\omega,q}$. We develop the $\omega,q$-analogue of the classical Titchmarsh--Weyl theory for such equations. In other words, we study the existence of square-integrable solutions of the singular Hahn--Sturm--Liouville equation. Accordingly, first we define an appropriate Hilbert space in terms of Jackson--N\"{o}rlund integral and then we study families of regular Hahn--Sturm--Liouville problems on $[\omega_{0},q^{-n}]$, $n\in \mathbb{N}$. Then we define a family of circles that converge either to a point or a circle. Thus, we will define the limit-point, limit-circle cases in the Hahn calculus setting by using Titchmarsh's technique.

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14

Dehghan, Mohammad, and Angelo B. Mingarelli. "Fractional Sturm–Liouville Eigenvalue Problems, II." Fractal and Fractional 6, no. 9 (August 30, 2022): 487. http://dx.doi.org/10.3390/fractalfract6090487.

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We continue the study of a non-self-adjoint fractional three-term Sturm–Liouville boundary value problem (with a potential term) formed by the composition of a left Caputo and left Riemann–Liouville fractional integral under Dirichlet type boundary conditions. We study the existence and asymptotic behavior of the real eigenvalues and show that for certain values of the fractional differentiation parameter α, 0<α<1, there is a finite set of real eigenvalues and that, for α near 1/2, there may be none at all. As α→1− we show that their number becomes infinite and that the problem then approaches a standard Dirichlet Sturm–Liouville problem with the composition of the operators becoming the operator of second order differentiation.

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15

Ozkan, Ahmet Sinan. "Ambarzumyan-type theorems on a time scale." Journal of Inverse and Ill-posed Problems 26, no. 5 (October 1, 2018): 633–37. http://dx.doi.org/10.1515/jiip-2017-0124.

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Abstract In this paper, we give Ambarzumyan-type theorems for a Sturm–Liouville dynamic equation with Robin boundary conditions on a time scale. Under certain conditions, we prove that the potential can be specified from only the first eigenvalue.

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16

Cernea, Aurelian. "On controllability for Sturm-Liouville type differential inclusions." Filomat 27, no. 7 (2013): 1321–27. http://dx.doi.org/10.2298/fil1307321c.

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17

Pachpatte, B. G. "On the Sturm-Liouville-type boundary value problem." Journal of Mathematical Analysis and Applications 108, no. 1 (May 1985): 92–98. http://dx.doi.org/10.1016/0022-247x(85)90010-1.

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18

Mukhtarov, Oktay, and Kadriye Aydemir. "The eigenvalue problem with interaction conditions at one interior singular point." Filomat 31, no. 17 (2017): 5411–20. http://dx.doi.org/10.2298/fil1717411m.

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Some physical processes, both classical physics and quantum physics reduced to eigenvalue problems for Sturm-Liouville equations. In the recent years there has been an increasing interest in discontinuous eigenvalue problems for various Sturm-Liouville type equations. Such problems are connected with heat transfer problems, vibrating string problems, diffraction problems and etc. In this study we shall investigate a class of two order eigenvalue problem with supplementary transmission conditions at one interior singular point. We give an operator-theoretic interpretation in suitable Hilbert space.

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19

Pikula, Milenko, Vladimir Vladicic, and Olivera Markovic. "A solution to the inverse problem for the Sturm-Liouville-type equation with a delay." Filomat 27, no. 7 (2013): 1237–45. http://dx.doi.org/10.2298/fil1307237p.

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The paper is devoted to study of the inverse problem of the boundary spectral assignment of the Sturm-Liouville with a delay. -y'(x) + q(x)y(? ? x) = ?y(x), q ? AS[0, ?], ? ? (0,1] (1) with separated boundary conditions: y(0) = y(?) = 0 (2) y(0) = y'(?) = 0 (3) It is argued that if the sequence of eigenvalues is given ?n(1) and ?n(2) tasks (1-2) and (1-3) respectively, then the delay factor ? ? (0,1) and the potential q ? AS[0, ?] are unambiguous. The potential q is composed by means of trigonometric Fourier coefficients. The method can be easily transferred to the case of ? = 1 i.e. to the classical Sturm-Liouville problem.

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20

Freitas, Pedro. "A nonlocal Sturm–Liouville eigenvalue problem." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 124, no. 1 (1994): 169–88. http://dx.doi.org/10.1017/s0308210500029279.

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A nonlocal eigenvalue problem of the form u″ + a(x)u + Bu = λu with homogeneous Dirichlet boundary conditions is considered, where B is a rank-one bounded linear operator and x belongs to some bounded interval on the real line. The behaviour of the eigenvalues is studied using methods of linear perturbation theory. In particular, some results are given which ensure that the spectrum remains real. A Sturm-type comparison result is obtained. Finally, these results are applied to the study of some nonlocal reaction–diffusion equations.

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21

Boumenir, A., and B. Chanane. "Computing eigenvalues of Sturm-Liouville systems of Bessel type." Proceedings of the Edinburgh Mathematical Society 42, no. 2 (June 1999): 257–65. http://dx.doi.org/10.1017/s001309150002023x.

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In this paper we shall develop a new method for the computation of eigenvalues of singular Sturm-Liouville problems of the Bessel type. This new method is based on the interpolation of a boundary function in Paley-Wiener spaces. Numerical results are provided to illustrate the method.

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22

ABOUELAZ, AHMED, AZZEDINE ACHAK, RADOUAN DAHER, and NAJAT SAFOUANE. "Quantitative Uncertainty Principle for Sturm-Liouville Transform." Kragujevac Journal of Mathematics 45, no. 03 (May 2021): 465–76. http://dx.doi.org/10.46793/kgjmat2103.465a.

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In this paper we consider the Sturm-Liouville transform ℱ(f) on ℝ+. We analyze the concentration of this transform on sets of ﬁnite measure. In particular, Donoho-Stark and Benedicks-type uncertainty principles are given.

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23

Clark, Steve, and Fritz Gesztesy. "On Povzner–Wienholtz-type self-adjointness results for matrix-valued Sturm–Liouville operators." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 133, no. 4 (August 2003): 747–58. http://dx.doi.org/10.1017/s0308210500002651.

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24

Farzana, Humaira, and Md Shafiqul Islam. "Computation of Some Second Order Sturm-Liouville Bvps using Chebyshev-Legendre Collocation Method." GANIT: Journal of Bangladesh Mathematical Society 35 (June 28, 2016): 95–112. http://dx.doi.org/10.3329/ganit.v35i0.28574.

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We propose Chebyshev-Legendre spectral collocation method for solving second order linear and nonlinear eigenvalue problems exploiting Legendre derivative matrix. The Sturm-Liouville (SLP) problems are formulated utilizing Chebyshev-Gauss-Lobatto (CGL) nodes instead of Legendre Gauss-Lobatto (LGL) nodes and Legendre polynomials are taken as basis function. We discuss, in details, the formulations of the present method for the Sturm-Liouville problems (SLP) with Dirichlet and mixed type boundary conditions. The accuracy of this method is demonstrated by computing eigenvalues of three regular and two singular SLP's. Nonlinear Bratu type problem is also tested in this article. The numerical results are in good agreement with the other available relevant studies.GANIT J. Bangladesh Math. Soc.Vol. 35 (2015) 95-112

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25

Binding, Paul, and Pável Drábek. "Sturm--Liouville theory for the p-Laplacian." Studia Scientiarum Mathematicarum Hungarica 40, no. 4 (October 1, 2003): 373–96. http://dx.doi.org/10.1556/sscmath.40.2003.4.1.

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A version of Sturm--Liouville theory is given for the one-dimensional p-Laplacian including the radial case. The treatment is modern but follows the strategy of Elbert's early work. Topics include a Prüfer-type transformation, eigenvalue existence, asymptotics and variational principles, and eigenfunction oscillation.

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26

Gül, Erdal. "On the regularized trace of a differential operator of Sturm-Liouville type." Filomat 36, no. 13 (2022): 4515–23. http://dx.doi.org/10.2298/fil2213515g.

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In this work, we study a spectral problem for the abstract Sturm-Liouville operator with a bounded operator coefficient V(t) and with periodic boundary conditions on the interval [0, ?], and we present a regularized trace formula for this operator.

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27

TOPSAKAL, Nilüfer, and Rauf AMİROV. "On GLM type integral equation for singular Sturm-Liouville operator which has discontinuous coefficient." Communications Faculty Of Science University of Ankara Series A1Mathematics and Statistics 71, no. 2 (June 30, 2022): 305–25. http://dx.doi.org/10.31801/cfsuasmas.923029.

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In this study, we derive Gelfand-Levitan-Marchenko type main integral equation of the inverse problem for singular Sturm-Liouville equation which has discontinuous coefficient. Then we prove the unique solvability of the main integral equation.

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28

Toprakseven, Şuayip. "Hartman-Wintner and Lyapunov-type inequalities for high order fractional boundary value problems." Filomat 34, no. 7 (2020): 2273–81. http://dx.doi.org/10.2298/fil2007273t.

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In this paper, we obtain Hartman-Wintner and Lyapunov-type inequalities for the three-point fractional boundary value problem of the fractional Liouville-Caputo differential equation of order ? 2 (2; 3]. The results presented in this work are sharper than the existing results in the literature. As an application of the results, the fractional Sturm-Liouville eigenvalue problems have also been presented. Moreover, we examine the nonexistence of the nontrivial solution of the fractional boundary value problem.

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29

Allahverdiev, Bilender P., and Hüseyin Tuna. "Spectral Theory of Singular Hahn Difference Equation of the Sturm-Liouville Type." Communications in Mathematics 28, no. 1 (June 1, 2020): 13–25. http://dx.doi.org/10.2478/cm-2020-0002.

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AbstractIn this work, we consider the singular Hahn difference equation of the Sturm-Liouville type. We prove the existence of the spectral function for this equation. We establish Parseval equality and an expansion formula for this equation on a semi-unbounded interval.

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30

Bayramov, Azad, and Erdoğan Şen. "On a Sturm-Liouville type problem with retarded argument." Mathematical Methods in the Applied Sciences 36, no. 1 (May 30, 2012): 39–48. http://dx.doi.org/10.1002/mma.2567.

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31

Müller-Pfeiffer, E. "HILLE-WINTNER Type Comparison Theorems for STURM-LIOUVILLE Equations." Mathematische Nachrichten 142, no. 1 (1989): 167–73. http://dx.doi.org/10.1002/mana.19891420111.

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32

Guan, Sheng-Yu, Chuan-Fu Yang, Natalia Bondarenko, Xiao-Chuan Xu, and Yi-Teng Hu. "On the Hochstadt–Lieberman type problem with eigenparameter dependent boundary condition." Journal of Inverse and Ill-posed Problems 28, no. 4 (August 1, 2020): 557–65. http://dx.doi.org/10.1515/jiip-2019-0072.

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AbstractThe half-inverse problem is studied for the Sturm–Liouville operator with an eigenparameter dependent boundary condition on a finite interval. We develop a reconstruction procedure and prove the existence theorem for solution of the inverse problem. Our method is based on interpolation of entire functions.

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33

Khosravian-Arab, Hassan, and Mohammad Reza Eslahchi. "Müntz sturm-liouville problems: Theory and numerical experiments." Fractional Calculus and Applied Analysis 24, no. 3 (June 1, 2021): 775–817. http://dx.doi.org/10.1515/fca-2021-0034.

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Abstract This paper presents two new classes of Müntz functions which are called Jacobi-Müntz functions of the first and second types. These newly generated functions satisfy in two self-adjoint fractional Sturm-Liouville problems and thus they have some spectral properties such as: orthogonality, completeness, three-term recurrence relations and so on. With respect to these functions two new orthogonal projections and their error bounds are derived. Also, two new Müntz type quadrature rules are introduced. As two applications of these basis functions some fractional ordinary and partial differential equations are considered and numerical results are given.

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34

CERNEA, AURELIAN. "On controllability for a class of second-order differential inclusions." Carpathian Journal of Mathematics 27, no. 1 (2011): 34–40. http://dx.doi.org/10.37193/cjm.2011.01.10.

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By using a suitable fixed point theorem a sufficient condition for controllability is obtained for a Sturm-Liouville type differential inclusion in the case when the right hand side has convex values.

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35

Pivovarchik, Vyacheslav. "On Ambarzumian type theorems for tree domains." Opuscula Mathematica 42, no. 3 (2022): 427–37. http://dx.doi.org/10.7494/opmath.2022.42.3.427.

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It is known that the spectrum of the spectral Sturm-Liouville problem on an equilateral tree with (generalized) Neumann's conditions at all vertices uniquely determines the potentials on the edges in the unperturbed case, i.e. case of the zero potentials on the edges (Ambarzumian's theorem). This case is exceptional, and in general case (when the Dirichlet conditions are imposed at some of the pendant vertices) even two spectra of spectral problems do not determine uniquely the potentials on the edges. We consider the spectral Sturm-Liouville problem on an equilateral tree rooted at its pendant vertex with (generalized) Neumann conditions at all vertices except of the root and the Dirichlet condition at the root. In this case Ambarzumian's theorem can't be applied. We show that if the spectrum of this problem is unperturbed, the spectrum of the Neumann-Dirichlet problem on the root edge is also unperturbed and the spectra of the problems on the complimentary subtrees with (generalized) Neumann conditions at all vertices except the subtrees' roots and the Dirichlet condition at the subtrees' roots are unperturbed then the potential on each edge of the tree is 0 almost everywhere.

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36

Bas, Erdal. "Fundamental Spectral Theory of Fractional Singular Sturm-Liouville Operator." Journal of Function Spaces and Applications 2013 (2013): 1–7. http://dx.doi.org/10.1155/2013/915830.

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We give the theory of spectral properties for eigenvalues and eigenfunctions of Bessel type of fractional singular Sturm-Liouville problem. We show that the eigenvalues and eigenfunctions of the problem are real and orthogonal, respectively. Furthermore, we prove new approximations about the topic.

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37

Mukhtarov, O. Sh, and K. Aydemir. "New Type of Sturm-Liouville Problems in Associated Hilbert Spaces." Journal of Function Spaces 2014 (2014): 1–7. http://dx.doi.org/10.1155/2014/606815.

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We introduce a new type of discontinuous Sturm-Liouville problems, involving an abstract linear operator in equation. By suggesting own approaches we define some new Hilbert spaces to establish such properties as isomorphism, coerciveness, and maximal decreasing of resolvent operator with respect to spectral parameter. Then we find sufficient conditions for discreteness of the spectrum and examine asymptotic behaviour of eigenvalues. Obtained results are new even for continuous case, that is, without transmission conditions.

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38

Sa'idu, Auwalu, and Hikmet Koyunbakan. "A Conformable Inverse Problem with Constant Delay." Journal of Advances in Applied & Computational Mathematics 10 (August 16, 2023): 26–38. http://dx.doi.org/10.15377/2409-5761.2023.10.3.

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This paper aims to express the solution of an inverse Sturm-Liouville problem with constant delay using a conformable derivative operator under mixed boundary conditions. For the problem, we stated and proved the specification of the spectrum. The asymptotics of the eigenvalues of the problem was obtained and the solutions were extended to the Regge-type boundary value problem. As such, a new result, as an extension of the classical Sturm-Liouville problem to the fractional phenomenon, has been achieved. The uniqueness theorem for the solution of the inverse problem is proved in different cases within the interval (0,π). The results in the classical case of this problem can be obtained at α=1. 2000 Mathematics Subject Classification. 34L20,34B24,34L30.

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39

Sun, Fu, and Jiangang Qi. "A priori bounds and existence of non-real eigenvalues for singular indefinite Sturm–Liouville problems with limit-circle type endpoints." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 150, no. 5 (June 7, 2019): 2607–19. http://dx.doi.org/10.1017/prm.2019.7.

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AbstractThe present paper deals with non-real eigenvalues of singular indefinite Sturm–Liouville problems with limit-circle type endpoints. A priori bounds and the existence of non-real eigenvalues of the problem associated with a special separated boundary condition are obtained.

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40

Bondarenko, Natalia. "An Inverse Spectral Problem for the Matrix Sturm-Liouville Operator with a Bessel-Type Singularity." International Journal of Differential Equations 2015 (2015): 1–4. http://dx.doi.org/10.1155/2015/647396.

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The inverse problem by the Weyl matrix is studied for the matrix Sturm-Liouville equation on a finite interval with a Bessel-type singularity in the end of the interval. We construct special fundamental systems of solutions for this equation and prove the uniqueness theorem of the inverse problem.

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41

ÖZKAN, A. Sinan, and Yaşar ÇAKMAK. "Ambarzumyan Type Theorems for a Class of Sturm-Liouville Problem." Cumhuriyet Science Journal 38, no. 3 (September 30, 2017): 396–99. http://dx.doi.org/10.17776/csj.340393.

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42

Cernea, Aurelian. "On a Sturm-Liouville type differential inclusion of fractional order." Fractional Differential Calculus, no. 2 (2017): 385–93. http://dx.doi.org/10.7153/fdc-2017-07-19.

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43

Yurko, V. "Inverse problems for Sturm–Liouville operators on bush-type graphs." Inverse Problems 25, no. 10 (September 16, 2009): 105008. http://dx.doi.org/10.1088/0266-5611/25/10/105008.

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44

张, 亮. "Inverse Sturm-Liouville Problems with Distribution Potentials of Atkinson Type." Advances in Applied Mathematics 07, no. 12 (2018): 1565–73. http://dx.doi.org/10.12677/aam.2018.712183.

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Liu, Yicheng, Jun Wu, and Zhixiang Li. "Impulsive Boundary Value Problems for Sturm-Liouville Type Differential Inclusions." Journal of Systems Science and Complexity 20, no. 3 (September 2007): 370–80. http://dx.doi.org/10.1007/s11424-007-9032-3.

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46

Yurko, V. A. "Inverse problem for Sturm-Liouville operators on hedgehog-type graphs." Mathematical Notes 89, no. 3-4 (April 2011): 438–49. http://dx.doi.org/10.1134/s000143461103014x.

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47

Mahmudov, Elimhan N. "Optimization of Mayer Problem with Sturm–Liouville-Type Differential Inclusions." Journal of Optimization Theory and Applications 177, no. 2 (March 7, 2018): 345–75. http://dx.doi.org/10.1007/s10957-018-1260-2.

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48

Ozarslan, Ramazan, Ahu Ercan, and Erdal Bas. "β −type fractional Sturm‐Liouville Coulomb operator and applied results." Mathematical Methods in the Applied Sciences 42, no. 18 (July 17, 2019): 6648–59. http://dx.doi.org/10.1002/mma.5769.

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49

Yang, Chuan-Fu, and Vjacheslav Yurko. "On the determination of differential pencils with nonlocal conditions." Journal of Inverse and Ill-posed Problems 26, no. 5 (October 1, 2018): 577–88. http://dx.doi.org/10.1515/jiip-2017-0076.

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Abstract:

Abstract Inverse problems for differential pencils with nonlocal conditions are considered. Uniqueness theorems of inverse problems from the Weyl-type function and spectra are proved, which are generalizations of the well-known Weyl function and Borg’s inverse problem for the classical Sturm–Liouville operators.

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50

Štikonas, Artūras. "Asymptotic analysis of Sturm-Liouville problem with Robin and two-point boundary conditions." Lietuvos matematikos rinkinys 63 (December 10, 2022): 9–18. http://dx.doi.org/10.15388/lmr.2022.29692.

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Abstract:

We analyze the initial value problem and get asymptotic expansions for solution. We investigate the characteristic equation for Sturm-Liouville problem with one classical Robin type boundary condition and another two-point nonlocal boundary condition. Finally, we obtain asymptotic expansions for eigenvalues and eigenfunctions.

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