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Mathematics
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Boundary Value Problems, Dynamical Systems and Inverse Spectral Problems
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Boundary Value Problems, Dynamical Systems and Inverse Spectral Problems

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Difference and Differential Equations".

Deadline for manuscript submissions: closed (31 March 2024) | Viewed by 6099

Special Issue Editors

Prof. Dr. Jiangang Qi

E-Mail Website
Guest Editor
Department of Mathematics, Shandong University, Weihai 264209, China
Interests: Sturm–Liouville problems; singular spectral problems; eigenvalues; gap and ratio; inverse problems; green functions; Laplace operators; Mercer theorem
Prof. Dr. Xu Zhang

E-Mail Website
Guest Editor
Department of Mathematics, Shandong University, Weihai 264209, China
Interests: differential equations and dynamical systems

Special Issue Information

Dear Colleagues,

Sturm–Liouville problems represent an important tool to study classical mechanics. In particular, the spectrum of singular problems has become a powerful tool to understand and explain quantum phenomena, which has attracted the attention of many mathematicians and physicists. In the mathematical theory itself, the spectral theory of differential operators has also become an important part of operator theory, harmonic analysis, and other research directions, and provides a solid basic theoretical tool for solving the basic problems of differential equations.

Dynamical systems began originally in Newton's study of the two-body problem, i.e., the problem of calculating the motion of the earth around the sun. However, the extension of Newton's study to the three-body problem, i.e., the problem of the motion of sun, earth, and moon, turned out to be much more difficult to solve. The breakthrough was made by Poincaré's work by introducing a powerful geometric approach. Later, this area was greatly influenced by Birkhoff, van del Pol, Andronov, Littlewood, Cartwright, Levinson, Smale, Kolmogorov, Arnold, Moser, and so on. Nowadays, dynamical system can be thought of as an interdisciplinary subject, applicable in many fields.

This Special Issue aims to collect original and significant contributions on boundary value problems, spectral theorem, and dynamical systems. The Special Issue can also serve as a platform for exchanging ideas between scholars interested in differential equations and dynamical systems.

Prof. Dr. Jiangang Qi
Prof. Dr. Xu Zhang
Guest Editors

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Keywords

  • sturm–Liouville problems
  • singular spectral problems
  • boundary value problems
  • eigenvalues
  • gap and ratio
  • inverse problems
  • green functions
  • Laplace operators
  • Mercer theorem
  • self-adjoint operators
  • symmetric
  • non-symmetric
  • differential equations
  • dynamical systems

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Published Papers (6 papers)

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Research

11 pages, 250 KiB  
Article
Bounds of Eigenvalues for Complex q-Sturm–Liouville Problem
by Xiaoxue Han
Mathematics 2024, 12(17), 2646; https://doi.org/10.3390/math12172646 - 26 Aug 2024
Viewed by 401
Abstract
The eigenvalues of complex q-Sturm–Liouville boundary value problems are the focus of this paper. The coefficients of the corresponding q-Sturm–Liouville equation provide the lower bounds on the real parts of all eigenvalues, and the real part of the eigenvalue and the [...] Read more.
The eigenvalues of complex q-Sturm–Liouville boundary value problems are the focus of this paper. The coefficients of the corresponding q-Sturm–Liouville equation provide the lower bounds on the real parts of all eigenvalues, and the real part of the eigenvalue and the coefficients of this q-Sturm–Liouville equation provide the bounds on the imaginary part of each eigenvalue. Full article
(This article belongs to the Special Issue Boundary Value Problems, Dynamical Systems and Inverse Spectral Problems)
6 pages, 243 KiB  
Article
Determination of the Impulsive Dirac Systems from a Set of Eigenvalues
by Ran Zhang, Chuanfu Yang and Kai Wang
Mathematics 2023, 11(19), 4086; https://doi.org/10.3390/math11194086 - 26 Sep 2023
Viewed by 641
Abstract
In this work, we consider the inverse spectral problem for the impulsive Dirac systems on (0,π) with the jump condition at the point π2. We conclude that the matrix potential Q(x) on the whole [...] Read more.
In this work, we consider the inverse spectral problem for the impulsive Dirac systems on (0,π) with the jump condition at the point π2. We conclude that the matrix potential Q(x) on the whole interval can be uniquely determined by a set of eigenvalues for two cases: (i) the matrix potential Q(x) is given on 0,(1+α)π4; (ii) the matrix potential Q(x) is given on (1+α)π4,π, where 0<α<1. Full article
(This article belongs to the Special Issue Boundary Value Problems, Dynamical Systems and Inverse Spectral Problems)
12 pages, 277 KiB  
Article
Boundary Value Problem for a Loaded Pseudoparabolic Equation with a Fractional Caputo Operator
by Serik Aitzhanov, Kymbat Bekenayeva and Zamira Abdikalikova
Mathematics 2023, 11(18), 3987; https://doi.org/10.3390/math11183987 - 20 Sep 2023
Cited by 2 | Viewed by 966
Abstract
Differential equations containing fractional derivatives, for both time and spatial variables, have now begun to attract the attention of mathematicians and physicists; they are used in connection with these equations as mathematical models of various processes. The fractional derivative equation tool plays a [...] Read more.
Differential equations containing fractional derivatives, for both time and spatial variables, have now begun to attract the attention of mathematicians and physicists; they are used in connection with these equations as mathematical models of various processes. The fractional derivative equation tool plays a crucial role in describing plenty of natural processes concerning physics, biology, geology, and so on. In this paper, we studied a loaded equation in relation to a spatial variable for a linear pseudoparabolic equation, with an initial and second boundary value condition (the Neumann condition), and a fractional Caputo derivative. A distinctive feature of the considered problem is that the load at the point is in the higher partial derivatives of the solution. The problem is reduced to a loaded equation with a nonlocal boundary value condition. A way to solve the considered problem is by using the method of energy inequalities, so that a priori estimates of solutions for non-local boundary value problems are obtained. To prove that this nonlocal problem is solvable, we used the method of continuation with parameters. The existence and uniqueness theorems for regular solutions are proven. Full article
(This article belongs to the Special Issue Boundary Value Problems, Dynamical Systems and Inverse Spectral Problems)
18 pages, 336 KiB  
Article
Backward Stackelberg Games with Delay and Related Forward–Backward Stochastic Differential Equations
by Li Chen, Peipei Zhou and Hua Xiao
Mathematics 2023, 11(13), 2898; https://doi.org/10.3390/math11132898 - 28 Jun 2023
Cited by 1 | Viewed by 924
Abstract
In this paper, we study a kind of Stackelberg game where the controlled systems are described by backward stochastic differential delayed equations (BSDDEs). By introducing a new kind of adjoint equation, we establish the sufficient verification theorem for the optimal strategies of the [...] Read more.
In this paper, we study a kind of Stackelberg game where the controlled systems are described by backward stochastic differential delayed equations (BSDDEs). By introducing a new kind of adjoint equation, we establish the sufficient verification theorem for the optimal strategies of the leader and the follower in a general case. Then, we focus on the linear–quadratic (LQ) backward Stackelberg game with delay. The backward Stackelberg equilibrium is presented by the generalized fully coupled anticipated forward–backward stochastic differential delayed Equation (AFBSDDE), which is composed of anticipated stochastic differential equations (ASDEs) and BSDDEs. Moreover, we obtain the unique solvability of the AFBSDDE using the continuation method. As an application of the theoretical results, the pension fund problem with delay effect is considered. Full article
(This article belongs to the Special Issue Boundary Value Problems, Dynamical Systems and Inverse Spectral Problems)
19 pages, 824 KiB  
Article
Transcendence and the Expression of the Spectral Series of a Class of Higher Order Differential Operators
by Bing Xie, Jing Li and Jiangang Qi
Mathematics 2023, 11(3), 636; https://doi.org/10.3390/math11030636 - 27 Jan 2023
Viewed by 1115
Abstract
In this paper, a relationship between the spectral zeta series of a class of higher order self-adjoint differential operators on the unit circle and the integral of Green functions is established by Mercer’s Theorem. Furthermore, the explicit expression and the transcendental nature of [...] Read more.
In this paper, a relationship between the spectral zeta series of a class of higher order self-adjoint differential operators on the unit circle and the integral of Green functions is established by Mercer’s Theorem. Furthermore, the explicit expression and the transcendental nature of the spectral series are obtained by the integral representation. Finally, several applications in physics about differential operators’ spectral theory, yellow some further works, and the transcendental nature of some zeta series are listed. Full article
(This article belongs to the Special Issue Boundary Value Problems, Dynamical Systems and Inverse Spectral Problems)
17 pages, 307 KiB  
Article
A Class of Singular Sturm–Liouville Problems with Discontinuity and an Eigenparameter-Dependent Boundary Condition
by Jinming Cai, Zhaowen Zheng and Kun Li
Mathematics 2022, 10(23), 4430; https://doi.org/10.3390/math10234430 - 24 Nov 2022
Viewed by 1191
Abstract
In this paper, we study a singular Sturm–Liouville problem with an eigenparameter-dependent boundary condition and transmission conditions at two interior points. Using an operator-theoretical formulation, we transfer the problem to an operator in an appropriate Hilbert space. It is proved that the operator [...] Read more.
In this paper, we study a singular Sturm–Liouville problem with an eigenparameter-dependent boundary condition and transmission conditions at two interior points. Using an operator-theoretical formulation, we transfer the problem to an operator in an appropriate Hilbert space. It is proved that the operator is self-adjoint. We also give the asymptotic formulas of the eigenvalues of the problem. Moreover, Green’s function is also discussed. Full article
(This article belongs to the Special Issue Boundary Value Problems, Dynamical Systems and Inverse Spectral Problems)
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