scholarly journals Principal solution in Weyl-Titchmarsh theory for second order Sturm-Liouville equation on time scales

2017 ◽  
pp. 1-18 ◽  
Author(s):  
Petr Zemánek
Keyword(s):  
Time Scales ◽  
Liouville Equation ◽  
Second Order ◽  
Principal Solution ◽  
Sturm Liouville

Diophantine equations for Morgan-Voyce and other modified orthogonal polynomials

2008 ◽  
Vol 58 (1) ◽  
Author(s):  
Thomas Stoll ◽  
Robert Tichy
Keyword(s):  
Orthogonal Polynomials ◽  
Liouville Equation ◽  
Diophantine Equation ◽  
Diophantine Equations ◽  
Second Order ◽  
Recurrence Equation ◽  
Classical Orthogonal Polynomials ◽  
The Third ◽  
Sturm Liouville

AbstractIt is well-known that Morgan-Voyce polynomials B n(x) and b n(x) satisfy both a Sturm-Liouville equation of second order and a three-term recurrence equation ([SWAMY, M.: Further properties of Morgan-Voyce polynomials, Fibonacci Quart. 6 (1968), 167–175]). We study Diophantine equations involving these polynomials as well as other modified classical orthogonal polynomials with this property. Let A, B, C ∈ ℚ and {pk(x)} be a sequence of polynomials defined by $$\begin{gathered} p_0 (x) = 1 \hfill \\ p_1 (x) = x - c_0 \hfill \\ p_{n + 1} (x) = (x - c_n )p_n (x) - d_n p_{n - 1} (x), n = 1,2,..., \hfill \\ \end{gathered} $$ with $$(c_0 ,c_n ,d_n ) \in \{ (A,A,B),(A + B,A,B^2 ),(A,Bn + A,\tfrac{1}{4}B^2 n^2 + Cn)\} $$ with A ≠ 0, B > 0 in the first, B ≠ 0 in the second and C > −¼B 2 in the third case. We show that the Diophantine equation with m > n ≥ 4, ≠ 0 has at most finitely many solutions in rational integers x, y.

scholarly journals On Basis Property of Root Functions For a Class Second Order Differential Operator

2020 ◽  
Vol 5 (1) ◽  
pp. 361-368
Author(s):  
Volkan Ala ◽  
Khanlar R. Mamedov
Keyword(s):  
Differential Operator ◽  
Boundary Conditions ◽  
Spectral Parameter ◽  
Liouville Equation ◽  
Basis Property ◽  
Second Order ◽  
Order Differential Operator ◽  
Root Functions ◽  
Sturm Liouville

AbstractIn this work we investigate the completeness, minimality and basis properties of the eigenfunctions of one class discontinuous Sturm-Liouville equation with a spectral parameter in boundary conditions.

scholarly journals Inverse Problems for Sturm-Liouville Differential Operators on Closed Sets

2019 ◽  
Vol 50 (3) ◽  
pp. 199-206 ◽  
Author(s):  
V. Yurko
Keyword(s):  
Inverse Problems ◽  
Time Scales ◽  
Differential Operators ◽  
Spectral Characteristics ◽  
Second Order ◽  
Constructive Algorithms ◽  
Closed Sets ◽  
Sturm Liouville

Second-order differential operators on closed sets (time scales) are considered. Properties of their spectral characteristics are obtained and inverse problems are studied, which consists in recovering the operators from their spectral characteristics. We establish the uniqueness and develop constructive algorithms for the solution of the inverse problems.

INTEGRAL CHARACTERIZATION OF THE PRINCIPAL SOLUTION O HALF-LINEAR SECOND ORDER DIF ERENTIAL EQUATIONS

2000 ◽  
Vol 36 (3-4) ◽  
pp. 455-469 ◽  
Author(s):  
O. DoŠlÝ ◽  
A. Elbert
Keyword(s):  
Nontrivial Solution ◽  
Second Order ◽  
Principal Solution ◽  
Sturm Liouville

Under some restrictions on the functions r (t)and c (t),a nontrivial solution x (t)of the nonoscillatory half-linear second order diferential equation: is principal if and only if… In case p =2 the criterion (*)reduces to a Hartman result on the principal solutions of the Sturm –Liouville di .erential equations.

scholarly journals Finite spectrum of Sturm-Liouville problems with eigenparameter-dependent boundary conditions on time scales

Filomat ◽  
2019 ◽  
Vol 33 (6) ◽  
pp. 1747-1757
Author(s):  
Ji-Jun Ao ◽  
Juan Wang
Keyword(s):  
Spectral Analysis ◽  
Boundary Conditions ◽  
Time Scales ◽  
Liouville Equation ◽  
Time Scale ◽  
The Self ◽  
Finite Spectrum ◽  
Sturm Liouville

The spectral analysis of a class of Sturm-Liouville problems with eigenparameter-dependent boundary conditions on bounded time scales is investigated. By partitioning the bounded time scale such that the coefficients of Sturm-Liouville equation satisfy certain conditions on the adjacent subintervals, the finite eigenvalue results are obtained. The results show that the number of eigenvalues not only depend on the partition of the bounded time scale, but also depend on the eigenparameter-dependent boundary conditions. Both of the self-adjoint and non-self-adjoint cases are considered in this paper.

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