Abstract We consider discrete Sturm–Liouville-type equations of the form $$\begin{aligned} \varDelta (r_n\varDelta x_n)=a_nf(x_{\sigma (n)})+b_n. \end{aligned}$$ ? ( r n x )</mml:...

Time scale theory helps us to combine differential equations with difference equations. Especially in models such as biology, medicine, and economics, since the independent variable is handled discrete, it requires analyze discrete clusters. In these cases, defined $\mathbb{Z}$ are considered. Boundary value problems (BVP's) used solve model many physical areas. this study, we examined spectral...

The Sturm–Liouville hierarchy of evolution equations was introduced in [Adv. Nonlinear Stud. 11 (2011), 555–591] and includes the Korteweg–de Vries and the Camassa– Holm hierarchies. We discuss some solutions of this hierarchy which are obtained as limits of algebro-geometric solutions. The initial data of our solutions are (generalized) reflectionless Sturm–Liouville potentials [Stoch. Dyn. 8 ...

Sturm-Liouville Theory Christopher J. Adkins Master of Science Graduate Department of Mathematics University of Toronto 2014 A basic introduction into Sturm-Liouville Theory. We mostly deal with the general 2ndorder ODE in self-adjoint form. There are a number of things covered including: basic asymptotics, properties of the spectrum, interlacing of zeros, transformation arguments...

In this article, we consider dissipative Sturm–Liouville operators in the limit-circle case on time scales. Then, using the Livšic’s Theorem, we prove the completeness of the system of root vectors for dissipative Sturm–Liouville operators. 2013 Elsevier Inc. All rights reserved.

This paper deals with the discrete system being finite-difference approximation of Sturm–Liouville problem frozen argument. The inverse theory is developed for this system. We describe two principal cases: degenerate and non-degenerate. For these cases, appropriate problems statements are provided, uniqueness theorems proved, reconstruction algorithms obtained. Moreover, relationship between ei...