The Strum-Liouville equation is expressed in Hamiltonian form. A simple generating function is derived which defines a large class of canonical transformations and reduces the Sturm-Liouville equation to the solution of a first order equation with a single unknown. The method is developed with particular reference to the wave equation. The procedure unifies many apparently diverse treatments an...

On q-Di®erence Equations Moemen H. Abu-Risha Department of Mathematics Cairo University, Eygpt E-mail: [email protected] We give an investigation of some properties of solutions of linear q¡di®erence equations de ̄ned on an interval [a;1): The cases a = 0 and a > 0 are essentially di®erent. The study of the latter case is necessary for the derivation of asymptotic formulae and oscil...

A regularized trace formula of first order for the matrix Sturm-Liouville equation with eigenparameter in the boundary conditions is obtained.

We study the Al-Salam–Chihara polynomials when q > 1. Several solutions of the associated moment problem are found, and the orthogonality relations lead to explicit evaluations of several integrals. The polynomials are shown to have raising and lowering operators and a second order operator equation of Sturm-Liouville type whose eigenvalues are found explicitly. We also derive new measures with...

In this paper, we study the nonlinear second-order impulsive qk-difference equations with Sturm-Liouville type, in which nonlinear team and impulsive teams are dependent on first-order qk-derivatives. We obtain the existence and uniqueness results of solutions for the problem by Banach’s contraction mapping principle and Schaefer’s fixed point theorems. Finally, we give two examples to demonstr...

We define and study the properties of Darboux-type transformations between Sturm–Liouville problems with boundary conditions containing rational Herglotz–Nevanlinna functions of the eigenvalue parameter (including the Dirichlet boundary conditions). Using these transformations, we obtain various direct and inverse spectral results for these problems in a unified manner, such as asymptotics of e...

Abstract The goal of this study is to analyse the eigenvalues and weak eigenfunctions a new type multi-interval Sturm-Liouville problem (MISLP) which differs from standard problems (SLPs) in that Strum-Liouville equation defined on finite number non-intersecting subintervals boundary conditions are set not only at endpoints but also internal points interaction. For self-adjoint treatment consid...