Improvements on the Leighton oscillation theorem for second-order dynamic equations
-
Aǧacik Zafer
and
Sibel Doǧru Akgöl
Abstract
The time scales version of the Leighton oscillation theorem states that if
where a and p are rd-continuous with a(t) > 0 for t ≥ t0, then every solution of the second-order self-adjoint dynamic equation
is oscillatory. The theorem turns into the famous Leighton oscillation theorem when the time scale is taken as the set of real numbers, and its discrete version when the time scale is taken as the set of integers. The divergence of the first improper integral in (∗) means that the dynamic equation is in canonical form. The equation is called noncanonical when the integral is convergent. In this study, we establish an improved version of the Leighton oscillation theorem on time scales that can be applied to both canonical and noncanonical types of dynamic equations. Furthermore, we allow the second improper integral in (∗) to be convergent. In the special case, we derive completely new Leighton-type oscillation theorems for second-order self-adjoint difference equations
where Δ is the forward difference operator, defined by Δxk = xk+1 − xk (the derivative). Examples are given to illustrate the significance of these theorems.
(Communicated by Jozef Dz̆urina)
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Articles in the same Issue
- Generalized Sasaki mappings in d0-Algebras
- On a theorem of Nathanson on Diophantine approximation
- Constructing infinite families of number fields with given indices from quintinomials
- Partitions into two Lehmer numbers in ℤq
- Fundamental systems of solutions of some linear differential equations of higher order
- On k-Circulant matrices involving the Lucas numbers of even index
- Explicit formulae for the Drazin inverse of the sum of two matrices
- On nonoscillation of fractional order functional differential equations with forcing term and distributed delays
- Novel generalized tempered fractional integral inequalities for convexity property and applications
- Convergence of α-Bernstein-Durrmeyer operators about a collection of measures
- The problem of finding eigenvalues and eigenfunctions of boundary value problems for an equation of mixed type
- Orbital Hausdorff dependence and stability of the solution to differential equations with variable structure and non-instantaneous impulses
- Improvements on the Leighton oscillation theorem for second-order dynamic equations
- Topogenous orders on forms
- Comparison of topologies on fundamental groups with subgroup topology viewpoint
- An elementary proof of the generalized Itô formula
- Asymptotic normality for kernel-based test of conditional mean independence in Hilbert space
- Advancing reliability and medical data analysis through novel statistical distribution exploration
- Historical notes on the 75th volume of Mathematica Slovaca - Authors of the first issue from 1951
