Sturm-Picone Comparison Theorem of Second-Order Linear Equations on Time Scales

Sturm-Picone Comparison Theorem of Second-Order Linear Equations on Time Scales

A Nonlinear Sturm–Picone Comparison Theorem for Dynamic Equations on Time Scales

The authors derive an analog of the well-known Picone identity but for nonlinear dynamic equations on time scales. As a consequence, they obtain a nonlinear comparison theorem in the spirit of the classical Sturm–Picone comparison theorem. Comparison results yielding the nonoscillation of all solutions of nonlinear equations are also obtained. AMS subject classification: 34C10, 34C15, 39A11.

First order linear fuzzy dynamic equations on time scales

In this paper, we study the concept of generalized differentiability for fuzzy-valued functions on time scales. Usingthe derivative of the product of two functions, we provide solutions to first order linear fuzzy dynamic equations. Wepresent some examples to illustrate our results.

Overview of Weyl–Titchmarsh Theory for Second Order Sturm–Liouville Equations on Time Scales

In this paper we present an overview of the basic Weyl–Titchmarsh theory for second order Sturm–Liouville equations on time scales. We construct the m(λ)function, the Weyl solution, and the Weyl disk. We justify the terminology “disk” by its geometric properties, show explicitly the coordinates of the center of the disk, and calculate its radius. We show that the dichotomy regarding the squarei...

New comparison and oscillation theorems for second-order half-linear dynamic equations on time scales

∧ second-order half-linear dynamic equation (r(t)(x(t))) + p(t)x(σ (t)) = 0, where r(t) > 0, p(t) are continuous, ∫ ∞ t0 (r(t)) 1 α ∆t = ∞, α is a quotient of odd positive integers. In particular, no explicit sign assumptions aremadewith respect to the coefficient p(t). We give conditions under which every positive solution of the equations is strictly increasing. For α = 1, T = R, the result i...