Off-campus UNL users: To download campus access dissertations, please use the following link to log into our proxy server with your NU ID and password. When you are done browsing please remember to return to this page and log out.
Non-UNL users: Please talk to your librarian about requesting this dissertation through interlibrary loan.
Boundary value problems, oscillation theory, and the Cauchy functions for dynamic equations on a measure chain
Abstract
In Chapter 1 we briefly discuss the calculus on measure chains which was developed by Stefan Hilger. For functions [special characters omitted] we introduce the delta-derivative and the delta-integral and state fundamental results. We then state some known results which we will use to prove our main results in Chapters 2 and 3. In Chapter 2 we prove existence and uniqueness theorems for solutions of the boundary value problem [special characters omitted] for t in a measure chain [special characters omitted]. The main result in this chapter is to show that if we have upper and lower solutions of [special characters omitted]with the lower solution α(t) below the upper solution β(t) and if α(a) ≤ A ≤ β(a) and α(σ2 (b)) ≤ B ≤ β(σ2 (b)), then our boundary value problem has a solution and this solution stays between α(t) and β( t). We then use this result to show other existence-uniqueness theorems. At the beginning of Chapter 3 we are concerned with proving various properties of an exponential function for a time scale. One of our main results in this chapter is to completely determine the sign of this exponential function. This then determines when first order linear homogeneous dynamic equations and their adjoints are oscillatory or nonoscillatory. In the last section of this chapter we derive the characteristic equation of a higher order linear dynamic equation on a time scale and give oscillation criteria for this dynamic equation on a time scale. In Chapter 4 we consider the n-th order linear dynamic equation [special characters omitted] where pi(t), 0 ≤ i ≤ n, are real-valued functions defined on [special characters omitted]. In this chapter we only consider time scales [special characters omitted] such that every point in [special characters omitted] is isolated. We define the Cauchy function K( t, s) for this dynamic equation and then we prove a variation of constants formula. One of our main concerns is to see how the Cauchy function for an equation is related to the Cauchy functions for the factored parts of the operator P. Finally we consider the equation [special characters omitted] where each of the pi's is a constant and obtain a formula for the Cauchy function.
Subject Area
Mathematics
Recommended Citation
Akin, Elvan, "Boundary value problems, oscillation theory, and the Cauchy functions for dynamic equations on a measure chain" (2000). ETD collection for University of Nebraska-Lincoln. AAI9997003.
https://digitalcommons.unl.edu/dissertations/AAI9997003