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# 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 f:T R 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 x^{DD}t =f&parl0;t,x^{s}t &parr0;,xa=A,x&parl0;s^{ 2}b&parr0;=B for *t* in a measure chain T . The main result in this chapter is to show that if we have upper and lower solutions of x^{DD}t =f&parl0;t,x^{s}t&parr0; with the lower solution α(*t*) below the upper solution β(*t*) and if α(*a*) ≤ * A* ≤ β(*a*) and α(σ^{2} (*b*)) ≤ *B* ≤ β(σ^{2 } (*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 Pxt=S^{ n}i=0pi tx&parl0;s^{i} t&parr0;=0 where *p _{i}*(

*t*), 0 ≤

*i*≤

*n*, are real-valued functions defined on T . In this chapter we only consider time scales T such that every point in T 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 Pxt=S

^{ n}i=0pix&parl0; s

^{i}t&parr0; =0 where each of the

*p*'s is a constant and obtain a formula for the Cauchy function. ^

_{i}#### 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.

http://digitalcommons.unl.edu/dissertations/AAI9997003