Date of Award

12-17-2024

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Applied Science

First Advisor

Eric Kaufmann

Abstract

In this dissertation, we study the fourth-order iterative differential equation \begin{displaymath} x^{(4)}(t) = \f(t, x(t), x^{[2]}(t), \dots, x^{[m]}(t)) \end{displaymath} where $x^{[2]}(t) = x(x(t))$ and $x^{[j]}(t) = x(x^{[j - 1]}(t))$ for $j > 2$. We consider the above equation with multiple sets of boundary conditions, and we state results on the existence and uniqueness of solutions for each set of boundary conditions. In Chapter 2, the boundary conditions are conjugate boundary conditions, \begin{align*} &x(-a) = -a, \ x'(-a) = b, \ x''(-a) = c, \ x(a) = a \\ &x(-a) = -a, \ x(a) = a, \ x'(a) = b, \ x''(a) = c. \end{align*} In Chapter 3, we again are concerned with conjugate boundary conditions. However, our results from Chapter 2 force the function $f$ to be of one sign. In order to remove this restriction, we modify the boundary conditions to \begin{align*} &x(a) = c, \ x'(a) = 0, \ x''(a) = 0, \ x(b) = d \\ &x(a) = c, \ x(b) = d, \ x'(a) = 0, \ x''(a) = 0, \end{align*} where $a < c < d < b$. Finally, Chapter 4 is concerned with non-local conditions, \begin{align*} &x(a) = x'(a) = x''(a) = 0 \\ &\alpha \int_{a}^{\eta} x(t) \, ds = x(b), \ \eta \in (a, b). \end{align*} In addition to results on existence and uniqueness of solutions, we also state a result on continuous dependence on the function $f$. Our process is the same for all three problems. We first invert the differential equation into an integral equation. We then define an integral operator based off of the integral equation, which we can use to obtain our results. The main tools used for our results are the Schauder's fixed point theorem and a result on the difference of iterative functions.

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