## Doctoral Dissertations

#### Abstract

"The concept of periodic functions defined on the real numbers or on the integers is a classical topic and has been studied intensively, yielding numerous applications in every kind of science. It is of importance that the real numbers and the integers are closed with respect to addition. However, for a number q > 1, the so-called q-time scale, i.e., the set of nonnegative integer powers of q, is not closed with respect to addition, and therefore it was not possible to define periodic functions on the q-time scale in an obvious way. In this thesis, this important open problem has been resolved and the definition of periodic functions defined on the q-time scale is given. Using this new definition of periodic functions defined on the q-time scale, five distinct results involving periodic solutions of various kinds of q-difference equations are presented, namely as follows. First, Floquet theory for q-difference equations is established. Second, the Cushing-Henson conjecture is proved for periodic solutions of the Beverton-Holt q-difference equation, resulting in applications in the study of biology, in particular population models. Third, stability for Hamiltonian q-difference systems is investigated. Fourth, the existence of periodic solutions of a q-difference boundary value problem is examined by applying the well-known Mountain Pass theorem. Fifth, the existence of positive periodic solutions of higher-order functional q-difference equations is studied by applying the well-known fixed-point theorem in a cone. Besides these five research papers that are based on the newly introduced definition of periodic functions on the q-time scale, this thesis also contains an introduction, a section on time scales calculus, a section on quantum calculus, and a conclusion"--Abstract, page v.

#### Advisor(s)

Bohner, Martin, 1966-

#### Committee Member(s)

Akin, Elvan

Le, Vy Khoi

Dosly, Ondrej

Clark, Stephen L.

#### Department(s)

Mathematics and Statistics

#### Degree Name

Ph. D. in Mathematics

#### Publisher

Missouri University of Science and Technology

#### Publication Date

Summer 2012

#### Journal article titles appearing in thesis/dissertation

- Floquet theory for 1-difference equations
- Beverton-Holt q-difference equation
- Stability for Hamiltonian q-difference systems
- Existence of periodic solutions of a q-difference boundary value problem
- Positive periodic solutions of higher-order functional q-difference equations

#### Pagination

xi, 116 pages

#### Note about bibliography

Includes bibliographical references.

#### Rights

© 2012 Rotchana Chieochan, All rights reserved.

#### Document Type

Dissertation - Open Access

#### File Type

text

#### Language

English

#### Library of Congress Subject Headings

Differential equations

Floquet theory

Hamiltonian systems

Periodic functions

q-series

#### Thesis Number

T 10023

#### Print OCLC #

815766027

#### Electronic OCLC #

801823153

#### Link to Catalog Record

#### Recommended Citation

Chieochan, Rotchana, "Periodic q-difference equations" (2012). *Doctoral Dissertations*. 1967.

https://scholarsmine.mst.edu/doctoral_dissertations/1967