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# SOLUTIONS TO DIFFERENTIAL EQUATIONS VIA FIXED POINT APPROACHES: NEW MATHEMATICAL FOUNDATIONS AND APPLICATIONS

Published online by Cambridge University Press:  09 June 2022

## MSC classification

Type
Abstracts of Australasian PhD Theses
Information
© The Author(s), 2022. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

The central aim of this thesis is to construct a fuller and firmer mathematical foundation for the solutions to various classes of nonlinear differential equations than that currently available in the literature. This includes boundary value problems (BVPs) that involve ordinary differential equations, and initial value problems (IVPs) for fractional differential equations. In particular, we establish new conditions that guarantee the existence, uniqueness and approximation of solutions to second-order BVPs, third-order BVPs and fourth-order BVPs for ordinary differential equations. The results enable us, in turn, to shed new light on problems from applied mathematics, engineering and physics, such as: the Emden and Thomas–Fermi equations; the bending of elastic beams through an application of our general theories; and laminar flow in channels with porous walls. We also ensure the existence, uniqueness and approximation of solutions to some IVPs for fractional differential equations. An understanding of the existence, uniqueness and approximation of solutions to these problems is fundamental from both pure and applied points of view. Our methods involve an analysis of nonlinear operators through fixed-point theory in new and interesting ways. Part of the novelty involves generating new conditions under which these operators are contractive, invariant and/or establishing new a priori bounds on potential solutions. As such, we draw on Banach fixed-point theorem, Schauder fixed-point theorem, Rus’s contraction mapping theorem, and a continuation theorem due to A. Granas and its constructive version known as the continuation method for contractive maps. The ideas in this thesis break new ground at the intersection of pure and applied mathematics. Thus, this work will be of interest to those who are researching the theoretical aspects of differential equations, and those who are interested in better understanding their applications.

Some of the research from the thesis has been published in [Reference Almuthaybiri and Tisdell1Reference Almuthaybiri and Tisdell6].

## Footnotes

Thesis submitted to the University of New South Wales in June 2021; degree approved on 2 November 2021; supervisors Christopher Tisdell and Chris Goodrich.

## References

Almuthaybiri, S. S. and Tisdell, C. C., ‘Global existence theory for fractional differential equations: new advances via continuation methods for contractive maps’, Analysis (Berlin) 39(4) (2019), 117128.Google Scholar
Almuthaybiri, S. S. and Tisdell, C. C., ‘Establishing existence and uniqueness of solutions to the boundary value problem involving a generalized Emden equation, embracing Thomas–Fermi-like theories’, J. Engrg. Math. 124 (2020), 110.CrossRefGoogle Scholar
Almuthaybiri, S. S. and Tisdell, C. C., ‘Existence and uniqueness of solutions to third-order boundary value problems: analysis in closed and bounded sets’, Differ. Equ. Appl. 12(12) (2020), 291312.Google Scholar
Almuthaybiri, S. S. and Tisdell, C. C., ‘Sharper existence and uniqueness results for solutions to third-order boundary value problems’, Math. Model. Anal. 25(3) (2020), 409420.CrossRefGoogle Scholar
Almuthaybiri, S. S. and Tisdell, C. C., ‘Sharper existence and uniqueness results for solutions to fourth-order boundary value problems and elastic beam analysis’, Open Math. 18(1) (2020), 10061024.Google Scholar
Almuthaybiri, S. S. and Tisdell, C. C., ‘Uniqueness of solutions for a coupled system of nonlinear fractional differential equations via weighted norms’, Commun. Appl. Nonlinear Anal. 28(1) (2021), 6576.Google Scholar