Date on Master's Thesis/Doctoral Dissertation

8-2023

Document Type

Doctoral Dissertation

Degree Name

Ph. D.

Department

Mathematics

Degree Program

Applied and Industrial Mathematics, PhD

Committee Chair

Riedel, Thomas

Committee Co-Chair (if applicable)

Powers, Robert

Committee Member

Powers, Robert

Committee Member

Larson, Lee

Committee Member

Himes, Paul

Author's Keywords

Functional equations; stability; Cauchy's equation; probabilistic metric space; fuzzy logic; analysis

Abstract

The most famous functional equation f(x+y)=f(x)+f(y) known as Cauchy's equation due to its appearance in the seminal analysis text Cours d'Analyse (Cauchy 1821), was used to understand fundamental aspects of the real numbers and the importance of regularity assumptions in mathematical analysis. Since then, the equation has been abstracted and examined in many contexts. One such examination, introduced by Stanislaw Ulam and furthered by Donald Hyers, was that of stability. Hyers demonstrated that Cauchy's equation exhibited stability over Banach Spaces in the following sense: functions that approximately satisfy Cauchy's equation are approximated with the same level of error by functions that are solutions of Cauchy's equation, namely linear maps. Here we pose the question of the stability of Cauchy's equation for functions defined on the monoid known as Delta Plus, the space of cumulative distribution functions. We present stability results analogous to those of Hyer's and Ulam and results involving a new perspective on stability. We furnish a connection between the two perspectives and examples of the need for some regularity assumptions.

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