Date of Award
2026
Document Type
Dissertation
Degree Name
Doctor of Philosophy (PhD)
Department
Mathematical Sciences
Committee Chair
Toka Diagana
Committee Member
Charles A. Hester
Committee Member
Ian Knowles
Committee Member
Simon Bortz
Committee Member
Dongsheng Wu
Research Advisor
Claudio H. Morales
Subject(s)
Fixed point theory, Vector spaces, Convex sets
Abstract
A geometric theory of seminorms is introduced and exposited. The concept of a G-Space, which is a Frechet which holds the sequence of seminorms that generate its topology as intrinsic to its definition, is introduced. Abstract results which involve concluding, from the existence of common pseudo-fixed points of a family of operators on a G-Space, the existence of a common fixed point for that family, are presented. These results are exploited to prove the existence of fixed points of some operators defined on totally closed, convex subsets of certain G-Spaces. Three specific fixed point scenarios are addressed. The first is the case of a coherently asymptotic nonexpansive type, iteratively continuous operator which permits a bounded orbit self map under suitable convexity conditions of the underlying G-Space. In this case, it is also shown that under modest additional assumptions, a commuting family of such operators permits a fixed point. Second, a coherently demicontinuous pseudocontractive operator on a bounded subset of a suitable G-Space is shown to possess a fixed-point. Finally a sufficient conditions for a coherently continuous pseudocontractive operator possessing the weakly inward condition to have a fixed point are presented.
Recommended Citation
McKinney, Quinten Ryan, "The geometry of seminormed spaces with applications to the fixed point theory of operators defined on geometric Fréchet spaces" (2026). Dissertations. 483.
https://louis.uah.edu/uah-dissertations/483