The Department of Mathematics discussed a Master’s thesis submitted by researcher Intisar Kareem Atiya Duraib, focusing on the topological structure of Quasi-2-Normed Spaces and the development of Best Proximity Point Theory, with the establishment of new existence and uniqueness results under suitable contraction conditions.
The thesis was discussed at the Late Professor Dr. Aribi Al-Zubai Hall under the supervision of Professor Dr. Salwa Salman .
The thesis aimed to investigate the topological structure of Quasi-2-Normed Spaces and to develop the theory of best proximity points within this framework by establishing existence and uniqueness results for certain classes of non-self mappings that satisfy appropriate contraction conditions.
Quasi-2-normed spaces constitute a generalization of classical 2-normed spaces, in which the conventional triangle inequality is replaced by a quasi-triangle inequality involving a positive constant greater than or equal to one. This generalization broadens the class of mathematical spaces that can be investigated while preserving several fundamental analytical properties, thereby providing a wider framework for the development of nonlinear functional analysis.
The first part of the thesis focused on investigating the topological structure of Quasi-2-Normed Spaces. A topology generated by a suitable separating family of quasi-2-seminorms was constructed and examined in detail.
The study established the compatibility of this topology with the quasi-2-norm structure, providing the underlying space with the structure of a topological vector space. The induced topology and its fundamental properties were also investigated, together with the conditions under which Quasi-2-Normed Spaces can be normable and metrizable.
These structural results provide an important mathematical foundation for extending analytical techniques developed in normed spaces and locally convex spaces to the broader setting of Quasi-2-Normed Spaces.
In the second part of the thesis, the researcher developed, within the established topological framework, the theory of best proximity points in Quasi-2-Normed Spaces.
The thesis established new existence and uniqueness results for specific classes of non-self mappings satisfying suitable contraction conditions. These results extend several classical results from metric spaces, normed spaces, and quasi-normed settings to the more general framework of Quasi-2-Normed Spaces.
The study also demonstrated that when the sets under consideration have a nonempty intersection, the corresponding best proximity point theorems naturally reduce to fixed point theorems. This provides a unified framework for studying both best proximity point theory and fixed point theory within Quasi-2-Normed Spaces.
The significance of the thesis lies in its contribution to the structural and topological study of Quasi-2-Normed Spaces, as well as in extending the applicability of best proximity point and fixed point theories to a more general mathematical setting.
For future research, the study recommends extending best proximity point theory in Quasi-2-Normed Spaces to multivalued mappings, in addition to investigating more general types of contraction conditions. Such extensions may open new avenues of research in nonlinear functional analysis and fixed point theory.
The thesis is directly aligned with Sustainable Development Goal 4 (SDG 4): Quality Education, through its contribution to advanced mathematical research, the development of specialized scientific knowledge, and the advancement of postgraduate research capabilities in mathematics and functional analysis.
The research also contributes indirectly to Sustainable Development Goal 9 (SDG 9): Industry, Innovation and Infrastructure, through the development of abstract mathematical tools and theories that may provide foundational knowledge for scientific and technological innovation, particularly in mathematical analysis, applied mathematics, and the modeling of complex problems.
The thesis highlights the importance of fundamental mathematical research as a scientific foundation for expanding knowledge and developing theoretical frameworks that support progress in science, engineering, and modern technologies.
