Fixed Point Theory and Topological and Geometry Structure

2023 Fiscal Year Final Research Report

• Project/Area Number: 16K05207
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Kyushu Institute of Technology
• Principal Investigator: Suzuki Tomonari 九州工業大学, 大学院工学研究院, 教授 (00303173)
• Co-Investigator(Kenkyū-buntansha): 加藤 幹雄 信州大学, 工学部, 非常勤講師 (50090551)
• Project Period (FY): 2016-04-01 – 2024-03-31
• Keywords: 不動点 / contraction / ν-generalized metric / τ-distance / semimetric space / semicompleteness

Outline of Final Research Achievements

We studied fixed point theory in metric spaces. For example, we proved a new generalization of both Bogin's fixed point theorem and Ciric's fixed point theorem in complete metric spaces. In the proof, we gave a new method. Also, we introduced the concept of τ'-distance which is a very slight generalization of τ-distance and which is more natural than τ-distance. Using this concept, we proved some related theorems. In metric fixed point theory, there are several contractive conditions. We discussed these conditions by a unified method.
We studied several spaces which do not have rich structure. For example, we prove that a ν-generalized metric space has both strongly compatible topology and the strongest sequentially compatible topology. Also, we proved Caristi's fixed point theorem in (Σ,≠)-complete semimetric spaces.

Free Research Field

不動点理論

Academic Significance and Societal Importance of the Research Achievements

完備距離空間における不動点理論は1960年代と1970年代に大きく発展した。その後、本質的に新しい不動点定理は証明されなかったように思える。今回得られた不動点定理は証明の手法もとても新しく、今後の発展が期待できる。τ'-distance の導入や縮小条件に関する結果は、今後の議論展開を容易にするため、新たな発展へ寄与できると思われる。
最近、 semimetric space や ν-generalized metric space といった弱い構造しか持たない空間での研究が盛んに行われている。本研究もその一部であり、得られた成果を用いて、さらに精密な研究が行われると期待できる。