SavedAdd your journal

Journals / Kafkas Üniversitesi Fen Bilimleri Enstitüsü Dergisi / 2025 / Cilt: 18 Sayı: 2

New Asymptotic and Functional Properties for Detecting the Failure of the Fixed Point Property: A Unified Framework Beyond Classical c_0 and N_1 Structures

Sabit Nokta Özelliğinin Sağlanamamasını Belirlemeye Yönelik Asimptotik ve Fonksiyonel Çerçeveler: AI–c₀, N₁ ve Fonksiyonel Sandviç Özelliği

Veysel Nezir(KAFKAS UNIVERSITY)0000-0001-9640-8526
Pages
38–46
Publication Date
2025-12-26
🏛️Bibliographic metadata and full text access for this article are provided via DergiPark.
DergiPark Record ↗
✨

AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

We provide implications, examples, counterexamples, and Hahn-Banach based constructions, together with figures and comparative tables.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:We refine analytic and geometric tools for detecting the failure of the Fixed Point Property (FPP) in Banach spaces.
📊
Key Findings & Data:Classical detectors include asymptotically isometric copies of c_0 (AI-c_0), due to Dowling-Lennard-Turett, and the analytic N_1 property of Álvaro–Cembranos–Mendoza.
💡
Conclusion & Impact:We provide implications, examples, counterexamples, and Hahn-Banach based constructions, together with figures and comparative tables.
🔬Design:Experimental Field Trial
📍Scope:Comprehensive Sample
⚙️Variables:Sabit nokta özelliği · genişlemeyen fonksiyon · asimptotik izometrik kopya

Abstract

We refine analytic and geometric tools for detecting the failure of the Fixed Point Property (FPP) in Banach spaces. Classical detectors include asymptotically isometric copies of c_0 (AI-c_0), due to Dowling-Lennard-Turett, and the analytic N_1 property of Álvaro–Cembranos–Mendoza. Importantly, AI-c_0 implies N_1, while the converse fails; thus there exist spaces with N_1 but without any asymptotically isometric copy of c_0. Building on these, we introduce sup-dominated and functional-sandwiched frameworks (FSP, sFSP) that still detect FPP failure in classes where both a geometric AI-c_0 copy is absent and the classical N_1 estimate is too weak. We provide implications, examples, counterexamples, and Hahn-Banach based constructions, together with figures and comparative tables.

Özet

Banach uzaylarında Sabit Nokta Özelliği’nin (FPP) başarısızlığını saptamak için kullanılan analitik ve geometrik araçları iyileştirip geliştiriyoruz. Klasik belirleyiciler arasında, Dowling–Lennard–Turett tarafından ortaya konan asimptotik olarak izometrik c_0 kopyaları (AI-c_0) ile Álvaro–Cembranos–Mendoza’nın analitik N_1özelliği yer alır. Álvaro–Cembranos–Mendoza’nın önemli sonuçlarından birisi AI-c_0 ⇒ N_1 geçerli iken tersi doğru değildir; dolayısıyla N_1 özelliğine sahip olup, asimptotik olarak izometrik bir c_0 kopya içermeyen uzaylar mevcuttur. Bu gerçekten daha güçlü bir sonuç elde etmek üzerine, hem geometrik bir AI-c_0 kopyasının bulunmadığı hem de klasik N_1kestiriminin yetersiz kaldığı sınıflarda dahi FPP başarısızlığını yakalamayı sürdüren “üstten baskın” (sup-dominated) ve “fonksiyonel sandviç” (FSP, sFSP) çerçevelerini tanıtıyoruz. Yapıların ima yönlerini, örnekler, karşı-örnekler ve Hahn–Banach temelli kurulumlarla birlikte şekiller ve karşılaştırmalı tablolar sunuyoruz.

🔖

Cite This Article

Nezir, V. (2025). New Asymptotic and Functional Properties for Detecting the Failure of the Fixed Point Property: A Unified Framework Beyond Classical c_0 and N_1 Structures. Kafkas Üniversitesi Fen Bilimleri Enstitüsü Dergisi, Cilt: 18 Sayı: 2, 38–46. https://doi.org/10.58688/kujs.1820324

📄 Full Text PDF Viewer

📄

PDF görüntüleyici hazırlanıyor...

References (17)

  1. Álvaro, J. M., Cembranos, P., & Mendoza, J. (2017). Renormings of c0 and the fixed point property. Journal of Mathematical Analysis and Applications, 454(2), 1106-1113.
  2. Browder, F. E. (1965). Nonexpansive nonlinear operators in a Banach space. Proceedings of the National Academy of Sciences, 54(4), 1041-1044.
  3. Das, S., Nezir, V., Güven, A. (2025). New asymptotically isometric properties that imply the failure of the fixed point property in copies of ℓ1. Miskolc Mathematical Notes, 26(1), 181-193.
  4. Dowling, P., Johnson, W., Lennard, C., & Turett, B. (1997). The optimality of James’s distortion theorems. Proceedings of the American Mathematical Society, 125(1), 167-174.
  5. Dowling, P., & Lennard, C. (1997). Every nonreflexive subspace of 𝐿₁ [0, 1] fails the fixed point property. Proceedings of the American Mathematical Society, 125(2), 443-446.
  6. Dowling, P. N., Lennard, C. J., & Turett, B. (1996). Reflexivity and the fixed-point property for nonexpansive maps. Journal of mathematical analysis and applications, 200(3), 653-662.
  7. Dowling, P. N., Lennard, C. J., & Turett, B. (1998). Asymptotically isometric copies of c0 in Banach Spaces. Journal of mathematical analysis and applications, 219(2), 377-391.
  8. Dowling, P. N., Lennard, C. J., & Turett, B. (2000). Some fixed point results in l1 and c0. Nonlinear Analysis: Theory, Methods & Applications, 39(7), 929-936.
  9. Dowling, P. N., Lennard, C. J., & Turett, B. (2001). Renormings of ℓ1 and c0 and fixed point properties. In Handbook of metric fixed point theory (pp. 269-297). Dordrecht: Springer Netherlands.
  10. Dowling, P. N., Lennard, C. J., & Turett, B. (2002). The fixed point property for subsets of some classical Banach spaces. Nonlinear Analysis: Theory, Methods & Applications, 49(1), 141-145.
  11. Goebel, K., & Kirk, W. A. (1990). Topics in metric fixed point theory (Vol. 28). Cambridge university press. James, R. C. (1964). Uniformly non-square Banach spaces. Annals of Mathematics, 80(3), 542-550.
  12. Kirk, W. A. (1965). A fixed point theorem for mappings which do not increase distances. The American mathematical monthly, 72(9), 1004-1006.
  13. Nezir, V. (2020). Asymptotically isometric copies of l^(1⨁0). Hacettepe Journal of Mathematics And Statistics, 49(3), 984-997.
  14. Nezir, V., Mustafa, N. (2019). Exploring asymptotically isometric properties that imply the failure of the fixed point property in copies of c0. MAS International Conference on Mathematics-Engineering-Natural Medical Sciences-V, 454-461.
  15. Nezir, V., Mustafa, N. (2019). Exploring asymptotically isometric properties that imply the failure of the fixed point property in copies of l^1. MAS International Conference on Mathematics-Engineering-Natural Medical Sciences-V, 462-469.
  16. Nezir, V., Mustafa, N. (2023). On alternatives to the notions of c0 and l^1 copies and failure of fixed point property. 4. International Istanbul Current Scientific Research Congress, 481-492.
  17. Nezir, V., Mustafa, N., & Güven, A. (2020). Yet some other alternative asymptotically isometric properties inside copies of l^1 and their implication of failure of fixed point property. 3rd International Conference on Mathematical and Related Sciences: Current Trends and Developments, 29-35.