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

Yıl 2025, Cilt: 18 Sayı: 2, 38 - 46, 26.12.2025

Veysel Nezir

https://doi.org/10.58688/kujs.1820324

Öz

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.

Anahtar Kelimeler

Sabit nokta özelliği , genişlemeyen fonksiyon , asimptotik izometrik kopya , N_1 özelliği

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

Öz

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.

Anahtar Kelimeler

Fixed point property , nonexpansive mappings , asymptotically isometric copy , N_1 property

Kaynakça

• 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.
• Browder, F. E. (1965). Nonexpansive nonlinear operators in a Banach space. Proceedings of the National Academy of Sciences, 54(4), 1041-1044.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• Kirk, W. A. (1965). A fixed point theorem for mappings which do not increase distances. The American mathematical monthly, 72(9), 1004-1006.
• Nezir, V. (2020). Asymptotically isometric copies of l^(1⨁0). Hacettepe Journal of Mathematics And Statistics, 49(3), 984-997.
• 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.
• 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.
• 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.
• 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.