Approximate quasi-periodic solutions for nonlinear fractional equations with finite temporal translations
AIMS Mathematics, cilt.11, sa.9, ss.30689-30713, 2026 (SCI-Expanded, Scopus)
- Yayın Türü: Makale / Tam Makale
- Cilt numarası: 11 Sayı: 9
- Basım Tarihi: 2026
- Doi Numarası: 10.3934/math.20261216
- Dergi Adı: AIMS Mathematics
- Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus
- Sayfa Sayıları: ss.30689-30713
- Anahtar Kelimeler: Besicovitch spaces, finite temporal translations, memory effects, nonlinear fractional equations, nonlocal temporal operators, quasi-periodic dynamics, variational methods
- Açık Arşiv Koleksiyonu: AVESİS Açık Erişim Koleksiyonu
- İstanbul Medipol Üniversitesi Adresli: Hayır
Özet
We investigate a class of nonlinear fractional lattice equations with finite temporal translation interactions under quasi-periodic external forcing. The model combines a fractional time operator, nonlinear interactions generated by a potential function, and finite translations of the unknown function in the temporal variable. In particular, the translation terms are represented by the finite-difference operator (∆τu)(t) = u(t + τ) − u(t), and are not interpreted as a conventional time-delay differential equation. The present formulation should not be confused with a conventional time-delay differential equation. Working in a quasi-periodic functional framework associated with finitely generated frequency modules, we study approximate quasi-periodic response states. Using a variational approach based on averaged energy functionals and Ekeland's variational principle, we construct finite-frequency quasi-periodic solutions corresponding to approximations of the external forcing. We also analyze the lack of compactness inherent in the Besicovitch framework. The results provide a variational description of approximate quasi-periodic responses in nonlinear fractional systems with finite temporal translations and multi-frequency excitation.