Approximate quasi-periodic solutions for nonlinear fractional equations with finite temporal translations

Dhaou Lassoued; Cemil Tunç; Dhaou Lassoued; Cemil Tunç

doi:10.3934/math.20261216

AIMS Mathematics

2026, Volume 11, Issue 9: 30689-30713. doi: 10.3934/math.20261216

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Research article

Approximate quasi-periodic solutions for nonlinear fractional equations with finite temporal translations

Dhaou Lassoued ¹,
Cemil Tunç ^{2
,
,}

1.
Mathematics and Applications Laboratory LR17ES11, Department of Mathematics, Faculty of Sciences of Gabès, Cité Erriadh 6072 Gabès, Tunisia
2.
School of Engineering and Natural Sciences, Istanbul Medipol University 34810, Istanbul, Turkey

Received: 29 May 2026 Revised: 22 August 2026 Accepted: 11 September 2026 Published: 20 September 2026
MSC : 34K37, 37K10, 37K60, 26A33, 70K75

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 $ (\Delta_\tau u)(t) = u(t+\tau)-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.
- nonlinear fractional equations,
- quasi-periodic dynamics,
- finite temporal translations,
- nonlocal temporal operators,
- memory effects,
- variational methods,
- Besicovitch spaces
Citation: Dhaou Lassoued, Cemil Tunç. Approximate quasi-periodic solutions for nonlinear fractional equations with finite temporal translations[J]. AIMS Mathematics, 2026, 11(9): 30689-30713. doi: 10.3934/math.20261216

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Abstract

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 $ (\Delta_\tau u)(t) = u(t+\tau)-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.

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