Variable-exponent modular delays in state-dependent delay differential equations: an abstract semigroup formulation

Mostafa Bachar; Mostafa Bachar

doi:10.3934/era.2026379

Electronic Research Archive

2026, Volume 34, Issue 12: 8997-9032. doi: 10.3934/era.2026379

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Research article

Variable-exponent modular delays in state-dependent delay differential equations: an abstract semigroup formulation

Mostafa Bachar ^,

Department of Mathematics, College of Science, King Saud University, Riyadh 11451, Saudi Arabia

Received: 04 July 2026 Revised: 07 October 2026 Accepted: 10 October 2026 Published: 26 October 2026

We study state-dependent delay differential equations whose delays are generated by variable-exponent modular functionals. Because translations are not, in general, bounded on $ L^{p(\cdot)}(-h_0, 0;\mathbb{R}^n) $, the classical delay shift semigroup cannot be constructed directly on the variable-exponent space. We therefore combine the Ito-Kappel semigroup framework with the modular structure of $ L^{p(\cdot)} $. We establish an equivalent abstract semilinear formulation, prove local well-posedness, and obtain well-posedness on finite intervals under stronger Lipschitz and growth assumptions. For modular state-dependent delays, we verify the required local Lipschitz properties and derive sufficient conditions for decay relative to the zero equilibrium and for conditional incremental contraction. Numerical experiments illustrate the role of the variable exponent through the modular delay and the distinction between these two sufficient estimates.
- state-dependent delay equations,
- variable-exponent modular delays,
- variable-exponent Lebesgue spaces,
- Ito-Kappel framework,
- semigroup formulation,
- well-posedness
Citation: Mostafa Bachar. Variable-exponent modular delays in state-dependent delay differential equations: an abstract semigroup formulation[J]. Electronic Research Archive, 2026, 34(12): 8997-9032. doi: 10.3934/era.2026379

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Abstract

We study state-dependent delay differential equations whose delays are generated by variable-exponent modular functionals. Because translations are not, in general, bounded on $ L^{p(\cdot)}(-h_0, 0;\mathbb{R}^n) $, the classical delay shift semigroup cannot be constructed directly on the variable-exponent space. We therefore combine the Ito-Kappel semigroup framework with the modular structure of $ L^{p(\cdot)} $. We establish an equivalent abstract semilinear formulation, prove local well-posedness, and obtain well-posedness on finite intervals under stronger Lipschitz and growth assumptions. For modular state-dependent delays, we verify the required local Lipschitz properties and derive sufficient conditions for decay relative to the zero equilibrium and for conditional incremental contraction. Numerical experiments illustrate the role of the variable exponent through the modular delay and the distinction between these two sufficient estimates.

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