Explicit estimates for coupled weakly singular retarded integral inequalities and applications to fractional Volterra-type systems

Liqiang Chen; Liqiang Chen

doi:10.3934/math.20261334

AIMS Mathematics

2026, Volume 11, Issue 10: 34193-34214. doi: 10.3934/math.20261334

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Research article

Explicit estimates for coupled weakly singular retarded integral inequalities and applications to fractional Volterra-type systems

Liqiang Chen ^,

School of Mathematics and Physics, Hechi University, Hechi 546300, China

Received: 18 May 2026 Revised: 10 July 2026 Accepted: 27 July 2026 Published: 21 October 2026
MSC : 26D10, 26D15, 26A33, 34A12, 45J05

This paper establishes a unified comparison framework for coupled weakly singular retarded integral inequalities arising in fractional Volterra-type systems. The inequalities contain powered unknowns, several weakly singular kernels of different orders, delayed arguments, nonconstant outside multipliers, and piecewise continuous inhomogeneous data. In addition to a closed Mittag–Leffler majorant, a variable-coefficient resolvent estimate is derived that retains the coefficient envelopes inside the fractional integral. A further extension treats state-dependent impulse amplitudes at prescribed times and gives a product bound for their cumulative amplification. The proofs use running maxima and Volterra resolvents. Applications yield a priori bounds, continuous dependence, uniqueness, and a classical Caputo formulation in an explicitly stated solution class, and a numerical illustration compares the computed solution with the analytical majorant.
- Gronwall–Bellman inequality,
- weakly singular integral inequality,
- retarded argument,
- coupled integral system,
- explicit estimate,
- Mittag–Leffler function,
- fractional Volterra-type system
Citation: Liqiang Chen. Explicit estimates for coupled weakly singular retarded integral inequalities and applications to fractional Volterra-type systems[J]. AIMS Mathematics, 2026, 11(10): 34193-34214. doi: 10.3934/math.20261334

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Abstract

This paper establishes a unified comparison framework for coupled weakly singular retarded integral inequalities arising in fractional Volterra-type systems. The inequalities contain powered unknowns, several weakly singular kernels of different orders, delayed arguments, nonconstant outside multipliers, and piecewise continuous inhomogeneous data. In addition to a closed Mittag–Leffler majorant, a variable-coefficient resolvent estimate is derived that retains the coefficient envelopes inside the fractional integral. A further extension treats state-dependent impulse amplitudes at prescribed times and gives a product bound for their cumulative amplification. The proofs use running maxima and Volterra resolvents. Applications yield a priori bounds, continuous dependence, uniqueness, and a classical Caputo formulation in an explicitly stated solution class, and a numerical illustration compares the computed solution with the analytical majorant.

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