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Peer-reviewedCC BY 4.0Axioms

Modified Gronwall–Bellman-Type Retarded Integral Inequalities and Applications

  • Khaled Boukerrioua1
  • Mohsen Dlala2,3
  • Taoufik Ghrissi4
  • Mohamed Ali Hammami5
5 affiliations
  1. 1Department of Mathematics, Badji Mokhtar-Annaba University, P.O. Box 12, Annaba 23000, Algeria
  2. 2Department of Mathematics, College of Science, Qassim University, Buraydah 51452, Saudi Arabia
  3. 3Department of Mathematics and Physics, Sfax Preparatory Institute for Engineering Studies (IPEIS), University of Sfax, Sfax 3072, Tunisia
  4. 4Department of Mathematics-Physics, Gafsa Preparatory Institute for Engineering Studies (IPEIG), University of Gafsa, Gafsa 2112, Tunisia
  5. 5Department of Mathematics, Faculty of Sciences, University of Sfax, Sfax 3000, Tunisia
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Abstract

We establish new nonlinear retarded integral inequalities of Gronwall–Bihari type involving time-dependent kernels, variable delays, nonconstant free terms, and iterated Volterra-type contributions. The first class of results provides explicit estimates for inequalities with kernels depending on both the observation time and the integration variable. A second class treats composite nonlinearities through successive comparison transformations and extends known retarded inequalities to the presence of an additional iterated kernel. The obtained estimates are then applied to nonlinear Volterra integral equations and neutral functional differential equations. For the Volterra problem, sufficient conditions for uniqueness, global existence, and boundedness are derived under Osgood-type assumptions. For neutral functional differential equations, the retarded estimates are combined with Driver’s continuation framework to obtain a global existence criterion. A delayed Liénard system is included as an illustrative application. These results extend several classical Gronwall–Bihari and retarded integral inequalities and provide a flexible tool for the qualitative analysis of nonlinear integral and functional differential equations.

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Cite

BibTeXRIS
BibTeX
@article{boukerrioua2026modified,
  title = {{Modified Gronwall–Bellman-Type Retarded Integral Inequalities and Applications}},
  author = {Khaled Boukerrioua and Mohsen Dlala and Taoufik Ghrissi and Mohamed Ali Hammami},
  journal = {Axioms},
  year = {2026},
  month = oct,
  doi = {10.3390/axioms15110795},
  url = {https://www.mdpi.com/2075-1680/15/11/795},
  note = {CC BY 4.0},
}
RIS
TY  - JOUR
TI  - Modified Gronwall–Bellman-Type Retarded Integral Inequalities and Applications
AU  - Khaled Boukerrioua
AU  - Mohsen Dlala
AU  - Taoufik Ghrissi
AU  - Mohamed Ali Hammami
PY  - 2026
DA  - 2026/10/23
JO  - Axioms
DO  - 10.3390/axioms15110795
UR  - https://www.mdpi.com/2075-1680/15/11/795
AB  - We establish new nonlinear retarded integral inequalities of Gronwall–Bihari type involving time-dependent kernels, variable delays, nonconstant free terms, and iterated Volterra-type contributions. The first class of results provides explicit estimates for inequalities with kernels depending on both the observation time and the integration variable. A second class treats composite nonlinearities through successive comparison transformations and extends known retarded inequalities to the presence of an additional iterated kernel. The obtained estimates are then applied to nonlinear Volterra integral equations and neutral functional differential equations. For the Volterra problem, sufficient conditions for uniqueness, global existence, and boundedness are derived under Osgood-type assumptions. For neutral functional differential equations, the retarded estimates are combined with Driver’s continuation framework to obtain a global existence criterion. A delayed Liénard system is included as an illustrative application. These results extend several classical Gronwall–Bihari and retarded integral inequalities and provide a flexible tool for the qualitative analysis of nonlinear integral and functional differential equations.
SN  - 2075-1680
ER  -