Gronwall inequality for the discrete generalized proportional fractional derivative

Martin Bohner; Rajrani Gupta; Martin Bohner; Rajrani Gupta

doi:10.3934/era.2026240

Electronic Research Archive

2026, Volume 34, Issue 8: 5380-5394. doi: 10.3934/era.2026240

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Gronwall inequality for the discrete generalized proportional fractional derivative

Martin Bohner ^,,
Rajrani Gupta

Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO 65409, USA

Received: 06 April 2026 Revised: 05 June 2026 Accepted: 17 June 2026 Published: 26 June 2026

We established a new Gronwall inequality for the discrete generalized proportional fractional derivative. The result extended known discrete fractional inequalities and provided explicit bounds, uniqueness criteria, and Ulam–Hyers stability for nonlinear delayed fractional systems. An example was provided to illustrate the theory.
Citation: Martin Bohner, Rajrani Gupta. Gronwall inequality for the discrete generalized proportional fractional derivative[J]. Electronic Research Archive, 2026, 34(8): 5380-5394. doi: 10.3934/era.2026240

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Abstract

We established a new Gronwall inequality for the discrete generalized proportional fractional derivative. The result extended known discrete fractional inequalities and provided explicit bounds, uniqueness criteria, and Ulam–Hyers stability for nonlinear delayed fractional systems. An example was provided to illustrate the theory.

References

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