Application of an extended Gronwall-type inequality for a \((k,\psi)\)-Hilfer proportional fractional Ambartsumian system via nonlocal integral conditions

Volume 41, Issue 4, pp 456--486 https://dx.doi.org/10.22436/jmcs.041.04.02

Publication Date: November 21, 2025 Submission Date: January 22, 2025 Revision Date: July 24, 2025 Accteptance Date: September 12, 2025

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Authors

W. Sudsutad - Department of Statistics, Faculty of Science, Ramkhamhaeng University, Bangkok 10240, Thailand. C. Thaiprayoon - Research Group of Theoretical and Computation in Applied Science, Department of Mathematics, Faculty of Science, Burapha University, Chonburi 20131, Thailand. J. Kongson - Research Group of Theoretical and Computation in Applied Science, Department of Mathematics, Faculty of Science, Burapha University, Chonburi 20131, Thailand. A. Aphithana - Division of Mathematics, Department of Mathematics and Computer Science, Faculty of Science and Technology, Rajamangala University of Technology Krungthep, Bangkok 10120, Thailand.

Abstract

This work obtained a novel nonlinear coupled Ambartsumian system subject to nonlocal integral conditions with the \((k,\psi)\)-Hilfer proportional fractional operator. According to this study, we presented an extended \((k,\psi)\)-Hilfer proportional fractional Gronwall inequality. The results concerning existence and uniqueness were established by employing the fixed point theory of Banach's and Krasnosel'skii's types. Furthermore, a variety of stability in the context of Ulam-Hyers-Mittag-Leffler and Ulam-Hyers-Rassias-Mittag-Leffler were investigated. In addition, two numerical examples were demonstrated to illustrate and apply the main results by using a novel numerical technique based on decomposition formula.

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Keywords

• \((k,\psi)\)-Hilfer proportional fractional operator
• Gronwall inequality
• existence and uniqueness
• fixed point theorem
• Ulam-Hyers-Mittag-Leffler stability

MSC

•  26A33
•  26D10
•  34A08
•  34B10
•  33E12

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