Weerawat Sudsutad, Chatthai Thaiprayoon, Jutarat Kongson, Aphirak Aphithana, On extended Gronwall-Bellman-Bihari type integral inequalities via $(k,\psi)$-Hilfer proportional fractional operators and their applications, Vol. 2025 (2025), No. 26, pp. 1- 17

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DOI: 10.23952/jnfa.2025.26

Received June 11, 2025; Accepted September 1, 2025; Published September 25, 2025

Abstract. In the framework of the $(k,\psi)$ -Hilfer proportional fractional operator ( $(k,\psi)$ -HPFO), we investigate a new type of integral inequalities of Gronwall-Bellman-Bihari. In addition to covering current results in fractional calculus, these results provide a generalized framework that connects them. The integral inequalities are a helpful tool for examining how solutions to the nonlinear $(k,\psi)$ -Hilfer proportional fractional differential equations behave, especially in terms of their stability, uniqueness, and boundedness. The framework produces additional generalizations and reproduces a number of known results as special cases by choosing appropriate parameter values and kernel functions. Several examples are given to demonstrate how our method works for fractional-order systems with memory-dependent dynamics.

How to Cite this Article:
W. Sudsutad, C. Thaiprayoon, J. Kongson, A. Aphithana, On extended Gronwall-Bellman-Bihari type integral inequalities via $(k,\psi)$ -Hilfer proportional fractional operators and their applications, J. Nonlinear Funct. Anal. 2025 (2025) 26.