On the generalized coupled Hadamard-Gronwall-Bellman-type inequalities with applications to fractional delay systems

Abstract

The Gronwall-Bellman inequality is a primary tool for proving various types of stability. For this importance, the present paper focuses on the generalized forms of the well-known Gronwall-Bellman inequality in the context of the Hadamard fractional calculus. We prove and generalize the coupled version of the Hadamard-Gronwall-Bellman inequality and then, generalize its extended form with the sum of two non-decreasing functions. In the sequel, the applicability of these inequalities is established in proving the existence and Ulam-Hyers stability of a Caputo-Hadamard coupled delay system and a Caputo-Hadamard damped initial value problem, which are appeared in population dynamic problems.