Mixed Caputo integro-fractional delay system with non-instantaneous impulses: analysis of existence, uniqueness, and finite-time stability

We introduce a novel class of mixed integral Caputo fractional delay dynamic systems incorporating non-instantaneous impulsive effects. The proposed model includes neutral terms and involves time-dependent delays within the integral limits. Our primary aim is to explore the qualitative behaviour of the system, with particular attention to the existence and uniqueness of solutions, as well as finite-time stability analysis. To derive our results, we employ analytical methods such as fixed point theorems, Gronwall-type inequalities, and Ho¨lder's inequality. Moreover, we define a delayed Mittag–Leffler type matrix to effectively capture memory and delay effects. An illustrative example is provided to demonstrate the applicability of the theoretical findings.