This paper examined the controllability of fractional impulsive neutral Volterra-Fredholm integro-differential equations with state-dependent delay, employing the Caputo fractional derivative and a semigroup of compact and analytic operators. Controllability results were established by using Schauder's fixed point theorem, addressing the complexities arising from fractional dynamics combined with state-dependent delays. The theoretical findings were further confirmed through a detailed example and numerical simulations that show convergence of the solutions. Also, the role of artificial intelligence in the analysis and control of such systems governed by these equations was investigated, thereby opening up opportunities for machine learning to be coupled with fractional calculus for a better predictive solution and better control of systems. These results provide some insight into stability as well as controllability of systems governed by fractional differential equations with impulsive and state-dependent behaviors.

In this paper, we establish a new fixed point result for Hardy-Rogers type contractions in double controlled metric-like spaces. Our result generalizes many important theorems in the literature. We will provide an example to illustrate our results.

In this publication, our objective was to introduce and establish the concepts of ${\kappa }_{G_m}$-contraction and generalized $(\alpha ,{\kappa }_{G_m})$-contraction in complete $G_m$-metric spaces, which led to the discovery of novel fixed points, coincidence points, and common fixed points. Additionally, we demonstrated the usefulness of our main results by applying it to the investigation of the integral equation. Also, we presenting a noteworthy example demonstrating the practicality of our primary hypothesis.