Solvability and stability of mean-field stochastic differential equations driven by time-changed Lévy noise

In this article, we focused on a class of mean-field stochastic differential equations driven by time-changed Lévy noise. We first discussed the existence and uniqueness of solutions under the non-Lipschitz case with the Lipschitz condition as the special case by adopting the Carathéodory approximation. To prove our results, we established a new time-changed retarded integral inequality, which is easy to apply in practice and can be considered as a more general tool in some situations. Then, the classical Itô formula was extended to that for mean-field stochastic differential equations driven by time-changed Lévy noise. As an application of Itô's formula, we showed that the trivial solution is $p$-th moment asymptotically stable, stable in probability, asymptotically stable in probability, and globally asymptotically stable in probability based on the Lyapunov function. Finally, an example was presented to validate the produced results.