Abstract
In this letter, we defined a weighted Caputo-type differential operator which was used to character relaxation and diffusion models in two different types. Then, one of the weighted Caputo-type integral operators by solving the related linear differential equation was also defined. Subsequently, we studied their main properties. Lastly, the existence of a unique solution of the related nonlinear differential equation through the Banach fixed point theorem was presented. These basic facts lay a solid foundation for the concrete practice in the future.