Not sure whether this is really helpful, but the general direction at which  you can look is the theory of attractive fixed points of discrete non-autonomous dynamical systems. For example look at [this][1] or [this][2], or [this][3] (just to indicate a few among many works). 


  [1]: https://www.sciencedirect.com/science/article/pii/S0960077921007670?casa_token=eftiPyUWqFQAAAAA:4LEJ89Pd67zemlMavWkg8EaAY7zyxFbkE8ArNDyjAr_z_jlKQxn2C1kuN2ta7BBS05q6bb2B
  [2]: https://www.tandfonline.com/doi/full/10.1080/10236190500424274
  [3]: https://link.springer.com/article/10.1007/BF03322551