Fractional Subscripts? What is the physical meaning when we have an empirical formula with fractional subscripts such as $\ce{LiMn_{1-x}Fe_{x}PO4}$ $(0 \le x \le 1)$? 

This an example of the paper the notation above is used:

Bai, J.; Hong, J.; Chen, H.; Graetz, J.; Wang, F. J. Phys. Chem. C 2015, 119 (5), 2266–2276. DOI 10.1021/jp508600u.
These indicate "doping" or non-stoichiometric compounds.

In this case, there's some fraction of Fe, and some fraction of Mn in a lithium iron phosphate battery. If I remember correctly, you need to dope a different ion to increase the conductivity of $\ce{LiFePO4}$.

See also here: How are non-Stoichiometric compounds determined?