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The study of small data Sobolev solutions to the Cauchy problem for weakly coupled systems of semi-linear fractional σ – evolution equations with different power nonlinearities is of interest to us in this research. These solutions must exist globally (in time). We explain the relationships between the admissible range of exponents p 1 and p 2 symmetrically in our main modeland the regularity assumptions for the data by using L m − L q estimates of Sobolev solutions to related linear models with a vanishing right-hand side and some fixed point argument. This allows us to prove the global (in time) existence of small data Sobolev solutions. [ABSTRACT FROM AUTHOR]