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We consider the relationship between when the r-times composition of two maps commute, and the concepts of compatible mappings of type (A), faintly compatible mappings, compatible mappings of type (R), compatible mappings of type (P), and compatible mappings of type (K), respectively, and obtain some higher-order fixed point theorems in the sense of [Clement Ampadu, Fixed Point Theory for Higher-Order Mappings. ISBN: 5800118959925, lulu.com, 2016]