In particular, firmly nonexpansive mappings are averaged.
That is, if each of the mappings ({T_{i}}^{N}_{i=1}) is averaged, then so is the composite (T_{1}cdots T_{N}).
That is, if each of the mappings { T i } i = 1 N is averaged, then so is the composite T 1 ⋯ T N. In particular, if T 1 is α 1 -averaged and T 2 is α 2 -averaged, where α 1, α 2 ∈ ( 0, 1 ), then both T 1 T 2 and T 2 T 1 are α-averaged, where α = α 1 + α 2 − α 1 α 2. (ii) If the mappings { T i } i = 1 N are averaged and have a common fixed point, then ⋂ i = 1 N Fix ( T i ) = Fix ( T 1 ⋯ T N ). .
That is, if each of the mappings { T i } i = 1 N is averaged, then so is the composite T 1 ⋯ T N. In particular, if T 1 is α 1 -averaged and T 2 is α 2 -averaged, where α 1, α 2 ∈ ( 0, 1 ), then the composite T 1 T 2 is α-averaged, where α = α 1 + α 2 − α 1 α 2. (iii) If the mappings { T i } i = 1 N are averaged and have a common fixed point, then ⋂ i = 1 N Fix ( T i ) = Fix ( T 1 ⋯ T N ). .
That is, if each of the mappings { T i } i = 1 N is averaged, then so is the composite T 1, …, T N. In particular, if T 1 is α 1 -averaged and T 2 is α 2 -averaged, where α 1, α 2 ∈ ( 0, 1 ), then the composite T 1 T 2 is α-averaged, where α = α 1 + α 2 − α 1 α 2. If the mappings { T i } i = 1 N are averaged and have a common fixed point, then ⋂ i = 1 N F i x ( T i ) = F i x ( T 1 ⋯ T N ).
