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Then by Lemma 2.5, we have Fix ( W ) = Fix ( J r ) ∩ Fix ( S ) ∩ Fix ( G ) = F.
Note that if T1,...,T N are nonexpansive mappings with a common fixed point and satisfy the property (S), then we have Fix ( T 1 ∘ ⋯ ∘ T N ) = ⋂ 1 ≤ i ≤ N Fix ( T i ).
So don't make these mistakes and if you already have, fix them before it's too late.
That is, if X F ∪ X H ≠ ∅, then we have Fix F ∩ Fix H ≠ ∅, which follows easily from 1-3.
Then we have Fix ( t ) ∩ T ( x ) ≠ ∅ for all x ∈ Fix ( t ) since t satisfies condition ( K ).
By the above arguments, we have Fix d ( L ) = 〈 a 〉, and so Fix d ( L ) is a principal ideal.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com