Existence of bounded C^n solutions with bounded derivatives is established for second-order iterative functional equations via the Fiber Contraction Theorem and Faà di Bruno's formula.
Balcerowski: On the functional equation x + f (y + f (x)) = y + f (x + f (x))
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Higher regularity of solutions of an iterative functional equation
Existence of bounded C^n solutions with bounded derivatives is established for second-order iterative functional equations via the Fiber Contraction Theorem and Faà di Bruno's formula.