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On the formal first cocycle equation for iteration groups of type II

Harald Fripertinger, Ludwig Reich (2012)

ESAIM: Proceedings

Let x be an indeterminate over ℂ. We investigate solutions α ( s , x ) = n 0 α n ( s ) x n , αn : ℂ → ℂ, n ≥ 0, of the first cocycle equation α ( s + t , x ) = α ( s , x ) α t , F ( s , x ) , s , t , ( Co 1 ) in ℂ [[x]], the ring of formal power series over ℂ, where (F(s,x))s ∈ ℂ is an iteration group of type II, i.e. it is a solution of the translation equation F ( s + t , x ) = F ( s , F ( t , x ) ) , s , t , ( T ) of the form F(s,x) ≡ x + ck(s)xk mod xk+1, where k ≥ 2 and ck ≠ 0 is necessarily an additive function. It is easy to prove that the coefficient functions αn(s) of α ( s , x ) = 1 + n 1 α n ( s ) x n are polynomials in ck(s).It is possible to replace...

On the functional equation defined by Lie's product formula

Gerd Herzog, Christoph Schmoeger (2006)

Studia Mathematica

Let E be a real normed space and a complex Banach algebra with unit. We characterize the continuous solutions f: E → of the functional equation f ( x + y ) = l i m n ( f ( x / n ) f ( y / n ) ) .

On the functional equation H [tau (F,G), chi (F,G)] = H (F,G).

Mónica Sánchez Soler (1988)

Stochastica

In this paper we solve the functional equationH [tau(F,G), chi (F,G)] = H (F,G)where the unknowns tau and chi are two semigroups on a space of distribution functions, and H is a given pointwise binary operation on this space satisfying some regularity conditions.

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