 |  |  | Solutions of the inhomogeneous Cauchy functional equation and
applications on the stability of additive functions |
Solutions of the inhomogeneous Cauchy functional equation and
applications on the stability of additive functions
Consider the solvability and the solutions g:R -> R
of the inhomogeneous Cauchy functional equation
g(x+y)-g(x)-g(y)=d(x,y), (x,yÎR) (IC)
for a given function d:R´R -> R.
(That kind of inhomogeneous Cauchy functional equations appear for instance
when solving the iteration problem in power series rings.
Cf. [24].) If (IC) is solvable for a given d (for instance
when d is a symmetric solution of the cocycle equation) and if |d| is
bounded on R then (IC) has the solution
h0(x)=-åi=0¥ (1)/(2i+1)d(2ix,2ix).
(See for instance [10][9][11].)
This result implies the Hyers-Ulam stability of additive functions
(cf. [11] pages 153 - 154).
It is interesting to determine under which assumptions on d (inclusive
an algebraic integrability condition for (IC)) limits different from h0
provide descriptions of solutions of (IC).
Formal calculations suggest expressions of the form
h1(x)=-limk -> ¥ (1)/(mk)åj=0mk-1 d(jx,x)
where mk is an unbounded strictly monotonic increasing sequence.
Or for m³2
h2(x)=-åj=0¥ (1)/(mj+1)sm(mjx),
where sm(x)=d(x,x)+d(2x,x)+...+d((m-1)x,x).
We would like to investigate whether these solutions lead to new stability
theorems for additive functions.
harald.fripertinger@kfunigraz.ac.at,
last changed: February 9, 2001
| | | Solutions of the inhomogeneous Cauchy functional equation and
applications on the stability of additive functions |