Let be the Banach space of all real functions which are defined, bounded and continuous on with the norm. Let be an operator transforming the space into itself and such that
for all functions and for any , and is a continuous function such that .
Further, assume that is a solution of the operator equation
In the paper under review, the following result is proved: The function is an asymptotically stable solution of equation (*) if for any there exists such that for every and for every other solution of equation (*) the inequality holds.
As an application, the functional-integral equation is studied.