On a functional equation related to a problem of G. Derfel

Two years ago, during the 21st European Conference on Iteration Theory, Gregory Derfel asked: " Does there exist a non-trivial bounded continuous solution of the equation 2f( x) = f( x-1)+ f(-2x)?" He repeated the question during the 55th International Symposium on Functional Equations. In this paper we present a partial solution of a more general problem, connected to the functional equation f( x) = M f( x + t1), f( x + t2),..., f( x + tn-1), f( ax) , where n. N, t1, t2,..., tn-1. R\{0}, a. (-8, 0) and M is a given function in n variables satisfying some additional properties. In particular, M can be a weighted quasi-arithmetic mean in n variables.