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∈(-∞,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. [ABSTRACT FROM AUTHOR]