I 18 Summaries AEO. MATH.

which generalize the sine and cosine (d'Alembert) equations. A conditional d'Alem- bert-Wilson equation and some related unconditional equations in more than two variables are also considered.

An iff fixed point criterion for continuous self-mappings on a complete metric space

J. JACHYMSKI

Let f be a self-mapping on a metric space (X, d). We give necessary and sufficient conditions for the sequences { f " x } (x ~ X ) to be equivalent Cauchy. As a typical application we get the following result. Let f be continuous and (X, d) be complete. If, for any x, y ~ X d( f n x , f~y) ~ 0 and for some c > O, this convergence is uniform for all x, y in X with d(x, y) < c then f h a s a unique fixed point p, and f~x ~ p , for each x in X.

This theorem includes among others results of Angelov, Browder, Edelstein, Hicks and Matkowski.

Complementary permutations for abelian groups

J. H. B. KEMPERMAN AND TEUNIS J. OTT

Let G be an additively written abelian group and let h: G--* G be a given function. M. Hall Jr. (1952) and L. Fuchs (1958) already answered the following question. For what functions h: G-- ,G does the functional equation a ( x ) + z ( x ) = h(x)

(x ~ G) have as its solution a pair of permutations a and z of G? In this paper, we give explicit constructions of such a pair tr, T in a number of cases, in particular when h(x) m X and G is finite. We further determine the finite groups G where the latter a , , can be chosen to be automorphisms.

In the case where G is an infinite topological group, we study in how far o- and , can be chosen as Borel measurable permutations, given that h: G ---, G itself is Borel measurable.