Families of conditionally approximately convex functions

Abstract

Let $(G,+)$ be an Abelian topological group, which is also a $T_{0}$-space and a Baire space simultaneously, D be an open connected subset of G and $\alpha : D-D \rightarrow {\mathbb R}$ be a function continuous at zero and such that $\alpha (0)=0$. We show that if $(f_n)$ is a sequence of continuous functions $f_n : D \rightarrow {\mathbb R}$ such that $f_n(z) \le \frac{1}{2} f_n(x)+\frac{1}{2}f(y)+\alpha (x-y)$ for $n\in {\mathbb N}$ and $x,y,z\in D$ such that $2z=x+y$ and if $(f_n)$ is pointwise convergent [bounded] then it is convergent uniformly on compact subsets of D [in the case when G is additionally a separable space, it contains a subsequence which is convergent on compact subsets of D].