Asymptotic stability of the Cauchy and Jensen functional equations
  The aim of this note is to investigate the asymptotic stability behaviour of the Cauchy and Jensen functional equations. Our main results show that if these equations hold for large arguments with small error, then they are also valid everywhere with a new error term which is a constant multiple of the original error term. As consequences, we also obtain results of hyperstability character for these two functional equations. 
Anna Bahyrycz and Zsolt P\'ales and Magdalena Piszczek
math.CA