On the solutions of two generalized Cauchy functional equations (Q1089695)

Certain characterization problems in the area of damage model theory can be reduced to problems of solving functional equations. This paper deals with two such problems examined earlier by the first author [J. Appl. Probab. 17, 563-569 (1980; Zbl 0428.60019), and Bull., Calcutta Stat. Assoc. 31, 77-83 (1982; Zbl 0495.62019)]. The corresponding functional equations studied are \[ f[xg(\phi)+\phi]=f(x)f(\phi)\quad 0\leq x,\quad \phi \leq 1 \] where f(.) is continuous and g(.) is positive in [0,1], and \[ f(x+dx\phi +\phi)=k(\phi)f(x)+f(\phi) \] where f(.) is continuous on \([0,1]\), k(.) is also continuous on \([0,1]\) such that \(k(\phi)\geq 1\) for all \(\phi\in [0,1]\), and d is a real number such that \(| d| <1\forall d<0.\) The above equations are solved when both x and \(\phi\) are variables in \([0,1]\) and also when \(\phi\) is a constant in \((0,1]\).