Mountain climbing: an alternate proof.
It is proved that the solution of the multiplicative Cauchy functional equation on the Lorentz cone of dimension greater than two is a power function of the determinant. The equation is solved in full generality, i.e. no smoothness assumptions on the unknown function are imposed. Also the functional equation of ratios, of a similar nature, is solved in full generality.
We solve the multiplicative Cauchy equation for real functions of symmetric positive definite matrices under the differentiability restriction. The specialty of the problem lies in the symmetry of the multiplication.
Let be complex vector spaces. Recently, Park and Th.M. Rassias showed that if a mapping satisfies for all , , then the mapping satisfies for all , . Furthermore, they proved the generalized Hyers-Ulam stability of the functional equation () in complex Banach spaces. In this paper, we will adopt the idea of Park and Th. M. Rassias to prove the stability of a quadratic functional equation with complex involution via fixed point method.