On a General n-Linear Functional Equation

Let X, Y be linear spaces over fields K and F, respectively, and let f : X-n -> Y. For some fixed a(ji) is an element of K \ {0}, C-i1...i(n) is an element of F, for j is an element of {1, ..., n}, i, i(j) is an element of {1, 2}, we consider the equation f(a(11)x(11) + a(12)x(12), ..., a(n1)x(n1) + a(n2)x(n2)) = Sigma(1 <= i1, ...,in <= 2) C-i1 ... i(n) f(x(1i1), ...,x(nin)), (*) for all x(jij) is an element of X, j is an element of {1, ..., n}, i(j) is an element of {1, 2}. We determine the general solution of (*) and as special cases we obtain some results for multi-Cauchy, multi-Jensen and multi-Cauchy-Jensen equations.

Automated summary

This passage discusses equations between linear spaces and linear mappings, and obtains the general solution and results for specific cases.