In this paper the functional equation F(x + G(t(H(y-x))) = S(A(t)K(x), B(t)K(y)), x,y ε I, x ≤ y, t ε [0,1], is solved under monotonicity and injectivity assumptions for the unkown functions involved. This equation turned out to be useful in the axiomatic treatment of various joint-receipt utility structures. After several reduction steps, the solution of the above equation is reduced to that of Δ-1(Δ(h(u)/H(u + v)) · H(x)/H(x+u+v)) - H(u)H(x+u+v)/H(u+v)H(x+u), x,u,v > 0, X+u+v ε I. Assuming that H and Δ are strictly monotonic, it is shown that H is a log-concave function, and as a consequence, using Lebesgue's theorem, the everywhere-differentiability of F is deduced. Then, differentiating this equation, a functional differential equation for H is derived. The solutions of this new equation are then determined and the form of Δ is described. Thus, the general solution of the original equation is also obtained.
- Choice models
- Composite functional equations
- Utility models
ASJC Scopus subject areas
- Discrete Mathematics and Combinatorics
- Applied Mathematics