4- https://doi.org/10.1007/s000100050127
On a functional equation arising from joint-receipt utility models
- May 1, 2000
- Aequationes Mathematicae
- Gy Maksa +2 more
In this paper the functional equation¶¶\( F\Bigl(x + G(t(H(y - x))\Bigr) = S\Bigl(A(t)K(x),B(t)K(y)\Bigr), \qquad x,y \in I, x \le y, t \in [0,1] \),¶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¶¶\( \Gamma^{-1}\bigl(\Gamma\bigl(\frac{H(u)}{H(u + v)}\bigr) \cdot \frac{H(x)}{H(x + u + v)}\bigr) = \frac{H(u)H(x + u + v)}{H(u + v)H(x + u)},\qquad x,u,v > 0,\,x + u + v\in I \).¶Assuming that H and \( \Gamma \) 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 \( \Gamma \) 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 \( \Gamma \) is described. Thus, the general solution of the original equation is also obtained.
- # Functional Equation
- # Lebesgue's Theorem
- # Log-concave Function
- # Monotonicity Assumptions
- # Solution Of Equation
- # Axiomatic Treatment
- # Functional Differential Equation
- # Original Equation
- # Utility Models
- # General Solution
