### Abstract

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.

Original language | English |
---|---|

Pages (from-to) | 273-286 |

Number of pages | 14 |

Journal | Aequationes Mathematicae |

Volume | 59 |

Issue number | 3 |

Publication status | Published - 2000 |

### Fingerprint

### Keywords

- Choice models
- Composite functional equations
- Utility models

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Aequationes Mathematicae*,

*59*(3), 273-286.

**On a functional equation arising from joint-receipt utility models.** / Maksa, Gyula; Marley, A. A J; Páles, Zsolt.

Research output: Contribution to journal › Article

*Aequationes Mathematicae*, vol. 59, no. 3, pp. 273-286.

}

TY - JOUR

T1 - On a functional equation arising from joint-receipt utility models

AU - Maksa, Gyula

AU - Marley, A. A J

AU - Páles, Zsolt

PY - 2000

Y1 - 2000

N2 - 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.

AB - 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.

KW - Choice models

KW - Composite functional equations

KW - Utility models

UR - http://www.scopus.com/inward/record.url?scp=0000071467&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0000071467&partnerID=8YFLogxK

M3 - Article

VL - 59

SP - 273

EP - 286

JO - Aequationes Mathematicae

JF - Aequationes Mathematicae

SN - 0001-9054

IS - 3

ER -