On a functional equation of Luce

Summary In a recent communication to J. Aczél, R. Duncan Luce asked about the functional equationU(x)U(G(x)F(y)) = U(G(x))U(xy) forx, y > 0, (1) which has arisen in his research on certainty equivalents of gambles. He was particularly interested in cases in which the unknowns (U, F andG) are strictly increasing functions from (0, + ) into (0, + ). In this paper we solve (1) in the case whereU, F andG are continuously differentiable with everywhere positive first derivatives. Our solution is perhaps novel in that in certain cases (1) reduces to a functional equation in a single variable and in other cases to a functional equation in several variables; see [1] for the terminology.     

On a generalization of the Cauchy functional equation

Summary While looking for solutions of some functional equations and systems of functional equations introduced by S. Midura and their generalizations, we came across the problem of solving the equationg(ax + by) = Ag(x) + Bg(y) + L(x, y) (1) in the class of functions mapping a non-empty subsetP of a linear spaceX over a commutative fieldK, satisfying the conditionaP + bP P, into a linear spaceY over a commutative fieldF, whereL: X × X Y is biadditive,a, b K\{0}, andA, B F\{0}. Theorem.Suppose that K is either R or C, F is of characteristic zero, there exist A ₁,A ₂,B ₁,B ₂, F\ {0}with L(ax, y) = A ₁ L(x, y), L(x, ay) = A ₂ L(x, y), L(bx, y) = B ₁ L(x, y), and L(x, by) = B ₂ L(x, y) for x, y X, and P has a non-empty convex and algebraically open subset. Then the functional equation (1)has a solution in the class of functions g: P Y iff the following two conditions hold: L(x, y) = L(y, x) for x, y X, (2)if L 0, then A ₁ =A ₂,B ₁ =B ₂,A = A ₁ ² ,and B = B ₁ ² . (3) Furthermore, if conditions (2)and (3)are valid, then a function g: P Y satisfies the equation (1)iff there exist a y ₀ Y and an additive function h: X Y such that if A + B 1, then y ₀ = 0;h(ax) = Ah(x), h(bx) =Bh(x) for x X; g(x) = h(x) + y ₀ + 1/2A ₁ ^-1 B ₁ ^-1 L(x, x)for x P.     

Une généralisation d'un résultat de J. Aczél et M. Hosszú sur l'équation de translation

Résumé En généralisant un résultat de J. Aczél et M. Hosszú on donne des conditions nécessaires et suffisantes pour qu'une solution de l'équation de translationF(F(, x), y) = F(, xy), oùF: × G , est un ensemble arbitraire,G forme un groupe, soit de la formeF(, x) = f ^–1(f()·1(x)), oùf est une bijection de au groupeG ₁ isomorphe avecG et 1 est un homomorphisme deG àG ₁. On considère aussi le cas oùG forme un espace vectoriel sur le corps des nombres rationels.Si est un intervalle ayant plus qu'un point etG = R ^m avec l'addition comme l'opération on trouve des conditions pour que la fonction continueF soit de la formeF(, x ₁,, x _m ) =f ^–1(f() + c ₁ x ₁ + +c _m x _m ), oùf est une homéomorphie de àR et (c ₁,,c _m ) R ^m .

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On the Baxter functional equation

Summary A new shorter proof is given for the Theorem of P. Volkmann and H. Weigel determining the continuous solutionsf:R R of the Baxter functional equationf(f(x)y + f(y)x – xy) = f(x)f(y). The proof is based on the well known theorem of J. Aczél describing the continuous, associative, and cancellative binary operations on a real interval.     

Fundamental equation of information revisited

Summary This paper presents a new, shorter and more direct proof of the following result of J. Aczél and C. T. Ng: IfM: J R (J =]0, 1[ ^k ) is both multiplicative and additive, then the general solution: J R of(x) + M(1 – x)(y/1 – x) = (y) + M(1 – y)(x/1 – y) (x, y, x + y J) is given by(x) = ifM = 0,(x) = M(x)[L(x) + ] + M(1 – x)L(1 – x) ifM 0,where is an arbitrary constant andL: J R is an arbitrary solution of the logarithmic functional equationL(xy) = L(x) + L(y) (x, y J). Also, some extensions of this result to fields more general than the reals are given.     

Generalized sine equations,III

Summary We solve the equationf(x + y)f(x – y) = P(f(x), f(y)) under various conditions on the unknown functionsf, P.     

Multiplicative symmetry and related functional equations

Summary Let (G, *) be a commutative monoid. Following J. G. Dhombres, we shall say that a functionf: G G is multiplicative symmetric on (G, *) if it satisfies the functional equationf(x * f(y)) = f(y * f(x)) for allx, y inG. (1)Equivalently, iff: G G satisfies a functional equation of the following type:f(x * f(y)) = F(x, y) (x, y G), whereF: G × G G is a symmetric function (possibly depending onf), thenf is multiplicative symmetric on (G, *).In Section I, we recall the results obtained for various monoidsG by J. G. Dhombres and others concerning the functional equation (1) and some functional equations of the formf(x * f(y)) = F(x, y) (x, y G), (E) whereF: G × G G may depend onf. We complete these results, in particular in the case whereG is the field of complex numbers, and we generalize also some results by considering more general functionsF. In Section II, we consider some functional equations of the formf(x * f(y)) + f(y * f(x)) = 2F(x, y) (x, y K), where (K, +, ·) is a commutative field of characteristic zero, * is either + or · andF: K × K K is some symmetric function which has already been considered in Section I for the functional equation (E). We investigate here the following problem: which conditions guarantee that all solutionsf: K K of such equations are multiplicative symmetric either on (K, +) or on (K, ·)? Under such conditions, these equations are equivalent to some functional equations of the form (E) for which the solutions have been given in Section I. This is a partial answer to a question asked by J. G. Dhombres in 1973.     

On functions having Cauchy differences in some prescribed sets

Summary AssumeE is a real topological vector space,F is a real Banach space,K is a discrete subgroup ofF andC is a symmetric, convex and compact subset ofF such thatK (6C) = {0}. If a functionh:E F is continuous at at least one point andh(x + y) – h(x) – h(y) K + C for allx, y E, then there exists a continuous linear functiona:E F such thath(x) – a(x) K + C for everyx E.     

Holomorphic solutions of an inhomogeneous Cauchy equation

Summary We consider the functional equation(x + y) – (x) – (y) = f(x)f(y)h(x + y) and we find all its homomorphic solutionsf, h, defined in a neighbourhood of the origin.     

Wilson's functional equation on C

Summary We find the complete set of continuous solutionsf, g of Wilson's functional equation _{n = 0} ^{N – 1} f(x + wⁿy) = Nf(x)g(y), x, y C, given a primitiveN ^th rootw of unity.Disregarding the trivial solutionf = 0 andg any complex function, it is known thatg satisfies a version of d'Alembert's functional equation and so has the formg(z) = g (z) = N^–1 _{n = 0} ^{N – 1} E(wⁿz) for some C². HereE ₍₁, ₂)(x + iy) = exp( ₁x + ₂).For fixedg = g the space of solutionsf of Wilson's functional equation can be decomposed into theN isotypic subspaces for the action of Z _N on the continuous functions on C. We prove that ther ^th component, wherer {0, 1, ,N – 1}, of any solution satisfies the signed functional equation _{n = 0} ^{N – 1} f(x + wⁿy)w^nr = Ng(x)f(y), x, y C. We compute the solution spaces of each of these signed equations: They are 1-dimensional and spanned byz _{n = 0} ^{N – 1} w^nr E(wⁿz), except forg = 1 andr 0 where they are spanned by andz ^{N – r}. Adding the components we get the solution of Wilson's equation. Analogous results are obtained with the action ofZ _N on C replaced by that ofSO(2).The case ofg = 0 in the signed equations is special and solved separately both for Z _N andSO(2).     

Orthogonality and additivity modulo a subgroup

Summary LetE be a real inner product space of dimension at least 2,F a topological Abelian group, andK a discrete subgroup ofF. Assume also thatF is continuously divisible by 2 (that is, the functionu 2u is a homeomorphism ofF ontoF). Iff: E F fulfils the conditionf(x + y) – f(x) – f(y) K for all orthogonalx, y E and is continuous at the origin then there exist continuous additive functionsa: R F andA: E F such thatf(x) – a(x ²)– A(x) K for everyx E. Dedicated to the memory of Alexander M. Ostrowski on the occasion of the 100th anniversary of his birth.     

On an alternative cauchy equation in two unknown functions. Some classes of solutions

Summary In this paper we consider the alternative Cauchy functional equationg(xy) g(x)g(y) impliesf(xy) = f(x)f(y) wheref, g are functions from a topological group (X, ·) into a group (S,·). First we prove that, ifS is a Hausdorff topological group andX satisfies some weak additional hypotheses, then (f, g) is a continuous solution if and only if eitherf org is a homomorphism. Then we describe a more general class of solutions forX =R ⁿ .Partially supported by M.U.R.S.T. Research funds (40%)Dedicated to the memory of Alexander M. Ostrowski on the occasion of the 100th anniversary of his birth.     

Conditional functional equations and orthogonal additivity

Summary Some examples of classes of conditional equations coming from information theory, geometry and from the social and behavioral sciences are presented. Then the classical case of the Cauchy equation on a restricted domain is extensively discussed. Some results concerning the extension of local homomorphisms and the implication -additivity implies global additivity are illustrated. Problems concerning the equations[cf(x + y) – af(x) – bf(y) – d][f(x + y) – f(x – f(y)] = 0[g(x + y) – g(x) – g(y)][f(x + y) – f(x) – f(y)] = 0f(x + y) – f(x) – f(y) V (a suitable subset of the range) are presented.The consideration of the conditional Cauchy equation is subsequently focused on the case when it makes sense to interpret as a binary relation (orthogonality):f: (X, +, ) (Y, +);f(x + z) = f(x) + f(z) (x, z Z; x z). A brief sketch on solutions under regularity conditions is given. It is then shown that all regularity conditions can be removed. Finally, several applications (also to physics and to the actuarial sciences) are discussed. In all these cases the attention is focused on open problems and possible extensions of previous results.     

D'Alembert's functional equation on restricted domains

Summary In the class of functionalsf:X , whereX is an inner product space with dimX 3, we study the D'Alembert functional equationf(x + y) + f(x – y) = 2f(x)f(y) (1) on the restricted domainsX ₁ = {(x, y) X ²/x, y = 0} andX ₂ = {(x, y) X ²/x = y}. In this paper we prove that the equation (1) restricted toX ₁ is not equivalent to (1) on the whole spaceX. We also succeed in characterizing all common solutions if we add the conditionf(2x) = 2f²(x) – 1. Using this result, we prove the equivalence between (1) restricted toX ₂ and (1) on the whole spaceX. This research follows similar previous studies concerning the additive, exponential and quadratic functional equations.     

On an exponential-cosine functional equation

The exponential cosine functional equationf(x + y) + (2f ²(y) – f(2y))f(x – y) = 2f(x)f(y) is studied in some detail whenf is a complex valued function defined on a Banach space. We supply conditions which ensure continuity off everywhere under the hypothesis thatf is continuous at a point. We also find solutions of the functional equation which are continuous at some point.     

On generalized cocycle equations

Summary Motivated by results on the classical cocycle equation, we solved the more general equationF ₁ (x + y, z) + F ₂ (y + z, x) + F ₃ (z + x, y) + F ₄ (x, y) + F ₅ (y, z) + F ₆ (z, x) = 0 for six unknown functions mapping ordered pairs from an abelian group into a vector space over the rationals.Dedicated to the memory of Alexander M. Ostrowski on the occasion of the 100th anniversary of his birth.     

On a characterization of polynomials by divided differences

Summary We consider the functional equationf[x ₁,x ₂,, x _n ] =h(x ₁ + +x _n ) (x ₁,,x _n K, x _j x _k forj k), (D) wheref[x ₁,x ₂,,x _n ] denotes the (n – 1)-st divided difference off and prove Theorem. Let n be an integer, n 2, let K be a field, char(K) 2, with # K 8(n – 2) + 2. Let, furthermore, f, h: K K be functions. Then we have that f, h fulfil (D) if, and only if, there are constants a_j K, 0 j n (a := a_n, b := a_{n – 1}) such thatf = ax ⁿ +bx ^{n – 1} + +a ₀ and h = ax + b.     

Existence of a smooth solution of one boundary-value problem

We study a periodic boundary-value problem for the quasilinear equationu _tt–u_xx=F[u, u_t], u(0, t)=u(, t)=0,u(x, t+2)=u(x, t). We establish conditions that guarantee the validity of the uniqueness theorem.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 12, pp. 1717–1719, December, 1995.     

On the general solution of a nonsymmetric partial difference functional equation analogous to the wave equation

We give the general solution of the nonsymmetric partial difference functional equationf(x + t,y) + f(x – t,y) – 2f(x,y)/t ² =f(x,y + s) + f(x,y – s) – 2f(x,y)/s ² (N) analogous to the well-known wave equation ( ²/x ² – ²/y ²)f(x,y) = 0 with the aid of generalized polynomials when no regularity assumptions are imposed onf. The result is as follows. Theorem.Let R be the set of all real numbers. A function f: R × R R satisfies the functional equation (N)for all x, y R, s, t R\{0}, and s t if and only if there exist