Regularity properties of functional equations and inequalities

Summary By a well-known theorem of Lebesgue and Fréchet every measurable additive real function is continuous. This result was improved by Ostrowski who showed that a (Jensen-) convex real function must be continuous if it is bounded above on a set of positive Lebesgue measure. Recently, R. Trautner provided a short and elegant proof of the Lebesgue—Fréchet theorem based on a representation theorem for sequences on the real line.We consider here a locally compact topological groupX with some Haar measure. Then the following generalizes Trautner's theorem: Theorem.Let M be a measurable subset of X of positive finite Haar measure. Then there is a neighbourhood W of the identity e such that for each sequence (z _n )in W there is a subsequence (z _nk )and points y and x _k in M with z _nk =x _k ·y ^–1 for k . Using this theorem we obtain the following extensions of the theorems of Lebesgue and Fréchet and of Ostrowski. Theorem.Let R and T be topological spaces. Suppose that R has a countable base and that X is metrizable. If g: X R and H: R × X T are mappings where g is measurable on a set M of positive finite Haar measure and H is continuous in its first variable, then any solution f: X T of f(x · y) = H(g)(x), y) for x, yX is continuous. Theorem.Let G: X × X be a mapping. If there is a subset M of X of positive finite Haar measure such that for each yX the mapping x G(x, y) is bounded above on M, then any solution f: x of f(x · y) G(x, y) for x, yX is locally bounded above. We also prove category analogues of the above results and obtain similar results for general binary mappings in place of the group operation in the argument off.     

Über die Funktionalungleichungf(x + y) + f(xy) ⩾ f(x) + f(y) + f(x)f(y)

Summary The functional inequalityf(x + y) + f(xy) f(x) + f(y) + f(x)f(y), solved for a real continuous function, differentiable at zero.

     

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 stability of the Cauchy equation on semigroups

Summary We say that Hyers's theorem holds for the class of all complex-valued functions defined on a semigroup (S, +) (not necessarily commutative) if for anyf:S such that the set {f(x + y) – f(x) – f(y): x, y S} is bounded, there exists an additive functiona:S for which the functionf – a is bounded.Recently L. Székelyhidi (C. R. Math. Rep. Acad. Sci. Canada8 (1986) has proved that the validity of Hyers's theorem for the class of complex-valued functions onS implies its validity for functions mappingS into a semi-reflexive locally convex linear topological spaceX. We improve this result by assuming sequential completeness of the spaceX instead of its semi-reflexiveness. Our assumption onX is essentially weaker than that of Székelyhidi. Theorem.Suppose that Hyers's theorem holds for the class of all complex-valued functions on a semigroup (S, +) and let X be a sequentially complete locally convex linear topological (Hausdorff) space. If F: S X is a function for which the mapping (x, y) F(x + y) – F(x) – F(y) is bounded, then there exists an additive function A : S X such that F — A is bounded.     

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.     

Measurable solutions of a (2, 2)-type sum form functional equation

Summary In this paper we find the general measurable solutions of the functional equationF(xy) + F(x(1 – y)) – F((1 – x)y) – F((1 – x)(1 – y)) = G(x)H(y) (x, y ]0, 1[) whereF, G, H:]0, 1[ C are unknown functions. The solution of this equation is part of our program to determine the measurable solutions of the functional equationF ₁₁ (xy) + F ₁₂ (x(1 – y)) + F ₂₁ ((1 – x)y) + F ₂₂ ((1 – x)(1 – y)) = G(x)H(y) (x, y ]0, 1[). Our method of solution is based on the structure theorem of sum form equations of (2, 2)-type and on a result of B. Ebanks and the author concerning the linear independence of certain functions.     

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.     

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.     

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.     

Semigroup-valued solutions of some composite equations

Let X be a linear space over the field K of real or complex numbers and (S, °) be a semigroup. We determine all solutions of the functional equation $$f(x+g(x)y)=f(x)\circ f(y)\quad \text{for}\quad x,y\in X$$ in the class of pairs of functions (f,g) such that f : X → S and g : X → K satisfies some regularity assumptions. Several consequences of this result are presented.     

On functions with the cauchy difference bounded by a functional. III

We are going to discuss special cases of a conditional functional inequality

On Jensen's functional equation

Summary Letf be a map from a groupG into an abelian groupH satisfyingf(xy) + f(xy ^–1) = 2f(x), f(e) = 0, wherex, y G ande is the identity inG. A set of necessary and sufficient conditions forS(G, H) = Hom(G, H) is given whenG is abelian, whereS(G, H) denotes all the solutions of the functional equation. The case whenG is non-abelian is also discussed.     

Additive selections of superadditive set-valued functions

Summary A set-valued functionF from a coneC with a cone-basis of a topological vector spaceX into the family of all non-empty compact convex subsets of a locally convex spaceY is called superadditive provided thatF(x) + F(y) F(x + y), for allx, y C. We show that every superadditive set-valued function admits an additive selection.Dedicated to Professor Otto Haupt on his 100th birthday