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.     

Some functional equations in the spaces of generalized functions

Summary. Making use of the fundamental solution of the heat equation we find the solutions of some functional equations such as the Cauchy equations, Pexider equations, quadratic functional equations and dAlembert equations in the spaces of Schwartz distributions and Sato hyperfunctions.     

Nonlinear difference equations investigated via critical point methods

The multiplicity of solutions for nonlinear difference equations involving the p$p$-Laplacian is investigated. The approach is based on critical point theorems in the setting of finite dimensional Banach spaces.     

Analysis of general quadrature methods for integral equations of the second kind

Summary This paper is concerned with a class of approximation methods for integral equations of the form , wherea andb are finite,f andy are continuous and the kernelk may be weakly singular. The methods are characterized by approximate equations of the form ; such methods include the Nyström method and a variety of product-integration methods. A general convergence theory is developed for methods of this type. In suitable cases it has the feature that its application to a specific method depends only on a knowledge of convergence properties of the underlying quadrature rule. The theory is used to deduce convergence results, some of them new, for a number of specific methods.Work supported by the U.S. Department of Energy     

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 an application of hypoellipticity to solutions of functional equations

Summary We study the regularity of solutions of functional equations of a generalized mean value type. In this paper we give sufficient conditions for the regularity by using hypoellipticity which is a concept of the theory of partial differential equations. We also give an affirmative answer to a conjecture of H. wiatak. A part of the results was announced in the comprehensive paper [8] on our joint works. To prove the regularity of solutions of functional equations is one of the central problems in the theory of functional equations (see [1]).     

Supernumary polylogarithmic ladders and related functional equations

Summary The nature of the polylogarithmic ladder is briefly reviewed, and its close relationship to the associated cyclotomic equation explained. Generic results for the base determined by the family of equationsu ^p +u ^q = 1 are developed, and many new supernumary ladders, existing for particular values ofp andq, are discussed in relation to theirad hoc cyclotomic equations. Results for ordersn from 6 through 9, for which no relevant functional equations are known, are reviewed; and new results for the base , where ³ + = 1, are developed through the sixth order.Special results for the exponentp from 4 through 6 are determined whenever a new cyclotomic equation can be constructed. Only the equationu ⁵+u ³ = 1 has so far resisted this process. The need for the constraint (p,q) = 1 is briefly considered if redundant formulas are to be avoided.The equationu ^6m+1 +u ^6r–1 = 1 is discussed and some valid results deduced. This equation is divisible byu ² –u + 1, and the quotient polynomial is useful for constructing cyclotomic equations. The casem = 1,r = 2 is the first example encountered for which no valid ladders have yet been found.New functional equations to give the supernumary -ladders of index 24 are developed, but their construction runs into difficulty at the third order, apparently requiring the introduction of an adjoint set of variables that blocks the extension to the fourth order.A demonstration, based on the indices of existing accessible and supernumary ladders, indicates that functional equations based on arguments ±z ^m (1–z) ^r (1 +z) ^s are not capable of extension to the sixth order.There are some miscellaneous supernumary ladders that seem incapable, at this time, of analytic proof, and these are briefly discussed. In conclusion, applications of ladders are considered, and attention drawn to the existence of ladders with the base on the unit circle giving rise to Clausenfunction formulas which may play an important role inK-theory.     

On a general principle in geometry that leads to functional equations

Summary Suppose that an invariant (or an invariant notion) of some geometry is given, like the distance between two points, the cross ratio of four points, the tangential distance between two spheres (or like the notion of orthogonality, of order, of a circle). One may ask what are the functions preserving (or preserving partially) that invariant (invariant notion). Originating from this principle some functional equation problems are formulated, namely the functional equations of distance, of area, of angle preservance.Dedicated to the memory of Alexander M. Ostrowski on the occasion of the 100th anniversary of his birth.