Jens Schwaiger

Continuous Solutions of the Go a ,b-Schinzel Equation on the Nonnegative Reals andon Related Domains

We determine all continuous solutions f X R 3 R of the functional equation f x y f x f x f y on r... more We determine all continuous solutions f X R 3 R of the functional equation f x y f x f x f y on restricted domains like fx ! 0Y y ! 0g or fx b 0Y y b 0gX 0 The Goøa , b-Schinzel equation f x y f x f x f yY 1 originating in the theory of continuous groups [3,4,6], has a rich literature (see e.g. [1^10]). In most of these works particular a¤ne semigroups on R are considered, particular in the sense that elements of the form (0,) are permitted.

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Distributional implications of equal sacrifice rules

Social Choice and Welfare, 1988

A conclusive story can be made out of equal absolute and equal proportional sacrifice rules if on... more

Sincov's and other functional equations and negative interest rates

arXiv (Cornell University), Oct 5, 2022

Investigating the future value (Ã × Ø) of a capital Ã invested at date Ë at date Ø the "natural" ... more Investigating the future value (Ã × Ø) of a capital Ã invested at date Ë at date Ø the "natural" condition (Ã × Ø) ≥ Ã has lost its naturality because of the strange fact of negative interest rates. This leads to the task of describing the possible solutions of the multiplicative Sincov equation (× Ù) = (× Ø) (Ø Ù) for × ≤ Ø ≤ Ù where (× Ø) = 0 may happen. In this paper we solve this task and discuss connections to the theory of investments.

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On the existence of m-norms in vector spaces over valued fields

Aequationes mathematicae

Gähler (Untersuchungen über verallgemeinerte m-metrische Räume. I”. German. Math Nachr 40, 165–18... more Gähler (Untersuchungen über verallgemeinerte m-metrische Räume. I”. German. Math Nachr 40, 165–189, 1969) investigated m-metric spaces and in particular m-normed spaces over the field of reals. Here the existence of m-norms will be investigated and this in the more general setting of vector spaces over arbitrary non-trivial valued fields.

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On the stability of derivations of higher order

On the stability of derivations of higher order Herrn Professor Zenon Moszner mit den besten Wüns... more On the stability of derivations of higher order Herrn Professor Zenon Moszner mit den besten Wünschen für die Zukunft zum 70. Geburtstag gewidmet Abstract. Derivations of order n as defined by L. Reich are additive and nonlinear functions / : R-> • R with / (1) = 0 which satisfy the func tional equation 8ai ° Sa2 ° ■ ■ ■ ° <U"+i / = 0 for all a i,a 2 , ■ ■ ■ ,a n+ i £ R, where 5af (x) := f(a x)-a f(x). Here we prove several stability results concerning this (and similar) functional equations.

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Sincov’s and other functional equations and negative interest rates

Aequationes mathematicae

Investigating the future value F(K, s, t) of a capital K invested between dates s and t, the “nat... more Investigating the future value F(K, s, t) of a capital K invested between dates s and t, the “natural” condition $$F(K,s,t)\ge K$$ F ( K , s , t ) ≥ K has lost its naturality because of the strange fact of negative interest rates. This leads to the task of describing the possible solutions of the multiplicative Sincov equation $$f(s,u)=f(s,t)f(t,u)$$ f ( s , u ) = f ( s , t ) f ( t , u ) for $$s\le t\le u$$ s ≤ t ≤ u where $$f(s,t)=0$$ f ( s , t ) = 0 may happen. In this paper we solve this task and discuss connections to the theory of investments.

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On the Construction of the Field of Reals by Means of Functional Equations and Their Stability and Related Topics

Developments in Functional Equations and Related Topics, 2017

There are certain approaches to the construction of the field of real numbers which do not refer ... more There are certain approaches to the construction of the field of real numbers which do not refer to the field of rationals. Two of these ideas are closely related to stability investigations for the Cauchy equation and for some homogeneity equation. The a priory different subgroups of $\mathbb{Z}^{\mathbb{Z}}$ used are shown to be more or less identical. Extension of these investigations shows that given a commutative semigroup G and a normed space X with completion X c the group Hom(G, X c ) is isomorphic to $\mathcal{A}(G,X)/\mathcal{B}(G,X)$ where $\mathcal{B}(G,X)$ is the subgroup of X G of all bounded functions and $\mathcal{A}(G,X)$ the subgroup of those f: G → X for which the Cauchy difference (x, y) ↦ f(x + y) − f(x) − f(y) is bounded.

A characterization of damped and undamped harmonic oscillations by a superposition property II

Mathematica Pannonica, 2005

On capital stocks related to reciprocal shareholdings

Stability of the Homogeneity and Completeness By

It is known that the classical homogeneity equation f ðxÞ f ðxÞ under rather weak conditions is s... more It is known that the classical homogeneity equation f ðxÞ f ðxÞ under rather weak conditions is stable (cf. [4, 8]). Following J. SCHWAIGER [14] we will consider here the stability of the -homogeneity equation f ðxÞ ðÞ f ðxÞ. We will prove, without assumption that is a homomorphism, the stability of the -homogeneity equation under some conditions. One of the necessary conditions is that the target space of the function f be se-quentially complete. We will also prove that the converse is true, that is we will show that stability of the -homogeneity implies the fact that if the target space is a normed space then it has to be a Banach space. 1.

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More on Rootless Matrices By

durch das w. M. Ludwig Reich) YOOD in [5] considered the problem of finding ‘‘rootless’ ’ matrice... more durch das w. M. Ludwig Reich) YOOD in [5] considered the problem of finding ‘‘rootless’ ’ matrices, i.e., complex nn-matrices A such that there is no matrix S and no positive integer r2 such that Sr A. Among others it was shown there that any nilpotent nn-matrix A (At 0 for some t1) with maximal rank (n 1) is rootless. In connection with this result the question was raised, whether there are nilpotent matrices of rank less than n 1 which still are rootless. Using results from [1, Kapitel 8.6, 8.7] (English version [2]) we will show that there are nilpotent rootless matrices of all (reasonable) ranks. The main tool we will use is the concept of JORDAN normal forms [1, Kapitel 7], [2, chapter 7]. Compare also [3], where even the infinite-dimensional case is covered. Let be a complex number and n a positive integer. Then the nn-matrix

$is called a Jordan block of length n associated with the eigenvalue 4. Note that J,,(A) is regular, if \ 4 0, and nilpotent, if \ = 0. For short we write J, for Jn(0): Jn := Jn(0).$

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On the stability of derivations of higher order

Derivations of order n as defined by L. Reich are additive and nonlinear functions / : R —>• R... more Derivations of order n as defined by L. Reich are additive and nonlinear functions / : R —>• R with / ( 1 ) = 0 which satisfy the func tional equation 8ai ° Sa2 ° ■ ■ ■ ° <U„+i / = 0 for all a i,a 2 , ■ ■ ■ ,a n+i £ R, where 5af(x ) := f(a x) — a f(x ). Here we prove several stability results concerning this (and similar) functional equations. 1. In [K], Chap. XIV a derivation f : M —> M is defined to be an additive mapping which additionally satisfies the Leibniz rule f(x y ) = x f(y ) + y f(x ), In [R92] the operators Sa are introduced. For functions / : M —> M and reals a we have $a(f)(x) ■ = f(a x ) a f(x ) . In [R98] it was shown that an additive function f is a derivation if and only if and / ( 1) = 0 and (Sa oSt,)f = 0, a, f e e l . Moreover in the same paper (Satz 2) and in [UR] this leads to the following generalization. D e f i n i t i o n 1 A function / : M —> M is called a derivation of order n (e No) if and only if / is additive with / ( 1) = 0 and if / ...

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Stability of the Homogeneity and Completeness By

It is known that the classical homogeneity equation f ðxÞ f ðxÞ under rather weak conditions is s... more It is known that the classical homogeneity equation f ðxÞ f ðxÞ under rather weak conditions is stable (cf. [4, 8]). Following J. SCHWAIGER [14] we will consider here the stability of the -homogeneity equation f ðxÞ ðÞ f ðxÞ. We will prove, without assumption that is a homomorphism, the stability of the -homogeneity equation under some conditions. One of the necessary conditions is that the target space of the function f be se-quentially complete. We will also prove that the converse is true, that is we will show that stability of the -homogeneity implies the fact that if the target space is a normed space then it has to be a Banach space. 1.

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On functions continuous on certain classes of `thin' sets

Continuity properties of functions with continuous restrictions to certain classes of `thin' ... more Continuity properties of functions with continuous restrictions to certain classes of `thin' sets are considered. By construction of an explicit counterexample it is shown that continuity of the restriction to neither of the following classes of sets is su cient to guarantee continuity of a real-valued function on the plane: (real) analytic images of compact intervals, zero sets of real analytic functions in two variables and regular C -images of compact intervals. However, continuity of a real-valued function in two variables along all regular C-curves implies its continuity. In addition it is shown that a function which is continuous on the boundary of some ball of positive radius in a Hilbert space, or even | in the nite dimensional case | on the boundary of a bounded open set must be continuous if it satis es a Cauchy-type functional equation. In dimension two the same result can be obtained if the set on which continuity is required is connected and contains three noncollin...

More on Rootless Matrices By

durch das w. M. Ludwig Reich) YOOD in [5] considered the problem of finding ‘‘rootless’ ’ matrice... more durch das w. M. Ludwig Reich) YOOD in [5] considered the problem of finding ‘‘rootless’ ’ matrices, i.e., complex nn-matrices A such that there is no matrix S and no positive integer r2 such that Sr A. Among others it was shown there that any nilpotent nn-matrix A (At 0 for some t1) with maximal rank (n 1) is rootless. In connection with this result the question was raised, whether there are nilpotent matrices of rank less than n 1 which still are rootless. Using results from [1, Kapitel 8.6, 8.7] (English version [2]) we will show that there are nilpotent rootless matrices of all (reasonable) ranks. The main tool we will use is the concept of JORDAN normal forms [1, Kapitel 7], [2, chapter 7]. Compare also [3], where even the infinite-dimensional case is covered. Let be a complex number and n a positive integer. Then the nn-matrix

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On the stability of derivations of higher order

Derivations of order n as defined by L. Reich are additive and nonlinear functions / : R —>• R... more Derivations of order n as defined by L. Reich are additive and nonlinear functions / : R —>• R with / ( 1 ) = 0 which satisfy the func tional equation 8ai ° Sa2 ° ■ ■ ■ ° <U„+i / = 0 for all a i,a 2 , ■ ■ ■ ,a n+i £ R, where 5af(x ) := f(a x) — a f(x ). Here we prove several stability results concerning this (and similar) functional equations. 1. In [K], Chap. XIV a derivation f : M —> M is defined to be an additive mapping which additionally satisfies the Leibniz rule f(x y ) = x f(y ) + y f(x ), In [R92] the operators Sa are introduced. For functions / : M —> M and reals a we have $a(f)(x) ■ = f(a x ) a f(x ) . In [R98] it was shown that an additive function f is a derivation if and only if and / ( 1) = 0 and (Sa oSt,)f = 0, a, f e e l . Moreover in the same paper (Satz 2) and in [UR] this leads to the following generalization. D e f i n i t i o n 1 A function / : M —> M is called a derivation of order n (e No) if and only if / is additive with / ( 1) = 0 and if / ...

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On functions continuous on certain classes of `thin' sets

Continuity properties of functions with continuous restrictions to certain classes of `thin' ... more Continuity properties of functions with continuous restrictions to certain classes of `thin' sets are considered. By construction of an explicit counterexample it is shown that continuity of the restriction to neither of the following classes of sets is su cient to guarantee continuity of a real-valued function on the plane: (real) analytic images of compact intervals, zero sets of real analytic functions in two variables and regular C -images of compact intervals. However, continuity of a real-valued function in two variables along all regular C-curves implies its continuity. In addition it is shown that a function which is continuous on the boundary of some ball of positive radius in a Hilbert space, or even | in the nite dimensional case | on the boundary of a bounded open set must be continuous if it satis es a Cauchy-type functional equation. In dimension two the same result can be obtained if the set on which continuity is required is connected and contains three noncollin...

Generalized Polynomials in One and in Several Variables

In earlier papers the authors considered relations between general- ized polynomials p of degree ... more In earlier papers the authors considered relations between general- ized polynomials p of degree ≤ n and functions P in n variables being Jensen in each variable such that p is the diagonalization of P. Jensen in each variable means that P is a generalized polynomial of degree ≤ 1 in each variable. Here we derive analogous results connecting functions of several variables which are generalized polynomials of degree ≤ �i in the i-th variable and generalized polynomials (in one variable) of degree ≤ Pi. We also discuss the question whether a function being a polynomial separately in each variable has to be a polynomial jointly in all variables.

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Generalized Polynomials in One and in Several Variables

In earlier papers the authors considered relations between general- ized polynomials p of degree ... more In earlier papers the authors considered relations between general- ized polynomials p of degree ≤ n and functions P in n variables being Jensen in each variable such that p is the diagonalization of P. Jensen in each variable means that P is a generalized polynomial of degree ≤ 1 in each variable. Here we derive analogous results connecting functions of several variables which are generalized polynomials of degree ≤ �i in the i-th variable and generalized polynomials (in one variable) of degree ≤ Pi. We also discuss the question whether a function being a polynomial separately in each variable has to be a polynomial jointly in all variables.

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Investigations on the Hyers–Ulam stability of generalized radical functional equations

Aequationes mathematicae, 2019

In (Brzdęk and Schwaiger in Aeq Math 92: 975-991, 2018) solutions of far reaching generalizations... more In (Brzdęk and Schwaiger in Aeq Math 92: 975-991, 2018) solutions of far reaching generalizations of the so-called radical functional equation f (p(π(x) + π(y))) = f (x) + f (y) have been investigated. These investigations are continued here by analysing the corresponding stability results, which have been the main subject of several recent papers. We propose a very general and uniform approach.

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