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In this paper, a computer program developed in the computer algebra system Maple is presented, which investigates alienness and strong alienness of linear functional equations. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to prove that if on a commutative hypergroup an exponential monomial has the property that the linear subspace of all sine functions in its variety is one dimensional, then this exponential monomial is a linear combination of generalized moment functions. [ABSTRACT FROM AUTHOR]

In the present paper, a general class of linear functional equations is considered and a computer program is described, which determines the exact solutions of systems of equations belonging to this class. [ABSTRACT FROM AUTHOR]

In an earlier paper we discussed the composition of derivations of order 1 on a commutative ring R, showing that (i) the composition of n derivations of order 1 yields a derivation of order at most n, and (ii) under additional conditions on R the composition of n derivations of order exactly 1 forms a derivation of order exactly n. In the present paper we consider the composition of derivations of any orders on rings. We show that on any commutative ring R the composition of a derivation of order at most n with a derivation of order at most m results in a derivation of order at most n + m . If R is an integral domain of sufficiently large characteristic, then the composition of a derivation of order exactly n with a derivation of order exactly m results in a derivation of order exactly n + m . As in the previous paper, the results are proved using Leibniz difference operators. [ABSTRACT FROM AUTHOR]

This article has two aims. First, we provide the solution to a problem posed by the author in a previous paper. Second, we consider a problem posed by Kannappan and Kurepa (Aequat Math 4: 163-175, 1970). Our results show that additive functions linked by certain types of functional equations are combinations of linear functions and derivations of various orders. We show that this is not generally the case for the problem of Kannappan and Kurepa, and we modify their problem accordingly. [ABSTRACT FROM AUTHOR]

The general solution of the multiplicative symmetric functional equation f (f (x) y) = f (x f (y)) , in abelian groups, was given by Dhombres. But it is still an open problem in non-abelian groups and abelian quasi-groups. In this paper, by using a new approach introduced by the author, we give a vast class of its solutions in arbitrary semigroups. It also works for non-abelian groups, and gives its general solution in abelian groups with a different representation from Dhombres's solution. Also, we introduce some related equations and closed relationships between decomposer type and multiplicative symmetric equations. It is worth noting that the new approach and methods can be used for studying the equation on magmas (groupoids) as the most general algebraic structures for the topic. At last we pose a conjecture that guesses the general solution in arbitrary groups (and possibly semigroups). [ABSTRACT FROM AUTHOR]

In this paper we consider a class of conjugate equations, which generalizes de Rham's functional equations. We give sufficient conditions for the existence and uniqueness of solutions under two different series of assumptions. We consider regularity of solutions. In our framework, two iterated function systems are associated with a series of conjugate equations. We state local regularity by using the invariant measures of the two iterated function systems with a common probability vector. We give several examples, especially an example such that infinitely many solutions exists, and a new class of fractal functions on the two-dimensional standard Sierpiński gasket which are not harmonic functions or fractal interpolation functions. We also consider a certain kind of stability. [ABSTRACT FROM AUTHOR]

In this note we provide the solution to a problem posed by the first author in a previous paper. In particular, we prove a result relating the number of nonzero coefficients of a certain functional equation to the order of any derivation satisfying that equation. [ABSTRACT FROM AUTHOR]

The present article is devoted to generalized Salem functions, the generalized shift operator, and certain related problems. A description of further investigations of the author of this article is given. These investigations (in terms of various representations of real numbers) include generalized Salem functions and generalizations of the Gauss–Kuzmin problem. [ABSTRACT FROM AUTHOR]

Recently, Dong (Aequ Math 86:269-277, ) has proved the generalized stability of the functional equation $${\|f(x + y)\| = \|f(x) + f(y)\|}$$ under the assumption that X, the domain of f, is an Abelian group. In this paper, we prove a generalization of this result by removing the commutativity assumption of X. [ABSTRACT FROM AUTHOR]

Various functional equations satisfied by one or two ( N × N)-matrices $${\mathbf{F}(z) }$$ and $${\mathbf{G}(z) }$$ depending on the scalar variable z are investigated, with N an arbitrary positive integer. Some of these functional equations are generalizations to the matrix case ( N > 1) of well-known functional equations valid in the scalar ( N = 1) case, such as $${\mathbf{F}(x) \, \mathbf{F}(y) = \, \mathbf{F}(x y) \, \rm and \, \mathbf{G}({\it x}) \, \mathbf{G}({\it y}) = \mathbf{G}({\it x+y}) }$$ ; others-such as $${\mathbf{G}(y) \, \mathbf{F}(x) = \mathbf{F}(x) \, \mathbf{G}(xy) }$$ -possess nontrivial solutions only in the matrix case ( N > 1), namely their scalar ( N = 1) counterparts only feature quite trivial solutions. It is also pointed out that if two ( N × N)-matrices $${\mathbf{F}(x) \, \rm and \, \mathbf{G}({\it y})}$$ satisfy the triplet of functional equations written above-and nontrivial examples of such matrices are exhibited- then they also satisfy an endless hierarchy of matrix functional relations involving an increasing number of scalar independent variables, the first items of which read $${\mathbf{F}(x_{1}) \, \mathbf{G}(y_{1}) \, \mathbf{F} (x_{2}) = \mathbf{F}(x_{1} x_{2}) \, \mathbf{G } (x_{2} y_{1}) \, \rm and \, \mathbf{G}({\it y}_{1}) \, \mathbf{F} ({\it x}_{1}) \, \mathbf{G} ({\it y}_{2}) = \mathbf{F} ({\it x}_{1}) \, \mathbf{G} ({\it x}_{1} {\it y}_{1}+{\it y}_{2}) }$$ . [ABSTRACT FROM AUTHOR]

Let X, Y be linear spaces over a field K . Assume that f : X 2 → Y satisfies the general linear equation with respect to the first and with respect to the second variables, that is, for all x , x i , y , y i ∈ X and with a i , b i ∈ K \ { 0 } , A i , B i ∈ K ( i ∈ { 1 , 2 } ). It is easy to see that such a function satisfies the functional equation for all x i , y i ∈ X ( i ∈ { 1 , 2 } ), where C 1 : = A 1 B 1 , C 2 : = A 1 B 2 , C 3 : = A 2 B 1 , C 4 : = A 2 B 2 . We describe the form of solutions and study relations between (∗) and (∗ ∗) . [ABSTRACT FROM AUTHOR]

We investigate the class of bisymmetric and quasitrivial binary operations on a given set and provide various characterizations of this class as well as the subclass of bisymmetric, quasitrivial, and order-preserving binary operations. We also determine explicitly the sizes of these classes when the set is finite. [ABSTRACT FROM AUTHOR]

Define on the set G:=R+×R the operation (t,a)∗(s,b)=(ts,tb+a). (G,∗) is a non-commutative group with the neutral element (1, 0). We consider a non-commutative translation equation F(η,F(ξ,x))=F(η∗ξ,x), η,ξ∈G, x∈I, F(1,0)=id, where I is an open interval and F:G×I→I is a continuous mapping. This equation can be written in the form: F((t,a),F((s,b),x))=F((ts,tb+a),x), t,s∈R+, x∈I. For t=1 the family {F(t,a)} defines an additive iteration group, however for a=0 it defines a multiplicative iteration group. We show that if F(t, 0) for some t≠1 has exactly one fixed point xt, (F(t,0)-id)(xt-id)≥0 and for an a>0F(1,a)>id, then there exists a unique homeomorphism φ:I→R such that F((s,b),x)=φ-1(sφ(x)+b) for s∈R+ and b∈R. [ABSTRACT FROM AUTHOR]

Given a set T⊂(0,+∞), a function c:T→R and a real number p we study continuous solutions φ of the simultaneous equations φ(tx)=φ(x)+c(t)xp,t∈T.Here φ is defined on an interval I⊂(0,+∞), so the equations are postulated on a restricted domain: for any fixed t∈T we assume that x∈I is such that tx∈I. In the case when T is large in a sense, we determine the form of φ on a non-trivial subinterval of I. The research is a continuation of that of "non-restricted", where I=(0,+∞), made in Jarczyk (Ann Univ Sci Budapest Sect Comp 40:353-362, 2013). [ABSTRACT FROM AUTHOR]

We continue the study of additive functions fk:R→F(1≤k≤n) linked by an equation of the form ∑k=1npk(x)fk(qk(x))=0, where the pk and qk are polynomials, R is an integral domain of characteristic 0, and F is the fraction field of R. A method is presented for solving all such equations. We also consider the special case ∑k=1nxmkfk(xjk)=0 in which the pk and qk are monomials. In this case we show that if there is no duplication, i.e. if (mk,jk)≠(mp,jp) for k≠p, then each fk is the sum of a linear function and a derivation of order at most n-1. Furthermore, if this functional equation is not homogeneous then the maximal orders of the derivations are reduced in a specified way. [ABSTRACT FROM AUTHOR]

In 1990, Benz asked whether a real additive mapping satisfying $$xf(y)=yf(x)$$ for all points ( x, y) on the unit circle must be linear. In 2005, Boros and Erdei showed that it must be so. Here we generalize the problem to a pair of additive functions f, g related by the functional equation $$xf(y)=yg(x)$$ for all points ( x, y) on a specified curve. We find that for many (but not all) types of curves this forces f and g to be equal and linear. [ABSTRACT FROM AUTHOR]

Solution sets of systems of linear equations over fields are characterized as being affine subspaces. But what can we say about the 'shape' of the set of all solutions of other systems of equations? We study solution sets over arbitrary algebraic structures, and we give a necessary condition for a set of n-tuples to be the set of solutions of a system of equations in n unknowns over a given algebra. In the case of Boolean equations we obtain a complete characterization, and we also characterize solution sets of systems of Boolean functional equations. [ABSTRACT FROM AUTHOR]

Let $${(S,\cdot)}$$ be a semigroup, ( H, +) an abelian group and $${f: S \to H}$$ . The first and second order Cauchy differences of f are Higher order Cauchy differences C f are defined recursively. In the case of H = R, a ring where multiplication is distributive over addition, we show that functions $${f: S\to R}$$ with vanishing finite Cauchy differences are closed under multiplication. The equation C f = 0 is considered for cyclic groups, free abelian groups and other selected quotients of the free groups. [ABSTRACT FROM AUTHOR]

Given real nonzero coefficients a, A, b, B, a real parameter c and an inhomogeneity $${\varphi : \mathbb{R}\times\mathbb{R} \rightarrow \mathbb{R}}$$ , we present necessary and sufficient conditions for the existence and uniqueness of solutions of the equation $${f(ax+by+c) - Af(x) - Bf(y) = \varphi(x,y)}$$ . The solutions of this equation are given explicitly for each of the various cases emerging from the possibilities concerning the arithmetic nature of the four coefficients and the algebraic relationships between them. [ABSTRACT FROM AUTHOR]

We provide a unifying framework for the treatment of equations of the form for additive maps f and integers p, q (1 ≤ k ≤ n). We show how to solve many equations of this type, and we present some open problems. In general our unknown functions map an integral domain of characteristic zero into itself. When negative exponents appear, we restrict our attention to fields of characteristic zero. All of the results could be formulated for integral domains or fields of sufficiently large characteristic as well. [ABSTRACT FROM AUTHOR]