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In this paper, we present the central solutions f , g : G × G → C of the functional Eq. f (x 1 y 1 , x 2 y 2) + g (x 1 y 1 - 1 , x 2) + f (x 1 , x 2 y 2 - 1) = f (x 1 y 1 - 1 , x 2 y 2 - 1) + g (x 1 y 1 , x 2) + f (x 1 , x 2 y 2) for all x 1 , x 2 , y 1 , y 2 ∈ G , where G is a 2-divisible group. [ABSTRACT FROM AUTHOR]

The classical result of L. Székelyhidi states that (under some assumptions) every solution of a general linear equation must be a polynomial function. It is known that Székelyhidi's result may be generalized to equations where some occurrences of the unknown functions are multiplied by a linear combination of the variables. In this paper we study the equations where two such combinations appear. The simplest nontrivial example of such a case is given by the equation F (x + y) - F (x) - F (y) = y f (x) + x f (y) considered by Fechner and Gselmann (Publ Math Debrecen 80(1–2):143–154, 2012). In the present paper we prove several results concerning the systematic approach to the generalizations of this equation. [ABSTRACT FROM AUTHOR]

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]

Let S be a monoid (= semigroup with identity), and let σ : S → S be a homomorphism such that σ ∘ σ = i d . In an earlier paper we solved the Pexiderized d'Alembert functional equation (PDFE) f (x y) + g (σ (y) x) = h (x) k (y) for unknown f , g , h , k : S → C , assuming that S is either regular or generated by its squares and that one of the unknown functions is central. The present paper has two main results. The first describes the solutions of PDFE on a general monoid in terms of multiplicative functions, solutions of a special case of the sine subtraction law, and solutions of other functional equations with just one unknown function. The second main result uses the first one to give a more detailed solution of PDFE on a larger class of monoids than has been treated previously. We also find the continuous solutions on topological monoids. Examples are given to illustrate the results. [ABSTRACT FROM AUTHOR]

In this paper, we determine the central solutions f : G × G → C of the functional equation f (x 1 σ y 1 , x 2 τ y 2) - f (x 1 σ y 1 , x 2) - f (x 1 , x 2 τ y 2) = f (x 1 y 1 , x 2 y 2) - f (x 1 y 1 , x 2) - f (x 1 , x 2 y 2) for all x 1 , x 2 , y 1 , y 2 ∈ G , where G is a group, σ , τ : G → G are involutions. [ABSTRACT FROM AUTHOR]

The equality problem of two-variable Bajraktarević means can be expressed as the functional equation f g - 1 ( f (x) + f (y) g (x) + g (y) ) = h k - 1 ( h (x) + h (y) k (x) + k (y) ) (x , y ∈ I) , where I is a nonempty open real interval, f , g , h , k : I → R are continuous functions, g, k are positive and f/g, h/k are strictly monotone. This functional equation, for the first time, was solved by Losonczi in 1999 under 6th-order continuous differentiability assumptions. Additional and new characterizations of this equality problem have been found recently by Losonczi, Páles and Zakaria under the same regularity assumptions in 2021. In this paper it is shown that the same conclusion can be obtained under substantially weaker regularity conditions, namely, assuming only first-order differentiability. [ABSTRACT FROM AUTHOR]

We deal with an alienation problem for an Euler–Lagrange type functional equation f (α x + β y) + f (α x - β y) = 2 α 2 f (x) + 2 β 2 f (y) assumed for fixed nonzero real numbers α , β , 1 ≠ α 2 ≠ β 2 , and the classic quadratic functional equation g (x + y) + g (x - y) = 2 g (x) + 2 g (y). We were inspired by papers of Kim et al. (Abstract and applied analysis, vol. 2013, Hindawi Publishing Corporation, 2013) and Gordji and Khodaei (Abstract and applied analysis, vol. 2009, Hindawi Publishing Corporation, 2009), where the special case g = γ f was examined. [ABSTRACT FROM AUTHOR]

In the present paper we consider the problem of characterization of maps which can be expressed as an affine difference i.e. a three-place map of the form t f (x) + (1 - t) f (y) - f (t x + (1 - t) y). We give a general solution of a functional equation associated with this problem. [ABSTRACT FROM AUTHOR]

In this paper we find the solutions of the functional equation f (x y) = g (x) h (y) + ∑ j = 1 n g j (x) h j (y) , x , y ∈ M , where M is a monoid, n ≥ 2 , and g j (for j = 1 , ... , n ) are linear combinations of at least 2 distinct nonzero multiplicative functions. [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]

Consider a real vector space denoted as X, and let cc(Y) represent the collection of all convex and compact subsets of a real Hausdorff topological vector space Y. This paper investigates set-valued solutions of the Pexiderized quadratic functional equation f1(x+y)+f2(x-y)=f3(x)+f4(y),\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\begin{aligned} f_1(x+y)+f_2(x-y)=f_3(x)+f_4(y), \end{aligned}$$\end{document}for unknown functions f1,f2,f3,f4:X→cc(Y)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$f_1,f_2,f_3,f_4:X\rightarrow cc(Y)$$\end{document}. This functional equation incorporates many functional equations including the quadratic, Cauchy’s and Drygas’ equations. A characterization for set-valued solutions of this functional equation is presented in this paper. [ABSTRACT FROM AUTHOR]

Given two functions f , g : I → R and a probability measure μ on the Borel subsets of [0, 1], the two-variable mean M f , g ; μ : I 2 → I is defined by M f , g ; μ (x , y) : = (f g ) - 1 ∫ 0 1 f (t x + (1 - t) y) d μ (t) ∫ 0 1 g (t x + (1 - t) y) d μ (t) (x , y ∈ I). This class of means includes quasiarithmetic as well as Cauchy and Bajraktarević means. The aim of this paper is, for a fixed probability measure μ , to study their equality problem, i.e., to characterize those pairs of functions (f, g) and (F, G) for which M f , g ; μ (x , y) = M F , G ; μ (x , y) (x , y ∈ I) holds. Under at most sixth-order differentiability assumptions for the unknown functions f, g and F, G, we obtain several necessary conditions in terms of ordinary differential equations for the solutions of the above equation. For two particular measures, a complete description is obtained. These latter results offer eight equivalent conditions for the equality of Bajraktarević means and of Cauchy means. [ABSTRACT FROM AUTHOR]

In the present paper we are concerned with the problem of characterization of maps which can be expressed as an affine difference i.e. a map of the form t f (x) + (1 - t) f (y) - f (t x + (1 - t) y) , where t ∈ (0 , 1) is a given number. We give a general solution of the functional equation associated with this problem. [ABSTRACT FROM AUTHOR]

Several results concerning the homomorphism functional equation f x ∨ y = f x ∨ f y , in the class of semilattices, as well as other similar equations, are presented in the paper. [ABSTRACT FROM AUTHOR]

This paper examines various aspects related to the Cauchy functional equation $$f(x+y)=f(x)+f(y)$$ , a fundamental equation in the theory of functional equations. In particular, it considers its solvability and its stability relative to subsets of multi-dimensional Euclidean spaces and tori. Several new types of regularity conditions are introduced, such as one in which a complex exponent of the unknown function is locally measurable. An initial value approach to analyzing this equation is considered too and it yields a few by-products, such as the existence of a non-constant real function having an uncountable set of periods which are linearly independent over the rationals. The analysis is extended to related equations such as the Jensen equation, the multiplicative Cauchy equation, and the Pexider equation. The paper also includes a rather comprehensive survey of the history of the Cauchy equation. [ABSTRACT FROM AUTHOR]

In this paper, we determine the complex-valued solutions of the functional equation f (x α σ (y)) + h (τ (y) x) = 2 f (x) k (y) for all x , y ∈ G , where G is a group, α ∈ G is a fixed element and σ , τ : G → G are involutions. [ABSTRACT FROM AUTHOR]

The aim of this work is to study the conditional distributivity given by the following functional equation: F (x , U (y , z)) = U (F (x , y) , F (x , z)) , x , y , z ∈ [ 0 , 1 ] , 0 < U (y , z) < 1 where F is a nullnorm and U is a uninorm. The paper discusses one case: F is continuous and U ∈ U max ( U min ) with continuous underlying operators. [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]

Let G be an abelian group, $${\mathbb{C}}$$ be the field of complex numbers, $${\alpha \in G}$$ be any fixed element and $${\sigma : G \to G}$$ be an involution. In this paper, we determine the general solution $${f, g : G \to \mathbb{C}}$$ of the functional equation $${f(x + \sigma y + \alpha) + g(x + y + \alpha) = 2f(x)f(y)}$$ for all $${x, y \in G}$$ . [ABSTRACT FROM AUTHOR]

In this paper we prove that if a generalized polynomial function f satisfies the condition f( x) f( y) = 0 for all solutions of the equation x + y = 1, then f is identically equal to 0. [ABSTRACT FROM AUTHOR]

The connected door space is an enigmatic topological space in which every proper nonempty subset is either open or closed, but not both. This paper provides an elementary proof of the classification theorem of connected door spaces. More importantly, we show that connected door topologies can be viewed as solutions of the valuation f(A)+f(B)=f(A∪B)+f(A∩B) and the equation f(A)+f(B)=f(A∪B). In addition, some special solutions, which can be regarded as a union of connected door spaces, are provided. [ABSTRACT FROM AUTHOR]

Cain, Clark and Rose defined a function $${{f\colon\mathbb{R}^n \to \mathbb{R}}}$$ to be vertically rigid if graph( cf) is isometric to graph( f) for every c ≠ 0. It is horizontally rigid if graph $${(f(c \vec{x}))}$$ is isometric to graph( f) for every c ≠ 0 (see Cain et al., Real Anal. Exch. 31:515-518, ). In Balka and Elekes (J. Math. Anal. Appl. 345:880-888, ) the authors of the present paper settled Janković's conjecture by showing that a continuous function of one variable is vertically rigid if and only if it is of the form a + bx or $${{a+be^{kx} (a,b,k \in \mathbb{R})}}$$. Later they proved in Balka and Elekes (Real. Anal. Exch. 35:139-156, ) that a continuous function of two variables is vertically rigid if and only if after a suitable rotation around the z-axis it is of the form $${a + bx + dy, a +s(y)e^{kx}}$$ or $${{a + be^{kx} + dy (a,b,d,k \in \mathbb{R}, k \neq 0, s\colon \mathbb{R}\to \mathbb{R}\,{\rm is\, continuous})}}$$. The problem remained open in higher dimensions. The characterization in the case of horizontal rigidity is surprisingly simpler. Richter (Real Anal. Exch. 35:343-354, ) proved that a continuous function of one variable is horizontally rigid if and only if it is of the form $${{a+bx (a,b\in \mathbb{R})}}$$. The goal of the present paper is to prove that a continuous function of two variables is horizontally rigid if and only if it is of the form $${{a+ bx + dy (a,b,d \in \mathbb{R})}}$$. This problem also remains open in higher dimensions. The main new ingredient of the present paper is the use of functional equations. [ABSTRACT FROM AUTHOR]

A derivation (of order 1) satisfies the reduction formula f (x k) = k x k - 1 f (x) for any integer k. In this article we find corresponding reduction formulas for derivations of higher order on commutative rings. More precisely, for every derivation f of order n and every positive integer k we find an explicit formula for f (x k) as a linear combination of x k - 1 f (x) , x k - 2 f (x 2) , ... , x k - n f (x n) . The proof hinges on the hypergeometric identity ∑ k ≥ 0 (- 1) k n k n + d + 1 - k n - j d + j - k j = n j for any positive integer n, nonnegative integer d, and integer j satisfying 0 ≤ j ≤ n . We prove this identity via the WZ-method. [ABSTRACT FROM AUTHOR]

This paper studies the solutions of the integral Van Vleck's functional equation for the sine and of the integral Kannappan's functional equation. Topics discussed include complex-valued solutions of the functional equation, d'Alembert's functional equation, and generalization of the lemmas of H. Stetkær.

In the present paper, we determine the complex-valued solutions ( f, g) of the functional equation in the setting of groups and monoids that need not be abelian, where $${\sigma,\tau}$$ are involutive automorphisms. We prove that their solutions can be expressed in terms of multiplicative and additive functions. [ABSTRACT FROM AUTHOR]

In this paper we prove some stability theorems for functional equations of the form g[F(x, y)]=H[g(x), g(y), x, y]. As special cases we obtain well known results for Cauchy and Jensen equations and for functional equations in a single variable. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the system of functional equations, where is a complex *-algebra and are unknown additive functions. This problem arises naturally in connection with the question of representability of quadratic functionals via sesquilinear forms on modules over. We give a fairly complete solution of the problem and examine the case of some particular algebras. [ABSTRACT FROM AUTHOR]

The paper is a survey about Hyers-Ulam stability of functional equations and systems in several variables. Its content is divided in the following chapters: The bibliography contains 120 items. [ABSTRACT FROM AUTHOR]

In this paper, we obtain hyperstability results for the $${\sigma}$$ -Drygas functional equation and the inhomogeneous $${\sigma}$$ -Drygas functional equation on semigroups. Namely, we show that a function satisfying the $${\sigma}$$ -Drygas equation approximately must be exactly the solution of it. [ABSTRACT FROM AUTHOR]

Let G be a group, $${\mathbb{C}}$$ be the field of complex numbers, z be any fixed, nonzero element in the center Z( G) of the group G, and $${\sigma : G \to G}$$ be an involution. The main goals of this paper are to study the functional equations $${f(x{\sigma}yz_{0}) - f(xyz_{0}) = 2f(x)f(y)}$$ and $${f(x{\sigma}yz_{0}) + f(xyz_{0}) = 2f(x)f(y)}$$ for all $${x, y \in G}$$ and some fixed element z in the center Z( G) of the group G. [ABSTRACT FROM AUTHOR]

This paper treats a functional equation of d'Alembert, namely f( x + y) − f( x − y) = g( x) h( y), in the setting of groups and monoids that need not be abelian. Of the several possibilities for generalizing x − y to the non-abelian case, here we replace it by σ( y) x where σ is an involutive automorphism, and find the complex-valued solutions ( f, g, h) of f( xy) − f( σ( y) x) = g( x) h( y) in terms of multiplicative and additive functions. [ABSTRACT FROM AUTHOR]

For any set X we let F (X) denote the complex vector space of functions f : X → C . Let X = S be a magma, and let V be a subspace of F (S) , which is invariant under left or right translations. It is known for an abelian group S that if p 1 χ 1 , ⋯ , p n χ n ∈ F (S) are nonzero exponential polynomials with distinct exponentials χ 1 , ⋯ , χ n then p 1 χ 1 + ⋯ + p n χ n ∈ V ⇒ p 1 χ 1 , ⋯ , p n χ n ∈ V and χ 1 , ⋯ , χ n ∈ V . We extend this to magmas. Our results imply that any exponential polynomial solution f ∈ F (S) of f (x y) = f (x) χ (y) + χ (x) f (y) where χ ∈ F (S) is an exponential, has the form f = a χ where a ∈ F (S) is additive, even when χ has zeros. [ABSTRACT FROM AUTHOR]

Aichinger's equation is used to give simple proofs of several well-known characterizations of polynomial functions as solutions of certain functional equations. Concretely, we use that Aichinger's equation characterizes polynomial functions to solve, for arbitrary commutative groups, Ghurye–Olkin's functional equation, Wilson's functional equation, the Kakutani–Nagumo–Walsh functional equation, and a general version of Fréchet's unmixed functional equation. [ABSTRACT FROM AUTHOR]

In our paper we find all functions $${f : \mathbb {R} \times \mathbb {R}^{3} \rightarrow \mathbb {H}}$$ and $${g : \mathbb {R}^{3} \rightarrow \mathbb {H}}$$ satisfying $${f (r, {\bf v}) f (s, {\bf w}) = -\langle{\bf v},{\bf w}\rangle + f (rs, s{\bf v} + r{\bf w} + {\bf v} \times {\bf w})}$$ $${(r, s \in \mathbb {R}, {\bf v},{\bf w} \in \mathbb {R}^{3})}$$ , and $${g({\bf v})g({\bf w}) = -\langle{\bf v}, {\bf w}\rangle + g({\bf v} \times {\bf w})}$$ $$({{\bf v},{\bf w} \in \mathbb {R}^{3}})$$ . These functional equations were motivated by the well-known identities for vector products and quaternions, which can be obtained from the solutions f ( r, ( v, v, v)) = r + v i + v j + v k and g(( v , v, v)) = v i + v j + v k. [ABSTRACT FROM AUTHOR]

In this paper, we solve the ( k, s)-Fibonacci functional equation Moreover, we prove the Hyers-Ulam stability of the ( k, s)-Fibonacci functional equation in β-normed spaces. [ABSTRACT FROM AUTHOR]

We provide three large classes of control functions that ensure the hyperstability of the Cauchy equation on restricted domains included in various types of commutative semigroups. Among other consequences, we obtain significant improvements on similar results known from the literature for several Aoki-Rassias-type control functions. [ABSTRACT FROM AUTHOR]

Given a probability space (Ω , A , P) , a complete separable Banach space X with the σ -algebra B (X) of all its Borel subsets, an operator Λ : Ω → L (X , X) and ξ : Ω → X we consider the B (X) ⊗ A -measurable function f : X × Ω → X given by f (x , ω) = Λ (ω) x + ξ (ω) and investigate the continuous dependence of a weak limit π f of the sequence of iterates (f n (x , ·)) n ∈ N of f, defined by f 0 (x , ω) = x , f n + 1 (x , ω) = f (f n (x , ω) , ω n + 1) for x ∈ X and ω = (ω 1 , ω 2 , ⋯) . Moreover for X taken as a Hilbert space we characterize π f via the functional equation φ f (u) = ∫ Ω φ f (Λ (ω) u) φ ξ (u) P (d ω) with the aid of its characteristic function φ f . We also indicate the continuous dependence of a solution of that equation. [ABSTRACT FROM AUTHOR]

Our main result is that we describe the solutions g , f : S → C of the functional equation g (x σ (y)) = g (x) g (y) - f (x) f (y) + α f (x σ (y)) , x , y ∈ S , where S is a semigroup, α ∈ C is a fixed constant and σ : S → S an involutive automorphism. [ABSTRACT FROM AUTHOR]

Let C be an affine plane curve. We consider additive functions f:K→K for which f(x)f(y)=0, whenever (x,y)∈C. We show that if K=R and C is the hyperbola with defining equation xy=1, then there exist nonzero additive functions with this property. Moreover, we show that such a nonzero f exists for a field K if and only if K is transcendental over Q or over Fp, the finite field with p elements. We also consider the general question when K is a finite field. We show that if the degree of the curve C is large enough compared to the characteristic of K, then f must be identically zero. [ABSTRACT FROM AUTHOR]

The goal of this paper is to give a survey of all important characterizations of sum form information measures that depend upon k discrete complete probability distributions (without zero probabilities) of length n and which satisfy a generalized additivity property. It turns out that most of the problems have been solved, but some open problems lead to the very simple looking functional equations and where f, g: ]0, , 1[ k → ℝ and k ∈ ℕ. Moreover new entropies analogous to the Shannon entropy, entropies of degree α, entropies of degree (α, β) are introduced for χ, β ∈ ℕ. [ABSTRACT FROM AUTHOR]

In the first part of the present paper we prove some necessary conditions satisfied by the solutions of a system of functional equations related to Plurality Functions. In the second part we describe a geometric-combinatorial procedure for the construction of the solutions of that system. This procedure yields all possible solutions. [ABSTRACT FROM AUTHOR]

In the class of functionals f: X → ℝ, where X is an inner product space with dim X ≥ 3, we study the D'Alembert functional equation f(x + y) + f(x − y) = 2f(x)f(y) (1) on the restricted domains X = {( x, y) ∈ X/〈 x, y〉 = 0} and X = {( x, y) ∈ X/∥ x∥ = ∥ y∥}. In this paper we prove that the equation (1) restricted to X is not equivalent to (1) on the whole space X. We also succeed in characterizing all common solutions if we add the condition f(2x) = 2f(x) − 1. Using this result, we prove the equivalence between (1) restricted to X and (1) on the whole space X. This research follows similar previous studies concerning the additive, exponential and quadratic functional equations. [ABSTRACT FROM AUTHOR]

This paper presents a new, shorter and more direct proof of the following result of J. Aczél and C. T. Ng: If M: J → R ( J = ]0, 1[) is both multiplicative and additive, then the general solution ϕ: J → R of ϕ(x) + M(1 − x)ϕ(y/1 − x) = ϕ(y) + M(1 − y)ϕ(x/1 − y) ( x, y, x + y ∈ J) is given by ϕ(x) = α if M = 0, ϕ(x) = M(x)[L(x) + α] + M(1 − x)L(1 − x) if M ≠ 0, where α is an arbitrary constant and L: J → R is an arbitrary solution of the logarithmic functional equation L(xy) = L(x) + L(y) (x, y ∈ J). Also, some extensions of this result to fields more general than the reals are given. [ABSTRACT FROM AUTHOR]

In this paper we solve the equations which generalize the sine and cosine (d'Alembert) equations. A conditional d'Alembert-Wilson equation and some related unconditional equations in more than two variables are also considered. [ABSTRACT FROM AUTHOR]

Functional equations of the form are investigated on ℝ and on more general domains. The paper is divided into five sections. After two introductory sections we solve in Section 3 two special cases of the equation In Section 4 we prove that the solutions of (FE) under certain conditions are exponential polynomials. Finally, the general solutions of the equation are determined in Section 5. The last equation contains as special cases the equations of d'Alembert, Cauchy, Lobachevskii, Jensen, Pexider, Wilson, the sine equation, etc. [ABSTRACT FROM AUTHOR]

This paper begins with another proof of a theorem of W. Benz [2] concerning dilations in normed linear spaces. Our proof motivates several questions which are addressed thereafter. For instance it is shown that, if I is an open interval in ℝ, γ: I → ℝ, γ is continuously differentiable and there exist a,..., a ∈ I such that { γ′(a,..., γ′(a)} is linearly independent, then { γ(t): t ∈ I} contains a Hamel basis for ℝ over ℚ. [ABSTRACT FROM AUTHOR]

We study the functional equation for unknown functions f , g , h , k , ℓ : S → C , where S is a semigroup and m 1 , m 2 : S → C are multiplicative functions. The study is divided into two main parts: m 1 = m 2 and m 1 ≠ m 2 . In some cases we assume that one or more of the unknown functions is central and/or that S is a monoid. The solutions are found in all cases under the umbrella assumption that S is a commutative monoid. The continuous solutions on topological semigroups are also found. [ABSTRACT FROM AUTHOR]

Let G be a locally compact Hausdorff group and ψ : G → G is a continuous anti-endomorphism that need not be involutive. Our main goal is to extend previous results about Kannappan's functional equation g (x y z 0) + g (x ψ (y) z 0) = g (x) g (y) , where z 0 is an arbitrarily fixed element in G. We introduce and study the continuous solutions h , f , g : G → C of the integral-functional equation ∫ G { h (x y t) + μ (y) h (x ψ (y) t) } d ν (t) = f (x) g (y) , x , y ∈ G , where μ : G → C is a continuous multiplicative function satisfying μ (x ψ (x)) = 1 for all x ∈ G , and ν is a regular, compactly supported, complex-valued Borel measure on G. [ABSTRACT FROM AUTHOR]

We consider a functional equation which is a generalization of Whitehead's functional equation on groups. We present its general solution on free groups. Solutions on other selected groups are also given. [ABSTRACT FROM AUTHOR]

Let S be a semigroup, H a 2-torsion free, abelian group and $$C^2f$$ the second order Cauchy difference of a function $$f:S \rightarrow H$$ . Assuming that H is uniquely 2-divisible or S is generated by its squares we prove that the solutions f of $$C^2f = 0$$ are the functions of the form $$f(x) = j(x) + B(x,x)$$ , where j is a solution of the symmetrized additive Cauchy equation and B is bi-additive. Under certain conditions we prove that the terms j and B are continuous, if f is. We relate the solutions f of $$C^2f = 0$$ to Fréchet's functional equation and to polynomials of degree less than or equal to 2. [ABSTRACT FROM AUTHOR]