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The authors of [KMM] consider a system of two functional inequalities for a function $$f : {\mathbb{R}} \rightarrow {\mathbb{R}}$$ , and they show that, if certain arithmetical conditions and inequalities for certain parameters are fulfilled, f has to be a polynomial provided that f is continuous at some point x0. This result is derived here under the weaker condition that for some x0 the limit $${\rm lim}_{x \rightarrow x_0} f(x)$$ exists. Moreover, another system of inequalities is given leading to the same result on the nature of f. The methods used also give natural explanations for the fact that Bernoulli numbers play an important role in this context.

A remarkable contribution to the theory of differential equations was made by Adolph Steen in 1874. His results were presented in Danish in a journal containing few articles on mathematics and were wholly ignored by the mathematical community for more than a century. During this time much of his work was rediscovered and extended by numerous authors; in fact the relevant literature embraces five languages in addition to the original Danish and is scattered over correspondingly many countries. In Part I we outline the historical development of Steen's ideas and in Part II we reproduce his pioneering article in an English translation. Though correction of a historical injustice is the primary goal, not everything in his paper has been rediscovered by others. Hence some of the results retain, even at this late date, more than merely historical interest.

Zannier proved that for a Laurent series f(x) satisfying the functional equation of type f(x m ) = P(x, f(x)), where \( P(x, y) \in {\Bbb C}(x)[y] \), if f(x) is not rational the set of such m consists of the powers of a single integer. He mentioned that the case f(x) = P(x, f(x m )) should be proved in a similar way. In this paper we first verify this statement and second we show a theorem which is useful for proving the transcendence of a Laurent series satisfying a certain functional equation. This theorem is a generalization of the result that a Laurent series which satisfiesf(x m ) = P(x, f(x)), where\( P(x, y) \in {\Bbb C}(x, y),\,m \geq 2 \)cannot represent an algebraic function unless it is rational (Ke. Nishioka [1], Zannier [8], Section 3).

A recent note of Ih-Ching Hsu poses an unsolved problem, to wit, the general solution of the functional equation g(x1, x2) + g(φ1(x1), φ2(x2)) = g(x1, φ2(x2)) + g(φ1(x1),x2), where the φi are given functions. This short paper obtains the general solution. It gives conditions which imply that anycontinuous solution has form g1(x1) + g2(x2).

The aim of this paper is to fill in the gaps in the formulation and the proof of a theorem contained in the paper by K.Ozeki (Aequ Math 25:247–252, 1982) published in this journal. We also give a short proof of this theorem and use it to obtain certain information about the factorization of polynomials of the form $$f(x)-f(y)$$ .

In the present paper we describe the family of all closed convex plane curves of class $$C^1$$ C 1 which have circles as their isoptics. In the first part of the paper we give a certain characterization of all ellipses based on the notion of isoptic and we give a geometric characterization of curves whose orthoptics are circles. The second part of the paper contains considerations on curves which have circles as their isoptics and we show the form of support functions of all considered curves.

Non-monotonicity height is an important index to describe the complexity of dynamics for piecewise monotone functions. Although it is used extensively in the theory of iterative roots, its calculation is still difficult especially in the infinite case. In this paper, by introducing the concept of spanning interval, we first present a sufficient condition for piecewise monotone functions to have height infinity and then an algorithm for finding the spanning intervals is given. We further investigate the density of all piecewise monotone functions with infinite and finite height, respectively, and the results indicate the instability of height. At the end of this paper, the variance of height under composition, especially for functions of height 1 and infinity, are also discussed.

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$$\begin{aligned} F(x + y) - F(x) - F(y) = yf(x) + xf(y) \end{aligned}$$F(x+y)-F(x)-F(y)=yf(x)+xf(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.

A sequence $$(v_1,\ldots ,v_k)$$ of vertices in a graph G without isolated vertices is called a total dominating sequence if every vertex $$v_i$$ in the sequence totally dominates at least one vertex that was not totally dominated by $$\{v_1,\ldots , v_{i-1}\}$$ and $$\{v_1,\ldots ,v_k\}$$ is a total dominating set of G. The length of a shortest such sequence is the total domination number of G ( $$\gamma _{t}(G)$$ ), while the length of a longest such sequence is the Grundy total domination number of G ( $$\gamma _{gr}^t(G)$$ ). In this paper we study graphs with equal total and Grundy total domination numbers. We characterize bipartite graphs with both total and Grundy total dominations number equal to 4, and show that there is no connected chordal graph G with $$\gamma _{t}(G)=\gamma _{gr}^t(G)=4$$ . The main result of the paper is a characterization of bipartite graphs with $$\gamma _{t}(G)=\gamma _{gr}^t(G)=6$$ proved by establishing a surprising correspondence between the existence of such graphs and a classical but still open problem of the existence of certain finite projective planes.

Given a graph $G$ and a nondecreasing sequence $S=(s_1,\ldots,s_k)$ of positive integers, the mapping $c:V(G)\longrightarrow \{1,\ldots,k\}$ is called an $S$-packing coloring of $G$ if for any two distinct vertices $x$ and $y$ in $c^{-1}(i)$, the distance between $x$ and $y$ is greater than $s_i$. The smallest integer $k$ such that there exists a $(1,2,\ldots,k)$-packing coloring of a graph $G$ is called the packing chromatic number of $G$, denoted $\chi_{\rho}(G)$. The question of boundedness of the packing chromatic number in the class of subcubic (planar) graphs was investigated in several earlier papers; recently it was established that the invariant is unbounded in the class of all subcubic graphs. In this paper, we prove that the packing chromatic number of any 2-connected bipartite subcubic outerplanar graph is bounded by $7$. Furthermore, we prove that every subcubic triangle-free outerplanar graph has a $(1,2,2,2)$-packing coloring, and that there exists a subcubic outerplanar graph with a triangle that does not admit a $(1,2,2,2)$-packing coloring. In addition, there exists a subcubic triangle-free outerplanar graph that does not admit a $(1,2,2,3)$-packing coloring. A similar dichotomy is shown for bipartite outerplanar graphs: every such graph admits an $S$-packing coloring for $S=(1,3,\ldots,3)$, where $3$ appears $\Delta$ times ($\Delta$ being the maximum degree of vertices), and this property does not hold if one of the integers $3$ is replaced by $4$ in the sequence $S$., Comment: 24 pages

The aim of this paper is twofold. Firstly, we study the non-existence of finite order meromorphic solutions to the Cubic type of Fermat functional equation $$f(z)^3-3\tau f(z)f(z+c)+f(z+c)^3=1$$. In addition, the paper is concerned with the description of finite order entire solutions of the Quadratic type of Fermat functional equation $$f(z)^2-2\mu f(z)f(z+c)+f(z+c)^2=1$$.

The Hermite–Hadamard inequality was extended using iterated integrals by Retkes [Acta Sci Math (Szeged) 74:95–106, 2008]. In this paper we further extend the main results of the above paper for convex and also for s-convex functions in the second sense.

In this paper we study Gruss type inequalities for real and complex valued functions in probability spaces. Some earlier Gruss type inequalities are extended and refined. Our approach leads to new integral inequalities which are interesting in their own right. As an application, we give a Gruss type inequality for normal operators in a Hilbert space. Similar results are obtained only for self-adjoint operators in earlier papers.

The paper deals with F-normed functions and sequence spaces. First, some general results on such spaces are presented. But most of the results in this paper concern various monotonicity properties and various Kadec–Klee properties of F-normed Orlicz functions and sequence spaces and their subspaces of elements with order continuous norm, when they are generated by monotone Orlicz functions on $${\mathbb {R}}_{+}$$ and equipped with the classical Mazur–Orlicz F-norm. Strict monotonicity, lower (and upper) local uniform monotonicity and uniform monotonicity in the classical sense as well as their orthogonal counterparts are considered. It follows from the criteria that are presented for these properties that all the above classical monotonicity properties except for uniform monotonicity differ from their orthogonal counterparts [in contrast to Kothe spaces (see Hudzik et al. in Rocky Mt J Math 30(3):933–950, 2000)]. The Kadec–Klee properties that are considered in this paper correspond to various kinds of convergence: convergence locally in measure and convergence globally in measure for function spaces, uniform convergence and coordinatewise convergence in the case of sequence spaces.

The study of quantum integers and their operations is closely related to the studies of symmetries of roots of polynomials and of fundamental questions of decompositions in Additive Number Theory. In his papers on quantum arithmetics, Melvyn Nathanson raises the question of classifying solutions of functional equations arising from the multiplication of quantum integers, starting with polynomial solutions and then generalizing to rational function solutions. The classification of polynomial solutions with fields of coefficients of characteristic zero and support base P has been completed. In a paper concerning the Grothendieck group associated to the collection of polynomial solutions, Nathanson poses a problem which asks whether the set of rational function solutions strictly contains the set of ratios of polynomial solutions. It is now known that there are infinitely many rational function solutions $$\Gamma $$ with fields of coefficients of characteristic zero not constructible as ratios of polynomial solutions, even in the purely cyclotomic case, which is the case most similar to the polynomial solution case. The classification of polynomial solutions is thus not sufficient, in essential ways, to resolve the classification problem of all rational function solutions with fields of coefficients of characteristic zero. In this paper we study symmetries of roots of rational functions and the classification of the important class-the last and main obstruction to the classification problem-of rational function solutions, the purely cyclotomic, purely nonrational primitive solutions with fields of coefficients of characteristic zero and support base P, which allows us to complete the classification problem raised by Nathanson.

[EN] In this paper, we introduce a new generalized inverse, called weak core inverse (or, in short, WC inverse) of a complex square matrix. This new inverse extends the notion of the core inverse defined by Baksalary and Trenkler (Linear Multilinear Algebra 58(6):681-697, 2010). We investigate characterizations, representations, and properties for this generalized inverse. In addition, we introduce weak core matrices (or, in short, WC matrices) and we show that these matrices form a more general class than that given by the known weak group matrices, recently investigated by H. Wang and X. Liu., In what follows, we detail the acknowledgements. D.E. Ferreyra, F.E. Levis, A.N. Priori - Partially supported by Universidad Nacional de Rio Cuarto (Grant PPI 18/C559) and CONICET (Grant PIP 112-201501-00433CO). D.E. Ferreyra F.E. Levis - Partially supported by ANPCyT (Grant PICT 201803492). D.E. Ferreyra, N. Thome -Partially supported by Universidad Nacional de La Pampa, Facultad de Ingenieria (Grant Resol. Nro. 135/19). N. Thome -Partially supported by Ministerio de Economia, Industria y Competitividad of Spain (Grant Red de Excelencia MTM2017-90682-REDT) and by Universidad Nacional del Sur of Argentina (Grant 24/L108). We would like to thank the Referees for their valuable comments and suggestions which helped us to considerably improve the presentation of the paper

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.

Quasimetric spaces have been an object of thorough investigation since Frink’s paper appeared in 1937 and various generalisations of the axioms of metric spaces are now experiencing their well-deserved renaissance. The aim of this paper is to present two improvements of Frink’s metrization theorem along with some fixed point results for single-valued mappings on quasimetric spaces. Moreover, Cantor’s intersection theorem for sequences of sets which are not necessarily closed is established in a quasimetric setting. This enables us to give a new proof of a quasimetric version of the Banach Contraction Principle obtained by Bakhtin. We also provide error estimates for a sequence of iterates of a mapping, which seem to be new even in a metric setting.

The purpose of this paper is to investigate the invariance of the arithmetic mean with respect to two weighted Bajraktarevic means, i.e., to solve the functional equation $$\begin{aligned} \left( \frac{f}{g}\right) ^{\!\!-1}\!\!\left( \frac{tf(x)+sf(y)}{tg(x)+sg(y)}\right) +\left( \frac{h}{k}\right) ^{\!\!-1}\!\!\left( \frac{sh(x)+th(y)}{sk(x)+tk(y)}\right) =x+y \qquad (x,y\in I), \end{aligned}$$ where $$f,g,h,k:I\rightarrow \mathbb {R}$$ are unknown continuous functions such that g, k are nowhere zero on I, the ratio functions f / g, h / k are strictly monotone on I, and $$t,s\in \mathbb {R}_+$$ are constants different from each other. By the main result of this paper, the solutions of the above invariance equation can be expressed either in terms of hyperbolic functions or in terms of trigonometric functions and an additional weight function. For the necessity part of this result, we will assume that $$f,g,h,k:I\rightarrow \mathbb {R}$$ are four times continuously differentiable.

In the first part of the paper we give a new integral representation for the principal branch of the Lambert W function. Then we deduce two continued fraction expansions for this branch. At the end of the paper we study the numerical behavior of the approximants of these expansions.

In this paper we consider monic polynomials such that their coefficients coincide with their zeros. These polynomials were first introduced by S. Ulam. We combine methods of algebraic geometry and dynamical systems to prove several results. We obtain estimates on the number of Ulam polynomials of degree $N$. We provide additional methods to obtain algebraic identities satisfied by the zeros of Ulam polynomials, beyond the straightforward comparison of their zeros and coefficients. To address the question about existence of orthogonal Ulam polynomial sequences, we show that the only Ulam polynomial eigenfunctions of hypergeometric type differential operators are the trivial Ulam polynomials $\{x^N\}_{N=0}^\infty$. We propose a family of solvable $N$-body problems such that their stable equilibria are the zeros of certain Ulam polynomials., This version contains clarifications of the exposition to match the published version of the paper

In this paper our aim is to show some new inequalities of the Redheffer type for Bessel and modified Bessel functions of the first kind. The key tools in our proofs are some classical results on the monotonicity of quotients of differentiable functions as well as on the monotonicity of quotients of two power series. We also use some known results on the quotients of Bessel and modified Bessel functions of the first kind, and by using the monotonicity of the Dirichlet eta function we prove a sharp inequality for the tangent function. At the end of the paper a conjecture is stated, which may be of interest for further research.

The k-core, defined as the maximal subgraph of minimum degree at least k, of the random graph G(n,p) has been studied extensively. In a landmark paper Pittel, Wormald and Spencer [J Combin Theory Ser B 67 (1996), 111–151] determined the threshold dk for the appearance of an extensive k-core. The aim of the present paper is to describe how the k-core is “embedded” into the random graph in the following sense. Let k≥3 and fix d=np>dk. Colour each vertex that belongs to the k-core of G(n,p) in black and all remaining vertices in white. Here we derive a multi-type branching process that describes the local structure of this coloured random object as n tends to infinity. This generalises prior results on, e.g., the internal structure of the k-core. In the physics literature it was suggested to characterize the core by means of a message passing algorithm called Warning Propagation. Ibrahimi, Kanoria, Kraning and Montanari [Ann Appl Probab 25 (2015), 2743–2808] used this characterization to describe the 2-core of random hypergraphs. To derive our main result we use a similar approach. A key observation is that a bounded number of iterations of this algorithm is enough to give a good approximation of the k-core. Based on this the study of the k-core reduces to the analysis of Warning Propagation on a suitable Galton-Watson tree. © 2017 Wiley Periodicals, Inc. Random Struct. Alg., 00, 000–000, 2017

The goal of this paper is to point out that the results obtained in the recent papers (Chen and Song in Nonlinear Anal 72:1895–1901, 2010; Chu in J Math Anal Appl 327:1041–1045, 2007; Chu et al. in Nonlinear Anal 59:1001–1011, 2004a, J. Math Anal Appl 289:666–672, 2004b) can be seriously strengthened in the sense that we can significantly relax the assumptions of the main results so that we still get the same conclusions. In order to do this first, we prove that for $$n \ge 3$$ any transformation which preserves the n-norm of any n vectors is automatically plus-minus linear. This will give a re-proof of the well-known Mazur–Ulam-type result that every n-isometry is automatically affine ( $$n \ge 2$$ ) which was proven in several papers, e.g. in Chu et al. (Nonlinear Anal 70:1068–1074, 2009). Second, following the work of Rassias and Semrl (Proc Am Math Soc 118:919–925, 1993), we provide the solution of a natural Aleksandrov-type problem in n-normed spaces, namely, we show that every surjective transformation which preserves the unit n-distance in both directions ( $$n\ge 2$$ ) is automatically an n-isometry.

In the present paper we introduce a notion of the \(\mathbb {K}\)-Riemann integral as a natural generalization of a usual Riemann integral and study its properties. The aim of this paper is to extend the classical Hermite–Hadamard inequalities to the case when the usual Riemann integral is replaced by the \(\mathbb {K}\)-Riemann integral and the convexity notion is replaced by \(\mathbb {K}\)-convexity.

This paper gives simple proofs of Sylvester (` = 2) and Frobenius (` = 3) inequalities. Moreover, a new sufficient condition for the equality of the Frobenius inequality is provided. In addition, an extension for ` > 3 matrices and an application to generalized inverses are provided., This paper has been partially supported by Ministerio de Economia y Competitividad (Grant DGI MTM2013-43678P and Red de Excelencia MTM2015-68805-REDT). The author thanks the referees for their valuable suggestions.

A scientific law can be a faithful representation of the physical world only if its form is invariant with respect to changes in the unit or units. This is referred to as the ‘meaningfulness condition.’ This condition is powerful. If we require it, the mathematical form of a scientific or geometric law may be derivable from some abstract constraint, possibly verifiable by a thought experiment or a trivial argument. We discuss five examples of such abstract constraints in this paper: In each case, just one or a couple of meaningful mathematical representations are possible. In this paper, we derive the possible meaningful representations in five examples. These results are obtained under some general conditions, in addition to those listed in 1–5. Other meaningful representations may be possible under different additional conditions.

In the present paper a notion of delta (s, t)-convexity in the sense of Veselya nd Zajiucek is studied as a natural generalization of the classical (s, t)-convexity. The main result of this paper is a support theorem for delta (s, t)-convex mappings. Mathematics Subject Classification. 39B62, 26A51, 26B25.

Let $2 \leq a \leq b \leq c \in \mathbb{N}$ with $��=1/a+1/b+1/c$ be the corresponding hyperbolic triangle group. Many papers have been dedicated to the following question: what are the finite (simple) groups which appear as quotients of $T$? (Classically, for $(a,b,c)=(2,3,7)$ and more recently also for general $(a,b,c)$.) These papers have used either explicit constructive methods or probabilistic ones. The goal of this paper is to present a new approach based on the theory of representation varieties (via deformation theory). As a corollary we essentially prove a conjecture of Marion [21] showing that various finite simple groups are not quotients of $T$, as well as positive results showing that many finite simple groups are quotients of $T$.

In this paper, we prove the Hyers–Ulam stability theorem when $${f, g, h : \mathbb{R} \to \mathbb{R}}$$ satisfy $$|f(x + y) - g(x) - h(y)| \leq \epsilon$$ in a set $${\Gamma \subset \mathbb{R}^{2}}$$ of measure $${m(\Gamma) = 0}$$ , which refines a previous result in Chung (Aequat Math 83:313–320, 2012) and gives an affirmative answer to the question in the paper. As a direct consequence we obtain that if $${f, g, h : \mathbb{R} \to \mathbb{R}}$$ satisfy the Pexider equation $$f(x + y) - g(x) - h(y) = 0$$ in $${\Gamma}$$ , then the equation holds for all $${x, y \in \mathbb{R}}$$ . Using our method of construction of the set, we can find a set $${\Gamma \subset \mathbb{R}^{2n}}$$ of 2n-dimensional measure 0 and obtain the above result for the functions $${f, g, h : \mathbb{R}^{n} \to \mathbb{C}}$$ .

In the paper Brillouet-Belluot and Ebanks (Aequationes Math 60:233–242, 2000), the authors found all continuous functions f: [0, 1] → [0, + ∞) which verify f(0) = f(1) = 0 and the functional equation $$f(xy +c f(x) f(y)) = x f(y) + y f(x) +d \, f(x) f(y)$$ where c and d are given real numbers with c ≠ 0. In the present paper we obtain all continuous solutions \({f: \mathbb{R} \rightarrow \mathbb{R}}\) of the functional equation (1).

In this paper various notions of convexity of real functions with respect to Chebyshev systems defined over arbitrary subsets of the real line are introduced. As an auxiliary notion, a concept of a relevant divided difference and also a related lower Dinghas type derivative are also defined. The main results of the paper offer various characterizations of the convexity notions in terms of the nonnegativity of a generalized divided difference and the corresponding lower Dinghas type derivative.

Two natural extensions of Jensen’s functional equation on the real line are the equations f(xy) + f(xy−1) = 2f(x) and f(xy) + f(y−1x) = 2f(x), where f is a map from a multiplicative group G into an abelian additive group H. In a series of papers (see Ng in Aequationes Math 39:85–99, 1990; Ng in Aequationes Math 58:311–320, 1999; Ng in Aequationes Math 62:143–159, 2001), Ng solved these functional equations for the case where G is a free group and the linear group \({{GL_n(R), R=\mathbb{Z},\mathbb{R}}}\) , is a quadratically closed field or a finite field. He also mentioned, without a detailed proof, in the above papers and in (see Ng in Aequationes Math 70:131–153, 2005) that when G is the symmetric group Sn, the group of all solutions of these functional equations coincides with the group of all homomorphisms from (Sn, ·) to (H, + ). The aim of this paper is to give an elementary and direct proof of this fact.

In this second paper we continue our development of elementary methods for computing continuous complex-valued solutions of the 2-cocycle functional equation on locally compact groups. The first paper dealt primarily with solvable groups. Here we consider some classical motion groups of mathematical physics, including the Euclidean, Galilean, Lorentz and Poincare groups.

The paper deals with the polynomial-like iterative functional equation $$\lambda_1 f(x)+\lambda_2 f^2(x)+\cdots+\lambda_n f^n(x)=F(x).$$ By using Schauder’s fixed point theorem and a version of the uniform boundedness principle for families of convex (respectively higher order convex) functions as basic tools, the existence of nondecreasing convex (respectively higher order convex) solutions to this equation on open (possibly unbounded) intervals is investigated. The results of the paper complement similar ones established by other authors, concerning the existence of monotonic or convex solutions to the above equation on compact intervals. Some examples illustrating their applicability are provided.

Let $$f : {\mathbb{R}} \longrightarrow {\mathbb{R}}$$ denote the function given by $$f(t) = \sum \limits^{\infty}_{n=0} a_{n}t^{n}, t \in {\mathbb{R}}$$ , where $$a_n \geq 0, n \in {\mathbb{N}}$$ . Let K be a closed convex cone in a real Banach space and $$H : K \longrightarrow cc(K)$$ a linear and continuous set-valued function with nonempty, convex and compact values in K. The set-valued function $$F^t(x) := \sum \limits_{n=0}^{\infty} a_{n}t^{n}H^{n}(x), x \in K$$ is linear and continuous for all $$t \geq 0$$ and $$F^{t} \circ F^{s}(x) \subseteq \sum \limits_{n=0}^{\infty} c_{n}H^{n} (x), x \in K$$ where $$c_{n} = \sum \limits_{k=0}^{n} a_{k} a_{n-k}{{t^{k}}s^{n-k}}, t, s \geq 0$$ . The paper generalizes papers of J. Plewnia and M. Piszczek where f is an exponential and a cosine function respectively.

It is well known that there are no nonzero linear derivations on complex commutative semisimple Banach algebras. In this paper we prove the following extension of this result. Let A be a complex semisimple Banach algebra and let D : A → A be a linear mapping satisfying the relation D(x 2) = 2xD(x) for all x ∈ R. In this case D = 0. Throughout, R will represent an associative ring with center Z(R). A ring R is n-torsion free, where n > 1 is an integer, if nx = 0, x ∈ R implies x = 0. As usual the commutator xy − yx will be denoted by [x, y]. We shall use the commutator identities [xy, z] = [x, z] y + x [y, z] and [x, yz] = [x, y] z + y [x, z] for all x, y, z ∈ R. Recall that a ring R is prime if for a, b ∈ R, aRb = (0) implies that either a = 0 or b = 0, and is semiprime in case aRa = (0) implies that a = 0. An additive mapping D is called a derivation if D(xy) = D(x)y + xD(y) holds for all pairs x, y ∈ R, and is called a Jordan derivation in case D(x 2) = D(x)x + xD(x) is fulfilled for all x ∈ R. Obviously, any derivation is a Jordan derivation. The converse is in general not true. Herstein ([8]) has proved that any Jordan derivation on a 2-torsion free prime ring is a derivation (see also [1]). Cusack ([5]) has generalized Herstein’s result to 2-torsion free semiprime rings (see [2] for an alternative proof). An additive mapping D : R → R is called a left derivation if D(xy) = yD(x) + xD(y) holds for all pairs x, y ∈ R and is called a left Jordan derivation (or Jordan left derivation) in case D(x 2) = 2xD(x) is fulfilled for all x ∈ R. In this paper by a Banach algebra we mean a Banach algebra over the complex field.

The object of this paper is to investigate possibilities of extending the solution F : C → W of certain functional equations to V and of n-Jensen functions f : Cn → W to Vn, where C is a \({\mathbb{Q}}\) -convex subset of V and V, W are \({\mathbb{Q}}\) -vector spaces. Solutions of the considered functional equations defined on V and n-Jensen functions defined on Vn have been utilized in recent papers (e.g. [2], [4]) in order to characterize generalized polynomials P : V →W of degree ≤ n. Therefore these extension theorems may present a point of view for defining generalized polynomials on a \({\mathbb{Q}}\) -convex subset of a \({\mathbb{Q}}\) -vector space.

A real-valued function f defined on an open convex set $$D\subseteq X $$ is called (d, t)-convex if it satisfies $$f(tx + (1 - t)y)\leq tf(x)+(1 - t)f(y) + d(x, y)$$ for all $$x,\, y \in D$$ , where $$d : X {\times} X \rightarrow{\mathbb{R}}$$ is a given function and t $$\in$$ ]0, 1[ is a fixed parameter. The main result of the paper states that if f is locally bounded from above at a point of D and (d, t)-convex then it satisfies the convexity-type inequality (under some assumptions) $$f(sx + (1 - s)y)\leq sf(x) + (1 - s)f(y) +\varphi(s)d(x, y)$$ for all $$x, y \in D$$ and s $$\in$$ [0, 1], where $$\varphi : [0, 1] \rightarrow {\mathbb{R}}$$ is defined as the fixed point of a certain contraction. The main result of this paper offers a generalization of the celebrated Bernstein and Doetsch theorem and the recent results by Nikodem and Ng, Pales and the author.

In 1884, Konigs showed that when \(\varphi\)(z) is an analytic self-map of the unit disk fixing the origin, with 0 1. Under some natural hypotheses, they gave necessary and sufficient conditions for the existence of an analytic solution σ satisfying σ′(0) = I when \(\varphi\)′(0) is diagonalizable. In this paper, the problem when \(\varphi\)′(0) is not diagonalizable is considered. Both \(\varphi\)(z) and σ(z) will be regarded as vectors of purely formal power series, and it will not be assumed that \(\varphi\)(z) is analytic or that the series for \(\varphi\)(z) or σ(z) converge. Nevertheless, because of a process developed by Cowen and MacCluer, if the given \(\varphi\)(z) represents a map of the unit ball into itself of an appropriate form, then the results of this paper can be used to produce solutions of Schroder’s equation that are convergent power series, or (sometimes) to show that no such solution exists. A method of matrix completion is used.

In 1941 S. N. Bernstein proved the following characterization theorem. Let ξ1 and ξ2 be independent random variables. If ξ1 + ξ2 and ξ1 - ξ2 are independent, then ξ1 and ξ2 are Gaussian with equal dispersion. This paper is devoted to a group analogue of the Bernstein theorem. Let X be a locally compact separable abelian group and ξ1, ξ2 be independent random X-valued variables with distributions μ1, μ2. Denote by $$ \user1{\mathbb{T}} $$ and by ∑ a the group of rotations of the circle and the a-adic solenoid, respectively. In this paper we solve the following problem: In the case where either $$ X = \user1{\mathbb{R}} \times \user1{\mathbb{T}} $$ or X = ∑ a we describe all possible distributions μ1 and μ2 under the assumption that ξ1 + ξ2 and ξ1 - ξ2 are independent.

The issue of distributivity of aggregation operations is crucial for many different areas such as utility theory and integration theory. Of special interest are aggregation operations with annihilator. This paper is focused on the problem of distributivity for some so called associative, commutative aggregation operations with annihilator a, known as associative a-CAOA, and uninorms. The full characterization of distributive pairs for T-uninorms, S-uninorms and bi-uninorms is given.

The paper is dealing with determining the aggregation methods for problems of allocation. In the present paper we generalize results obtained by J. Aczel, C. T. Ng and C. Wagner by investigating the cases where the number of individuals assigning values to the decision variables is not necessarily finite, more exactly we admit aggregation of opinions being continuous functions or random variables. We also consider the case where the assigned values need not be numerical, but lie in a vector space.

Two inverse operations are considered in this paper: solving a given equation, i.e. finding its solution space, and constructing an equation from a given set, the elements of which being supposed as its solutions, i.e. finding a suitable representation for a given solution space. These equations and representations need not be unique, they may depend on our requirements, e.g. on smoothness or discreteness. These considerations are explained on differential and functional equations. This paper offers a new interpretation of results taken from the area of difference, differential and functional equations.

In the paper On covariant embeddings of a linear functional equation with respect to an analytic iteration group [3] the authors described the problem of covariant embeddings of a linear functional equation with respect to analytic iteration groups and solved it in the generic cases. However, some cases remained unsolved. In this paper we present the solution for these open cases.

In the paper On the mutual noncompatibility of homogeneous analytic non-power means (Aequationes Math. 45 (1993)) M. E. Kuczma considered analytic solutions of the functional equation $ x + g(y + f(x)) = y + g(x + f(y)) $ on the real line. Solutions in the class of twice differentiable functions are given in the author’s paper Differentiable solutions of a functional equation related to the non-power means (Aequationes Math. 55 (1998)). During the 38th International Symposium on Functional Equations N. Brillouet-Belluot presented the proof that differentiable solutions of this equation have the same form as in the previous cases. We present solutions of the equation in the class of monotonic Jensen convex or Jensen concave functions on the real line. This time we get already some new families of solutions.

In the present paper, we consider the linear functional equation¶¶\( \Phi (\alpha x) - \beta \Phi(x) = F(x) \qquad (x \in E) \),(1)¶where E and G are normed linear spaces over K, K is either \( {\Bbb R} \) or \( {\Bbb C} \)\( \alpha \) and \( \beta \) are given scalars in K, \( F : E \to G \) is a given function and \( \Phi: E \to G \) is the unknown function.¶In an earlier paper, we studied the case \( E = {\Bbb R} \) or \( [0,+\infty) \) or \( (0,+\infty), G = {\Bbb R} \), and we gave there the general solution of (1) and also its continuous and differentiable solutions by using elementary direct methods. The results presented here extend the previous ones to the general case.

On the occasion of his 80th birthday, Ludwig Reich, editor-in-chief of Aequationes Mathematicae from 1996 to 2008, gave an interview conducted via Zoom. This paper presents his answers to several questions regarding his education and academic life.