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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 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]

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 Šemrl (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. [ABSTRACT FROM AUTHOR]

Generalized Berwald manifolds are Finsler manifolds admitting linear connections such that the parallel transports preserve the Finslerian length of tangent vectors (compatibility condition). It is known (Vincze in J AMAPN 21:199–204, 2005) that such a linear connection must be metrical with respect to the averaged Riemannian metric given by integration of the Riemann-Finsler metric on the indicatrix hypersurfaces. Therefore the linear connection (preserving the Finslerian length of tangent vectors) is uniquely determined by its torsion. If the torsion is zero then we have a classical Berwald manifold. Otherwise, the torsion is some strange data we need to express in terms of the intrinsic quantities of the Finsler manifold. The paper presents the idea of the extremal compatible linear connection of a generalized Berwald manifold by minimizing the pointwise length of its torsion tensor. It is uniquely determined because the number of the Lagrange multipliers is equal to the number of the equations for the compatibility of the linear connection with the Finslerian metric. Using the reference element method, the extremal compatible linear connection can be expressed in terms of the canonical data as well. It is an intrinsic algorithm to check the existence of compatible linear connections on a Finsler manifold because it is equivalent to the existence of the extremal compatible linear connection. [ABSTRACT FROM AUTHOR]

In this paper, we consider the condition ∑ i = 0 n + 1 φ i (r i x + q i y) ∈ Z for real valued functions defined on a linear space V. We derive necessary and sufficient conditions for functions satisfying this condition to be decent in the following sense: there exist functions f i : V → R , g i : V → Z such that φ i = f i + g i , (i = 0 , ⋯ , n + 1) and ∑ i = 0 n + 1 f i (r i x + q i y) = 0 for all x , y ∈ V . [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 this paper, we show that the hyperstability of the general linear equation recently proved by Piszczek (Aequationes Math 88:163–168, 2014) is a direct consequence of that of the Cauchy equation proved earlier by Brzdȩk (Acta Math Hung 141:58–67, 2013). [ABSTRACT FROM AUTHOR]

The aim of this paper is to give an answer to a question posed by Alsina, Sikorska and Tomás. Namely, we show that, under suitable assumptions, a function f : X → Y from a normed space X into a normed space Y, satisfying the functional equation f y - ρ + ′ (x , y) ‖ x ‖ 2 x = f (y) - ρ + ′ (f (x) , f (y)) ‖ f (x) ‖ 2 f (x) , x , y ∈ X has to be a linear similarity (scalar multiple of a linear isometry). [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]

In this paper we study the functional equation where a, b, c are fixed complex numbers and $${f \colon \mathbb{C} \to \mathbb{C}}$$ is the unknown function. We show, that if there is i such that $${b_i / c_i \neq b_j /c_j}$$ holds for any $${1 \leq j \leq n,\ j \neq i}$$ , the functional equation has a nonconstant solution if and only if there are field automorphisms $${\phi_1, \ldots, \phi_k}$$ of $${\mathbb{C}}$$ such that $${\phi_1 \cdots \phi_k}$$ is a solution of the equation. [ABSTRACT FROM AUTHOR]