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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 main result of the present paper is about the solutions of the functional equation F ( x + y 2) + f 1 (x) + f 2 (y) = G (g 1 (x) + g 2 (y)) , x , y ∈ I , derived originally, in a natural way, from the invariance problem of generalized weighted quasi-arithmetic means, where F , f 1 , f 2 , g 1 , g 2 : I → R and G : g 1 (I) + g 2 (I) → R are the unknown functions assumed to be continuously differentiable with 0 ∉ g 1 ′ (I) ∪ g 2 ′ (I) , and the set I stands for a nonempty open subinterval of R . In addition to these, we will also touch upon solutions not necessarily regular. More precisely, we are going to solve the above equation assuming first that F is affine on I and g 1 and g 2 are continuous functions strictly monotone in the same sense, and secondly that g 1 and g 2 are invertible affine functions with a common additive part. [ABSTRACT FROM AUTHOR]

In this paper it is proved that, for a function f : X → E mapping from a normed linear space X into an inner product space E, the functional inequality ‖ 2 f (x) + 2 f (y) - f (x - y) ‖ ⩽ ‖ f (x + y) ‖ , ‖ x ‖ + ‖ y ‖ ⩾ d for some d > 0 , implies f is quadratic. Other types of functional inequalities related to the quadratic functional equation have also been investigated. Besides we establish the Hyers–Ulam stability on restricted domains, and we improve the bounds and thus the stability results obtained in Jung (J Math Anal Appl 222:126–137, 1998) and Rassias (J Math Anal Appl 276: 747–762, 2002). Finally we apply our recent results to the asymptotic behavior of quadratic functional equations of different types. [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 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]

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 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]

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]