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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]

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]