Orthogonalities and functional equations

Abstract

Abstract. In this survey we show how various notions of orthogonality appear in the theory of functional equations. After introducing some orthogonality relations, we give examples of functional equations postulated for orthogonal vectors only. We show their solutions as well as some applications. Then we discuss the problem of stability of some of them considering various aspects of the problem. In the sequel, we mention the orthogonality equation and the problem of preserving orthogonality. Last, but not least, in addition to presenting results, we state some open problems concerning these topics. Taking into account the big amount of results concerning functional equations postulated for orthogonal vectors which have appeared in the literature during the last decades, we restrict ourselves to the most classical equations.Mathematics Subject Classification. 39B52, 39B82, 46B20, 47B49, 15A86.

On The Continuous Dependence Of Solutions To Orthogonal Additivity Problem On Given Functions

“…. , N } which proves that a • c belongs to set (4). To see that a • c = 0 consider a β ∈ R with a(β) = 0.…”

“…The reader interested in further problems connected with orthogonal additivity is referred to the survey paper [4] by Justyna Sikorska.…”

On the set of orthogonally additive functions with orthogonally additive second iterate

“…The reader interested in further problems connected with orthogonal additivity is referred to the survey paper [7] by Justyna Sikorska.…”

Set-valued Orthogonal Additivity

“…With help of it we can give, e.g., several characterizations of inner product spaces among normed spaces as well as of Hilbert spaces among Banach spaces (see Rätz [25] or Sikorska [29] for more reference items). Equation (1) has its applications in physics, in the theory of ideal gas (see Aczél & Dhombres [1], Truesdell & Muncaster [32], Arkeryd & Cercignani [2]).…”