Functional equations motivated by the Lagrange's identity

Abstract

We solve two functional equations motivated by the following Lagrange's identity: i=1 nn = ai bi - (aibj - aj bi) , i=1 i=1 1^i<j^n which is valid for every n 6 N and each a1,... ,an,b1,bn from a commutative ring. The classical Lagrange's identity states that for every n G N = {1,2,...} and each ai, bi from a commutative ring R, where i = 1, .. . , n, we have: n 2 n n aibi = ai2 bi2 - i=1 i=1 i=1 (1) (aibj - ajbi)2 1^i<j^n or, if division by 2 is uniquely performable in R: n 2 n n n n (2) ($2 aibi) = (52a*2) (52b2)- 2 £ 52(aibj- abtfi= 1 i=1 i=1 i=1 j=1 These identities motivate the following two functional equations: n n n (3) f aibi = f(ai) f(bi) - f(aibj - ajbi) i=1 i=1 i=1 1^i<j^n and n n n n n (4) f (52 aibi) = (52 f (ai0(52 f (bi^ - 2^52 f (aibj - ajbi) ’ i=1 i=1 i=1 i=1 j=1 which can be discussed for an unknown mapping f acting between fields, rings or algebras. Dealing with equation (4) we need to assume additionally that the target space of f contains unit element 1 and 1. Observe also that equations (3) and (4) need not to be equivalent, unless f is even and f(0) = 0. It is worth to note that another functional equation related to the Lagrange's identity is already known. The following version of the Euler- Lagrange quadratic functional equation: n n n (5) Q aixi + Q(aixj - aj xi) = ai2 Q(xi) i=1 1^i<j^n i=1 i=1 and its several modifications jointly with the corresponding stability questions have been studied by J. M. Rassias [9], [10], [11], [12], H.-M. Kim, J. M. Rassias and Y.-S. Cho [5], H.-M. Kim and J. M. Rassias [6], M. J. Rassias and J. M. Rassias [13], A. Pietrzyk [8], among others. The main difference between (3) or (4) and (5) seems to lie in the nonlinearity of (3) and (4). Indeed, if Q1 and Q2 solve (5) then for each scalars Ai, A2 the map A1Q1 + A2Q2 provides a solution of (5). It is clear that (3) and (4) do not possess an analogical property and therefore one may expect a different behavior of these equations. The purpose of the present paper is to determine general solutions of (3) and (4). Therefore we provide an answer to the question whether, or to what extent, the Lagrange's identity characterizes the mapping x x2 on rings or algebras. Clearly, if the Lagrange's identity holds for a given n 2 then a simple substitution an = bn = 0 proves the validity of this identity for n - 1 (on a ring or on an arbitrary structure with adequate operations and the zero element). The converse implication is not straightforward and therefore one may ask if the Lagrange's identity assumed for n and then assumed for n-1 are equivalent (in a sense that corresponding functional equations have the same solutions). To settle the question we will solve corresponding functional equations (3) or (4), respectively assuming its validity for a fixed n only. Observe that if n = 1 then (3) reduces to the multiplicative Cauchy equation: (6) f(a1b1) = f(a1)f(b1) whereas (4) reduces to (7) f(a1M = f(a1)f(M - 2f(0)- The description of solutions of (6) is well known in the literature, see e.g. M. Kuczma [7, pp. 343-350]. Under some mild assumptions equation (7) can be easily solved by a reduction to (6).