Abstract
LetXbe a real normed space,Ya real Banach space, and letCn(X, Y) denote the space ofn-times continuously differentiable functionsf:X→Y. We prove that the classCnhas the double difference property, that is if Cf(x, y)≔f(x+y)−f(x)−f(y) belongs to the spaceCn(X×X, Y) then there exists an additive functionA:X→Ysuch thatf−A∈Cn(X, Y). Similar result is also obtained for the Jensen equation. As an application we show that the Cauchy and Jensen equations are stable with respect to large class of seminorms defined by means of derivatives.
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