### The Uniqueness of Solutions of a System of Functional Equations in Some Classes of Functions.

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Let ϕ be an arbitrary bijection of ${\mathbb{R}}_{+}$. We prove that if the two-place function ${\varphi}^{-1}[\varphi \left(s\right)+\varphi \left(t\right)]$ is subadditive in ${\mathbb{R}}_{+}^{2}$ then $\varphi $ must be a convex homeomorphism of ${\mathbb{R}}_{+}$. This is a partial converse of Mulholland’s inequality. Some new properties of subadditive bijections of ${\mathbb{R}}_{+}$ are also given. We apply the above results to obtain several converses of Minkowski’s inequality.

Equation [1] f(x+y) + f (f(x)+f(y)) = f (f(x+f(y)) + f(f(x)+y)) has been proposed by C. Alsina in the class of continuous and decreasing involutions of (0,+∞). General solution of [1] is not known yet. Nevertheless we give solutions of the following equations which may be derived from [1]: [2] f(x+1) + f (f(x)+1) = 1, [3] f(2x) + f(2f(x)) = f(2f(x + f(x))). Equation [3] leads to a Cauchy functional equation: [4]...

Assume that the generator of a Nemytskii composition operator is a function of three variables: the first two real and third in a closed convex subset of a normed space, with values in a real Banach space. We prove that if this operator maps a certain subset of the Banach space of functions of two real variables of bounded Wiener $\varphi $-variation into another Banach space of a similar type, and is uniformly continuous, then the one-sided regularizations of the generator are affine in the third variable....

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