Let (S, +) be a commutative semigroup, σ : S → S be an endomorphism with σ2 = id and let K be a field of characteristic different from 2. Inspired by the problem of strong alienation of the Jensen equation and the exponential Cauchy equation, we study the solutions f, g : S → K of the functional equation f(x+y)+f(x+σ(y))+g(x+y)=2f(x)+g(x)g(y) for x,y∈S. $$f(x + y) + f(x + \sigma (y)) + g(x + y) = 2f(x) + g(x)g(y)\;\;\;\;{\rm for}\;\;x,y \in S.$$ We also consider an analogous problem for the Jensen and the d’Alembert equations as well as for the d’Alembert and the exponential Cauchy equations.
We consider the Nemytskii operator, i.e., the operator of substitution, defined by (Nφ)(x) := G(x,φ(x)), where G is a given multifunction. It is shown that if N maps a Hölder space Hα into Hβ and N fulfils the Lipschitz condition then G(x,y) = A(x,y) + B(x), where A(x,·) is linear and A(·,y), B ∈ Hβ. Moreover, some conditions are given under which the Nemytskii operator generated by (1) maps Hα into Hβ and is Lipschitzian.
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In this paper we consider the Nemytskii operator (Hf) (t) = h(t, f (t)), generated by a given set-valued function h is considered. It is shown that if H is globally Lipschitzian and maps the space of functions of bounded p-variation (with respect to a weight function α) into the space of set-valued functions of bounded q-variation (with respect to α) ) 1 < q < p, then H is of the form (Hϕ)(t) = A(t)ϕ(t) + B(t). On the other hand, if 1 < p < q, then H is constant. It generalizes many earlier results of this type due to Chistyakov, Matkowski, Merentes-Nikodem, Merentes-Rivas, Smajdor-Smajdor and Zawadzka.
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Assuming that a Nemytskii operator maps a subset of the space of bounded variation functions in the sense of Riesz into another space of the same type, and is uniformly continuous, we prove that the generator of the operator is an affine function.
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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 φ-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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We consider the Nemytskij operator, i. e., the operator of substitution, defined by (N[...]x) := G(x,<[...](x)), where G is a given multifunction. It is shown that N maps C1 (I, C), the space of all continuously differentiable functions on the interval I with values in a cone C in a Banach space, into C1 (I, cc(Z)), the space of all continuously differentiable set-functions on I with compact and convex values in a Banach space Z and N fulfils the Lipschitz condition if and only if the generator G is of the form G(x,y)=A(x,y) + B(x) where A(x, •) is continuous, linear function, A(.,y) and B are continuously differentiable and the function x— > A(x, •) is Lipschitzian.
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We determine the solutions f : S → H of the following functional equation f(xy) + f(σ(y)x) = 2f(x), x,y∈S, and the solutions f1, f2, f3 : M → H of the functional equation f1(xy) + f2(σ(y)x) = 2f3(x), x,y∈M, where S is a semigroup, M is a monoid, H is an abelian group 2-torsion free, and σ is an involutive automorphism.
We consider the Nemytskij operator, defined by (Nφ)(x) ? G(x, φ(x)), where G is a given set-valued function. It is shown that if N maps AC(I, C), the space of all absolutely continuous functions on the interval I ? [0, 1] with values in a cone C in a reflexive Banach space, into AC(I, K), the space of all absolutely continuous set-valued functions on I with values in the set K, consisting of all compact intervals (including degenerate ones) on the real line R, and N is uniformly continuous, then the generator G is of the form G(x, y) = A(x)(y) + B(x), where the function A(x) is additive and uniformly continuous for every x ∈ I and, moreover, the functions x ? A(x)(y) and B are absolutely continuous. Moreover, a condition, under which the Nemytskij operator maps the space AC(I, C) into AC(I, K) and is Lipschitzian, is given.
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