Abstract
The aim of the paper is to generalize results by Sikorska on some functional equations for set-valued functions. In the paper, a tool is described for solving a generalized type of an integral-functional equation for a set-valued function F:X→cc(Y), where X is a real vector space and Y is a locally convex real linear metric space with an invariant metric. Most general results are described in the case of a compact topological group G equipped with the right-invariant Haar measure acting on X. Further results are found if the group G is finite or Y is Asplund space. The main results are applied to an example where X=R2 and Y=Rn, n∈N, and G is the unitary group U(1).
Highlights
Jordan and von Neumann in 1935 [1] gave characterizations of inner product spaces among normed vector spaces ( X, k · k) which lead to the functional equation in the form f ( x + y) + f ( x − y) = 2 f ( x ) + 2 f (y), x, y ∈ X
We proved in Theorem 2 that every function F, solution of (6) contains a function F, and a solution of (6), which can be expressed as the sum of a set-valued function A and a single-valued function h
We have attempted to generalize the study of the following functional equation: β+1 β−1
Summary
Jordan and von Neumann in 1935 [1] gave characterizations of inner product spaces among normed vector spaces ( X, k · k) which lead to the functional equation in the form f ( x + y) + f ( x − y) = 2 f ( x ) + 2 f (y), x, y ∈ X. Fréchet in 1935 [2] obtained a characterization of normed spaces with inner product in connection with the following functional equation:. Sikorska in [6] considers analogies of these equations for unknown set-valued functions. It has been shown in [6] that these equations can be transformed into the form which can be viewed as special cases of the equation
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