Set-Valued Solutions of a Generalized Popoviciu Functional Equation

Let X be a real vector space and cc(Y) be the collection of all convex and compact subsets of a real Hausdorff topological vector space Y. This paper investigates set-valued solutions of the functional equation \(\begin{aligned}&MF\left( \frac{x+y+z}{m}\right) +F(x)+F(y)+F(z)\\&\qquad =N\left[ F\left( \frac{x+y}{n}\right) +F\left( \frac{x+z}{n}\right) +F\left( \frac{y+z}{n}\right) \right] \end{aligned}\) $$\begin{array}{cc}& MF\left(\frac{x+y+z}{m}\right)+F(x)+F(y)+F(z)\hfill \\ \hfill & =N\left[F,\left(\frac{x+y}{n}\right),+,F,\left(\frac{x+z}{n}\right),+,F,\left(\frac{y+z}{n}\right)\right]\hfill \end{array}$$ for \(F:X\rightarrow cc(Y)\) $F:X\to cc(Y)$ , where M, N, m, n are fixed real numbers with \(N\ge 0\) $N\ge 0$ and \(m, n \ne 0\) $m,n\ne 0$ , as well as some other related functional equations. Representations and numerous properties of these solutions are given. We prove also that every set-valued solution of this functional equation has a selection satisfying this equation.