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Set-valued solutions for an equation of Jensen type. (English) Zbl 1075.39504
From the introduction: Let \(X\) be a vector space. We denote by \({\mathcal P}_{0}\) the collection of all non-empty subsets of \(X\). Let \(p\) be a real number, \(0<p<1\), \(X\), \(y\) be real vector spaces and \(K\) a convex cone in \(X\). In this paper, we are looking for solutions \(F\colon K \to {\mathcal P}_0(Y)\) of the equation \[ F((1-p)x+py)=(1-p)F(x)+pF(x).\tag{4} \] For \(p=1/2\), the equation (4) becomes the Jensen equation. It is well-known that real values functions that satisfy Jensen equation are of the form \(F=a+k\), where \(a\) is an additive function and \(k\) is a real number [M. Kuczma, “An introduction to the theory of functional equations and inequalities. Cauchy’s equation and Jensen’s inequality” (Prace Naukowe Uniwersytetu Slaskiego w Katowicach 489, Uniwersytet Slaski, Warszawa-Kraków-Katowice) (1985; Zbl 0555.39004)].
Z. Fifer [Rev. Roum. Math. Pures Appl. 31, 297-302 (1986; Zbl 0615.39006)] proved that an analogous representation holds for set-valued functions when \(K=[0,+\infty)\) and \(Y\) is a real Banach space. K. Nikodem [Rad. Mat. 3, 23-33 (1987; Zbl 0628.39013); “K-convex and K-concave set valued function” (Zeszyty Naukowe, No. 599, Lódz) (1989)] gave a characterization of the solution of Jensen equation for set-valued functions with compact convex values in a real topological vector space. In this paper, we prove that an analogous characterization holds for the equation (4).

39B52 Functional equations for functions with more general domains and/or ranges
47H04 Set-valued operators
54C60 Set-valued maps in general topology