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Note on Jensen and Pexider functional equations. (English) Zbl 0938.39026

The Jensen functional equation \[ 2f\left({x+y \over 2}\right) =f(x)+f(y) \tag{1} \] is solved in the case where \(f:M\to S\), \(M\) is an Abelian semigroup with the division by \(2,S\) is an abstract convex cone satisfying the cancellation law.
The result when applied to a set-valued version of (1) (where \(f\) takes values from the set \(ccl(X)\) of all nonempty, bounded, closed and convex subsets of a real normed space \(X\) and the operation + means the closure of the algebraic sum of the underlying sets) yields a generalization, in particular, of results by Z. Fifer [Rev. Roum. Math. Pures Appl. 31, 297-302 (1986; Zbl 0615.39006)] and K. Nikodem [Zeszyty Nauk. Politech. Łódz., Mat. 559 (1989)]. It is also proved that if the functions \(f,g,h: M\to S\) satisfy the Pexider equation \(f(x+y)= g(x)+h(y)\), then they are solutions to (1).

MSC:

39B52 Functional equations for functions with more general domains and/or ranges
47H30 Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.)
47H04 Set-valued operators
54C60 Set-valued maps in general topology

Citations:

Zbl 0615.39006
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