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Set-valued dynamics related to generalized Euler-Lagrange functional equations. (English) Zbl 1396.39020
Suppose that \(A\) is a convex cone in a real vector space \(X\) with \(0\in A\) and let \(S\) be a set-valued mapping from \(A-A\) (see the paper for the definition of the minus operation) to the family of all nonempty convex and closed subsets of a Banach space \(Y,\) containing the origin of \(Y.\) Using Minkowski addition and scalar multiplication, one can associate to every finite family \(x_{1},\dots,x_{n}\) of \(X\) the operators \(\biguplus_{x_{2}}S(x_{1})=S(x_{1}+x_{2})+S(x_{1}-x_{2})\) and \(\biguplus _{x_{2},\dots,x_{n}}S(x_{1})=\biguplus_{x_{n}}\left( \biguplus_{x_{2},\dots,x_{n-1} }S(x_{1})\right) \) for \(n\geq3.\)
The author provide an interesting characterization of the set-valued mappings \(S\) that verify (for some positive numbers \(a_{1}>1,\dots,a_{n}=1)\) the cubic inclusion \[ \biguplus_{a_{2}x_{2},\dots,a_{n}x_{n}}S(a_{1}x_{1})+2^{n-1}a_{1}\sum_{i=2} ^{n}a_{i}^{2}S(x_{i})\subseteq 2^{n-2}a_{1}\sum_{i=2}^{n}a_{i}^{2}\biguplus_{x_{i} }S(x_{1})+2^{n-1}a_{1}^{3}S(x_{1}), \] for all \(x_{1},\dots,x_{n}\in A,\) and also the boundedness condition \(\sup\left\{ \text{diam}(S(x)):x\in A\right\} <\infty.\) Precisely, the existence and uniqueness of a cubic mapping \(C:A-A\rightarrow Y\) such that \(C(x)\in S(x)\) for all \(x\in A\) is proved. A quartic analogue of this result is also included.

MSC:
39B52 Functional equations for functions with more general domains and/or ranges
39B62 Functional inequalities, including subadditivity, convexity, etc.
54C60 Set-valued maps in general topology
47H04 Set-valued operators
52A07 Convex sets in topological vector spaces (aspects of convex geometry)
39B82 Stability, separation, extension, and related topics for functional equations
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