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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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