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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.

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
Full Text: DOI
[1] Arrow, KJ; Debereu, G, Existence of an equilibrium for a competitive economy, Econometrica, 22, 265-290, (1954) · Zbl 0055.38007
[2] Aubin, J.P., Frankowska, H.: Set-Valued Analysis. Birkhäuser, Boston (1990) · Zbl 0713.49021
[3] Aumann, RJ, Integrals of set-valued functions, J. Math. Anal. Appl., 12, 1-12, (1965) · Zbl 0163.06301
[4] Brzdȩk, J; Popa, D; Xu, B, Selections of set-valued maps satisfying a linear inclusion in a single variable, Nonlinear Anal., 74, 324-330, (2011) · Zbl 1205.39025
[5] Debreu, G.: Integration of correspondences. In: Proceedings of Fifth Berkeley Symposium on Mathematical Statistics and Probability, vol. II, part. I, 1966, pp. 351-372 · Zbl 0163.06301
[6] Hess, C.: Set-valued integration and set-valued probability theory: an overview. In: Handbook of Measure Theory, vol. I-II, North-Holland, Amsterdam (2002) · Zbl 1022.60011
[7] Jun, KW; Kim, HM, On the stability of Euler-Lagrange type cubic mappings in quasi-Banach spaces, J. Math. Anal. Appl., 332, 1335-1350, (2007) · Zbl 1117.39017
[8] Jun, KW; Kim, HM; Chang, IS, On the Hyers-Ulam stability of an Euler-Lagrange type cubic functional equation, J. Comput. Anal. Appl., 7, 21-33, (2005) · Zbl 1087.39029
[9] Kang, D.S.: On the stability of generalized quartic mappings in quasi-\(β \)-normed spaces. J. Inequal. Appl. (2010) (Art. ID 198098) · Zbl 1187.39038
[10] Khodaei, H, On the stability of additive, quadratic, cubic and quartic set-valued functional equations, Results Math., 68, 1-10, (2015) · Zbl 1330.39029
[11] Kim, HM, On the stability problem for a mixed type of quartic and quadratic functional equation, J. Math. Anal. Appl., 324, 358-372, (2006) · Zbl 1106.39027
[12] Lee, Y.S., Chung, S.Y.: Stability of quartic functional equation in the spaces of generalized functions. Adv. Differ. Equ. (2009) (Art. ID 838347) · Zbl 1087.39029
[13] Lu, G; Park, C, Hyers-Ulam stability of additive set-valued functional equations, Appl. Math. Lett., 24, 1312-1316, (2011) · Zbl 1220.39030
[14] Nikodem, K.: \(K\)-Convex and \(K\)-Concave Set-Valued Functions. In: Zeszyty Naukowe Nr., vol. 559, Lodz (1989) · Zbl 0163.06301
[15] Nikodem, K, On quadratic set-valued functions, Publ. Math. Debr., 30, 297-301, (1984) · Zbl 0537.39002
[16] Nikodem, K; Popa, D, On selections of general linear inclusions, Publ. Math. Debr., 75, 239-249, (2009) · Zbl 1212.39041
[17] Nikodem, K; Popa, D, On single-valuedness of set-valued maps satisfying linear inclusions, Banach J. Math. Anal., 3, 44-51, (2009) · Zbl 1163.26353
[18] Park, C; O’Regan, D; Saadati, R, Stabiltiy of some set-valued functional equations, Appl. Math. Lett., 24, 1910-1914, (2011) · Zbl 1236.39034
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