# zbMATH — the first resource for mathematics

Stability of some set-valued functional equations. (English) Zbl 1236.39034
The authors consider the stability of some set-valued functional equations. Namely, they deal with mappings $$F$$ defined on a cone and with closed and bounded subsets of a real vector space as values. It is proved that if $$F$$ satisfies one of the inclusions: $F(x)+F(y)\subseteq 2F\left(\frac{x+y}{2}\right);$ $F(x+y)+F(x-y)\subseteq 2F(x)+2F(y);$ $F(2x+y)+F(2x-y)\subseteq 2F(x+y)+2F(x-y)+12F(x);$ $F(2x+y)+F(2x-y)+6F(y)\subseteq 4F(x+y)+4F(x-y)+24F(x),$ then it admits a unique additive, quadratic, cubic and quartic selection, respectively.

##### MSC:
 39B82 Stability, separation, extension, and related topics for functional equations
##### Keywords:
set-valued functional equation; cone; Hyers-Ulam stability
Full Text:
##### References:
 [1] Aumann, R.J., Integrals of set-valued functions, J. math. anal. appl., 12, 1-12, (1965) · Zbl 0163.06301 [2] Debreu, G., Integration of correspondences, (), 351-372, Part I · Zbl 0211.52803 [3] Arrow, K.J.; Debreu, G., Existence of an equilibrium for a competitive economy, Econometrica, 22, 265-290, (1954) · Zbl 0055.38007 [4] McKenzie, L.W., On the existence of general equilibrium for a competitive market, Econometrica, 27, 54-71, (1959) · Zbl 0095.34302 [5] Hindenbrand, W., Core and equilibria of a large economy, (1974), Princeton Univ. Press Princeton [6] Aubin, J.P.; Frankowska, H., Set-valued analysis, (1990), Birkhäuser Boston [7] Castaing, C.; Valadier, M., () [8] Klein, E.; Thompson, A., Theory of correspondence, (1984), Wiley New York [9] Hess, C., () [10] Rassias, Th.M., On the stability of the linear mapping in Banach spaces, Proc. amer. math. soc., 72, 297-300, (1978) · Zbl 0398.47040 [11] Ulam, S.M., Problems in modern mathematics, chapter VI, (1940), Wiley New York [12] Hyers, D.H., On the stability of the linear functional equation, Proc. natl. acad. sci. USA, 27, 222-224, (1941) · Zbl 0061.26403 [13] Gaˇvruta, P., A generalization of the hyers – ulam – rassias stability of approximately additive mappings, J. math. anal. appl., 184, 431-436, (1994) · Zbl 0818.46043 [14] Skof, F., Proprietà locali e approssimazione di operatori, Rend. sem. mat. fis. milano, 53, 113-129, (1983) [15] Cholewa, P.W., Remarks on the stability of functional equations, Aequationes math., 27, 76-86, (1984) · Zbl 0549.39006 [16] Czerwik, S., On the stability of the quadratic mapping in normed spaces, Abh. math. semin. univ. hamb., 62, 59-64, (1992) · Zbl 0779.39003 [17] Jun, K.; Kim, H., The generalized hyers – ulam – rassias stability of a cubic functional equation, J. math. anal. appl., 274, 867-878, (2002) · Zbl 1021.39014 [18] Lee, S.; Im, S.; Hwang, I., Quartic functional equations, J. math. anal. appl., 307, 387-394, (2005) · Zbl 1072.39024 [19] Nikodem, K., () [20] Lu, G.; Park, C., Hyers – ulam stability of additive set-valued functional equations, Appl. math. lett., 24, 1312-1316, (2011) · Zbl 1220.39030 [21] Nikodem, K., On quadratic set-valued functions, Publ. math. debrecen, 30, 297-301, (1984) · Zbl 0537.39002 [22] Nikodem, K., On jensen’s functional equation for set-valued functions, Radovi mat., 3, 23-33, (1987) · Zbl 0628.39013 [23] Nikodem, K., Set-valued solutions of the Pexider functional equation, Funkcia. ekvac., 31, 227-231, (1988) · Zbl 0698.39007 [24] Piao, Y.J., The existence and uniqueness of additive selection for $$(\alpha, \beta)$$-$$(\beta, \alpha)$$ type subadditive set-valued maps, J. northeast normal univ., 41, 38-40, (2009) [25] Popa, D., Additive selections of $$(\alpha, \beta)$$-subadditive set-valued maps, Glas. mat. ser. III, 36, 56, 11-16, (2001) · Zbl 1039.28013
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.