In this paper, we introduce set-valued additive functional equations and prove the Hyers-Ulam stability of the set-valued additive functional equations by using the fixed point method.
[1] T. Aoki: On the stability of the linear transformation in Banach spaces. J. Math. Soc. Japan 2 (1950), 64–66.
[2] K. J. Arrow and G. Debreu: Existence of an equilibrium for a competitive economy. Econometrica 22 (1954), 265–290.
[3] J. P. Aubin and H. Frankowska: Set-Valued Analysis. Birkhäuser, Boston, 1990.
[4] R. J. Aumann: Integrals of set-valued functions. J. Math. Anal. Appl. 12 (1965), 1–12.
[5] T. Cardinali, K. Nikodem and F. Papalini: Some results on stability and characterization of K-convexity of set-valued
functions. Ann. Polon. Math. 58 (1993), 185–192.
[6] T. Cascales and J. Rodrigeuz: Birkhoff integral for multi-valued functions. J. Math. Anal. Appl. 297 (2004), 540–560.
[7] C. Castaing and M. Valadier: Convex Analysis and Measurable Multifunctions. Lect. Notes in Math. 580, Springer,
Berlin, 1977.
[8] L. Cadariu and V. Radu: Fixed points and the stability of Jensen’s functional equation. J. Inequal. Pure Appl. Math. 4,
no. 1, Art. ID 4 (2003).
[9] L. Cadariu and V. Radu: On the stability of the Cauchy functional equation: a fixed point approach. Grazer Math. Ber.
346 (2004), 43–52.
[10] L. Cadariu and V. Radu: Fixed point methods for the generalized stability of functional equations in a single variable.
Fixed Point Theory Appl. 2008, Art. ID 749392 (2008).
[11] G. Debreu: Integration of correspondences. Proceedings of Fifth Berkeley Symposium on Mathematical Statistics and Probability, Vol. II, Part I (1966), 351–372.
[12] J. Diaz and B. Margolis: A fixed point theorem of the alternative for contractions on a generalized complete metric space. Bull. Am. Math. Soc. 74 (1968), 305–309.
[13] Iz. EL-Fassi: New stability results for the radical sextic functional equation related to quadratic mappings in (2, β)-Banach spaces. J. Fixed Point Theory Appl. 20 (2018), no. 4, Art. 138, 17 pp.
[14] M. Eshaghi Gordji, C. Park and M. B. Savadkouhi: The stability of a quartic type functional equation with the fixed point alternative. Fixed Point Theory 11 (2010), 265–272.
[17] C. Hess: Set-valued integration and set-valued probability theory: an overview, in Handbook of Measure Theory. Vols. I, II, North-Holland, Amsterdam, 2002.
[18] W. Hindenbrand: Core and Equilibria of a Large Economy. Princeton Univ. Press, Princeton, 1974.
[19] D. H. Hyers: On the stability of the linear functional equation. Proc. Nat. Acad. Sci. USA 27 (1941), 222–224.
[20] G. IsacandTh.M.Rassias:OntheHyers-Ulamstabilityofψ-additivemappings.J.Approx.Theory72(1993),131–137.
[21] G. Isac and Th. M. Rassias: Stability of ψ-additive mappings: Applications to nonlinear analysis. Int. J. Math. Math. Sci.
19 (1996), 219–228.
[22] E. Klein and A. Thompson: Theory of Correspondence. Wiley, New York, 1984.
[23] K. Lee: Stability of functional equations related to set-valued functions (preprint).
[24] L. W. McKenzie: On the existence of general equilibrium for a competitive market. Econometrica 27 (1959), 54–71.
[25] D. Mihet and V. Radu: On the stability of the additive Cauchy functional equation in random normed spaces. J. Math.
Anal. Appl. 343 (2008), 567–572.
[26] M. Mirzavaziri and M. S. Moslehian: A fixed point approach to stability of a quadratic equation. Bull. Braz. Math. Soc.
37 (2006), 361–376.
[27] K. Nikodem: On quadratic set-valued functions. Publ. Math. Debrecen 30 (1984), 297–301.
[28] K. Nikodem: On Jensen’s functional equation for set-valued functions. Radovi Mat. 3 (1987), 23–33.
[29] K. Nikodem: Set-valued solutions of the Pexider functional equation. Funkcialaj Ekvacioj 31 (1988), 227–231.
[30] K. Nikodem: K-Convex and K-Concave Set-Valued Functions. Zeszyty Naukowe Nr. 559, Lodz, 1989.
[31] Y. J. Piao: The existence and uniqueness of additive selection for (α, β)-(β, α) type subadditive set-valued maps. J. North-
east Normal University 41 (2009), 38–40.
[32] S. Pinelas, V. Govindan and K. Tamilvanan: Stability of a quartic functional equation. J. Fixed Point Theory Appl. 20
(2018), no. 4, Art. 148, 10 pp.
[33] D. Popa: Additive selections of (α, β)-subadditive set-valued maps. Glas. Mat. Ser. III, 36 (56) (2001), 11–16.
[34] V. Radu: The fixed point alternative and the stability of functional equations. Fixed Point Theory 4 (2003), 91–96.
[35] Th. M. Rassias: On the stability of the linear mapping in Banach spaces. Proc. Am. Math. Soc. 72 (1978), 297–300.
[36] L. Székelyhidi: Superstability of functional equations related to spherical functions. Open Math. 15 (2017), 427–432.
[37] S. M. Ulam: Problems in Modern Mathematics. Chapter VI, Science ed., Wiley, New York, 1940.
[1] T. Aoki: On the stability of the linear transformation in Banach spaces. J. Math. Soc. Japan 2 (1950), 64–66.
[2] K. J. Arrow and G. Debreu: Existence of an equilibrium for a competitive economy. Econometrica 22 (1954), 265–290.
[3] J. P. Aubin and H. Frankowska: Set-Valued Analysis. Birkhäuser, Boston, 1990.
[4] R. J. Aumann: Integrals of set-valued functions. J. Math. Anal. Appl. 12 (1965), 1–12.
[5] T. Cardinali, K. Nikodem and F. Papalini: Some results on stability and characterization of K-convexity of set-valued
functions. Ann. Polon. Math. 58 (1993), 185–192.
[6] T. Cascales and J. Rodrigeuz: Birkhoff integral for multi-valued functions. J. Math. Anal. Appl. 297 (2004), 540–560.
[7] C. Castaing and M. Valadier: Convex Analysis and Measurable Multifunctions. Lect. Notes in Math. 580, Springer,
Berlin, 1977.
[8] L. Cadariu and V. Radu: Fixed points and the stability of Jensen’s functional equation. J. Inequal. Pure Appl. Math. 4,
no. 1, Art. ID 4 (2003).
[9] L. Cadariu and V. Radu: On the stability of the Cauchy functional equation: a fixed point approach. Grazer Math. Ber.
346 (2004), 43–52.
[10] L. Cadariu and V. Radu: Fixed point methods for the generalized stability of functional equations in a single variable.
Fixed Point Theory Appl. 2008, Art. ID 749392 (2008).
[11] G. Debreu: Integration of correspondences. Proceedings of Fifth Berkeley Symposium on Mathematical Statistics and Probability, Vol. II, Part I (1966), 351–372.
[12] J. Diaz and B. Margolis: A fixed point theorem of the alternative for contractions on a generalized complete metric space. Bull. Am. Math. Soc. 74 (1968), 305–309.
[13] Iz. EL-Fassi: New stability results for the radical sextic functional equation related to quadratic mappings in (2, β)-Banach spaces. J. Fixed Point Theory Appl. 20 (2018), no. 4, Art. 138, 17 pp.
[14] M. Eshaghi Gordji, C. Park and M. B. Savadkouhi: The stability of a quartic type functional equation with the fixed point alternative. Fixed Point Theory 11 (2010), 265–272.
[17] C. Hess: Set-valued integration and set-valued probability theory: an overview, in Handbook of Measure Theory. Vols. I, II, North-Holland, Amsterdam, 2002.
[18] W. Hindenbrand: Core and Equilibria of a Large Economy. Princeton Univ. Press, Princeton, 1974.
[19] D. H. Hyers: On the stability of the linear functional equation. Proc. Nat. Acad. Sci. USA 27 (1941), 222–224.
[20] G. IsacandTh.M.Rassias:OntheHyers-Ulamstabilityofψ-additivemappings.J.Approx.Theory72(1993),131–137.
[21] G. Isac and Th. M. Rassias: Stability of ψ-additive mappings: Applications to nonlinear analysis. Int. J. Math. Math. Sci.
19 (1996), 219–228.
[22] E. Klein and A. Thompson: Theory of Correspondence. Wiley, New York, 1984.
[23] K. Lee: Stability of functional equations related to set-valued functions (preprint).
[24] L. W. McKenzie: On the existence of general equilibrium for a competitive market. Econometrica 27 (1959), 54–71.
[25] D. Mihet and V. Radu: On the stability of the additive Cauchy functional equation in random normed spaces. J. Math.
Anal. Appl. 343 (2008), 567–572.
[26] M. Mirzavaziri and M. S. Moslehian: A fixed point approach to stability of a quadratic equation. Bull. Braz. Math. Soc.
37 (2006), 361–376.
[27] K. Nikodem: On quadratic set-valued functions. Publ. Math. Debrecen 30 (1984), 297–301.
[28] K. Nikodem: On Jensen’s functional equation for set-valued functions. Radovi Mat. 3 (1987), 23–33.
[29] K. Nikodem: Set-valued solutions of the Pexider functional equation. Funkcialaj Ekvacioj 31 (1988), 227–231.
[30] K. Nikodem: K-Convex and K-Concave Set-Valued Functions. Zeszyty Naukowe Nr. 559, Lodz, 1989.
[31] Y. J. Piao: The existence and uniqueness of additive selection for (α, β)-(β, α) type subadditive set-valued maps. J. North-
east Normal University 41 (2009), 38–40.
[32] S. Pinelas, V. Govindan and K. Tamilvanan: Stability of a quartic functional equation. J. Fixed Point Theory Appl. 20
(2018), no. 4, Art. 148, 10 pp.
[33] D. Popa: Additive selections of (α, β)-subadditive set-valued maps. Glas. Mat. Ser. III, 36 (56) (2001), 11–16.
[34] V. Radu: The fixed point alternative and the stability of functional equations. Fixed Point Theory 4 (2003), 91–96.
[35] Th. M. Rassias: On the stability of the linear mapping in Banach spaces. Proc. Am. Math. Soc. 72 (1978), 297–300.
[36] L. Székelyhidi: Superstability of functional equations related to spherical functions. Open Math. 15 (2017), 427–432.
[37] S. M. Ulam: Problems in Modern Mathematics. Chapter VI, Science ed., Wiley, New York, 1940.
Park, C., Yun, S., Lee, J. R., Shın, D. Y. (2019). Set-Valued Additive Functional Equations. Constructive Mathematical Analysis, 2(2), 89-97. https://doi.org/10.33205/cma.528182
AMA
Park C, Yun S, Lee JR, Shın DY. Set-Valued Additive Functional Equations. CMA. June 2019;2(2):89-97. doi:10.33205/cma.528182