Set-Valued Additive Functional Equations

Yıl 2019, Cilt: 2 Sayı: 2, 89 - 97, 01.06.2019

Choonkil Park Sungsik Yun Jung Rye Lee Dong Yun Shın

https://doi.org/10.33205/cma.528182

Cited By: 1

Öz

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.

Anahtar Kelimeler

Hyers-Ulam stability, set-valued additive functional equation, fixed point

Kaynakça

• [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.
• [15] M.EshaghiGordjiandM.B.Savadkouhi:Stabilityofamixedtypecubic-quarticfunctionalequationinnon-Archimedean spaces. Appl. Math. Letters 23 (2010), 1198–1202.
• [16] P.Gavruta:AgeneralizationoftheHyers-Ulam-Rassiasstabilityofapproximatelyadditivemappings.J.Math.Anal.Appl. 184 (1994), 431–436.
• [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.