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Set-Valued Additive Functional Equations

Year 2019, , 89 - 97, 01.06.2019
https://doi.org/10.33205/cma.528182

Abstract

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.

References

  • [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.
Year 2019, , 89 - 97, 01.06.2019
https://doi.org/10.33205/cma.528182

Abstract

References

  • [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.
There are 37 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Choonkil Park 0000-0001-6329-8228

Sungsik Yun This is me

Jung Rye Lee

Dong Yun Shın This is me

Publication Date June 1, 2019
Published in Issue Year 2019

Cite

APA 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
Chicago Park, Choonkil, Sungsik Yun, Jung Rye Lee, and Dong Yun Shın. “Set-Valued Additive Functional Equations”. Constructive Mathematical Analysis 2, no. 2 (June 2019): 89-97. https://doi.org/10.33205/cma.528182.
EndNote Park C, Yun S, Lee JR, Shın DY (June 1, 2019) Set-Valued Additive Functional Equations. Constructive Mathematical Analysis 2 2 89–97.
IEEE C. Park, S. Yun, J. R. Lee, and D. Y. Shın, “Set-Valued Additive Functional Equations”, CMA, vol. 2, no. 2, pp. 89–97, 2019, doi: 10.33205/cma.528182.
ISNAD Park, Choonkil et al. “Set-Valued Additive Functional Equations”. Constructive Mathematical Analysis 2/2 (June 2019), 89-97. https://doi.org/10.33205/cma.528182.
JAMA Park C, Yun S, Lee JR, Shın DY. Set-Valued Additive Functional Equations. CMA. 2019;2:89–97.
MLA Park, Choonkil et al. “Set-Valued Additive Functional Equations”. Constructive Mathematical Analysis, vol. 2, no. 2, 2019, pp. 89-97, doi:10.33205/cma.528182.
Vancouver Park C, Yun S, Lee JR, Shın DY. Set-Valued Additive Functional Equations. CMA. 2019;2(2):89-97.