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Fixed point method for set-valued functional equations

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Abstract

In this paper, we introduce a set-valued cubic functional equation and a set-valued quartic functional equation and prove the Hyers-Ulam stability of the set-valued cubic functional equation and the set-valued quartic functional equation by using the fixed point method.

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References

  1. Aczél, J., Dhombres, J.: Functional Equations in Several Variables. Cambridge Univ. Press, Cambridge (1989)

    Book  MATH  Google Scholar 

  2. Aoki, T.: On the stability of the linear transformation in Banach spaces. J. Math. Soc. Jpn. 2, 64–66 (1950)

    Article  MATH  MathSciNet  Google Scholar 

  3. Arrow, K.J., Debreu, G.: Existence of an equilibrium for a competitive economy. Econometrica 22, 265–290 (1954)

    Article  MATH  MathSciNet  Google Scholar 

  4. Aubin, J.P., Frankowska, H.: Set-Valued Analysis. Birkhäuser, Boston (1990)

    MATH  Google Scholar 

  5. Aumann, R.J.: Integrals of set-valued functions. J. Math. Anal. Appl. 12, 1–12 (1965)

    Article  MATH  MathSciNet  Google Scholar 

  6. Cardinali, T., Nikodem, K., Papalini, F.: Some results on stability and characterization of \(K\)-convexity of set-valued functions. Ann. Polon. Math. 58, 185–192 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  7. Cascales, T., Rodrigeuz, J.: Birkhoff integral for multi-valued functions. J. Math. Anal. Appl. 297, 540–560 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  8. Castaing, C., Valadier, M.: Convex Analysis and Measurable Multifunctions, Lect. Notes in Math., vol. 580. Springer, Berlin (1977)

  9. Cădariu, L., Radu, V.: Fixed points and the stability of Jensen’s functional equation. J. Inequal. Pure Appl. Math. 4(1), Art. ID 4 (2003)

  10. Cădariu, L., Radu, V.: On the stability of the Cauchy functional equation: a fixed point approach. Grazer Math. Ber. 346, 43–52 (2004)

    MATH  MathSciNet  Google Scholar 

  11. Cădariu, L., Radu, V.: Fixed point methods for the generalized stability of functional equations in a single variable. Fixed Point Theory Appl. 2008, Art. ID 749392 (2008)

  12. Debreu, G.: Integration of correspondences. In: Proceedings of Fifth Berkeley Symposium on Mathematical Statistics and Probability, Vol. II, Part I, pp. 351–372 (1966)

  13. Diaz, J., Margolis, B.: A fixed point theorem of the alternative for contractions on a generalized complete metric space. Bull. Am. Math. Soc. 74, 305–309 (1968)

    Article  MATH  MathSciNet  Google Scholar 

  14. Eshaghi Gordji, M., Khodaei, H., Rassias, J.M.: Fixed point methods for the stability of general quadratic functional equation. Fixed Point Theory 12, 71–82 (2011)

    MATH  MathSciNet  Google Scholar 

  15. Eshaghi Gordji, M., Park, C.: The stability of a quartic type functional equation with the fixed point alternative. Fixed Point Theory 11, 265–272 (2010)

    MATH  MathSciNet  Google Scholar 

  16. Eshaghi Gordji, M., Savadkouhi, M.B.: Stability of a mixed type cubic-quartic functional equation in non-Archimedean spaces. Appl. Math. Lett. 23, 1198–1202 (2010)

  17. Gǎvruta, P.: A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings. J. Math. Anal. Appl. 184, 431–436 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  18. Hess, C.: Set-valued integration and set-valued probability theory: an overview. In: Handbook of Measure Theory, vols. I, II, North-Holland, Amsterdam (2002)

  19. Hindenbrand, W.: Core and Equilibria of a Large Economy. Princeton Univ. Press, Princeton (1974)

    Google Scholar 

  20. Hyers, D.H.: On the stability of the linear functional equation. Proc. Nat. Acad. Sci. USA 27, 222–224 (1941)

    Article  MATH  Google Scholar 

  21. Hyers, D.H., Isac, G., Rassias, ThM: Stability of Functional Equations in Several Variables. Birkhäuser, Basel (1998)

    Book  MATH  Google Scholar 

  22. Isac, G., Rassias, ThM: On the Hyers-Ulam stability of \(\psi \)-additive mappings. J. Approx. Theory 72, 131–137 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  23. Isac, G., Rassias, ThM: Stability of \(\psi \)-additive mappings: applications to nonlinear analysis. Internat. J. Math. Math. Sci. 19, 219–228 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  24. Jun, K., Kim, H.: The generalized Hyers-Ulam-Rassias stability of a cubic functional equation. J. Math. Anal. Appl. 274, 867–878 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  25. Klein, E., Thompson, A.: Theory of Correspondence. Wiley, New York (1984)

    Google Scholar 

  26. Lee, K.: Stability of functional equations related to set-valued functions (preprint)

  27. Lee, S., Im, S., Hwang, I.: Quartic functional equations. J. Math. Anal. Appl. 307, 387–394 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  28. McKenzie, L.W.: On the existence of general equilibrium for a competitive market. Econometrica 27, 54–71 (1959)

    Article  MATH  MathSciNet  Google Scholar 

  29. Miheţ, D., Radu, V.: On the stability of the additive Cauchy functional equation in random normed spaces. J. Math. Anal. Appl. 343, 567–572 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  30. Mirzavaziri, M., Moslehian, M.S.: A fixed point approach to stability of a quadratic equation. Bull. Braz. Math. Soc. 37, 361–376 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  31. Moslehian, M.S., Najati, A.: An application of a fixed point theorem to a functional inequality. Fixed Point Theory 10, 141–149 (2009)

    MATH  MathSciNet  Google Scholar 

  32. Nikodem, K.: On quadratic set-valued functions. Publ. Math. Debrecen 30, 297–301 (1984)

    MATH  MathSciNet  Google Scholar 

  33. Nikodem, K.: On Jensen’s functional equation for set-valued functions. Radovi Mat. 3, 23–33 (1987)

    MATH  MathSciNet  Google Scholar 

  34. Nikodem, K.: Set-valued solutions of the Pexider functional equation. Funkcialaj Ekvacioj 31, 227–231 (1988)

    MATH  MathSciNet  Google Scholar 

  35. Nikodem, K.: \(K\)-Convex and \(K\)-Concave Set-Valued Functions. Zeszyty Naukowe Nr., vol. 559, Lodz (1989)

  36. Park, C.: A fixed point approach to the stability of additive functional inequalities in \(RN\)-spaces. Fixed Point Theory 12, 429–442 (2011)

    MATH  MathSciNet  Google Scholar 

  37. Park, C.: Generalized Hyers-Ulam-Rassias stability of quadratic functional equations: a fixed point approach. Fixed Point Theory Appl. 2008, Art. ID 493751 (2008)

  38. Piao, Y.J.: The existence and uniqueness of additive selection for \((\alpha, \beta )\)-\((\beta,\alpha )\) type subadditive set-valued maps. J. Northeast Norm. Univ. 41, 38–40 (2009)

    MathSciNet  Google Scholar 

  39. Popa, D.: Additive selections of \((\alpha , \beta )\)-subadditive set-valued maps. Glas. Mat. Ser. III 36(56), 11–16 (2001)

  40. Radu, V.: The fixed point alternative and the stability of functional equations. Fixed Point Theory 4, 91–96 (2003)

    MATH  MathSciNet  Google Scholar 

  41. Rassias, J.M.: On approximation of approximately linear mappings by linear mappings. Bull. Sci. Math. 108, 445–446 (1984)

    MATH  MathSciNet  Google Scholar 

  42. Rassias, J.M.: Refined Hyers-Ulam approximation of approximately Jensen type mappings. Bull. Sci. Math. 131, 89–98 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  43. Rassias, J.M., Rassias, M.J.: Asymptotic behavior of alternative Jensen and Jensen type functional equations. Bull. Sci. Math. 129, 545–558 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  44. Rassias, ThM: On the stability of the linear mapping in Banach spaces. Proc. Am. Math. Soc. 72, 297–300 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  45. Rassias, ThM (ed.): Functional Equations and Inequalities. Kluwer Academic, Dordrecht (2000)

    MATH  Google Scholar 

  46. Rassias, ThM: On the stability of functional equations in Banach spaces. J. Math. Anal. Appl. 251, 264–284 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  47. Rassias, ThM: On the stability of functional equations and a problem of Ulam. Acta Math. Appl. 62, 23–130 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  48. Ulam, S.M.: Problems in Modern Mathematics, Chapter VI, Science edn. Wiley, New York (1940)

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  1. Research Institute for Natural Sciences, Hanyang University, Seoul, 04763, Republic of Korea

    Choonkil Park

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  1. Choonkil Park
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Correspondence to Choonkil Park.

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Park, C. Fixed point method for set-valued functional equations. J. Fixed Point Theory Appl. 19, 2297–2308 (2017). https://doi.org/10.1007/s11784-017-0418-0

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