A Note on the Stability of Some Functional Equations on Certain Groupoids

Year 2020, Volume: 3 Issue: 2, 96 - 103, 01.06.2020

Jedsada Senasukh Satit Saejung

https://doi.org/10.33205/cma.729765

Cited By: 2

Abstract

In this paper, we show that the stability of Cauchy set-valued functional equations and of Jensen set-valued functional equations can be derived from the stability of the corresponding equations in single-valued version.

Keywords

Stability, Cauchy set-valued functional equation, Jensen set-valued functional equation

References

• T. Aoki: On the stability of the linear transformation in Banach spaces. J. Math. Soc. Japan 2 (1950), 64–66.
• K. J. Arrow, G. Debreu: Existence of an equilibrium for a competitive economy. Econometrica 22 (1954), 265–290.
• R. J. Aumann: Integrals of set-valued functions. J. Math. Anal. Appl. 12 (1965), 1–12.
• J. Brzd˛ek: Stability of additivity and fixed point methods. Fixed Point Theory Appl. 2013 (2013), Article ID 401756, 9 pages.
• C. Castaing, M. Valadier: Convex analysis and measurable multifunctions: Lec. Notes in Math. Springer, Berlin, 1977.
• J. B. Diaz, B. Margolis: A fixed point theorem of the alternative for contractions on a generalized complete metric space. Bull. Amer. Math. Soc. 74 (1968), 305–309.
• G. L. Forti: An existence and stability theorems for a class of functional equations. Stochastica 4 (1) (1980), 23-30.
• Z. Gajda: On stability of additive mappings. Internat. J. Math. Math. Sci. 14(3) (1991), 431–434.
• P. Gˇavru¸ta: A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings. J. Math. Anal. Appl. 184(3) (1994), 431–436.
• D. H. Hyers: On the stability of the linear functional equation. Proc. Nat. Acad. Sci. U.S.A. 27 (1941), 222–224.
• G. H. Kim: Addendum to ‘On the stability of functional equations on square-symmetric groupoid. Nonlinear Anal. 62 (2005), 365–381.
• Z. Páles, P. Volkman and R. D. Luce: Hyers–Ulam stability of functional equations with a square-symmetric operation. Proc. Natl. Acad. Sci. USA, 95 (1998), 12772–12275.
• C. Park, S. Yun, J. Lee and D. Shin: Set-valued additive functional equations. Constr. Math. Anal. 2(2) (2019), 89–97.
• H. Przybycie ´ n: A note on closedness of algebraic sum of sets. Tbilisi Math. J. 9 (2) (2016), 71-74.
• Th. M. Rassias: On the stability of the linear mapping in Banach spaces. Proc. Amer. Math. Soc. 72 (1978), 297–300.
• S. Saejung, J. Senasukh: On stability and hyperstability of additive equations on a commutative semigroup. Acta Math. Hungar. 159(2) (2019), 358–373.
• S. M. Ulam: A collection of mathematical problems. Interscience Publishers, New York-London, 1960.