Abstract
In this paper, we introduce set-valued additive \(\rho \)-functional inequalities and prove the Hyers–Ulam stability of the set-valued additive \(\rho \)-functional inequalities by using the fixed point method.
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References
Aczel, J., Dhombres, J.: Functional Equations in Several Variables. Cambridge University Press, Cambridge (1989)
Aoki, T.: On the stability of the linear transformation in Banach spaces. J. Math. Soc. Japan 2, 64–66 (1950)
Arrow, K.J., Debreu, G.: Existence of an equilibrium for a competitive economy. Econometrica 22, 265–290 (1954)
Aubin, J.P., Frankow, H.: Set-Valued Analysis. Birkhäuser, Boston (1990)
Aumann, R.J.: Integrals of set-valued functions. J. Math. Anal. Appl. 12, 1–12 (1965)
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)
Cascales, T., Rodrigeuz, J.: Birkhoff integral for multi-valued functions. J. Math. Anal. Appl. 297, 540–560 (2004)
Castaing, C., Valadier, M.: Convex Analysis and Measurable Multifunctions, Lect. Notes in Math. Springer, Berlin (1977)
Cădariu, L., Radu, V.: Fixed points and the stability of Jensen’s functional equation. J. Inequal. Pure Appl. Math. 4(1), 4 (2003)
Cădariu, L., Radu, V.: On the stability of the Cauchy functional equation: a fixed point approach. Grazer Math. Ber. 346, 43–52 (2004)
Cădariu, L., Radu, V.: Fixed point methods for the generalized stability of functional equations in a single variable. Fixed Point Theory Appl. 2008, 749392 (2008)
Debreu, G.: Integration of correspondences. In: Proceedings of Fifth Berkeley Symposium on Mathematical Statistics and Probability, vol. II, Part I, pp. 351–372 (1966)
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)
Eshaghi Gordji, M., Park, C., Savadkouhi, M.B.: The stability of a quartic type functional equation with the fixed point alternative. Fixed Point Theory 11, 265–272 (2010)
Eshaghi, M.: Gordji and M.B. Savadkouhi, Stability of a mixed type cubic-quartic functional equation in non-Archimedean spaces. Appl. Math. Lett. 23, 1198–1202 (2010)
Gǎvruta, P.: A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings. J. Math. Anal. Appl. 184, 431–436 (1994)
Hess, C.: Set-valued integration and set-valued probability theory: an overview. Handbook of Measure Theory, vols. I, II. North-Holland, Amsterdam (2002)
Hindenbrand, W.: Core and Equilibria of a Large Economy. Princeton University Press, Princeton (1974)
Hyers, D.H.: On the stability of the linear functional equation. Proc. Nat. Acad. Sci. USA 27, 222–224 (1941)
Hyers, D.H., Isac, G., Rassias, ThM: Stability of Functional Equations in Several Variables. Birkhäuser, Basel (1998)
Isac, G., Rassias, T.H.: On the Hyers-Ulam stability of \(\psi \)-additive mappings. J. Approx. Theory 72, 131–137 (1993)
Isac, G., Rassias, T.H.: Stability of \(\psi \)-additive mappings: Applications to nonlinear analysis. Internat. J. Math. Math. Sci. 19, 219–228 (1996)
Khodaei, H., Rassias, T.M.: Set-valued dynamics related to generalized Euler-Lagrange functional equations. J. Fixed Point Theory Appl. 20, 20–32 (2018)
Klein, E., Thompson, A.: Theory of Correspondence. Wiley, New York (1984)
Lee, K.: Stability of functional equations related to set-valued functions (preprint)
McKenzie, L.W.: On the existence of general equilibrium for a competitive market. Econometrica 27, 54–71 (1959)
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)
Mirzavaziri, M., Moslehian, M.S.: A fixed point approach to stability of a quadratic equation. Bull. Braz. Math. Soc. 37, 361–376 (2006)
Nikodem, K.: On quadratic set-valued functions. Publ. Math. Debrecen 30, 297–301 (1984)
Nikodem, K.: On Jensen’s functional equation for set-valued functions. Radovi Mat. 3, 23–33 (1987)
Nikodem, K.: Set-valued solutions of the Pexider functional equation. Funkcialaj Ekvacioj 31, 227–231 (1988)
Nikodem, K.: K-convex and K-concave set-valued functions. Zeszyty Naukowe Nr. Lodz 559, 1–75 (1989)
Park, C.: Fixed point method for set-valued functional equations. J. Fixed Point Theory Appl. 19, 2297–2308 (2017)
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)
Popa, D.: Additive selections of \((\alpha, \beta )\)-subadditive set-valued maps. Glas. Mat. Ser. III 36(56), 11–16 (2001)
Radu, V.: The fixed point alternative and the stability of functional equations. Fixed Point Theory 4, 91–96 (2003)
Rassias, T.M.: On the stability of the linear mapping in Banach spaces. Proc. Am. Math. Soc. 72, 297–300 (1978)
Rassias, T.H. (ed.): Functional Equations and Inequalities. Kluwer Academic, Dordrecht (2000)
Rassias, ThM: On the stability of functional equations in Banach spaces. J. Math. Anal. Appl. 251, 264–284 (2000)
Rassias, T.H.: On the stability of functional equations and a problem of Ulam. Acta Math. Appl. 62, 23–130 (2000)
Ulam, S.M.: Problems in Modern Mathematics, Chap. VI, Science edn. Wiley, New York (1940)
Acknowledgements
C. Park was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2017R1D1A1B04032937).
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Park, C. Set-valued additive \(\rho \)-functional inequalities. J. Fixed Point Theory Appl. 20, 70 (2018). https://doi.org/10.1007/s11784-018-0547-0
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DOI: https://doi.org/10.1007/s11784-018-0547-0