Non-Archimedean hyperstability of Cauchy-Jensen functional equations on a restricted domain

• Autorzy: EL-Fassi I.

Identyfikatory

DOI

10.1515/jaa-2018-0015

• Języki publikacji: EN

Abstrakty

EN

Let X be a normed space, U ⊂ X \ {0} a non-empty subset, and (G, +) a commutative group equipped with a complete ultrametric d that is invariant (i.e., d(x + z, y + z) = d(x, y) for x, y, z ∈ G). Under some weak natural assumptions on U and on the function γ : U3 → [0, ∞), we study the new generalized hyperstability results when f : U → G satisfies the inequality d(αf( x + y / α + z), αf(z) + f(y) + f(x)) ≤ γ(x, y, z) for all x, y, z ∈ U, where x+y α + z ∈ U and α ≥ 2 is a fixed positive integer. The method is based on a quite recent fixed point theorem (Theorem 1 in [J. Brzdęk and K. Ciepliński, A fixed point approach to the stability of functional equations in non-Archimedean metric spaces, Nonlinear Anal. 74 (2011), no. 18, 6861-6867]) in some functions spaces.

Słowa kluczowe

EN

non-Archimedean hyperstability; Cauchy-Jensen functional equation; fixed point theorem

PL

hiperstateczność niearchimedejska; równanie funkcyjne Cauchy'ego-Jensena; twierdzenie o punkcie stałym

• Wydawca: De Gruyter
• Czasopismo: Journal of Applied Analysis
• Rocznik: 2018
• Tom: Vol. 24, nr 2
• Strony: 155--165
• Opis fizyczny: Bibliogr.32 poz.

Twórcy

autor

EL-Fassi I.

• Department of Mathematics, Faculty of Sciences, Ibn Tofail University, BP 133, Kenitra, Morocco

Bibliografia

• [1] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64-66.
• [2] C. Baak, Cauchy-Rassias stability of Cauchy-Jensen additive mappings in Banach spaces, Acta Math. Sin. (Engl. Ser.) 22 (2006), no. 6, 1789-1796.
• [3] A. Bahyrycz and M. Piszczek, Hyperstability of the Jensen functional equation, Acta Math. Hungar. 142 (2014), no. 2, 353-365.
• [4] D. G. Bourgin, Approximately isometric and multiplicative transformations on continuous function rings, Duke Math. J. 16 (1949), 385-397.
• [5] D. G. Bourgin, Classes of transformations and bordering transformations, Bull. Amer. Math. Soc. 57 (1951), 223-237.
• [6] J. Brzdęk, Hyperstability of the Cauchy equation on restricted domains, Acta Math. Hungar. 141 (2013), no. 1-2, 58-67.
• [7] J. Brzdęk, Remarks on hyperstability of the Cauchy functional equation, Aequationes Math. 86 (2013), no. 3, 255-267.
• [8] J. Brzdęk and K. Ciepliński, A fixed point approach to the stability of functional equations in non-Archimedean metric spaces, Nonlinear Anal. 74 (2011), no. 18, 6861-6867.
• [9] J. Brzdęk and K. Ciepliński, Hyperstability and superstability, Abstr. Appl. Anal. 2013 (2013), Article ID 401756.
• [10] J. Brzdęk, Stability of additivity and fixed point methods, Fixed Point Theory Appl. 2013 (2013), Paper No. 285.
• [11] J. Brzdęk, A hyperstability result for the Cauchy equation, Bull. Aust. Math. Soc. 89 (2014), no. 1, 33-40.
• [12] J. Brzdęk, Remarks on stability of some inhomogeneous functional equations, Aequationes Math. 89 (2015), no. 1, 83-96.
• [13] I.-I. EL-Fassi and S. Kabbaj, On the hyperstability of a Cauchy-Jensen type functional equation in Banach spaces, Proyecciones 34 (2015), no. 4, 359-375.
• [14] I.-I. EL-Fassi, S. Kabbaj and A. Charifi, Hyperstability of Cauchy-Jensen functional equations, Indag. Math. (N.S.) 27 (2016), no. 3, 855-867.
• [15] Z. Gajda, On stability of additive mappings, Internat. J. Math. Math. Sci. 14 (1991), no. 3, 431-434.
• [16] Z.-X. Gao, H.-X. Cao, W.-T. Zheng and L. Xu, Generalized Hyers-Ulam-Rassias stability of functional inequalities and functional equations, J. Math. Inequal. 3 (2009), no. 1, 63-77.
• [17] E. Gselmann, Hyperstability of a functional equation, Acta Math. Hungar. 124 (2009), no. 1-2, 179-188.
• [18] K. Hensel, Über eine neue Begründung der Theorie der algebraischen Zahlen, Jahresber. Dtsch. Math.-Ver. 6 (1899), 83-88.
• [19] D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. USA 27 (1941), 222-224.
• [20] K.-W. Jun, H.-M. Kim and J. M. Rassias, Extended Hyers-Ulam stability for Cauchy-Jensen mappings, J. Difference Equ. Appl. 13 (2007), no. 12, 1139-1153.
• [21] K.-W. Jun, H.-M. Kim and E. Y. Son, Generalized Hyers-Ulam stability of Cauchy-Jensen functional equations, in: Nonlinear Analysis, Springer Optim. Appl. 68, Springer, New York (2012), 343-352.
• [22] A. Khrennikov, Non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems and Biological Models, Math. Appl. 427, Kluwer Academic Publishers, Dordrecht, 1997.
• [23] Y.-H. Lee, On the stability of the monomial functional equation, Bull. Korean Math. Soc. 45 (2008), no. 2, 397-403.
• [24] Y.-H. Lee and K.-W. Jun, A generalization of the Hyers-Ulam-Rassias stability of Jensen’s equation, J. Math. Anal. Appl. 238 (1999), no. 1, 305-315.
• [25] G. Maksa and Z. Páles, Hyperstability of a class of linear functional equations, Acta Math. Acad. Paedagog. Nyházi. (N.S.) 17 (2001), no. 2, 107-112.
• [26] M. S. Moslehian and T. M. Rassias, Stability of functional equations in non-Archimedean spaces, Appl. Anal. Discrete Math. 1 (2007), no. 2, 325-334.
• [27] C. Park, Fixed points and Hyers-Ulam-Rassias stability of Cauchy-Jensen functional equations in Banach algebras, Fixed Point Theory Appl. 2007 (2007), Article ID 50175.
• [28] J. M. Rassias and M. J. Rassias, On the Ulam stability of Jensen and Jensen type mappings on restricted domains, J. Math. Anal. Appl. 281 (2003), no. 2, 516-524.
• [29] T. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), no. 2, 297-300.
• [30] T. M. Rassias, On a modified Hyers-Ulam sequence, J. Math. Anal. Appl. 158 (1991), no. 1, 106-113.
• [31] T. M. Rassias and P. Šemrl, On the behavior of mappings which do not satisfy Hyers-Ulam stability, Proc. Amer. Math. Soc. 114 (1992), no. 4, 989-993.
• [32] S. M. Ulam, A Collection of Mathematical Problems, Interscience Publishers, New York, 1960; reprinted as Problems in Modern Mathematics, John Wiley & Sons, Inc., New York, 1964.

Uwagi

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2018).