An Instructive Treatment of a Generalization of Găvruţă’s Stability Theorem
DOI:
https://doi.org/10.5644/SJM.06.1.01Keywords:
Groupoids and normed spaces, approximately additive and homogeneous functions, Găvruţă 's type stability theoremsAbstract
We prove several useful theorems on Hyers sequences and their pointwise limits in quite natural ways which make a straightforward generalization of Găvruţă''s stability theorem rather plausible.
2000 Mathematics Subject Classification. 20M15, 39B52, 39B82
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References
T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan, 2 (1950), 64–66.
R. Badora, Note on the superstability of the Cauchy functional equation, Publ. Math. Debrecen, 57 (2000), 421–424.
R. Badora, On the Hahn–Banach theorem for groups, Arch. Math., 86 (2006), 517–528.
K. Baron and P. Volkmann, On functions close to homomorphisms between square symmetric structures, Seminar LV, 14 (2002), 1–12 (electronic).
D. G. Bourgin, Classes of transformations and bordering transformations, Bull. Amer. Math. Soc., 57 (1951), 223–237.
S. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific, London, 2002.
V. A. Fa˘ıziev, Th. M. Rassias and P. K. Sahoo, The space of (ψ , γ )–additive mappings on semigroups, Trans. Amer. Math. Soc., 354 (1985), 4455-4472.
G. L. Forti, The stability of homomorphisms and amenability, with applications to functional equations, Abh. Math. Sem. Uiv. Hamburg, 57 (1987), 215–226.
G. L. Forti, Hyers–Ulam stability of functional equations in several variables, Aequationes Math., 50 (1995), 143–190.
G. L. Forti An existence and stability theorem for a class of functional equations, Stochastica, 4 (1980), 23–30.
Z. Gajda, On stability of additive mappings, Internat J. Math. Sci., 14 (1991), 431–434.
P. Gˇavrut¸ˇa, A generalization of the Hyers–Ulam–Rassias stability of approximately additive mappings, J. Math. Anal. Appl., 184 (1994), 431–436.
P. Gˇavrut¸ˇa, On a problem of G. Isac and Th. M. Rassias concerning the stability of mappings, J. Math. Anal. Appl., 261 (2001), 543–553.
P. Gˇavrut¸ˇa, M. Hosszu, D. Popescu and C. Cˇaprˇau, On the stability of Mappings and an answer to a problem of Th. M. Rassias, Ann. Math. Blaise Pascal, 2 (1995), 55–60.
R. Ger, The singular case in the stability behaviour of linear mappings, Grazer Math. Ber., 316 (1992), 59–70.
R. Ger, A survey of recent results on stability of functional equations, Proceedings of the 4th International Conference on Functional Equations and Inequalities Pedagogical University of Cracow, 1994, 5–36.
R. Ger and P. Volkmann, On sums of linear and bounded mappings, Abh. Math. Sem. Univ. Hamburg, 68 (1998), 103–108.
A. Gil´anyi, Z. Kaiser and Zs. P´ales, Estimates to the stability of functional equations, Aequationes Math., 73 (2007), 125–143.
A. Grabiec The generalized Hyers–Ulam stability of a class of functional equations, Publ. Math. Debrecen, 48 (1996), 217–235.
D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A, 27 (1941), 222–224.
D. H. Hyers, G. Isac and Th. M. Rassias, Stability of Functional Equations in Several Variables, Birkh¨auser, Boston, 1998
D. H. Hyers and Th. M. Rassias, Approximate homomorphisms, Aequationes Math., 44 (1992), 125–153.
G. Isac and Th. M. Rassias, On the Hyers-Ulam stability of ψ additive mappings, J. Approx. Theory, 72 (1993), 131–137.
G. Isac and Th. M. Rassias,Functional inequalities for approximately additive mappings, In: Th. M. Rassias and Jo. Tabor (Eds.), Stability of mappings of Hyers–Ulam type, Hadronic Press Collect. Orig. Art., Palm Harbor, FL, 1994, 117–125.
S.-M. Jung, Hyers–Ulam–Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press, Palm Harbor, 2001.
Z. Kaiser and Zs. P´ales, An example of a stable functional equation when the Hyers method does not work, J. Ineq. Pure Appl. Math., 6 (2005), 1–11.
G. H. Kim, On the stability of functional equations with square-symmetric operation, Math. Ineq. Appl., 4 (2001), 257–266.
M. Kuczma, An Introduction to the Theory of Functional Equations and Inequalities, Polish Sci. Publ. and Univ. Slaski, Warszawa, 1985. [29] Y.-H. Lee and K.-W. Jun, On the stability of approximately additive mappings, Proc. Amer. Math. Soc., 128 (1999), 1361–1369.
Zs. P´ales, Generalized stability of the Cauchy functional equations, Aequationes Math., 56 (1998), 222–232.
Zs. P´ales, Hyers–Ulam stability of the Cauchy functional equation on squaresymmetric groupoids, Publ. Math. Debrecen, 58 (2001), 651–666.
Zs. P´ales, P. Volkmann and R. D. Luce, Hyers–Ulam stability of functional equations with a square-symmetric operation, Proc. Natl. Acad. Sci. USA, 95 (1998), 12772–12775.
C.-G. Park, On the stability of the linear mapping in Banach modules, J. Math. Anal. Appl., 275 (2002), 711–720.
Gy. P´olya and G. Szeg˝o, Problems and Theorems in Analysis I, Springer Verlag, Berlin, 1972.
J. M. Rassias, On approximation of approximately linear mappings by linear mappings, J. Funct. Anal., 46 (1982), 126–130.
J. M. Rassias, Solution of a problem of Ulam, J. Approximation Theory, 57 (1989), 268–273.
Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72 (1978), 297–300.
Th. M. Rassias, On a modified Hyers-Ulam sequence, J. Math. Anal. Appl., 158 (1991), 106–113.
Th. M. Rassias, On the stability of functional equations and a problem of Ulam, Acta Applicandae Math., 62 (2000), 23–130.
Th. M. Rassias, The problem of S. M. Ulam for approximately multiplicative mappings, J. Math. Anal. Appl., 246 (2000), 353–378.
Th. M. Rassias On the stability of functional equations originated by a problem of Ulam, Mathematica (Cluj), 44 (2002), 39–75.
Th. M. Rassias and P. Semrl, On the behavior of mappings which do not satisfy Hyers–Ulam stability, Proc. Amer. Math. Soc., 114 (1992), 989–993.
Th. M. Rassias and P. Semrl, On the Hyers–Ulam stability of linear mappings, J. Math. Anal. Appl., 173 (1993), 325–338.
Th. M. Rassias and J. Tabor, On approximately additive mappings in Banach spaces, In: Th. M. Rassias and Jo. Tabor (Eds.), Stability of mappings of Hyers–Ulam type, Hadronic Press Collect. Orig. Art., Palm Harbor, FL, 1994, 123–137.
V. Radu, The fixed point alternative and the stability of functional equations, Fixed Point Theory, 4 (2003), 91–96.
J. R¨atz, On approximately additive mappings, In: E. F. Beckenbach (Ed.), General Inequalities 2 (Oberwolfach, 1978), Int. Ser. Num. Math. (Birkh¨auser, Basel-Boston), 47 (1980), 233–251.
J. Spakula and P. Zlatoˇs ˇ Almost homomorphisms of compact groups, Illinois J. Math., 48 (2004), 1183–1189.
A. Sz´az ´ An extension of an additive selection theorem of Z. Gajda and R. Ger, Tech. Rep., Inst. Math., Univ. Debrecen, 8 (2006), 1–24.
A. Sz´az, ´ A instructive treatment of a generalization of Hyers’s stability theorem, In: Th. M. Rassias and D. Andrica (Eds.), Inequalities and Applications, Cluj University Press, Romania, 2008, 245–271.
A. Sz´az ´ Applications of relations and relators in the extensions of stability theorems for homogeneous and additive functions, Aust. J. Math. Anal. Appl., 6 (2009), 1-66.
L. Sz´ekelyhidi, Remark 17, Aequationes Math., 29 (1985), 95–96.
