Addendum to ‘On the stability of functional equations on square-symmetric groupoid’. (English) Zbl 1112.39022

Summary: Let \((X,\diamond)\) be a square-symmetric groupoid, and \((Y,*,d)\) a complete metric divisible square-symmetric groupoid. In this paper, we investigate the Hyers-Ulam stability problem, using the functional inequality \(d(g(x\diamond y),g(x)*g(y))\leq\epsilon(x,y)\) for approximate mapping \(g\colon X\to Y\) of the functional equation \(f(x\diamond y)=f(x)*f(y)\). In particular, we investigate the case of \(f(x)*f(y)=H(f(x)^{1/t},f(y)^{1/t})\) on some set \(Y\) in which \(H\:Y\times Y\to Y\) is a continuous homogeneous function of degree \(t\).


39B82 Stability, separation, extension, and related topics for functional equations
39B52 Functional equations for functions with more general domains and/or ranges
39B72 Systems of functional equations and inequalities
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[1] Borelli-Forti, C.; Forti, G.-L., On a general hyers – ulam stability result, Int. J. math. math. sci., 18, 229-236, (1995) · Zbl 0826.39009
[2] Gaˇvruta, P., A generalization of the hyers – ulam – rassias stability of approximately additive mappings, J. math. anal. appl., 184, 431-436, (1994) · Zbl 0818.46043
[3] Hyers, D.H., On the stability of the linear functional equation, Proc. natl. acad. sci., 27, 222-224, (1941) · Zbl 0061.26403
[4] Hyers, D.H.; Rassias, Th.M., Approximate homomorphisms, Aeqnat. math., 44, 125-153, (1992) · Zbl 0806.47056
[5] Hyers, D.H.; Isac, G.; Rassias, Th.M., Stability of the functional equations in several variables, (1998), Birkhäuser Verlag · Zbl 0894.39012
[6] Kim, G.H., On the stability of functional equations with square-symmetric operations, Math. ineq. appl., 4, 257-266, (2001) · Zbl 0990.39028
[7] Kim, G.H., On the stability of homogeneous functional equations with degree \(t\) and \(n\)-variables, Math. ineq. appl., 4, 675-688, (2003) · Zbl 1051.39029
[8] Páles, Z., Generalized stability of the Cauchy functional equation, Aequationes math., 56, 222-232, (1998) · Zbl 0922.39008
[9] Páles, Z., Hyers – ulam stability of the Cauchy functional equation on square-symmetric groupoids, Publ. math. debrecen, 58, 651-666, (2001) · Zbl 0980.39022
[10] Páles, Z.; Volkmann, P.; Luce, D., Stability of functional equations with square-symmetric operations, Proc. natl. acad. sci., 95, 22, 12772-12775, (1998) · Zbl 0930.39020
[11] Rassias, Th.M., On the stability of the linear mapping in Banach spaces, Proc. am. math. soc., 72, 297-300, (1978) · Zbl 0398.47040
[12] Rassias, Th.M., On the modified hyers – ulam sequence, J. math. anal. appl., 158, 106-113, (1991) · Zbl 0746.46038
[13] Rassias, Th.M.; Šemrl, P., On the behavior of mappings which do not satisfy hyers – ulam stability, Proc. am. math. soc., 114, 989-993, (1992) · Zbl 0761.47004
[14] Rassias, Th.M.; Tabor, J., What is left of hyers – ulam stability, J. nat. geometry, 1, 65-69, (1992) · Zbl 0757.47032
[15] Ulam, S.M., Problems in modern mathematics, (1964), Wiley New York, (Chapter VI) · Zbl 0137.24201
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