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Elementary remarks on Ulam-Hyers stability of linear functional equations. (English) Zbl 1111.39026
The author proves the Hyers-Ulam stability of the family of linear functional equations of the form \[ \sum_{i=1}^s b_iF\big(\sum_{k=1}^m a_{ik}x_k\big)=0, \] where \(F: S \to X\), \(S\) is a vector space over a field \({\mathbb K}\) of characterisitic zero, \(X\) is a complex Banach space, \(b_1, \cdots, b_s\) are nonzero complex numbers with \(\sum_{i=1}^s b_i \neq 0\), \(a_{ik}\in {\mathbb K} \quad (1 \leq i \leq s, 1 \leq k \leq m)\), \(x_k \in S \quad (1 \leq k \leq m)\). Several useful remarks and open problems are also given.

MSC:
39B82 Stability, separation, extension, and related topics for functional equations
39B52 Functional equations for functions with more general domains and/or ranges
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[1] Baak, C.; Hong, S.-K.; Kim, M.-J., Generalized quadratic mappings of r-type in several variables, J. math. anal. appl., 310, 116-127, (2005) · Zbl 1077.39025
[2] Bae, J.-H., On the stability of 3-dimensional quadratic functional equation, Bull. Korean math. soc., 37, 477-486, (2000) · Zbl 0967.39011
[3] Bae, J.-H., On the stability of n-dimensional quadratic functional equation, Commun. Korean math. soc., 16, 103-111, (2001) · Zbl 1101.39304
[4] Bae, J.-H.; Jun, K.-W., On the generalized hyers – ulam – rassias stability of a quadratic functional equation, Bull. Korean math. soc., 38, 325-336, (2001) · Zbl 0983.39014
[5] Bae, J.-H.; Jun, K.-W., On the generalized hyers – ulam – rassias stability of n-dimensional quadratic functional equation, J. math. anal. appl., 258, 183-193, (2001) · Zbl 0983.39013
[6] Chang, I.-S.; Kim, H.-M., On the hyers – ulam stability of a quadratic functional equation, J. inequal. pure appl. math., 3, (2002), article 33, 12 pp. (electronic)
[7] Chang, I.-S.; Kim, H.-M., Hyers – ulam – rassias stability of a quadratic functional equation, Kyungpook math. J., 42, 71-86, (2002) · Zbl 1014.39019
[8] Chang, I.-S.; Yung, Y.-S., Stability of a functional equation deriving from cubic and quadratic functions, J. math. anal. appl., 283, 491-500, (2003) · Zbl 1038.39015
[9] Chang, I.-S.; Lee, E.-H.; Kim, H.-M., On hyers – ulam – rassias stability of a quadratic functional equation, Math. inequal. appl., 6, 87-95, (2003) · Zbl 1024.39008
[10] Forti, G.-L., Comments on the core of the direct method for proving hyers – ulam stability of functional equations, J. math. anal. appl., 295, 127-133, (2004) · Zbl 1052.39031
[11] Hyers, D.H., On the stability of the linear functional equation, Proc. natl. acad. sci., 27, 222-224, (1941) · Zbl 0061.26403
[12] Hyers, D.H.; Isac, G.; Rassias, Th.M., Stability of functional equations in several variables, Progr. nonlinear differential equations appl., vol. 34, (1998), Birkhäser · Zbl 0894.39012
[13] Jun, K.-W.; Kim, H.-M., The generalized hyers – ulam – rassias stability of a cubic functional equation, J. math. anal. appl., 274, 867-878, (2002) · Zbl 1021.39014
[14] Jun, K.-W.; Kim, H.-M., Solution of Hyers-Ulam stability problem for generalized pappus’ equation, J. math. anal. appl., 299, 100-112, (2004) · Zbl 1070.39029
[15] Jun, K.-W.; Kim, H.-M., Ulam stability problem for generalized A-quadratic mappings, J. math. anal. appl., 305, 466-476, (2005) · Zbl 1069.39030
[16] Jun, K.-W.; Kim, H.-M., Ulam stability problem for quadratic mappings of euler – lagrange, Nonlinear anal., 61, A, 1093-1104, (2005) · Zbl 1074.39027
[17] Jun, K.-W.; Kim, H.-M., On the hyers – ulam stability of a generalized quadratic and additive functional equation, Bull. Korean math. soc., 42, 133-148, (2005) · Zbl 1071.39030
[18] Kang, J.-H.; Lee, C.-J.; Lee, Y.-H., A note on the hyers – ulam – rassias stability of a quadratic equation, Bull. Korean math. soc., 41, 541-557, (2004) · Zbl 1067.39039
[19] Kim, G.H., On the stability of the quadratic mapping in normed spaces, Int. J. math. math. sci., 25, 217-229, (2001) · Zbl 0986.39015
[20] Lee, E.H., On the solution and stability of the quadratic type functional equations, Commun. Korean math. soc., 19, 477-493, (2004) · Zbl 1101.39307
[21] Lee, S.H.; Im, S.M.; Hwang, I.S., Quartic functional equations, J. math. anal. appl., 307, 387-394, (2005) · Zbl 1072.39024
[22] Lee, S.H.; Jun, K.-W., Hyers – ulam – rassias stability of a quadratic type functional equation, Bull. Korean math. soc., 40, 183-193, (2003) · Zbl 1049.39030
[23] Lee, Y.W., A generalized stability of the general euler – lagrange functional equation, Commun. Korean math. soc., 16, 607-619, (2001) · Zbl 1101.39303
[24] Lee, Y.W., On the stability of a quadratic Jensen type functional equation, J. math. anal. appl., 270, 590-601, (2002) · Zbl 1007.39026
[25] Lee, Y.W., Stability of a generalized quadratic functional equation with Jensen type, Bull. Korean math. soc., 42, 57-73, (2005) · Zbl 1074.39029
[26] Park, C.-G., Generalized quadratic mappings in several variables, Nonlinear anal., 57, 713-722, (2004) · Zbl 1058.39024
[27] Park, K.-H.; Jung, Y.-S., Stability of a cubic functional equation on groups, Bull. Korean math. soc., 41, 347-357, (2004) · Zbl 1059.39023
[28] Rassias, J.M., On the stability of the multi-dimensional euler – lagrange functional equation, Geometry, analysis and mechanics, (1994), World Sci. Publ., pp. 275-285 · Zbl 0842.39013
[29] Rassias, J.M., On the stability of a multi-dimensional Cauchy type functional equation, Geometry, analysis and mechanics, (1994), World Sci. Publ., pp. 365-376 · Zbl 0842.39014
[30] Rassias, J.M., On the stability of the general euler – lagrange functional equation, Demonstratio math., 29, 755-766, (1996) · Zbl 0884.47040
[31] Rassias, J.M., Solution of the Ulam stability problem for euler – lagrange quadratic mappings, J. math. anal. appl., 220, 613-639, (1998) · Zbl 0928.39014
[32] Rassias, J.M., Solution of the Ulam stability problem for quartic mappings, Glas. math. ser. III, 34, 54, 243-252, (1999) · Zbl 0951.39008
[33] Rassias, J.M., Solution of the Ulam stability problem for cubic mappings, Glas. math. ser. III, 36, 56, 63-72, (2001) · Zbl 0984.39014
[34] Rassias, J.M., On approximation of approximately quadratic mappings by quadratic mappings, Ann. math. sil., 15, 67-78, (2001) · Zbl 1087.39518
[35] Rassias, J.M., Solution of a quadratic stability hyers – ulam type problem, Ricerche mat., 50, 9-17, (2001) · Zbl 1221.39039
[36] Rassias, J.M., Solution of the Ulam stability problem for an Euler type quadratic functional equation, Southeast Asian bull. math., 26, 101-112, (2002) · Zbl 1017.39011
[37] Rassias, J.M., On the hyers – ulam stability problem for quadratic multi – dimensional mappings, Aequationes math., 64, 62-69, (2002) · Zbl 1009.39024
[38] Rassias, J.M., The Ulam stability problem in approximation of approximately quadratic mappings by quadratic mappings, J. inequal. pure appl. math., 5, (2004), article 52, 9 pp. (electronic) · Zbl 1055.39041
[39] Rassias, J.M.; Rassias, M.J., On the Ulam stability of Jensen and Jensen type mappings on restricted domains, J. math. anal. appl., 281, 516-524, (2003) · Zbl 1028.39011
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