Stability of quadratic and orthogonally quadratic functional equations
DOI :
https://doi.org/10.5269/bspm.66246Résumé
In this study, the authors examine the generalized Hyers-Ulam stability of the following quadratic
functional equation
g(3x + 2y+z) + g(3x + 2y − z) + g(3x − 2y + z) + g(3x − 2y − z)
= 8[g(x + y) + g(x − y)] + 2[g(x + z) + g(x − z)] + 16g(x)
The preceding equation is changed and its generalized Hyers-Ulam stability for the following quadratic functional equation
f(3x + 2y+z) + f(3x + 2y − z) + f(3x − 2y + z) + f(3x − 2y − z)
= 8[f(x + y) + f(x − y)] + 2[f(x + z) + f(x − z)] + 16f(x)
for any x, y, z ∈ X with x ⊥ y, y ⊥ z and z ⊥ x is studied in orthogonality space in the sense of Ratz.
Références
[1] Aczel, J., and Dhombres, J., Functional Equations in Several Variables, Cambridge Univ, Press, (1989).
[2] Alonso, J., and Benitez, C., Orthogonality in normed linear spaces: survey I. Main properties, Extracta Math., 3, 1-15, (1988).
[3] Alonso, J., and Benitez, C., Orthogonality in normed linear spaces: a survey II. Relations between main orthogonalities, Extracta Math., 4, 121-131, (1989).
[4] Aoki, T., On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan, 2, 64-66, (1950).
[5] Balamurugan, K., Arunkumar, M., and Ravindiran, P., Stability of a Cubic and Orthogonally Cubic Functional Equations, Int. J. Appl. Eng. Res., Vol. 10, No. 72, (2015).
[6] K. Balamurugan, M. Arunkumar, P. Ravindiran, Generalized Hyers-Ulam stability for a mixed quadratic-quartic(QQ) Functional Equation in Quasi-Banach Spaces , Br. J. Math. Comput. Sci., 9(2): 122-140, 2015, Article no.BJMCS.2015.192, (2015), DOI : 10.9734/BJMCS/2015/17551.
[7] Chung, I.S., Kim, H.M., On the Hyers-Ulam stability of quadratic functional equations, J. Inequal. Pure Appl. Math., Vol. 3, Issue.3, Art. 33, (2002).
[8] Czerwik, S., Functional Equations and Inequalities in Several Variables, World Scientific, River Edge, NJ., (2002).
[9] Gavruta, P., A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings , J. Math. Anal. Appl., 184, 431-436, (1994).
[10] Ger, R., and Sikorska, J., Stability of the orthogonal additivity, Bull. Pol. Acad. Sci. Math. 43, 143-151, (1995).
[11] Gudder, S., and Strawther, D., Orthogonally additive and orthogonally increasing functions on vector spaces, Pacific J. Math. 58, 427-436, (1995).
[12] Hyers, D.H., On the stability of the linear functional equation, Proc.Nat. Acad.Sci.,U.S.A.,27, 222-224, (1941).
[13] Hyers, D.H., Isac, G., and Rassias, Th.M., Stability of functional equations in several variables, Birkhauser, Basel, (1998).
[14] Kannappan, Pl., Functional Equations and Inequalities with Applications, Springer Monographs in Mathematics, (2009).
[15] Park, C. G., On the stability of the orthogonally quartic functional, Bull. Iranian Math. Soc., Vol 3(1), 63-70, (2005).
[16] Rassias, Th.M., On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72, 297-300, (1978).
[17] Rassias, J.M., On approximately of approximately linear mappings by linear mappings, J. Funct. Anal. USA, 46, 126-130, (1982).
[18] Rassias, J.M., On approximately of approximately linear mappings by linear mappings, Bull. Sci. Math, 108, 445-446, (1984).
[19] Ratz, J., On orthogonally additive mappings, Aequationes Math., 28, 35-49, (1985).
[20] Skof, F., Proprieta locali e approssimazione di operatori ` , Seminario Mat. e. Fis. di Milano, Vol.53, 113-129, (1983), https://doi.org/10.1007/BF02924890
[21] Ulam, S.M., Problems in Modern Mathematics, Science Editions, Wiley, NewYork, (1964).
[22] Vajzovic, F., Uber das Funktional H mit der Eigenschaft: (x,y) = 0 ⇒ H(x+y)+H(x−y) = 2H(x)+ 2H(y), Glas. Mat. Ser. III, 2(22), 73-81, (1967).
[2] Alonso, J., and Benitez, C., Orthogonality in normed linear spaces: survey I. Main properties, Extracta Math., 3, 1-15, (1988).
[3] Alonso, J., and Benitez, C., Orthogonality in normed linear spaces: a survey II. Relations between main orthogonalities, Extracta Math., 4, 121-131, (1989).
[4] Aoki, T., On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan, 2, 64-66, (1950).
[5] Balamurugan, K., Arunkumar, M., and Ravindiran, P., Stability of a Cubic and Orthogonally Cubic Functional Equations, Int. J. Appl. Eng. Res., Vol. 10, No. 72, (2015).
[6] K. Balamurugan, M. Arunkumar, P. Ravindiran, Generalized Hyers-Ulam stability for a mixed quadratic-quartic(QQ) Functional Equation in Quasi-Banach Spaces , Br. J. Math. Comput. Sci., 9(2): 122-140, 2015, Article no.BJMCS.2015.192, (2015), DOI : 10.9734/BJMCS/2015/17551.
[7] Chung, I.S., Kim, H.M., On the Hyers-Ulam stability of quadratic functional equations, J. Inequal. Pure Appl. Math., Vol. 3, Issue.3, Art. 33, (2002).
[8] Czerwik, S., Functional Equations and Inequalities in Several Variables, World Scientific, River Edge, NJ., (2002).
[9] Gavruta, P., A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings , J. Math. Anal. Appl., 184, 431-436, (1994).
[10] Ger, R., and Sikorska, J., Stability of the orthogonal additivity, Bull. Pol. Acad. Sci. Math. 43, 143-151, (1995).
[11] Gudder, S., and Strawther, D., Orthogonally additive and orthogonally increasing functions on vector spaces, Pacific J. Math. 58, 427-436, (1995).
[12] Hyers, D.H., On the stability of the linear functional equation, Proc.Nat. Acad.Sci.,U.S.A.,27, 222-224, (1941).
[13] Hyers, D.H., Isac, G., and Rassias, Th.M., Stability of functional equations in several variables, Birkhauser, Basel, (1998).
[14] Kannappan, Pl., Functional Equations and Inequalities with Applications, Springer Monographs in Mathematics, (2009).
[15] Park, C. G., On the stability of the orthogonally quartic functional, Bull. Iranian Math. Soc., Vol 3(1), 63-70, (2005).
[16] Rassias, Th.M., On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72, 297-300, (1978).
[17] Rassias, J.M., On approximately of approximately linear mappings by linear mappings, J. Funct. Anal. USA, 46, 126-130, (1982).
[18] Rassias, J.M., On approximately of approximately linear mappings by linear mappings, Bull. Sci. Math, 108, 445-446, (1984).
[19] Ratz, J., On orthogonally additive mappings, Aequationes Math., 28, 35-49, (1985).
[20] Skof, F., Proprieta locali e approssimazione di operatori ` , Seminario Mat. e. Fis. di Milano, Vol.53, 113-129, (1983), https://doi.org/10.1007/BF02924890
[21] Ulam, S.M., Problems in Modern Mathematics, Science Editions, Wiley, NewYork, (1964).
[22] Vajzovic, F., Uber das Funktional H mit der Eigenschaft: (x,y) = 0 ⇒ H(x+y)+H(x−y) = 2H(x)+ 2H(y), Glas. Mat. Ser. III, 2(22), 73-81, (1967).
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Publié
2025-09-23
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Research Articles
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Comment citer
Yagachitradevi, G., Lakshminarayanan, S. . ., & Ravindiran, P. . (2025). Stability of quadratic and orthogonally quadratic functional equations. Boletim Da Sociedade Paranaense De Matemática, 43, 1-9. https://doi.org/10.5269/bspm.66246
