References
- J. Alonso and C. Benitez, Orthogonality in normed linear spaces: a survey I. Main properties, Extracta Math. 3 (1988), 1-15.
- J. Alonso and C. Benitez, Orthogonality in normed linear spaces: a survey II. Relations between main orthogonalities, Extracta Math. 4 (1989), 121-131.
- G. Birkhoff, Orthogonality in linear metric spaces, Duke Math. J. 1 (1935), 169-172. https://doi.org/10.1215/S0012-7094-35-00115-6
- S.O. Carlsson, Orthogonality in normed linear spaces, Ark. Mat. 4 (1962),297-318. https://doi.org/10.1007/BF02591506
- P.W. Cholewa, Remarks on the stability of functional equations, Aequationes Math. 27 (1984), 76-86. https://doi.org/10.1007/BF02192660
- S. Czerwik, On the stability of the quadratic mapping in normed spaces, Abh. Math. Sem. Univ. Hamburg 62 (1992), 59-64. https://doi.org/10.1007/BF02941618
- S. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific Publishing Company, New Jersey, London, Singapore and Hong Kong, 2002.
- S. Czerwik, Stability of Functional Equations of Ulam-Hyers-Rassias Type, Hadronic Press, Palm Harbor, Florida, 2003.
- D. Deses, On the representation of non-Archimedean objects, Topology Appl. 153 (2005), 774-785. https://doi.org/10.1016/j.topol.2005.01.010
- C.R. Diminnie, A new orthogonality relation for normed linear spaces, Math. Nachr. 114 (1983), 197-203. https://doi.org/10.1002/mana.19831140115
- F. Drljevic, On a functional which is quadratic on A-orthogonal vectors, Publ. Inst. Math. (Beograd) (N.S.) 54 (1986), 63-71.
- M. Fochi, Functional equations in A-orthogonal vectors, Aequationes Math. 38 (1989), 28-40. https://doi.org/10.1007/BF01839491
- R. Ger and J. Sikorska, Stability of the orthogonal additivity, Bull. Pol. Acad. Sci. Math. 43 (1995), 143-151.
- S. Gudder and D. Strawther, Orthogonally additive and orthogonally increasing functions on vector spaces, Pacific J. Math. 58 (1975), 427-436. https://doi.org/10.2140/pjm.1975.58.427
- K. Hensel, Ubereine news Begrundung der Theorie der algebraischen Zahlen, Jahresber. Deutsch. Math.-Verein 6 (1897), 83-88.
- D.H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. USA 27 (1941), 222-224. https://doi.org/10.1073/pnas.27.4.222
- D.H. Hyers, G. Isac and Th.M. Rassias, Stability of Functional Equations in Several Variables, Birkhauser, Basel, 1998.
- R.C. James, Orthogonality and linear functionals in normed linear spaces, Trans. Amer. Math. Soc. 61 (1947), 265-292. https://doi.org/10.1090/S0002-9947-1947-0021241-4
- S. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press, Palm Harbor, Florida, 2001.
- A.K. Katsaras and A. Beoyiannis, Tensor products of non-Archimedean weighted spaces of continuous functions, Georgian Math. J. 6 (1999), 33-44. https://doi.org/10.1023/A:1022926309318
- A. Khrennikov, Non-Archimedean analysis: quantum paradoxes, dynamical systems and biological models, Mathematics and its Applications 427, Kluwer Academic Publishers, Dordrecht, 1997.
- S. Lee, S. Im and I. Hwang, Quartic functional equations, J. Math. Anal. Appl. 307 (2005), 387-394. https://doi.org/10.1016/j.jmaa.2004.12.062
- M.S. Moslehian, On the orthogonal stability of the Pexiderized quadratic equation, J. Difference Equ. Appl. 11 (2005), 999-1004. https://doi.org/10.1080/10236190500273226
- M.S. Moslehian, On the stability of the orthogonal Pexiderized Cauchy equation, J. Math. Anal. Appl. 318, (2006), 211-223. https://doi.org/10.1016/j.jmaa.2005.05.052
- M.S. Moslehian and Th.M. Rassias, Orthogonal stability of additive type equations, Aequationes Math. 73 (2007), 249-259. https://doi.org/10.1007/s00010-006-2868-0
- M.S. Moslehian and Gh. Sadeghi, A Mazur-Ulam theorem in non-Archimedean normed spaces, Nonlinear Anal.-TMA 69 (2008), 3405-3408. https://doi.org/10.1016/j.na.2007.09.023
- P.J. Nyikos, On some non-Archimedean spaces of Alexandrof and Urysohn, Topology Appl. 91 (1999), 1-23. https://doi.org/10.1016/S0166-8641(97)00239-3
- L. Paganoni and J. Ratz, Conditional function equations and orthogonal additivity, Aequationes Math. 50 (1995), 135-142. https://doi.org/10.1007/BF01831116
- C. Park and J. Park, Generalized Hyers-Ulam stability of an Euler-Lagrange type additive mapping, J. Difference Equ. Appl. 12 (2006), 1277-1288. https://doi.org/10.1080/10236190600986925
- A.G. Pinsker, Sur une fonctionnelle dans l'espace de Hilbert, C. R. (Dokl.) Acad. Sci. URSS, n. Ser. 20 (1938), 411-414.
- Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297-300. https://doi.org/10.1090/S0002-9939-1978-0507327-1
- Th.M. Rassias, On the stability of the quadratic functional equation and its applications, Stud. Univ. Babes-Bolyai Math. 43 (1998), 89-124.
- Th.M. Rassias, The problem of S.M. Ulam for approximately multiplicative mappings, J. Math. Anal. Appl. 246 (2000), 352-378. https://doi.org/10.1006/jmaa.2000.6788
- Th.M. Rassias, On the stability of functional equations in Banach spaces, J. Math. Anal. Appl. 251 (2000), 264-284. https://doi.org/10.1006/jmaa.2000.7046
- Th.M. Rassias (ed.), Functional Equations, Inequalities and Applications, Kluwer Academic Publishers, Dordrecht, Boston and London, 2003.
- J. Ratz, On orthogonally additive mappings, Aequationes Math. 28 (1985), 35- 49. https://doi.org/10.1007/BF02189390
- J. Ratz and Gy. Szabo, On orthogonally additive mappings IV, Aequationes Math. 38 (1989), 73-85. https://doi.org/10.1007/BF01839496
- F. Skof, Proprieta locali e approssimazione di operatori, Rend. Sem. Mat. Fis. Milano 53 (1983), 113-129. https://doi.org/10.1007/BF02924890
- K. Sundaresan, Orthogonality and nonlinear functionals on Banach spaces, Proc. Amer. Math. Soc. 34 (1972), 187-190. https://doi.org/10.1090/S0002-9939-1972-0291835-X
- Gy. Szabo, Sesquilinear-orthogonally quadratic mappings, Aequationes Math. 40 (1990), 190-200. https://doi.org/10.1007/BF02112295
- S.M. Ulam, Problems in Modern Mathematics, Wiley, New York, 1960.
-
F. Vajzovic, Uber das Funktional H mit der Eigenschaft: (x, y) = 0
$\Rightarrow$ H(x + y) + H(x − y) = 2H(x) + 2H(y), Glasnik Mat. Ser. III 2 (22) (1967), 73-81.
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