Browse > Article
http://dx.doi.org/10.7468/jksmeb.2015.22.4.343

ORTHOGONALITY AND LINEAR MAPPINGS IN BANACH MODULES  

YUN, SUNGSIK (DEPARTMENT OF FINANCIAL MATHEMATICS, HANSHIN UNIVERITY)
Publication Information
The Pure and Applied Mathematics / v.22, no.4, 2015 , pp. 343-357 More about this Journal
Abstract
Using the fixed point method, we prove the Hyers-Ulam stability of lin- ear mappings in Banach modules over a unital C*-algebra and in non-Archimedean Banach modules over a unital C*-algebra associated with the orthogonally Cauchy- Jensen additive functional equation.
Keywords
Hyers-Ulam stability; orthogonally Cauchy-Jensen additive functional equation; fixed point; non-Archimedean Banach module over C*-algebra, orthogonality space;
Citations & Related Records
연도 인용수 순위
  • Reference
1 D. Deses: On the representation of non-Archimedean objects. Topology Appl. 153 (2005), 774-785.   DOI
2 ______: Stability of Functional Equations of Ulam-Hyers-Rassias Type. Hadronic Press, Palm Harbor, Florida, 2003.
3 ______: Functional Equations and Inequalities in Several Variables. World Scientific Publishing Company, New Jersey, London, Singapore and Hong Kong, 2002.
4 S. Czerwik: On the stability of the quadratic mapping in normed spaces. Abh. Math. Sem. Univ. Hamburg 62 (1992), 59-64.   DOI
5 P.W. Cholewa: Remarks on the stability of functional equations. Aequationes Math. 27 (1984), 76-86.   DOI
6 S.O. Carlsson: Orthogonality in normed linear spaces. Ark. Mat. 4 (1962), 297-318.   DOI
7 ______: Fixed point methods for the generalized stability of functional equations in a single variable. Fixed Point Theory and Applications 2008, Art. ID 749392 (2008).
8 ______: On the stability of the Cauchy functional equation: a fixed point approach. Grazer Math. Ber. 346 (2004), 43-52.
9 L. Cădariu & V. Radu: Fixed points and the stability of Jensen's functional equation, J. Inequal. Pure Appl. Math. 4, no. 1, Art. ID 4 (2003).
10 G. Birkhoff: Orthogonality in linear metric spaces. Duke Math. J. 1 (1935), 169-172.   DOI
11 ______: Orthogonality in normed linear spaces: a survey II. Relations between main orthogonalities. Extracta Math. 4 (1989), 121-131.
12 J. Alonso & C. Benítez: Orthogonality in normed linear spaces: a survey I. Main properties. Extracta Math. 3 (1988), 1-15.
13 ______: Orthogonality and linear functionals in normed linear spaces. Trans. Amer. Math. Soc. 61 (1947), 265-292.   DOI
14 R.C. James: Orthogonality in normed linear spaces. Duke Math. J. 12 (1945), 291-302.   DOI
15 G. Isac & Th.M. Rassias: Stability of ψ-additive mappings: Appications to nonlinear analysis. Internat. J. Math. Math. Sci. 19 (1996), 219-228.   DOI
16 D.H. Hyers, G. Isac & Th.M. Rassias: Stability of Functional Equations in Several Variables. Birkhäuser, Basel, 1998.
17 D.H. Hyers: On the stability of the linear functional equation. Proc. Nat. Acad. Sci. U.S.A. 27 (1941), 222-224.   DOI
18 K. Hensel: Ubereine news Begrundung der Theorie der algebraischen Zahlen. Jahresber. Deutsch. Math. Verein 6 (1897), 83-88.
19 S. Gudder & D. Strawther: Orthogonally additive and orthogonally increasing functions on vector spaces. Pacific J. Math. 58 (1975), 427-436.   DOI
20 R. Ger & J. Sikorska: Stability of the orthogonal additivity. Bull. Polish Acad. Sci. Math. 43 (1995), 143-151.
21 M. Fochi: Functional equations in A-orthogonal vectors. Aequationes Math. 38 (1989), 28-40.   DOI
22 F. Drljević: On a functional which is quadratic on A-orthogonal vectors. Publ. Inst. Math.(Beograd) 54 (1986), 63-71.
23 C.R. Diminnie: A new orthogonality relation for normed linear spaces. Math. Nachr. 114 (1983), 197-203.   DOI
24 J. Diaz & B. Margolis: A fixed point theorem of the alternative for contractions on a generalized complete metric space. Bull. Amer. Math. Soc. 74 (1968), 305-309.   DOI
25 P.J. Nyikos: On some non-Archimedean spaces of Alexandrof and Urysohn. Topology Appl. 91 (1999), 1-23.   DOI
26 M.S. Moslehian & Gh. Sadeghi: A Mazur-Ulam theorem in non-Archimedean normed spaces. Nonlinear Anal.-TMA 69 (2008), 3405-3408.   DOI
27 R.V. Kadison & J.R. Ringrose: Fundamentals of the Theory of Operator Algebras. Academic Press, New York, 1983.
28 S. Jung: Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis. Hadronic Press, Palm Harbor, Florida, 2001.
29 M.S. Moslehian & Th.M. Rassias: Orthogonal stability of additive type equations. Aequationes Math. 73 (2007), 249-259.   DOI
30 ______: On the stability of the orthogonal Pexiderized Cauchy equation. J. Math. Anal. Appl. 318 (2006), 211-223.   DOI
31 M.S. Moslehian: On the orthogonal stability of the Pexiderized quadratic equation. J. Difference Equat. Appl. 11 (2005), 999-1004.   DOI
32 M. Mirzavaziri & M.S. Moslehian: A fixed point approach to stability of a quadratic equation. Bull. Braz. Math. Soc. 37 (2006), 361-376.   DOI
33 D. Miheţ & V. Radu: On the stability of the additive Cauchy functional equation in random normed spaces. J. Math. Anal. Appl. 343 (2008), 567-572.   DOI
34 A. Khrennikov: Non-Archimedean analysis: quantum paradoxes, dynamical systems and biological models. Mathematics and its Applications 427, Kluwer Academic Publishers, Dordrecht, 1997.
35 A.K. Katsaras & A. Beoyiannis: Tensor products of non-Archimedean weighted spaces of continuous functions. Georgian Math. J. 6 (1999), 33-44.   DOI
36 Th.M. Rassias (ed.): Functional Equations, Inequalities and Applications. Kluwer Academic Publishers, Dordrecht, Boston and London, 2003.
37 ______: On the stability of functional equations in Banach spaces. J. Math. Anal. Appl. 251 (2000), 264-284.   DOI
38 ______: The problem of S.M. Ulam for approximately multiplicative mappings. J. Math. Anal. Appl. 246 (2000), 352-378.   DOI
39 Th.M. Rassias, On the stability of the quadratic functional equation and its applications, Studia Univ. Babeş-Bolyai Math. 43 (1998), 89-124.
40 Th.M. Rassias: On the stability of the linear mapping in Banach spaces. Proc. Amer. Math. Soc. 72 (1978), 297-300.   DOI
41 V. Radu: The fixed point alternative and the stability of functional equations. Fixed Point Theory 4 (2003), 91-96.
42 A.G. Pinsker: Sur une fonctionnelle dans l’espace de Hilbert. C. R. (Dokl.) Acad. Sci. URSS, n. Ser. 20 (1938), 411-414.
43 C. Park & J. Park: Generalized Hyers-Ulam stability of an Euler-Lagrange type additive mapping. J. Difference Equat. Appl. 12 (2006), 1277-1288.   DOI
44 ______: Generalized Hyers-Ulam-Rassias stability of quadratic functional equations: a fixed point approach. Fixed Point Theory and Applications 2008, Art. ID 493751(2008).
45 C. Park: Fixed points and Hyers-Ulam-Rassias stability of Cauchy-Jensen functional equations in Banach algebras. Fixed Point Theory and Applications 2007, Art. ID50175 (2007).
46 L. Paganoni & J. Rätz: Conditional function equations and orthogonal additivity. Aequationes Math. 50 (1995), 135-142.   DOI
47 F. Vajzović: Über das Funktional H mit der Eigenschaft: (x, y) = 0 ⇒ H(x + y) +H(x - y) = 2H(x) + 2H(y). Glasnik Mat. Ser. III 2 (1967), no. 22, 73-81.
48 S.M. Ulam: Problems in Modern Mathematics. Wiley, New York, 1960.
49 Gy. Szabó: Sesquilinear-orthogonally quadratic mappings. Aequationes Math. 40 (1990), 190-200.   DOI
50 K. Sundaresan: Orthogonality and nonlinear functionals on Banach spaces. Proc. Amer. Math. Soc. 34 (1972), 187-190.   DOI
51 F. Skof: Proprietà locali e approssimazione di operatori. Rend. Sem. Mat. Fis. Milano 53 (1983), 113-129.   DOI
52 J. Rätz & Gy. Szabó: On orthogonally additive mappings IV . Aequationes Math. 38(1989), 73-85.   DOI
53 J. Rätz: On orthogonally additive mappings. Aequationes Math. 28 (1985), 35-49.   DOI