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http://dx.doi.org/10.4134/CKMS.2013.28.2.319

APPROXIMATE EULER-LAGRANGE-JENSEN TYPE ADDITIVE MAPPING IN MULTI-BANACH SPACES: A FIXED POINT APPROACH  

Moradlou, Fridoun (Department of Mathematics Sahand University of Technology)
Publication Information
Communications of the Korean Mathematical Society / v.28, no.2, 2013 , pp. 319-333 More about this Journal
Abstract
Using the fixed point method, we prove the generalized Hyers-Ulam-Rassias stability of the following functional equation in multi-Banach spaces: $${\sum_{1{\leq}i_<j{\leq}n}}\;f(\frac{r_ix_i+r_jx_j}{k})=\frac{n-1}{k}{\sum_{i=1}^n}r_if(x_i)$$.
Keywords
fixed point method; Hyers-Ulam-Rassias stability; multi-Banach spaces; Euler-Lagrange mapping;
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1 Th. M. Rassias, The problem of S. M. Ulam for approximately multiplicative mappings, J. Math. Anal. Appl. 246 (2000), no. 2, 352-378.   DOI   ScienceOn
2 L. Cadariu and V. Radu, On the stability of the Cauchy functional equation: a fixed point approach, Iteration theory (ECIT '02), 43-52, Grazer Math. Ber., 346, Karl-Franzens-Univ. Graz, Graz, 2004.
3 S. Czerwik, The stability of the quadratic functional equation, Stability of mappings of Hyers-Ulam type, 8191, Hadronic Press Collect. Orig. Artic., Hadronic Press, Palm Harbor, FL, 1994.
4 H. G. Dales and M. S. Moslehian, Stability of mappings on multi-normed spaces, Glasg. Math. J. 49 (2007), no. 2, 321-332.   DOI   ScienceOn
5 H. G. Dales and M. E. Polyakov, Multi-normed spaces and multi-Banach algebras, arxiv:1112.5148v2 (2012).
6 J. Diaz and 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
7 M. Eshaghi Gordji and S. Abbaszadeh, Stability of Cauchy-Jensen inequalities in fuzzy Banach spaces, Appl. Comput. Math. 11 (2012), no. 1, 27-36.
8 M. Eshaghi Gordji and H. Khodaei, Solution and stability of generalized mixed type cubic, quadratic and additive functional equation in quasi-Banach spaces, Nonlinear Anal. 71 (2009), no. 11, 5629-5643.   DOI   ScienceOn
9 M. Eshaghi Gordji, H. Khodaei, and Th. M. Rassias, Fixed points and stability for quadratic mappings in ${\beta}$-normed left Banach modules on Banach algebras, Results Math. 61 (2012), no. 3-4, 393-400.   DOI
10 M. Eshaghi Gordji and F. Moradlou, Approximate Jordan derivations on Hilbert C*-modules, Fixed Point Theory, (to appear).
11 M. Eshaghi Gordji, A. Najati, and A. Ebadian, Stability and superstability of Jordan homomorphisms and Jordan derivations on Banach algebras and C*-algebras: a fixed point approach, Acta Math. Sci. Ser. B Engl. Ed. 31 (2011), no. 5, 1911-1922.
12 T. Zhou Xu, J. M. Rassias, and W. Xin Xu, Generalized Ulam-Hyers stability of a general mixed AQCQ-functional equation in multi-Banach spaces: a fixed point approach, Eur. J. Pure Appl. Math. 3 (2010), no. 6, 1032-1047.
13 Th. M. Rassias, On the stability of functional equations in Banach spaces, J. Math. Anal. Appl. 251 (2000), no. 1, 264-284.   DOI   ScienceOn
14 F. Skof, Local properties and approximations of operators, Rend. Sem. Mat. Fis. Milano 53 (1983), 113-129.   DOI   ScienceOn
15 S. M. Ulam, A Collection of Mathematical Problems, Interscience Publ. New York, 1960.
16 L. Maligranda, A result of Tosio Aoki about a generalization of Hyers-Ulam stability of additive functions - a question of priority, Aequationes Math. 75 (2008), no. 3, 289-296.   DOI   ScienceOn
17 F. Moradlou, A. Najati, and H. Vaezi, Stability of homomorphisms and derivations on C*-ternary rings associated to an Euler-Lagrange type additive mapping, Results Math. 55 (2009), no. 3-4, 469-486.   DOI
18 F. Moradlou, H. Vaezi, and G. Z. Eskandani, Hyers-Ulam-Rassias stability of a quadratic and additive functional equation in quasi-Banach spaces, Mediterr. J. Math. 6 (2009), no. 2, 233-248.   DOI
19 M. S. Moslehian, Approximate C*-ternary ring homomorphisms, Bull. Braz. Math. Soc. 38 (2007), no. 4, 611-622.   DOI
20 F. Moradlou, H. Vaezi, and C. Park, Fixed points and stability of an additive functional equation of n-Apollonius type in C*-algebras, Abstr. Appl. Anal. 2008 (2008), Article ID 672618, 13 pages, 2008. doi:10.1155/2008/672618.   DOI   ScienceOn
21 C. Park and Th. M. Rassias, Hyers-Ulam stability of a generalized Apollonius type quadratic mapping, J. Math. Anal. Appl. 322 (2006), no. 1, 371-381.   DOI   ScienceOn
22 M. S. Moslehian, K. Nikodem, and D. Popa, Asymptotic aspect of the quadratic functional equation in multi-normed spaces, J. Math. Anal. Appl. 355 (2009), no. 2, 717-724.   DOI   ScienceOn
23 C. Park, On the stability of the linear mapping in Banach modules, J. Math. Anal. Appl. 275 (2002), no. 2, 711-720.   DOI   ScienceOn
24 C. Park, Fixed points and Hyers-Ulam-Rassias stability of Cauchy-Jensen functional equations in Banach algebras, Fixed Point Theory Appl. 2007 (2007), Art. ID 50175, 15 pp.
25 C. Park, Stability of an Euler-Lagrange-Rassias type additive mapping, Int. J. Appl. Math. Stat. 7 (2007), 101-111.
26 C. Park, Hyers-Ulam-Rassias stability of a generalized Apollonius-Jensen type additive mapping and isomorphisms between C*-algebras, Math. Nachr. 281 (2008), no. 3, 402-411.   DOI   ScienceOn
27 A. Pietrzyk, Stability of the Euler-Lagrange-Rassias functional equation, Demonstratio Mathematica 39 (2006), no. 3, 523-530.
28 V. Radu, The fixed point alternative and stability of functional equations, Fixed Point Theory 4 (2003), no. 1, 91-96.
29 J. M. Rassias, On approximation of approximately linear mappings by linear mappings, J. Funct. Anal. 46 (1982), no. 1, 126-130.   DOI
30 J. M. Rassias, On approximation of approximately linear mappings by linear mappings, Bull. Sci. Math. 108 (1984), no. 4, 445-446.
31 J. M. Rassias and M. J. Rassias, Refined Ulam stability for Euler-Lagrange type mappings in Hilbert spaces, Int. J. Appl. Math. Stat. 7 (2007), 126-132.
32 Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), no. 2, 297-300.   DOI   ScienceOn
33 L. Cadariu and V. Radu, Fixed points and the stability of quadratic functional equations, An. Univ. Timisoara, Ser. Mat.-Inform. 41 (2003), no. 1, 25-48.
34 C. Borelli and G. L. Forti, On a general Hyers-Ulam stability result, Internat. J. Math. Math. Sci. 18 (1995), no. 2, 229-236.   DOI   ScienceOn
35 D. G. Bourgin, Classes of transformations and bordering transformations, Bull. Amer. Math. Soc. 57 (1951), 223-237.   DOI
36 L. Cadariu and V. Radu, Fixed points and the stability of Jensen's functional equation, J. Inequal. Pure Appl. Math. 4 (2003), no. 1, Art. 4, 7 pp.
37 D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A. 27 (1941), 222-224.   DOI   ScienceOn
38 M. Eshaghi Gordji and M. Bavand Savadkouhi, Stability of a mixed type cubic-quartic functional equation in non-Archimedean spaces, Appl. Math. Lett. 23 (2010), no. 10, 1198-1202.   DOI   ScienceOn
39 M. Eshaghi Gordji and M. Bavand Savadkouhi, Stability of cubic and quartic functional equations in non-Archimedean spaces, Acta Appl. Math. 110 (2010), no. 3, 1321-1329.   DOI   ScienceOn
40 P. Gavruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), no. 3, 431-436.   DOI   ScienceOn
41 S.-M. Jung and J. M. Rassias, A fixed point approach to the stability of a functional equation of the spiral of Theodorus, Fixed Point Theory and Applications, 2008 (2008), Art. ID: 945010, 7 pp.   DOI   ScienceOn
42 H.-M. Kim and J. M. Rassias, Generalization of Ulam stability problem for Euler-Lagrange quadratic mappings, J. Math. Anal. Appl. 336 (2007), no. 1, 277-296.   DOI   ScienceOn
43 Y.-S. Lee and S.-Y. Chung, Stability of an Euler-Lagrange-Rassias equation in the spaces of generalized functions, Appl. Math. Lett., 21 (2008), no. 7, 694-700.   DOI   ScienceOn
44 T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64-66.   DOI