General solution and generalized Ulam-Hyers stability of a generalized n- type additive quadratic functional equation in Banach space and Banach algebra: direct and fixed point methods

Authors

  • S. Murthy

    GOVERNMENT ARTS COLLEGE,TIRUVANNAMALAI-606 603,TAMILNADU,INDIA.

  • M. Arunkumar

    Annai Veilankanni's College of Arts and Science,

  • V. Govindan

    GOVERNMENT ARTS COLLEGE,TIRUVANNAMALAI-606 603,TAMILNADU,INDIA.

How to Cite

Murthy, S., Arunkumar, M., & Govindan, V. (2015). General solution and generalized Ulam-Hyers stability of a generalized n- type additive quadratic functional equation in Banach space and Banach algebra: direct and fixed point methods. International Journal of Advanced Mathematical Sciences, 3(1), 25-64. https://doi.org/10.14419/ijams.v3i1.4402

DOI:

https://doi.org/10.14419/ijams.v3i1.4402

Keywords:

Additive Functional Equations, Quadratic Functional Equations, Mixed Type Functional Equations, Banach Space, Banach Algebra, Generalized Ulam-Hyers Stability, Fixed Point.

Abstract

In this paper, the authors introduce and investigate the general solution and generalized Ulam-Hyers stability of a generalized n-type additive-quadratic functional equation.


g(x + 2y; u + 2v) + g(x ô€€€ 2y; u ô€€€ 2v) = 4[g(x + y; u + v) + g(x ô€€€ y; u ô€€€ v)] ô€€€ 6g(x; u)
+ g(2y; 2v) + g(ô€€€2y;ô€€€2v) ô€€€ 4g(y; v) ô€€€ 4g(ô€€€y;ô€€€v)

Where  is a positive integer with , in Banach Space and Banach Algebras using direct and fixed point methods.

References

  1. [1] J. Aczel and J. Dhombres, Functional Equations in Several Variables, CambridgeUniversityPress, 1989. http://dx.doi.org/10.1017/CBO9781139086578.

    [2] T. Aoki, on the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan, 2 (1950), 64-66. http://dx.doi.org/10.2969/jmsj/00210064.

    [3] M. Arunkumar, S. Jayanthi, S. Hema Latha, Stability of quadratic derivations of Arun- quadratic functional equation, International Journal Mathematical Sciences and Engineering Applications, Vol. 5 No. V, Sept. (2011), 433-443.

    [4] M. Arunkumar, S. Karthikeyan, Solution and stability of -dimensional mixed Type additive and quadratic functional equation, Far East Journal of Applied Mathematics, Volume 54, Number 1, (2011), 47-64.

    [5] M. Arunkumar, John M. Rassias, On the generalized Ulam-Hyers stability of an AQ- mixed type functional equation with counter examples, Far East Journal of Applied Mathematics, Volume 71, No. 2, (2012), 279-305.

    [6] M. Arunkumar, Solution and stability of modified additive and quadratic functional equation in generalized 2-normed spaces, International Journal Mathematical Sciences and Engineering Applications, Vol. 7 No. I (January, 2013), 383-391.

    [7] M. Arunkumar, Generalized Ulam - Hyers stability of derivations of a AQ - functional equation, "Cubo A Mathematical Journal" dedicated to Professor Gaston M.N'Guérékata on the occasion of his 60th Birthday Vol.15, No 01, (2013), 159–169.

    [8] M. Arunkumar, P. Agilan, Additive Quadratic functional equation are Stable in Banach space: A Direct Method, Far East Journal of Applied Mathematics, Volume 80, No. 1, (2013), 105 – 121.

    [9] M. Arunkumar, G.Shobana, S. Hemalatha, Ulam – Hyers, Ulam – Trassias, Ulam-Grassias, Ulam – Jrassias Stabilities of A Additive – Quadratic Mixed Type Functional Equation In Banach Spaces, Proceedings of International Conference on Mathematics and Computing, ICMCE – 2013, 604 – 611, ISBN 978-93-82338-89-5.

    [10] M. Arunkumar, P. Agilan, C. Devi Shyamala Mary, Permanence of A Generalized AQ Functional Equation In Quasi-Beta Normed Spaces, International Journal of pure and applied Mathematics(Accepted) .

    [11] M. Arunkumar, Perturbation of n Dimensional AQ - mixed type Functional Equation via Banach Spaces and Banach Algebra: Hyers Direct and Alternative Fixed Point Methods, International Journal of Advanced Mathematical Sciences (IJAMS), Vol. 2 (1), (2014), 34-56.

    [12] M. Arunkumar, P. Agilan, Random stability of a additive quadratic Functional equation, Proceedings of International Conference on Applied Mathematical Models, ICAMMA 2014, 271 – 278.

    [13] M. Arunkumar, P. Agilan, C. Devi Shyamala Mary, Permanence of A Generalized AQ Functional Equation In Quasi-Beta Normed Spaces, A Fixed Point Approach, Proceedings of the International Conference on Mathematical Methods and Computation, Jamal Academic Research Journal an Interdisciplinary, (February 2014), 315-324.

    [14] C. Baak, D. Boo, Th.M. Rassias, Generalized additive mapping in Banach modules and isomorphism between C*-algebras, J. Math. Anal. Appl. 314, (2006), 150-161. http://dx.doi.org/10.1016/j.jmaa.2005.03.099.

    [15] R. Badora, on approximate derivations, Math. Inequal. Appl. 9 (2006), no.1, 167-173.

    [16] C.Borelli, G.L.Forti, on a general Hyers-Ulam stability, Internat J.Math.Math.Sci, 18 (1995), 229-236. http://dx.doi.org/10.1155/S0161171295000287.

    [17] L. Brown and G. Pedersen, C*-algebras of real rank zero, J. Funct. Analysis, 99, (1991), 138-149. http://dx.doi.org/10.1016/0022-1236(91)90056-B.

    [18] I.S. Chang, E.H. Lee, H.M. Kim, On the Hyers-Ulam-Rassias stability of quadratic functional equations, Math. Ineq. Appl., 6(1) (2003), 87-95.

    [19] P.W.Cholewa, Remarks on the stability of functional equations, Aequationes Math., 27 (1984), 76-86. http://dx.doi.org/10.1007/BF02192660.

    [20] S.Czerwik, on the stability of the quadratic mappings in normed spaces, Abh.Math.Sem.Univ Hamburg., 62 (1992), 59-64. http://dx.doi.org/10.1007/BF02941618.

    [21] S. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific, River Edge, NJ, 2002.

    [22] M. Eshaghi Gordji, H. Khodaie, Solution and stability of generalized mixed type cubic, quadratic and additive functional equation in Quasi-Banach spaces, arxiv: 0812. 2939v1 Math FA, 15 Dec 2008.

    [23] M. Eshaghi Gordji, H. Khodaei, J.M. Rassias, Fixed point methods for the stability of general quadratic functional equation, Fixed Point Theory, 12 (2011), no. 1, 71-82.

    [24] P. Gavruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl., 184 (1994), 431-436. http://dx.doi.org/10.1006/jmaa.1994.1211.

    [25] M.E. Gordji, J.M. Rassias,N. Ghobadipour, Generalized Hyers-Ulam stability of the -derivations, Abs. Appl. Anal. Volume 2009, Article ID 437931.

    [26] M.E. Gordji, S. Kaboli Gharetapeh, M.B. Savadkouhi, M.Aghaei, T.Karimi, On Cubic Derivations, Int. Journal of Math. Analysis, Vol.4, 2010, no.51, 2501-2514.

    [27] D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci., U.S.A.,27 (1941), 222-224. http://dx.doi.org/10.1073/pnas.27.4.222.

    [28] D. H. Hyers, G. Isac and Th. M. Rassias, Stability of functional equations in several variables, Birkhauser, Basel, 1998. http://dx.doi.org/10.1007/978-1-4612-1790-9.

    [29] K.W. Jun andD.W.Park, Almost derivations on the Banach algebra Cn[0,1], Bull. Korean Math.Soc.vol 33, No.3 (1996), 359-366.

    [30] K.W. Jun, H.M. Kim, On the Hyers-Ulam-Rassias stability of a generalized quadratic and additive type functional equation, Bull. Korean Math. Soc. 42 (1) (2005), 133-148. http://dx.doi.org/10.4134/BKMS.2005.42.1.133.

    [31] K.W. Jun and H.M. Kim, on the stability of an n-dimensional quadratic and additive type functional equation, Math. Ineq. Appl 9 (1) (2006), 153-165.

    [32] S.M. Jung, On the Hyers-Ulam stability of the functional equations that have the quadratic property, J. Math. Anal. Appl. 222 (1998), 126-137. http://dx.doi.org/10.1006/jmaa.1998.5916.

    [33] S.M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press,PlanHarbor, 2001.

    [34] Y.S. Jung, The Hyers-Ulam-Rassias Stability of module left derivations, J. Math. Anal. Appl., doi: 10.1016/j.jmaa.2007.07.003, 1-9. http://dx.doi.org/10.1016/j.jmaa.2007.07.003.

    [35] Pl. Kannappan, Quadratic functional equation and inner product spaces, Results Math., 27 (1995), 368-372. http://dx.doi.org/10.1007/BF03322841.

    [36] Pl. Kannappan, Functional Equations and Inequalities with Applications, Springer Monographs in Mathematics, 2009. http://dx.doi.org/10.1007/978-0-387-89492-8.

    [37] B.Margoils, J.B.Diaz, A fixed point theorem of the alternative for contractions on a generalized complete metric space, Bull.Amer. Math. Soc., (1968), 305-309.

    [38] Matina J. Rassias, M. Arunkumar,S. Ramamoorthi, Stability of the Leibniz additive-quadratic functional equation in Quasi-Beta normed space: Direct and fixed point methods, Journal of Concrete and Applicable Mathematics (JCAAM), Vol. 14 No. 1-2, (2014), 22 – 46.

    [39] S. Murthy, M. Arunkumar, G. Ganapathy, P. Rajarethinam, Stability of mixed type additive quadratic functional equation in Random Normed space, International Journal of Applied Mathematics (IJAM), Vol. 26. No. 2 (2013), 123-136. http://dx.doi.org/10.12732/ijam.v26i2.3.

    [40] A. Najati and M.B. Moghimi, On the Stability of a quadratic and additive functional equation, J. Math. Aanl. Appl., 337 (2008), 399-415. http://dx.doi.org/10.1016/j.jmaa.2007.03.104.

    [41] C. Park, on the stability of the linear mapping in Banach modules, J. Math. Aanl. Appl., 275 (2002), 711-720. http://dx.doi.org/10.1016/S0022-247X(02)00386-4.

    [42] C. Park, Linear functional equation in Banach modules over a C*-algebra, Acta. Appl. Math. 77 (2003), 125-161. http://dx.doi.org/10.1023/A:1024014026789.

    [43] C. Park, Linear derivations on Banach algebras, Funct. Anal. Appl. 9 (2004), no.3, 359-368.

    [44] C. Park, Fixed points and Hyers-Ulam-Rassias stability of Cauchy-Jensen functional equation in Banach algebras, Fixed Point Theory and Applications, 2007, Art ID 50175.

    [45] C. Park and J.Hou, Homomorphism and derivations in C*-algebras, Abstract Appl. Anal. 2007, Art. Id 80630.

    [46] J.M. Rassias, on approximately of approximately linear mappings by linear mappings, J. Funct. Anal. USA, 46, (1982) 126-130. http://dx.doi.org/10.1016/0022-1236(82)90048-9.

    [47] J.M. Rassias, H.M. Kim, Generalized Hyers-Ulam stability for general additive functional equations in quasi- -normed spaces, J. Math. Anal. Appl., 356 (2009), no. 1, 302-309.

    [48] Th. M. Rassias, on the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72 (1978), 297-300. http://dx.doi.org/10.1090/S0002-9939-1978-0507327-1.

    [49] Th.M.Rassias, on the stability of the functional equations in Banach spaces, J. Math. Anal. Appl., 251, (2000), 264-284. http://dx.doi.org/10.1006/jmaa.2000.7046.

    [50] Th. M. Rassias, Functional Equations, Inequalities and Applications, Kluwer Acedamic Publishers, Dordrecht, Bostan London, 2003. http://dx.doi.org/10.1007/978-94-017-0225-6.

    [51] K. Ravi, M.Arunkumar and J.M. Rassias, On the Ulam stability for the orthogonally general Euler-Lagrange type functional equation, International Journal of Mathematical Sciences, 2008 Vol.3, No. 08, 36-47.

    [52] S. M. Ulam, Problems in Modern Mathematics, Science Editions, Wiley,New York, 1964.

    [53] G.Zamani Eskandani, Stability of a mixed additive and quadratic functional equation in non-Archimedean Banach modules, Taiwanese journal of mathematics, Vol.14, No.4,pp. 1309-1324 August 2010.

Downloads

How to Cite

Murthy, S., Arunkumar, M., & Govindan, V. (2015). General solution and generalized Ulam-Hyers stability of a generalized n- type additive quadratic functional equation in Banach space and Banach algebra: direct and fixed point methods. International Journal of Advanced Mathematical Sciences, 3(1), 25-64. https://doi.org/10.14419/ijams.v3i1.4402