Generalized Stabilities of Euler-Lagrange-Jensen (a,b)-Sextic Functional Equations in Quasi-β-Normed Spaces

Article Sidebar

PDF
Cite as:
John Michael Rassias, Krishnan Ravi, Beri Venkatachalapathy Senthil Kumar, Generalized Stabilities of Euler-Lagrange-Jensen (a,b)-Sextic Functional Equations in Quasi-β-Normed Spaces, Int. J. Anal. Appl., 14 (2) (2017), 167-174.

Main Article Content

John Michael Rassias, Krishnan Ravi, Beri Venkatachalapathy Senthil Kumar

Abstract

The aim of this paper is to investigate generalized Ulam-Hyers stabilities of the following Euler-Lagrange-Jensen-$(a,b)$-sextic functional equation

$$

f(ax+by)+f(bx+ay)+(a-b)^6\left[f\left(\frac{ax-by}{a-b}\right)+f\left(\frac{bx-ay}{b-a}\right)\right]\\

= 64(ab)^2\left(a^2+b^2\right)\left[f\left(\frac{x+y}{2}\right)+f\left(\frac{x-y}{2}\right)\right]\\

+2\left(a^2-b^2\right)\left(a^4-b^4\right)[f(x)+f(y)]

$$

where $a\neq b$, such that $k\in \mathbb{R}$; $k=a+b\neq 0,\pm1$ and $\lambda=1+(a-b)^6-2\left(a^6+b^6\right)-62(ab)^2\left(a^2+b^2\right)\neq 0$, in quasi-$\beta$-normed spaces by using fixed point method. In particular, we prove generalized stabilities involving the sum of powers of norms, product of powers of norms and the mixed product-sum of powers of norms of the above functional equation in quasi-$\beta$-normed spaces by using fixed point method. A counter-example for a singular case is also indicated.

Article Details

References

  1. J. Aczel, Lectures on Functional Equations and their Applications, Vol. 19, Academic Press, New York, 1966.
  2. J. Aczel, Functional Equations, History, Applications and Theory, D. Reidel Publ. Company, 1984.
  3. C. Alsina, On the stability of a functional equation, General Inequalities, Vol. 5, Oberwolfach, Birkhauser, Basel, (1987), 263-271.
  4. T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan, 2 (1950), 64-66.
  5. L. Cadariu and V. Radu, Fixed points and stability for functional equations in probabilistic metric and random normed spaces, Fixed Point Theory Appl. 2009 (2009), Art. ID 589143, 18 pages.
  6. B. Bouikhalene and E. Elquorachi, Ulam-Gavruta-Rassias stability of the Pexider functional equation, Int. J. Appl. Math. Stat., 7 (2007), 7-39.
  7. I. S. Chang and H. M. Kim, On the Hyers-Ulam stability of quadratic functional equations, J. Ineq. Appl. Math. 33 (2002), 1-12.
  8. I. S. Chang and Y. S. Jung, Stability of functional equations deriving from cubic and quadratic functions, J. Math. Anal. Appl., 283 (2003), 491-500.
  9. J. K. Chung and P. K. Sahoo, On the general solution of a quartic functional equation, Bull. Korean Math. Soc., 40 (4) (2003), 565-576.
  10. S. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific Publishing Company, New Jersey, London, Singapore and Hong Kong, 2002.
  11. M. Eshaghi Gordji, S. Zolfaghari, J. M. Rassias and M. B. Savadkouhi, Solution and stability of a mixed type cubic and quartic functional equation in quasi-Banach spaces, Abst. Appl. Anal., 2009 (2009), Art. ID 417473.
  12. Z. Gajda, On stability of additive mappings, Int. J. Math. Math. Sci. 14 (3) (1991), 431-434.
  13. N. Ghobadipour and C. Park, Cubic-quartic functional equations in fuzzy normed spaces, Int. J. Nonlinear Anal. Appl., 1 (2010), 12-21.
  14. P. Gˇ avrutˇ a, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl., 184 (1994), 431-436.
  15. Heejeong Koh and Dongseung Kang, Solution and stability of Euler-Lagrange-Rassias quartic functional equations in various quasi-normed spaces, Abstr. Appl. Anal., 2013 (2013), Art. ID 908168, 8 pages.
  16. D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci., U.S.A., 27 (1941), 222-224.
  17. D. H. Hyers, G. Isac and Th. M. Rassias, Stability of Functional Equations in Several Variables, Birkhauser, Basel, 1998.
  18. G. Isac and Th. M. Rassias, Stability of ψ-additive mappings: applications to nonlinear analysis, Int. J. Math. Math. Sci., 19(2) (1996), 219-228.
  19. K. W. Jun and H. M. Kim, On the stability of Euler-Lagrange type cubic mappings in quasi-Banach spaces, J. Math. Anal. Appl. 332(2) (2007), 1335-1350.
  20. S. M. Jung, Hyers-Ulam-Rassias stability of functional equations in Mathematical Analysis, Hardonic press, Palm Harbor, 2001.
  21. Pl. Kannappan, Quadratic Functional Equation and Inner Product Spaces, Results Math. 27(3-4) (1995), 368-372.
  22. J. R. Lee, D. Y. Shin and C. Park, Hyers-Ulam stability of functional equations in matrix normed spaces, J. Inequal. Appl. 2013 (2013), Art. ID 22.
  23. E. Movahednia, Fixed point and generalized Hyers-Ulam-Rassias stability of a quadratic functional equation, J. Math. Comput. Sci., 6 (2013), 72-78.
  24. A. Najati and C. Park, Cauchy-Jensen additive mappings in quasi-Banach algebras and its applications, J. Nonlinear Anal. Appl., 2013 (2013), Art. ID jnaa-00191.
  25. P. Nakmahachalasint, Hyers-Ulam-Rassias and Ulam-Gavruta-Rassias stabilities of additive functional equation in several variables, Int. J. Math. Math. Sci. 2007 (2007) Art. ID 13437, 6 pages.
  26. C. Park, Fixed points and the stability of an AQCQ-functional equation in non-Archimedean normed spaces, Abstr. Appl. Anal., 2010 (2010) Art. ID 849543, 15 pages.
  27. C. G. Park, Stability of an Euler-Lagrange-Rassias type additive mapping, Int. J. Appl. Math. Stat., 7 (2007), 101-111.
  28. A. Pietrzyk, Stability of the Euler-Lagrange-Rassias functional equation, Demonstr. Math., 39(3) (2006), 523 - 530.
  29. J. M. Rassias, On approximately of approximately linear mappings by linear mappings, J. Funct. Anal. 46 (1982), 126-130.
  30. J. M. Rassias, On approximately of approximately linear mappings by linear mappings, Bull. Sci. Math., 108 (4) (1984), 445-446.
  31. J. M. Rassias, On a new approximation of approximately linear mappings by linear mappings, Discuss. Math., 7 (1985), 193-196.
  32. J. M. Rassias, On the stability of the Euler-Lagrange functional equation, Chinese J. Math., 20 (1992), 185-190.
  33. J. M. Rassias, On the stability of the non-linear Euler-Lagrange functional equation in real normed linear spaces, J. Math. Phys. Sci., 28 (1994), 231-235.
  34. J .M. Rassias, On the stability of the general Euler-Lagrange functional equation, Demonstr. Math., 29 (1996), 755-766.
  35. J. M. Rassias, Solution of the Ulam stability problem for Euler-Lagrange quadratic mappings, J. Math. Anal. Appl., 220 (1998), 613-639.
  36. J. M. Rassias, On the stability of the multi-dimensional Euler-Lagrange functional equation, J. Indian Math. Soc., 66 (1999), 1-9.
  37. J. M. Rassias, Solution of the Ulam stability problem for quartic mappings, Glasnic Matematicki. Serija III, 34(2) (1999), 243-252.
  38. J. M. Rassias, Solution of the Ulam stablility problem for cubic mappings, Glasnik Matematicki. Serija III, 36(1) (2001), 63-72.
  39. K. Ravi, M. Arunkumar and J. M. Rassias, Ulam stability for the orthogonally general Euler-Lagrange type functional equation, Int. J. Math. Stat. 3(A08) (2008), 36-46.
  40. K. Ravi, J. M. Rassias, M. Arunkumar and R. Kodandan, Stability of a generalized mixed type additive, quadratic, cubic and quartic functional equation, J. Inequ. Pure Appl. Math., 10(4) (2009), 1-29.
  41. Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72 (1978), 297-300.
  42. S. M. Ulam, Problems in Modern Mathematics, Rend. Chap. VI, Wiley, New York, 1960.
  43. T. Z. Xu, J. M. Rassias and W. X. Xu, A fixed point approach to the stability of a general mixed AQCQ-functional equation in non-Archimedean normed spaces, Discrete Dyn. Nat. Soc. 2010 (2010) Art. ID 812545, 24 pages.
  44. T. Z. Xu, J. M. Rassias, M. J. Rassias and W. X. Xu, A fixed point approach to the stability of quintic and sextic functional equations in quasi-β-normed spaces,J. Inequal. Appl., 2010 (2010), Art. ID 423231.
  45. T. Z. Xu, J. M. Rassias and W. X. Xu, A fixed point approach to the stability of a general mixed additive-cubic functional equation in quasi fuzzy normed spaces, Int. J. Phys. Sci. 6(2) (2011), 313-324.