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In this chapter, we recall some definitions and results which will be used later on in the book.

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References

  1. Abellanas, L., Alonso, L.: A general setting for Casimir invariants. J. Math. Phys. 16, 1580–1584 (1975)

    MATH  Google Scholar 

  2. Aczél, J.: The general solution of two functional equations by reduction to functions additive in two variables and with aid of Hamel-bases. Glasnik Mat.-Fiz. Astronom. Drustvo Mat. Fiz. Hrvatske 20, 65–73 (1965)

    MATH  Google Scholar 

  3. Aczél, J.: Lectures on Functional equations and Their Applications. Academic, New York/London (1966)

    MATH  Google Scholar 

  4. Aczél, J., Dhombres, J.: Functional Equations in Several Variables. Cambridge University Press, Cambridge (1989)

    MATH  Google Scholar 

  5. Almira, J.M., Luther, U.: Inverse closedness of approximation algebras. J. Math. Anal. Appl. 314, 30–44 (2006)

    MATH  MathSciNet  Google Scholar 

  6. Andrews, L.C.: Special Functions for Engineers and Applied Mathematicians. MacMilan, New York (1985)

    Google Scholar 

  7. Antoine, J.P., Inoue, A., Trapani, C.: O -dynamical systems and ∗-derivations of unbounded operator algebras. Math. Nachr. 204, 5–28 (1999)

    MATH  MathSciNet  Google Scholar 

  8. Aoki, T.: On the stability of the linear transformation in Banach spaces. J. Math. Soc. Jpn. 2, 64–66 (1950)

    MATH  Google Scholar 

  9. Bae, J., Park, W.: Generalized Ulam-Hyers stability of C -ternary algebra 3-homomorphisms for a functional equation. J. Chungcheong Math. Soc. 24, 147–162 (2011)

    Google Scholar 

  10. Bagarello, F., Trapani, C.: CQ -algebras: structure properties. Publ. RIMS Kyoto Univ. 32, 85–116 (1996)

    MATH  MathSciNet  Google Scholar 

  11. Bagarello, F., Trapani, C.: Morphisms of certain Banach C -modules. Publ. RIMS Kyoto Univ. 36, 681–705 (2000)

    MATH  MathSciNet  Google Scholar 

  12. Bagarello, F., Inoue, A., Trapani, C.: Some classes of topological quasi ∗-algebras. Proc. Am. Math. Soc. 129, 2973–2980 (2001)

    MATH  MathSciNet  Google Scholar 

  13. Bagarello, F., Inoue, A., Trapani, C.: Exponentiating derivations of quasi-∗-algebras: possible approaches and applications. Int. J. Math. Math. Sci. 2005, 2805–2820 (2005)

    MATH  MathSciNet  Google Scholar 

  14. Bagarello, F., Trapani, C., Triolo, S.: Quasi-∗-algebras of measurable operators. Studia Math. 172, 289–305 (2006)

    MATH  MathSciNet  Google Scholar 

  15. Bahyrycz, A., Piszczek, M.: Hyperstability of the Jensen functional equation. Acta Math. Hung. 142, 353–365 (2014)

    MATH  MathSciNet  Google Scholar 

  16. Bahyrycz, A., Brzdȩk, J., Piszczek, M., Sikorska, J.: Hyperstability of the Fréchet equation and a characterization of inner product spaces. J. Funct. Spaces Appl. 2013, Article ID 496361, 6 pp. (2013)

    Google Scholar 

  17. Bahyrycz, A., Brzdȩk, J., Leśniak, Z.: On approximate solutions of the generalized Volterra integral equation. Nonlinear Anal. Real World Appl. 20, 59–66 (2014)

    MATH  MathSciNet  Google Scholar 

  18. Baker, J.A.: The stability of certain functional equations. Proc. Am. Math. Soc. 112, 729–732 (1991)

    MATH  Google Scholar 

  19. Baktash, E., Cho, Y.J., Jalili, M., Saadati, R., Vaezpour, S.M.: On the stability of cubic mappings and quadratic mappings in random normed spaces. J. Inequal. Appl. 2008, Article ID 902187, 11 pp. (2008)

    Google Scholar 

  20. Banach, S.: Sur les operations dans les ensembles abstraits et leur application aux euations integrales. Fundam. Math. 3, 133–181 (1922)

    MATH  Google Scholar 

  21. Bazunova, N., Borowiec, A., Kerner, R.: Universal differential calculus on ternary algebras. Lett. Math. Phys. 67, 195–206 (2004)

    MATH  MathSciNet  Google Scholar 

  22. Belaid, B., Elqorachi, E., Rassias, Th.M.: On the Hyers–Ulam stability of approximately Pexider mappings. Math. Inequal. Appl. 11, 805–818 (2008)

    MATH  MathSciNet  Google Scholar 

  23. Benyamini, Y., Lindenstrauss, J.: Geometric Nonlinear Functional Analysis, Vol. 1. Colloquium Publications, vol. 48. American Mathematical Society, Providence (2000)

    Google Scholar 

  24. Birkhoff, G.: Orthogonality in linear metric spaces. Duke Math. J. 1, 169–172 (1935)

    MathSciNet  Google Scholar 

  25. Brillouët-Belluot, N., Brzdek, J., Ciepliński, K.: On some developments in Ulam’s type stability. Abstr. Appl. Anal. 2012, Article ID 716936, 41 pp. (2012)

    Google Scholar 

  26. Brzdȩk, J.: Remarks on hyperstability of the Cauchy functional equation. Aequ. Math. 86, 255–267 (2013)

    Google Scholar 

  27. Brzdȩk, J., Ciepliński, K.: Hyperstability and superstability. Abstr. Appl. Anal. 2013, Article ID 401756, 13 pp. (2013)

    Google Scholar 

  28. Brzdȩk, J., Cădariu, L., Ciepliński, K.: Fixed point theory and the Ulam stability. J. Funct. Spaces 2014, Article ID 829419, 16 pp. (2014)

    Google Scholar 

  29. Brzdȩk, J., Ciepliński, K., Leśniak, Z.: On Ulam’s type stability of the linear equation and related issues. Discret. Dyn. Nat. Soc. 2014, Article ID 536791, 14 pp. (2014)

    Google Scholar 

  30. Cădariu, L., Radu, V.: Fixed points and the stability of Jensen’s functional equation. J. Inequal. Pure Appl. Math. 4(1), Article 4 (2003)

    Google Scholar 

  31. Cădariu, L., Radu, V.: Fixed point methods for the generalized stability of functional equations in a single variable. Fixed Point Theory Appl. 2008, Article ID 749392, 15 pp. (2008)

    Google Scholar 

  32. Carlsson, S.O.: Orthogonality in normed linear spaces. Ark. Mat. 4, 297–318 (1962)

    MATH  MathSciNet  Google Scholar 

  33. Castro, L.P., Ramos, A.: Hyers-Ulam-Rassias stability for a class of nonlinear Volterra integral equations. Banach J. Math. Anal. 3, 47–56 (2009)

    MathSciNet  Google Scholar 

  34. Cauchy, A.L.: Cours d’analyse de l’Ecole Polytechnique. Vol. 1, Analyse algebrique. Debure, Paris (1821)

    Google Scholar 

  35. Chahbi, A., Bounader, N.: On the generalized stability of d’Alembert functional equation. J. Nonlinear Sci. Appl. 6, 198–204 (2013)

    MATH  MathSciNet  Google Scholar 

  36. Cho, Y.J., Rassias, Th.M., Saadati, R.: Stability of Functional Equations in Random Normed Spaces. Springer, New York (2013)

    MATH  Google Scholar 

  37. Cho, Y.J., Saadati, R., Yang, Y.O.: Random-ternary algebras and application. J. Inequal. Appl. 2015, 26 (2015)

    Google Scholar 

  38. Cholewa, P.W.: Remarks on the stability of functional equations. Aequ. Math. 27, 76–86 (1984)

    MATH  MathSciNet  Google Scholar 

  39. Chu, H.Y., Kang, D.S., Rassias, Th.M.: On the stability of a mixed n-dimensional quadratic functional equation. Bull. Belg. Math. Soc.–Simon Stevin 15, 9–24 (2008)

    Google Scholar 

  40. Ciepliński, K.: Generalized stability of multi-additive mappings. Appl. Math. Lett. 23, 1291–1294 (2010)

    MATH  MathSciNet  Google Scholar 

  41. Ciepliński, K.: Stability of multi-additive mappings in β-Banach spaces. Nonlinear Anal. 75, 4205–4212 (2012)

    MATH  MathSciNet  Google Scholar 

  42. Ciepliński, K.: Applications of fixed point theorems to the Hyers-Ulam stability of functional equations-a Survey. Ann. Funct. Anal. 3, 151–164 (2012)

    MATH  MathSciNet  Google Scholar 

  43. Czerwik, S.: On the stability of the quadratic mapping in normed spaces. Abh. Math. Sem. Univ. Hambg. 62, 59–64 (1992)

    MATH  MathSciNet  Google Scholar 

  44. Dales, H.G.: Banach Algebras and Automatic Continuity. London Mathematical Society Monographs, New Series, vol. 24, Oxford University Press, Oxford (2000)

    Google Scholar 

  45. Dales, H.G., Moslehian, M.S.: Stability of mappings on multi-normed spaces. Glasg. Math. J. 49, 321–332 (2007)

    MATH  MathSciNet  Google Scholar 

  46. Dales, H.G., Polyakov, M.E.: Multi-normed spaces. Dissertationes Math. (Rozprawy Mat.) 488, 165 pp. (2012)

    Google Scholar 

  47. Diaz, J., Margolis, B.: A fixed point theorem of the alternative for contractions on a generalized complete metric space. Bull. Am. Math. Soc. 74, 305–309 (1968)

    MATH  MathSciNet  Google Scholar 

  48. Epifanio, G., Trapani, C.: Quasi-∗-algebras valued quantized fields. Ann. Inst. H. Poincaré 46, 175–185 (1987)

    MATH  MathSciNet  Google Scholar 

  49. Eshaghi Gordji, M., Khodaei, H.: Stability of Functional Equations. Lap Lambert Academic Publishing, Saarbrücken (2010)

    Google Scholar 

  50. Eshaghi Gordji, M., Ramezani, M., Cho, Y.J., Baghani, H.: Approximate Lie brackets: a fixed point approach. J. Inequal. Appl. 2012, 125 (2012)

    Google Scholar 

  51. Eskandani, G.Z., Gavruta, P.: Hyers-Ulam-Rassias stability of pexiderized Cauchy functional equation in 2-Banach spaces. J. Nonlinear Sci. Appl. 5(Special issue), 459–465 (2012)

    MATH  MathSciNet  Google Scholar 

  52. Eskandani, G.Z., Rassias, Th.M.: Hyers-Ulam-Rassias stability of derivations in proper JCQ -triples. Mediterr. J. Math. 10, 1391–1400 (2013)

    MATH  MathSciNet  Google Scholar 

  53. Faiziev, V.A., Rassias, Th.M., Sahoo, P.K.: The space of (ψ, γ)–additive mappings on semigroups. Trans. Am. Math. Soc. 354, 4455–4472 (2002)

    MATH  MathSciNet  Google Scholar 

  54. Forti, G.L.: An existence and stability theorem for a class of functional equations. Stochastica 4, 22–30 (1980)

    Google Scholar 

  55. Forti, G.L.: Comments on the core of the direct method for proving Hyers-Ulam stability of functional equations. J. Math. Anal. Appl. 295, 127–133 (2004)

    MATH  MathSciNet  Google Scholar 

  56. Fredenhagen, K., Hertel, J.: Local algebras of observables and pointlike localized fields. Commun. Math. Phys. 80, 555–561 (1981)

    MATH  MathSciNet  Google Scholar 

  57. Gajda, Z.: On stability of additive mappings. Int. J. Math. Math. Sci. 14, 431–434 (1991)

    MATH  MathSciNet  Google Scholar 

  58. Gao, Z.X., Cao, H.X., Zheng, W.T., Xu, L.: Generalized Hyers-Ulam-Rassias stability of functional inequalities and functional equations. J. Math. Inequal. 3, 63–77 (2009)

    MathSciNet  Google Scholar 

  59. Gauss, C.F.: Theoria moyus corporum caelestium. Perthes–Besser, Hamburg (1809)

    Google Scholar 

  60. Gǎvruta, P.: On the stability of some functional equations. In: Stability of Mappings of Hyers-Ulam Type, pp. 93–98. Hadronic Press. Palm Harbor (1994)

    Google Scholar 

  61. Gǎvruta, P.: On a problem of G. Isac and Th.M. Rassias concerning the stability of mappings. J. Math. Anal. Appl. 261, 543–553 (2001)

    Google Scholar 

  62. Grabiec, A.: The generalized Hyers-Ulam stability of a class of functional equations. Publ. Math. Debr. 48, 217–235 (1996)

    MATH  MathSciNet  Google Scholar 

  63. Hyers, D.H.: On the stability of the linear functional equation. Proc. Natl. Acad. Sci. USA 27, 222–224 (1941)

    MathSciNet  Google Scholar 

  64. Hyers, D.H.: The stability of homomorphisms and related topics. In: Rassias, Th.M. (ed.), Global Analysis Analysis on Manifolds. Teubner-Texte zur Mathematik, pp. 140–153. B.G. Teubner, Leipzig (1983)

    Google Scholar 

  65. Hyers, D.H., Rassias, Th.M.: Approximate homomorphisms. Aequ. Math. 44, 125–153 (1992)

    MATH  MathSciNet  Google Scholar 

  66. Hyers, D.H., Isac, G., Rassias, Th.M.: Stability of Functional Equations in Several Variables. Birkhäuser, Basel (1998)

    MATH  Google Scholar 

  67. Hyers, D.H., Isac, G., Rassias, Th.M.: On the asymptoticity aspect of Hyers-Ulam stability of mappings. Proc. Am. Math. Soc. 126, 425–430 (1998)

    MATH  MathSciNet  Google Scholar 

  68. Isac, G., Rassias, Th.M.: On the Hyers-Ulam stability of additive mappings. J. Approx. Theory 72, 131–137 (1993)

    MATH  MathSciNet  Google Scholar 

  69. Isac, G., Rassias, Th.M.: Stability of ψ-additive mappings: applications to nonlinear analysis. Int. J. Math. Math. Sci. 19, 219–228 (1996)

    MATH  MathSciNet  Google Scholar 

  70. Jacobson, N.: Lie Algebras. Dover, New York (1979)

    Google Scholar 

  71. Jesen, B., Karpf, J., Thorup, A.: Some functional equations in groups and rings. Math. Scand. 22, 257–265 (1968)

    MathSciNet  Google Scholar 

  72. Jung, S.-M.: On the Hyers-Ulam-Rassias stability of approximately additive mappings. J. Math. Anal. Appl. 204, 221–226 (1996)

    MATH  MathSciNet  Google Scholar 

  73. Jung, S.-M.: Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis. Hadronic Press, Palm Harbor (2001)

    MATH  Google Scholar 

  74. Jung, S.-M.: Hyers-Ulam stability of a system of first order linear differential equations with constant coefficients. J. Math. Anal. Appl. 320, 549–561 (2006)

    MATH  MathSciNet  Google Scholar 

  75. Jung, S.-M.: A fixed point approach to the stability of an equation of the square spiral. Banach J. Math. Anal. 1, 148–153 (2007)

    MATH  MathSciNet  Google Scholar 

  76. Jung, S.-M.: A fixed point approach to the stability of isometries. J. Math. Anal. Appl. 329, 879–890 (2007)

    MATH  MathSciNet  Google Scholar 

  77. Jung, S.-M.: A fixed point approach to the stability of a Volterra integral equation. Fixed Point theory Appl. 2007, Article ID 57064, 9 pp. (2007)

    Google Scholar 

  78. Jung, Y.S., Chang, I.S.: The stability of a cubic type functional equation with the fixed point alternative. J. Math. Anal. Appl. 306, 752–760 (2005)

    MATH  MathSciNet  Google Scholar 

  79. Jung, S.-M., Kim, T.S.: A fixed point approach to the stability of the cubic functional equation. Bol. Soc. Mat. Mexicana 12, 51–57 (2006)

    MATH  MathSciNet  Google Scholar 

  80. Jung, S.-M., Lee, Z.H.: A fixed point approach to the stability of quadratic functional equation with involution. Fixed Point Theory Appl. 2008, Article ID 732086, 11 pp. (2008)

    Google Scholar 

  81. Jung, S.-M., Rassias, Th.M.: Ulam’s problem for approximate homomorphisms in connection with Bernoulli’s differential equation. Appl. Math. Comput. 187, 223–227 (2007)

    MATH  MathSciNet  Google Scholar 

  82. Jung, S.-M.: Hyers-Ulam stability of linear differential equations of first order. Appl. Math. Lett. 22, 70–74 (2009)

    MATH  MathSciNet  Google Scholar 

  83. Jung, S.-M.: Hyers–Ulam–Rassias Stability of Functional Equations in Nonlinear Analysis. Springer Optimization and Its Applications, vol. 48. Springer, New York/Dordrecht/Heidelberg/London (2011)

    Google Scholar 

  84. Jung, S.-M., Rassias, J.M.: A fixed point approach to the stability of a functional equation of the spiral of Theodorus. Fixed Point Theory Appl. 2008, Article ID 945010, 7 pp. (2008)

    Google Scholar 

  85. Jung, S.-M., Rassias, M.Th.: A linear functional equation of third order associated to the Fibonacci numbers. Abstr. Appl. Anal. 2014, Article ID 137468, 7 pp. (2014)

    Google Scholar 

  86. Jung, S.-M., Kim, T.S., Lee, K.S.: A fixed point approach to the stability of quadratic functional equation. Bull. Korean Math. Soc. 43, 531–541 (2006)

    MATH  MathSciNet  Google Scholar 

  87. Jung, S.-M., Popa, D., Rassias, M.Th.: On the stability of the linear functional equation in a single variable on complete metric groups. J. Glob. Optim. 59, 165–171 (2014)

    MATH  MathSciNet  Google Scholar 

  88. Jung, S.-M., Rassias, M.Th., Mortici, C.: On a functional equation of trigonometric type. Appl. Math. Comput. 252, 294–303 (2015)

    MathSciNet  Google Scholar 

  89. Kenari, H.M., Saadati, R., Park, C.: Homomorphisms and derivations in C -ternary algebras via fixed point method. Adv. Differ. Equ. 2012, 13 pp. (2012)

    Google Scholar 

  90. Kerner, R.: The cubic chessboard. Geometry and physics. Class. Quantum Gravity 14(1A), A203–A225 (1997)

    MATH  MathSciNet  Google Scholar 

  91. Khodaei, H., Eshaghi Gordji, M., Kim, S.S., Cho, Y.J.: Approximation of radical functional equations related to quadratic and quartic mappings. J. Math. Anal. Appl. 397, 284–297 (2012)

    MathSciNet  Google Scholar 

  92. Kochanek, T., Lewicki, M.: Stability problem for number-theoretically multiplicative functions. Proc. Am. Math. Soc. 135, 2591–2597 (2007)

    MATH  MathSciNet  Google Scholar 

  93. Kuczma, M.: An Introduction to the Theory of Functional Equations and Inequalities. Uniwersytet Slaski, Warszawa/Krakow/Katowice (1985)

    MATH  Google Scholar 

  94. Lassner, G.: Algebras of unbounded operators and quantum dynamics. Physica 124 A, 471–480 (1984)

    Google Scholar 

  95. Lee, Y.H., Jung, S.-M., Rassias, M.Th.: On an n-dimensional mixed type additive and quadratic functional equation. Appl. Math. Comput. 228, 13–16 (2014)

    MathSciNet  Google Scholar 

  96. Legendre, A.M.: Elements de geometrie, Note II. Didot, Paris (1791)

    Google Scholar 

  97. Li, Y., Shen, Y.: Hyers-Ulam stability of nonhomogeneous linear differential equations of second order. Int. J. Math. Math. Sci. 2009, Article ID 576852, 7 pp. (2009)

    Google Scholar 

  98. Lu, G., Park, C.: Additive functional inequalities in Banach spaces. J. Inequal. Appl. 2012, 294 (2012)

    Google Scholar 

  99. Lungu, N., Popa, D.: Hyers-Ulam stability of a first order partial differential equation. J. Math. Anal. Appl. 385, 86–91 (2012)

    MATH  MathSciNet  Google Scholar 

  100. Luxemburg, W.A.J.: On the convergence of successive approximations in the theory of ordinary differential equations, II. Koninkl, Nederl. Akademie van Wetenschappen, Amsterdam, Proc. Ser. A (5) 61; Indag. Math. 20, 540–546 (1958)

    Google Scholar 

  101. Miheţ, D.: The fixed point method for fuzzy stability of the Jensen functional equation. Fuzzy Sets Syst. 160, 1663–1667 (2009)

    MATH  Google Scholar 

  102. Miheţ, D., Radu, V.: On the stability of the additive Cauchy functional equation in random normed spaces. J. Math. Anal. Appl. 343, 567–572 (2008)

    MATH  MathSciNet  Google Scholar 

  103. Mirzavaziri, M., Moslehian, M.S.: A fixed point approach to stability of a quadratic equation. Bull. Braz. Math. Soc. (N.S.) 37, 361–376 (2006)

    Google Scholar 

  104. Miura, T., Miyajima, S., Takahasi, S.E.: Hyers-Ulam stability of linear differential operator with constant coefficients. Math. Nachr. 258, 90–96 (2003)

    MATH  MathSciNet  Google Scholar 

  105. Miura, T., Jung, S.-M., Takahasi, S.E.: Hyers-Ulam-Rassias stability of the Banach space valued linear differential equations y′ = λ y. J. Korean Math. Soc. 41, 995–1005 (2004)

    Google Scholar 

  106. Mlesnite, O.: Existence and Ulam-Hyers stability results for coincidence problems. J. Nonlinear Sci. Appl. 6, 108–116 (2013)

    MATH  MathSciNet  Google Scholar 

  107. Moslehian, M.S.: Almost derivations on C -ternary rings. Bull. Belg. Math. Soc.-Simon Stevin 14, 135–142 (2007)

    MATH  MathSciNet  Google Scholar 

  108. Moslehian, M.S.: Superstability of higher derivations in multi-Banach algebras. Tamsui Oxf. J. Math. Sci. 24, 417–427 (2008)

    MATH  MathSciNet  Google Scholar 

  109. Moslehian, M.S., Rassias, Th.M.: Orthogonal stability of additive type equations. Aequ. Math. 73, 249–259 (2007)

    MATH  MathSciNet  Google Scholar 

  110. Moslehian, M.S., Rassias, Th.M.: Generalized Hyers-Ulam stability of mappings on normed Lie triple systems. Math. Inequal. Appl. 11, 371–380 (2008)

    MATH  MathSciNet  Google Scholar 

  111. Moslehian, M.S., Nikodem, K., Popa, D.: Asymptotic aspect of the quadratic functional equation in multi-normed spaces. J. Math. Anal. Appl. 355, 717–724 (2009)

    MATH  MathSciNet  Google Scholar 

  112. Moszner, Z.: On the stability of functional equations. Aequ. Math. 77, 33–38 (2009)

    MATH  MathSciNet  Google Scholar 

  113. Najati, A., Cho, Y.J.: Generalized Hyers-Ulam stability of the pexiderized Cauchy functional equation in non-Archimedean spaces. Fixed Point Theory Appl. 2011, Article ID 309026, 11 pp. (2011)

    Google Scholar 

  114. Najati, A., Rahimi, A.: A fixed point approach to the stability of a generalized Cauchy functional equation. Banach J. Math. Anal. 2, 105–112 (2008)

    MATH  MathSciNet  Google Scholar 

  115. Najati, A., Ranjbari, A.: Stability of homomorphisms for a 3D Cauchy-Jensen type functional equation on C -ternary algebras. J. Math. Anal. Appl. 341, 62–79 (2008)

    MATH  MathSciNet  Google Scholar 

  116. Najati, A., Rassias, Th.M.: Stability of homomorphisms and (θ, ϕ)-derivations. Appl. Anal. Discret. Math. 3, 264–281 (2009)

    MATH  MathSciNet  Google Scholar 

  117. Najati, A., Kang, J.I., Cho, Y.J.: Local stability of the pexiderized Cauchy and Jensen’s equations in fuzzy spaces. J. Inequal. Appl. 2011, 78 (2011)

    Google Scholar 

  118. Najatim, A., Rassias, Th.M.: Stability of a mixed functional equation in several variables on Banach modules. Nonlinear Anal. 72, 1755–1767 (2010)

    MathSciNet  Google Scholar 

  119. Nikoufar, E., Rassias, Th.M.: θ-centralizers on semiprime Banach *-algebras. Ukr. Math. J. 66, 300–310 (2014)

    Google Scholar 

  120. Novotný, P., Hrivnák, J.: On (α, β, γ)-derivations of Lie algebras and corresponding invariant functions. J. Geom. Phys. 58, 208–217 (2008)

    MATH  MathSciNet  Google Scholar 

  121. Pallu de la Barriére, R.: Algèbres unitaires et espaces d’Ambrose. Ann. Ecole Norm. Sup. 70, 381–401 (1953)

    Google Scholar 

  122. Park, C.: Linear functional equations in Banach modules over a C -algebra. Acta Appl. Math. 77, 125–161 (2003)

    MATH  MathSciNet  Google Scholar 

  123. Park, C.: On an approximate automorphism on a C -algebra. Proc. Am. Math. Soc. 132, 1739–1745 (2004)

    MATH  Google Scholar 

  124. Park, C.: Lie ∗-homomorphisms between Lie C -algebras and Lie ∗-derivations on Lie C -algebras. J. Math. Anal. Appl. 293, 419–434 (2004)

    MATH  MathSciNet  Google Scholar 

  125. Park, C.: Approximate homomorphisms on JB -triples. J. Math. Anal. Appl. 306, 375–381 (2005)

    MATH  MathSciNet  Google Scholar 

  126. Park, C.: Homomorphisms between Lie JC -algebras and Cauchy-Rassias stability of Lie JC -algebra derivations. J. Lie Theory 15, 393–414 (2005)

    MATH  MathSciNet  Google Scholar 

  127. Park, C.: Homomorphisms between Poisson JC -algebras. Bull. Braz. Math. Soc. 36, 79–97 (2005)

    MATH  MathSciNet  Google Scholar 

  128. Park, C.: Isomorphisms between unital C -algebras. J. Math. Anal. Appl. 307, 753–762 (2005)

    MATH  MathSciNet  Google Scholar 

  129. Park, C.: Isomorphisms between C -ternary algebras. J. Math. Phys. 47(10), 103512 (2006)

    MathSciNet  Google Scholar 

  130. Park, C.: Isomorphisms between C -ternary algebras. J. Math. Anal. Appl. 327, 101–115 (2007)

    MATH  MathSciNet  Google Scholar 

  131. Park, C.: Generalized Hyers-Ulam stability of functional equations: a fixed point approach. Taiwan. J. Math. 14, 1591–1608 (2010)

    MATH  Google Scholar 

  132. Park, C.: Orthogonal stability of a cubic-quartic functional equation. J. Nonlinear Sci. Appl. 5(Special issue), 28–36 (2012)

    MATH  MathSciNet  Google Scholar 

  133. Park, C., Cui, J.: Generalized stability of C -ternary quadratic mappings. Abstr. Appl. Anal. 2007, Article ID 23282 (2007)

    Google Scholar 

  134. Park, C., Najati, A.: Homomorphisms and derivations in C -algebras. Abstr. Appl. Anal. 2007, Article ID 80630 (2007)

    Google Scholar 

  135. Park, C., Rassias, Th.M.: Fixed points and generalized Hyers-Ulam stability of quadratic functional equations. J. Math. Inequal. 1, 515–528 (2007)

    MATH  MathSciNet  Google Scholar 

  136. Park, C., Rassias, Th.M.: Homomorphisms between JC*-algebras. Studia Univ. “Babes–Bolyai”, Math. 53, 43–55 (2008)

    Google Scholar 

  137. Park, C., Rassias, Th.M.: On the stability of orthogonal functional equations. Tamsui Oxf. J. Math. Sci. 24, 355–365 (2008)

    MATH  MathSciNet  Google Scholar 

  138. Park, C., Rassias, Th.M.: Fixed points and stability of functional equations. In: Pardalos, P.M., Rassias, Th.M., Khan, A.A. (eds.), Nonlinear Analysis and Variational Problems. Springer Optimization and Its Applications, vol. 35, pp. 125–134. Springer, New York (2010)

    Google Scholar 

  139. Park, C., Saadati, R.: Approximation of a generalized additive mapping in multi-Banach modules and isomorphisms in multi-C -algebras: a fixed-point approach. Adv. Differ. Equ. 2012, 162, 14 pp. (2012)

    Google Scholar 

  140. Park, C., Boo, D.H., Rassias, Th.M.: Approximately additive mappings over p-adic fields. J. Chungcheong Math. Soc. 21, 1–14 (2008)

    Google Scholar 

  141. Park, C., Lee, J.R., Shin, D.Y.: Fixed points and stability of functional equations associated with inner product spaces. In: Difference Equations and Applications, pp. 235–242, Ugur-Bahesehir University Publishing Company, Istanbul (2009)

    Google Scholar 

  142. Park, C., Lee, J.R., Rassias, Th.M., Saadati, R.: Fuzzy ∗-homomorphisms and fuzzy ∗-derivations in induced fuzzy C -algebras. Math. Comput. Model. 54, 2027–2039 (2011)

    MATH  MathSciNet  Google Scholar 

  143. Popa, D., Rasa, I.: On the Hyers-Ulam stability of the linear differential equation. J. Math. Anal. Appl. 381, 530–537 (2011)

    MATH  MathSciNet  Google Scholar 

  144. Popa, D., Rasa, I.: The Fréchet functional equation with application to the stability of certain operators. J. Approx. Theory 164, 138–144 (2012)

    MATH  MathSciNet  Google Scholar 

  145. Popovych, R., Boyko, V., Nesterenko, M., Lutfullin, M.: Realizations of real low–dimensional Lie algebras. J. Phys. A 36, 7337–7360 (2003)

    MATH  MathSciNet  Google Scholar 

  146. Pourpasha, M.M., Rassias, Th.M., Saadati, R., Vaezpour, S.M.: The stability of some differential equations. Math. Probl. Eng. 2011, Article ID 128479, 15 pp. (2011)

    Google Scholar 

  147. Prastaro, A., Rassias, Th.M.: Ulam stability in geometry of PDE’s. Nonlinear Funct. Anal. Appl. 8, 259–278 (2003)

    MATH  MathSciNet  Google Scholar 

  148. Radu, V.: The fixed point alternative and the stability of functional equations. Fixed Point Theory 4, 91–96 (2003)

    MATH  MathSciNet  Google Scholar 

  149. Rand, D., Winternitz, P., Zassenhaus, H.: On the identification of Lie algebra given by its structure constants I. Direct decompositions, Levi decompositions and nil radicals. Linear Algebra Appl. 109, 197–246 (1988)

    Google Scholar 

  150. Rassias, Th.M.: On the stability of the linear mapping in Banach spaces. Proc. Am. Math. Soc. 72, 297–300 (1978)

    MATH  Google Scholar 

  151. Rassias, J.M.: On approximation of approximately linear mappings by linear mappings. J. Funct. Anal. 46, 126–130 (1982)

    MATH  MathSciNet  Google Scholar 

  152. Rassias, J.M.: On approximation of approximately linear mappings by linear mappings. Bull. Sci. Math. 108, 445–446 (1984)

    MATH  MathSciNet  Google Scholar 

  153. Rassias, Th.M.: New characterizations of inner product spaces. Bulletin des Sciences Mathematiques, 2ed serie, Paris 108, 95–99 (1984)

    MATH  Google Scholar 

  154. Rassias, Th.M.: Problem 16: 2. In: Report of the 27th International Symposium on Functional Equations. Aequationes Mathematicae, Bielsko-Bia la, Poland, vol. 39, 292–293 (1990)

    Google Scholar 

  155. Rassias, Th.M.: On a modified Hyers-Ulam sequence. J. Math. Anal. Appl. 158, 106–113 (1991)

    MATH  MathSciNet  Google Scholar 

  156. Rassias, Th.M.: Functional Equations, Inequalities and Applications. Kluwer Academic Publishers, Dordrecht/Boston/London (2003)

    MATH  Google Scholar 

  157. Rassias, Th.M., Šemrl, P.: On the Hyers-Ulam stability of linear mappings. J. Math. Anal. Appl. 173, 325–338 (1993)

    MATH  MathSciNet  Google Scholar 

  158. Rassias, Th.M., Tabor, J.: What is left of Hyers–Ulam stability? J. Nat. Geom. 1, 65–69 (1992)

    MATH  MathSciNet  Google Scholar 

  159. Rassias, Th.M., Tabor, J.: Stability of Mappings of Hyers–Ulam Type. Hadronic Press, Palm Harbor (1994)

    MATH  Google Scholar 

  160. Rätz, J.: On approximately additive mappings. In: Beckenbach, E.F. (ed.), General Inequalities 2. International Series of Numerical Mathematics, vol. 47, pp. 233–251. Birkhäuser, Basel (1980)

    Google Scholar 

  161. Rolewicz, S.: Metric Linear Spaces. PWN-Polish Scientific Publishers/Reidel, Dordrecht (1984)

    Google Scholar 

  162. Saadati, R., Sadeghi, Gh., Rassias, Th.M.: Approximate generalized additive mappings in proper multi–CQ –algebras. Filomat 28, 677–694 (2014)

    Google Scholar 

  163. Saadati, R., Cho, Y.J., Vahidi, J.: The stability of the quartic functional equation in various spaces. Comput. Math. Appl. 60, 1994–2002 (2010)

    MATH  MathSciNet  Google Scholar 

  164. Saadati, R., Rassias, Th.M., Cho, Y.J.: Approximate (α, β, γ)-derivation on random Lie C -algebras. RACSAM 109, 1–10 (2015)

    MATH  MathSciNet  Google Scholar 

  165. Sahoo, P.K., Riedel, T.: Mean Value Theorem and Functional Equations. World Scientific Publishing Company, Singapore/River Edge/London/Hong Kong (1998)

    Google Scholar 

  166. Schwaiger, J.: Remark 12. In: Report on the 25th International Symposium on Functional Equations. Aequationes Mathematicae, Hamburg-Rissen, Germany, vol. 35, pp. 120–121 (1985)

    Google Scholar 

  167. Shilkret, N.: Non-Archimedian Banach algebras. Ph.D. thesis, Polytechnic University, ProQuest LLC (1968)

    Google Scholar 

  168. Shulman, E.V.: Group representations and stability of functional equations. J. Lond. Math. Soc. 54, 111–120 (1996)

    MATH  MathSciNet  Google Scholar 

  169. Smital, J.: On Functions and Functional Equations. Adam Hilger, Bristol/Philadelphia (1988)

    MATH  Google Scholar 

  170. Takhtajan, L.: On foundation of the generalized Nambu mechanics. Commun. Math. Phys. 160, 295–315 (1994)

    MATH  MathSciNet  Google Scholar 

  171. Trapani, C.: Quasi-∗-algebras of operators and their applications. Rev. Math. Phys. 7, 1303–1332 (1995)

    MATH  MathSciNet  Google Scholar 

  172. Trapani, C.: Bounded elements and spectrum in Banach quasi ∗-algebras. Studia Math. 172, 249–273 (2006)

    MATH  MathSciNet  Google Scholar 

  173. Ulam, S.M.: A Collection of Mathematical Problems. Interscience Publishers, New York (1960)

    MATH  Google Scholar 

  174. Ulam, S.M.: Problems in Modern Mathematics. Wiley, New York (1960)

    Google Scholar 

  175. Urs, C.: Ulam-Hyers stability for coupled fixed points of contractive type operators. J. Nonlinear Sci. Appl. 6, 124–136 (2013)

    MATH  MathSciNet  Google Scholar 

  176. Wang, G., Zhou, M., Sun, L.: Hyers-Ulam stability of linear differential equations of first order. Appl. Math. Lett. 21, 1024–1028 (2008)

    MATH  MathSciNet  Google Scholar 

  177. Wang, Z., Li, X., Rassias, Th.M.: Stability of an additive-cubic-quartic functional equation in multi-Banach spaces. Abstr. Appl. Anal. 2011, Article ID 536520, 11 pp. (2011)

    Google Scholar 

  178. Wyrobek, W.: Orthogonally additive functions modulo a discrete subgroup. Aequ. Math. 78, 63–69 (2009). Springer, New York (2009)

    Google Scholar 

  179. Zariski, O., Samuel, P.: Commutative Algebra. Van Nostrand, Princeton (1958)

    MATH  Google Scholar 

  180. Zeidler, E.: Nonlinear Functional Analysis and Its Applications I: Fixed-Point Theorems. Springer, New York (1986)

    MATH  Google Scholar 

  181. Zettl, H.: A characterization of ternary rings of operators. Adv. Math. 48, 117–143 (1983)

    MATH  MathSciNet  Google Scholar 

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Cho, Y.J., Park, C., Rassias, T.M., Saadati, R. (2015). Introduction. In: Stability of Functional Equations in Banach Algebras. Springer, Cham. https://doi.org/10.1007/978-3-319-18708-2_1

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