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On the Stability of Functional Equations and a Problem of Ulam

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Abstract

In this paper, we study the stability of functional equations that has its origins with S. M. Ulam, who posed the fundamental problem 60 years ago and with D. H. Hyers, who gave the first significant partial solution in 1941. In particular, during the last two decades, the notion of stability of functional equations has evolved into an area of continuing research from both pure and applied viewpoints. Both classical results and current research are presented in a unified and self-contained fashion. In addition, related problems are investigated. Some of the applications deal with nonlinear equations in Banach spaces and complementarity theory.

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References

  1. Aczél, J.: Lectures on Functional Equations and Their Applications, Academic Press, New York, 1966.

    Google Scholar 

  2. Aczél, J.: A Short Course on Functional Equations, D. Reidel, Dordrecht, 1987.

    Google Scholar 

  3. Aczél, J. and Dhombres, J.: Functional Equations in Several Variables, Cambridge Univ. Press, Cambridge, 1989.

    Google Scholar 

  4. Amann, H.: Fixed points of asymptotically linear maps in ordered Banach spaces, J. Funct. Anal. 14(1973), 162–171.

    Google Scholar 

  5. Badea, C.: On the Hyers–Ulam stability of mappings: the direct method, In: Th. M. Rassias and J. Tabor(eds), Stability of Mappings of Hyers–Ulam Type, Hadronic Press, Florida, 1994, pp. 7–13.

    Google Scholar 

  6. Baker, J.: The stability of the cosine equation, Proc. Amer. Math. Soc. 80(1980), 411–416.

    Google Scholar 

  7. Baker, J., Lawrence, J. and Zorzitto, F.: The stability of the equation f (x + y) = f (x)f (y), Proc. Amer. Math. Soc. 74(1979), 242–246.

    Google Scholar 

  8. Berruti, G. and Skof, F.: Risultati di equivalenza per un'equazione di Cauchy alternativa negli spazi normati, Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur. 125(1991), 154–167.

    Google Scholar 

  9. Bonsall, F. F. and Duncan, J.: Numerical Ranges of Operators on Normed Spaces and Elements of Normed Algebras, London Math. Soc. Lecture Notes Ser. 2, Cambridge Univ. Press, London, 1971.

    Google Scholar 

  10. Borelli, C.: On Hyers–Ulam stability of HosszÚ's functional equation, Results Math. 26 (1994g), 221–224.

  11. Borelli, C. and Forti, G. L.: On a general Hyers–Ulam stability result, Internat. J. Math. Math. Sci. 18(1995), 229–236.

    Google Scholar 

  12. Bourgin, D. G.: Classes of transformations and bordering transformations, Bull. Amer. Math. Soc. 57(1951), 223–237.

    Google Scholar 

  13. Browder, F.: Nonlinear functional analysis and nonlinear integral equations of Hammerstein and Urysohn type, In: E. Zarantonello (ed.), Contributions to Nonlinear Functional Analysis, Academic Press, New York, 1971, pp. 425–500.

    Google Scholar 

  14. Brzd¸ek, J.: A note on stability of additive mappings, In: Th. M. Rassias and J. Tabor (eds), Stability of Mappings of Hyers–Ulam Type, Hadronic Press, Florida, 1994, pp. 19–22.

    Google Scholar 

  15. Brzd¸ek, J.: On functionals which are orthogonally additive modulo Z, Results Math. 30 (1996), 25–38.

    Google Scholar 

  16. Brzd¸ek, J.: The Cauchy and Jensen differences on semigroups, Publ. Math. Debrecen 48(1996), 117–136.

    Google Scholar 

  17. Brzd¸ek, J.: On the Cauchy difference on normed spaces, Abh. Math. Sem. Univ. Hamburg 66(1996), 143–150.

    Google Scholar 

  18. Brzd¸ek, J.: On orthogonally exponential and orthogonally additive mappings, Proc. Amer. Math. Soc. 125(1997), 2127–2132.

    Google Scholar 

  19. Câc, N. P. and Gatica, J. A.: Fixed point theorems for mappings in ordered Banach spaces, J. Math. Anal. Appl. 71(1979), 547–557.

    Google Scholar 

  20. Castillo, E. and Ruiz-Cobo, M. R.: Functional Equations and Modelling in Science and Engineering, Dekker, New York, 1992.

    Google Scholar 

  21. Cauchy, A. L.: Cours d'analyse de l'École Polytechnique, Vol. I, Analyse algébrique, Debure, Paris, 1821.

    Google Scholar 

  22. Cholewa, P. W.: The stability of the sine equation, Proc. Amer. Math. Soc. 88(1983), 631–634.

    Google Scholar 

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

    Google Scholar 

  24. van der Corput, J. G.: Goniometrische functies gekarakteriseerd door een functionaal betrekking, Euclides 17(1940g), 55–75.

  25. Cottle, R. W., Pang, J. S. and Stone, R. E.: The Linear Complementarity Problem, Academic Press, 1992.

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

    Google Scholar 

  27. Czerwik, S.: On the stability of the homogeneous mapping, C.R. Math. Rep. Acad. Sci. Canada 14(1992), 268–272.

    Google Scholar 

  28. Czerwik, S.: The stability of the quadratic functional equation, In: Th. M. Rassias and J. Tabor(eds), Stability of Mappings of Hyers–Ulam Type, Hadronic Press, Florida, 1994, pp. 81–91.

    Google Scholar 

  29. Davison, T. M. K.: The complete solution of HosszÚ's functional equation over a field, Aequationes Math. 11(1974), 114–115.

    Google Scholar 

  30. Day, M. M.: Normed Linear Spaces, Ergeb. Math. Grenzgeb., Springer, Berlin, 1958.

    Google Scholar 

  31. Forti, G. L.: An existence and stability theorem for a class of functional equations, Stochastica 4(1980), 23–30.

    Google Scholar 

  32. Forti, G. L.: The stability of homomorphisms and amenability, with applications to functional equations, Abh. Math. Sem. Univ. Hamburg 57(1987), 215–226.

    Google Scholar 

  33. Forti, G. L.: Hyers–Ulam stability of functional equations in several variables, Aequationes Math. 50(1995), 143–190.

    Google Scholar 

  34. Förg-Rob, W. and Schwaiger, J.: On the stability of a system of functional equations characterizing generalized hyperbolic and trigonometric functions, Aequationes Math. 45(1993), 285–296.

    Google Scholar 

  35. Förg-Rob, W. and Schwaiger, J.: On the stability of some functional equations for generalized hyperbolic functions and for the generalized cosine equation, Results Math. 26 (1994), 274–280.

    Google Scholar 

  36. Gajda, Z.: On stability of the Cauchy equation on semigroups, Aequationes Math. 36 (1988), 76–79.

    Google Scholar 

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

    Google Scholar 

  38. Găvruta, P.: A generalization of the Hyers–Ulam–Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184(1994), 431–436.

    Google Scholar 

  39. Găvruta, P.: On the stability of some functional equations, In: Th. M. Rassias and J. Tabor(eds), Stability of Mappings of Hyers–Ulam Type, Hadronic Press, Florida, 1994, pp. 93–98.

    Google Scholar 

  40. Găvruta, P.: On the Hyers–Ulam–Rassias stability of mappings, J. Math. Anal. Appl., to appear.

  41. Găvruta, P., Hossu, M., Popescu, D. and Căprău, C.: On the stability of mappings, Bull. Appl. Math. Tech. Univ. Budapest 73(1994), 169–176.

    Google Scholar 

  42. Găvruta, P., Hossu, M., Popescu, D. and Căprău, C.: On the stability of mappings and an answer to a problem of Th. M. Rássias, Ann. Math. Blaise Pascal 2(1995), 55–60.

    Google Scholar 

  43. Ger, R.: Stability of addition formulae for trigonometric mappings, Zeszyty Nauk. Politech. Slasiej, Ser. Mat. Fiz. 64(1990), 75–84.

    Google Scholar 

  44. Ger, R.: Superstability is not natural, Rocznik Naukowo-Dydaktyczny WSP w Krakowie, Prace Mat. 159(1993), 109–123.

    Google Scholar 

  45. Ger, R. and Šemrl, P.: The stability of the exponential equation, Proc. Amer. Math. Soc. 124(1996), 779–787.

    Google Scholar 

  46. Greenleaf, F. P.: Invariant Means on Topological Groups, Van Nostrand Math. Stud. 16, New York, 1969.

  47. Haruki, H. and Rassias, Th. M.: New generalizations of Jensen's functional equation, Proc. Amer. Math. Soc. 123(1995), 495–503.

    Google Scholar 

  48. Haruki, H. and Rassias, Th. M.: A new functional equation of Pexider type related to the complex exponential function, Trans. Amer. Math. Soc. 347(1995), 3111–3119.

    Google Scholar 

  49. HosszÚ, M.: On the functional equation F(x)+y, z)+F(x, y)= F(x, y+z)+F (y, z), Period. Math. Hungar. 1(1971), 213–216.

    Google Scholar 

  50. Hyers, D. H.: On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A. 27(1941), 222–224.

    Google Scholar 

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

    Google Scholar 

  52. Hyers, D. H., Isac, G. and Rassias, Th. M.: Stability of Functional Equations in Several Variables, Birkhäuser, Boston, 1998.

    Google Scholar 

  53. Hyers, D. H. and Rassias, Th. M.: Approximate homomorphisms, Aequationes Math. 44 (1992), 125–153.

    Google Scholar 

  54. Isac, G.: Opérateurs asymptotiquement linéaires sur des espaces locallement convexes, Colloq. Math. 46(1982), 67–72.

    Google Scholar 

  55. Isac, G.: Complementarity Problems, Lecture Notes in Math. 1528, Springer, 1992.

  56. Isac, G.: The fold complementarity problem and the order complementarity problem, In: Topological Methods in Nonlinear Analysis, 1997.

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

    Google Scholar 

  58. Isac, G. and Rassias, Th. M.: Functional inequalities for approximately additive mappings, In: Th. M. Rassias and J. Tabor(eds), Stability of Mappings of Hyers–Ulam Type, Hadronic Press, Florida, 1994, pp. 117–125.

    Google Scholar 

  59. Isac, G. and Rassias, Th.M.: Stability of-additive mappings: Applications to nonlinear analysis, Internat. J. Math. Math. Sci. 19(1996), 219–228.

    Google Scholar 

  60. Jarosz, K.: Perturbations of Banach Algebras, Lecture Notes in Math. 1120, Springer, Berlin, 1985.

    Google Scholar 

  61. Johnson, B. E.: Approximately multiplicative functionals, J. London Math. Soc. (2) 34 (1986), 489–510.

    Google Scholar 

  62. Johnson, B. E.: Approximately multiplicative maps between Banach algebras, J. London Math. Soc. (2) 37(1988), 294–316.

    Google Scholar 

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

    Google Scholar 

  64. Jung, S.-M.: On solution and stability of functional equation f (x+y) 2 = af (x)f (y)+ bf (x) 2 + cf (y) 2, Bull. Korean Math. Soc. 34(1997), 561–571.

    Google Scholar 

  65. Jung, S.-M.: On the superstability of the functional equation f (x y) = yf (x), Abh. Math. Sem. Univ. Hamburg 67(1997), 315–322.

    Google Scholar 

  66. Jung, S.-M.: Hyers–Ulam–Rassias stability of functional equations, Dynamic Systems Appl. 6(1997), 541–566.

    Google Scholar 

  67. Jung, S.-M.: On the modified Hyers–Ulam–Rassias stability of the functional equation for gamma function, Mathematica 39 (62) (1997), 235–239.

    Google Scholar 

  68. Jung, S.-M.: On a modified Hyers–Ulam stability of homogeneous equation, Internat. J. Math. Math. Sci. 21(1998), 475–478.

    Google Scholar 

  69. Jung, S.-M.: On modified Hyers–Ulam–Rassias stability of a generalized Cauchy functional equation, Nonlinear Studies 5(1998), 59–67.

    Google Scholar 

  70. Jung, S.-M.: Hyers–Ulam–Rassias stability of Jensen's equation and its application, Proc. Amer. Math. Soc. 126(1998), 3137–3143.

    Google Scholar 

  71. Jung, S.-M.: On the stability of the functional equation for exponential function, Manuscript.

  72. Jung, S.-M.: On an asymptotic behavior of exponential functional equation, Studia Univ. Babes-Bolyai, to appear.

  73. Jung, S.-M.: On the stability of the functional equation f (x) y = f (x) y, Mathematica, to appear.

  74. Jung, S.-M.: Superstability of homogeneous functional equation, Kyungpook Math. J. 38(1998), 251–257.

    Google Scholar 

  75. Jung, S.-M.: On the superstability of the functional equation f (x) y = f (x) y, In: Th. M. Rassias (ed.), Functional Equations and Inequalities, Hadronic Press, Florida, to appear.

  76. Jung, S.-M.: On the Hyers–Ulam stability of a quadratic functional equation, In: Th. M. Rassias (ed.), New Approaches in Nonlinear Analysis, Hadronic Press, Florida, 1999, pp. 125–132.

    Google Scholar 

  77. Jung, S.-M.: On the Hyers–Ulam stability of the functional equations that have the quadratic property, J. Math. Anal. Appl. 222(1998), 126–137.

    Google Scholar 

  78. Jung, S.-M.: Hyers–Ulam–Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press, Florida, to appear.

  79. Jung, S.-M. and Kim, B.: On the stability of the quadratic functional equation on bounded domains, Abh. Math. Sem. Univ. Hamburg 69(1999), 293–308.

    Google Scholar 

  80. Kairies, H. H.: Die Gammafunktion als stetige Lösung eines Systems von Gauss-Funktionalgleichungen, Results Math. 26(1994), 306–315.

    Google Scholar 

  81. Kannappan, Pl.: Quadratic functional equation and inner product spaces, Results Math. 27(1995), 368–372.

    Google Scholar 

  82. Kominek, Z.: On a local stability of the Jensen functional equation, Demonstratio Math. 22(1989), 499–507.

    Google Scholar 

  83. Krasnoselskii, M. A.: Positive Solutions of Operator Equations, Nordhoff, Groningen, 1964.

  84. Krasnoselskii, M. A. and Zabreiko, P. P.: Geometrical Methods of Nonlinear Analysis, Springer, Berlin, 1984.

    Google Scholar 

  85. Kravvaritis, D.: Nonlinear random operators of monotone type in Banach spaces, J. Math. Anal. Appl. 78(1980), 488–496.

    Google Scholar 

  86. Kuczma, M.: An Introduction to the Theory of Functional Equations and Inequalities, Pa´nstwowe Wydawnictwo Naukowe-Uniwersytet ´ Sl¸aski, Warszawa, 1985.

    Google Scholar 

  87. Kurepa, S.: On the quadratic functional, Publ. Inst. Math. Acad. Serbe Sci. Beograd 13 (1959), 57–72.

    Google Scholar 

  88. Lajkó, K.: Applications of extensions of additive functions, Aequationes Math. 11 (1974), 68–76.

    Google Scholar 

  89. Losonczi, L.: On the stability of HosszÚ's functional equation, Results Math. 29(1996), 305–310.

    Google Scholar 

  90. Maksa, G.: Problems 18, In: Report on the 34th ISFE, Aequationes Math. 53, 1997, pp. 194.

  91. Mininni, M.: Coincidence degree and solvability of some nonlinear functional equations in normed spaces: A spectral approach, Nonlinear Anal. 1(1977), 105–122.

    Google Scholar 

  92. Ng, C. T.: Jensen's functional equation on groups, Aequationes Math. 39(1990), 85–90.

    Google Scholar 

  93. Opoitsev, V. I.: Nonlinear Statical Systems, Economics–Mathematics Library, Nauka, Moscow (Russian), 1986.

    Google Scholar 

  94. Páles, Z.: Remark 27, In: Report on the 34th ISFE, Aequationes Math. 53, 1997, pp. 200–201.

    Google Scholar 

  95. Parnami, J. C. and Vasudeva, H. L.: On Jensen's functional equation, Aequationes Math. 43(1992), 211–218.

    Google Scholar 

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

    Google Scholar 

  97. Rassias, J. M.: On a new approximation of approximately linear mappings by linear mappings, Discuss. Math 7(1985), 193–196.

    Google Scholar 

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

    Google Scholar 

  100. Rassias, Th. M.: On the stability of mappings, Rend. Sem. Mat. Fis. Milano 58(1988), 91–99.

    Google Scholar 

  101. Rassias, Th. M.: Seven problems in mathematical analysis, In: Th. M. Rassias (ed.), Topics in Mathematical Analysis, World Sci. Publ., Singapore, 1989.

    Google Scholar 

  102. Rassias, Th. M.: The stability of mappings and related topics, In: Report on the 27th ISFE, Aequationes Math. 39, 1990, pp. 292–293.

    Google Scholar 

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

    Google Scholar 

  104. Rassias, Th. M.: Problem 18, In: Report on the 31st ISFE, Aequationes Math. 47, 1994, pp. 312–313.

    Google Scholar 

  105. Rassias, Th. M.: Remark and problem 19, In: Report on the 31st ISFE, Aequationes Math. 47, 1994, pp. 313–314.

    Google Scholar 

  106. Rassias, Th.M.: On a problem of S.M. Ulam and the asymptotic stability of the Cauchy functional equation with applications, Internat. Ser. Numer. Math. 123(1997), 297–309.

    Google Scholar 

  107. Rassias, Th. M.: Inner Product Spaces and Applications, Longman, 1997.

  108. Rassias, Th.M.: On the stability of the quadratic functional equation and its applications, to appear.

  109. Rassias, Th. M.: On the stability of the quadratic functional equation, to appear.

  110. Rassias, Th. M.: On the stability of functional equations originated by a problem of Ulam, Studia Univ. Babes-Bolyai, to appear.

  111. Rassias, Th. M. and Šemrl, P.: On the behavior of mappings which do not satisfy Hyers– Ulam stability, Proc. Amer. Math. Soc. 114(1992), 989–993.

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  114. Rassias, Th. M. and Tabor, J.: On approximately additive mappings in Banach spaces, In: Th. M. Rassias and J. Tabor(eds), Stability of Mappings of Hyers–Ulam Type, Hadronic Press, Florida, 1994, pp. 127–134.

    Google Scholar 

  115. Rätz, J.: On approximately additive mappings, In: E. F. Beckenbach (ed.), General Inequalities 2, Birkhäuser, Basel, 1980, pp. 233–251.

    Google Scholar 

  116. Rudin, W.: Functional Analysis, McGraw-Hill, New York, 1991.

    Google Scholar 

  117. Schwaiger, J.: Remark 10, In: Report on the 30th ISFE, Aequationes Math. 46, 1993, pp. 289.

  118. Šemrl, P.: The stability of approximately additive functions, In: Th. M. Rassias and J. Tabor(eds), Stability of Mappings of Hyers–Ulam Type, Hadronic Press, Florida, 1994, pp. 135–140.

    Google Scholar 

  119. Šemrl, P.: The functional equation of multiplicative derivation is superstable on standard operator algebras, Integral Equations Operator Theory 18(1994), 118–122.

    Google Scholar 

  120. Skof, F.: Sull'approssimazione delle applicazioni localmente _-additive, Atti Accad. Sci. Torino 117(1983), 377–389.

    Google Scholar 

  121. Skof, F.: Proprietá locali e approssimazione di operatori, Rend. Sem. Mat. Fis. Milano 53(1983), 113–129.

    Google Scholar 

  122. Skof, F.: Approssimazione di funzioni _-quadratiche su dominio ristretto, Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur. 118(1984), 58–70.

    Google Scholar 

  123. Skof, F.: On two conditional forms of the equation kf.x C y/k D kf.x/ C f.y/k, Aequationes Math. 45(1993), 167–178.

    Google Scholar 

  124. Skof, F.: On the stability on functional equations on a restricted domain and a related topic, In: J. Tabor(eds), Stability of Mappings of Hyers–Ulam Type, Hadronic Press, Florida, 1994, pp. 141–151.

    Google Scholar 

  125. Skof, F. and Terracini, S.: Sulla stabilità dell'equazione funzionale quadratica su un dominio ristretto, Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur. 121(1987), 153–167.

    Google Scholar 

  126. Stetsenko, V. Y.: New two-sided estimates for the spectral radius of a linear positive operator, Dokl. Acad. Nauk Tadzhik SSSR 33(1991), 807–811.

    Google Scholar 

  127. Székelyhidi, L.: On a theorem of Baker, Lawrence and Zorzitto, Proc. Amer. Math. Soc. 84(1982), 95–96.

    Google Scholar 

  128. Székelyhidi, L.: Remark 17, In: Report on the 22nd ISFE, Aequationes Math. 29, 1985, pp. 95–96.

    Google Scholar 

  129. Székelyhidi, L.: Note on Hyers's theorem, C.R. Math. Rep. Acad. Sci. Canada 8(1986), 127–129.

    Google Scholar 

  130. Székelyhidi, L.: Fréchet equation and Hyers's theorem on noncommutative semigroups, Ann. Polon. Math. 48(1988), 183–189.

    Google Scholar 

  131. Székelyhidi, L.: The stability of the sine and cosine functional equations, Proc. Amer. Math. Soc. 110(1990), 109–115.

    Google Scholar 

  132. Tabor, J.: On approximate by linear mappings, Manuscript.

  133. Tabor, J.: On approximately linear mappings, In: Th. M. Rassias and J. Tabor(eds), Stability of Mappings of Hyers–Ulam Type, Hadronic Press, Florida, 1994, pp. 157–163.

    Google Scholar 

  134. Tabor, J.: HosszÚ's functional equation on the unit interval is not stable, Publ. Math. Debrecen 49(1996), 335–340.

    Google Scholar 

  135. Tabor, J.: Remark 20, In: Report on the 34th ISFE, Aequationes Math. 53, 1997, pp. 194–196.

    Google Scholar 

  136. Tabor, J. and Tabor, J.: Homogeneity is superstable, Publ. Math. Debrecen 45(1994), 123–130.

    Google Scholar 

  137. Ulam, S. M.: A Collection of Mathematical Problems, Interscience Publ., New York, 1960.

    Google Scholar 

  138. Villar, A.: Operator Theorems with Applications to Distributive Problems and Equilibrium Models, Lecture Notes in Econom. and Math. Systems 377, Springer, 1992.

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

    Google Scholar 

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    Themistocles M. Rassias

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Rassias, T.M. On the Stability of Functional Equations and a Problem of Ulam. Acta Applicandae Mathematicae 62, 23–130 (2000). https://doi.org/10.1023/A:1006499223572

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