Skip to main content
SpringerLink
Log in

Semi-Inner Products and Parapreseminorms on Groups and a Generalization of a Theorem of Maksa and Volkmann on Additive Functions

  • Chapter
  • First Online:
Ulam Type Stability

  • 347 Accesses

Abstract

By using inner products and paraprenorms on groups, we prove a natural generalization of a basic theorem of Gyula Maksa and Peter Volkmann on additive functions.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

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

    Book  MATH  Google Scholar 

  2. Alsina, C., Sikorska, J., Tomás, M. S.: Norm Derivatives and Characterizations of Inner Product Spaces. World Scientific, New Yersey (2010)

    MATH  Google Scholar 

  3. Amir, D: Characterizations of Inner Product Spaces. Birkhäuser, Besel (1986)

    Google Scholar 

  4. Antoine, J.-P., Grossmann, A.: Partial inner product spaces. I. General properties. J. Funct. Anal. 23, 369–378 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  5. Baker J.A.: On quadratic functionals continuous along rays. Glasnik Mat. 23, 215–229 (1968)

    MathSciNet  MATH  Google Scholar 

  6. Baron, K.: On additive involutions and Hamel bases. Aquationes Math. 87, 159–163 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  7. Baron, K.: Orthogonally additive bijections are additive. Aequationes Math. 89, 297–299 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  8. Batko, B, Tabor, J.: Stability of an alternative Cauchy equation on a restricted domain. Aequationes Math. 57, 221–232 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  9. Batko, B., Tabor, J.: Stability of the generalized alternative Cauchy equation. Abh. Math. Sem. Univ. Hamburg 69, 67–73 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bognár, J.: Indefinite Inner Product Spaces. Springer, Berlin (1974)

    Book  MATH  Google Scholar 

  11. Boros, Z.: Schwarz inequality over groups. In: Talk Held at the Conference on Inequalities and Applications, Hajdúszoboszló, Hungary (2016)

    Google Scholar 

  12. Boros, Z., Száz, Á.: Semi-inner products and their induced seminorms and semimetrics on groups, Technical Report, 2016/6, 11 pp. Institute of Mathematics, University of Debrecen, Debrecen (2016)

    Google Scholar 

  13. Boros, Z., Száz, Á.: A weak Schwarz inequality for semi-inner products on groupoids, Rostock. Math. Kolloq. 71, 28–40 (2016)

    MathSciNet  Google Scholar 

  14. Boros, Z, Száz, Á.: Generalized Schwarz inequalities for generalized semi-inner products on groupoids can be derived from an equality. Novi Sad J. Math. 47, 177–188 (2017)

    Article  MathSciNet  Google Scholar 

  15. Boros, Z., Száz, Á.: Infimum problems derived from the proofs of some generalized Schwarz inequalities, Teaching Math. Comput. Sci. 17, 41–57 (2019)

    Article  Google Scholar 

  16. Brzdek, J.: A note on stability of the Popoviciu functional equation on restricted domain. Demonstration Math. 43, 635–641 (2010)

    MathSciNet  MATH  Google Scholar 

  17. Brzdek, J., Ciepliński, K: Hyperstability and superstability. Abst. Appl. Anal. 2013, 13 (2013)

    Google Scholar 

  18. Burai, P., Száz, Á.: Relationships between homogeneity, subadditivity and convexity properties. Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. 16, 77–87 (2005)

    MathSciNet  MATH  Google Scholar 

  19. Chmieliński, J.: Normed spaces equivalent to inner product spaces and stability of functional equations. Aequationes Math. 87, 147–157 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  20. Chmieliński, J.: A note on a characterization of metrics generated by norms. Rocky Mt. J. Math. 45, 1801–1805 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  21. Chmieliński, J.: On functional equations related to additive mappings and isometries. Aequationes Math. 89, 97–105 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  22. Chung, S.-C., Lee, S.-B., Park, W.-G.: On the stability of an additive functional inequality. Int. J. Math. Anal. 6, 2647–2651 (2012)

    MathSciNet  MATH  Google Scholar 

  23. Dhombres, J.G.: Some Aspects of Functional Equations. Chulalongkorn University, Bangkok (1979)

    MATH  Google Scholar 

  24. Dong, Y.: Generalized stabilities of two functional equations. Aequationes Math. 86, 269–277 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  25. Dong, Y., Chen, L.: On generalized Hyers-Ulam stability of additive mappings on restricted domains of Banach spaces. Aequationes Math. 90, 871–878 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  26. Dong, Y., Zheng, B.: On hyperstability of additive mappings onto Banach spaces. Bull. Aust. Math. Soc. 91, 278–285 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  27. Dragomir, S.S.: Semi-Inner Products and Applications. Nova Science Publishers, Hauppauge (2004)

    MATH  Google Scholar 

  28. Drygas, H.: Quasi-inner products and their applications. In: Gupta, A.K. (Ed.) Advances in Multivariate Statistical Analysis. Theory and Decision Library (Series B: Mathematical and Statistical Methods), vol. 5, pp. 13–30. Reidel, Dordrecht (1987)

    Chapter  Google Scholar 

  29. Ebanks, B.R., Kannappan Pl., Sahoo, P.K.: A common generalization of functional equations characterizing normed and quasi-inner-product spaces. Canad. Math. Bull. 35, 321–327 (1992)

    Google Scholar 

  30. Elqorachi, E., Manar, Y., Rassias, Th.M.: Hyers-Ulam stability of the quadratic functional equation. In: Rassias, Th.M., Brdek, J. (Eds.) Functional Equations in Mathematical Analysis. Springer Optimization and Its Applications, vol. 52, pp. 97–105 (2012)

    Article  Google Scholar 

  31. Fechner, W.: Stability of a functional inequality associated with the Jordan-von Neumann functional equation. Aequationes Math. 71, 149–161 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  32. Fechner, W.: On the Hyers-Ulam stability of functional equations connected with additive and quadratic mappings. J. Math. Anal. Appl. 322, 774–786 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  33. Fechner, W.: Hlawka’s functional inequality. Aequationes Math. 87, 71–87 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  34. Fechner, W.: Hlawka’s functional inequality on topological groups. Banach J. Math. Anal. 11, 130–142 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  35. Fischer, P., Muszély, Gy.: Some generalizations of Cauchy’s functional equations (Hungarian). Mat. Lapok 16, 67–75 (1965)

    Google Scholar 

  36. Fischer, P., Muszély, Gy.: On some new generalizations of the functional equation of Cauchy. Can. Math. Bull. 10, 197–205 (1967)

    Google Scholar 

  37. Fochi, M.: An alternative functional equation on restrictef domain. Aequationes Math. 70, 201–212 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  38. Fréchet, M.: Sur lá definition axiomatique d’une classe d’espaces vectoriels distanciés applicables vectoriellement sur l’espace de Hilbert. Ann. Math. 36, 705–718 (1935)

    Article  MathSciNet  MATH  Google Scholar 

  39. Gǎvruta, P.: On the Hyers-Ulam-Rassias stability of the quadratic mappings. Nonlinear Funct. Anal. Appl. 9, 415–428 (2004)

    Google Scholar 

  40. Ger, R.: On a characterization of strictly convex spaces. Atti Acad. Sci. Torino Cl. Sci. Fis. Mat. Natur. 127, 131–138 (1993)

    MathSciNet  MATH  Google Scholar 

  41. Ger, R.: A Pexider-type equation in normed linear spaces. Sitzungsber. Abt. II 206, 291–303 (1997)

    MathSciNet  MATH  Google Scholar 

  42. Ger, R.: Fischer–Muszély additivity on Abelian groups. Comment Math. Tomus Specialis in Honorem Juliani Musielak, pp. 83–96 (2004)

    Google Scholar 

  43. Ger, R.: On a problem of Navid Safaei. In: Talk Held at the Conference on Inequalities and Applications, Hajdúszoboszló, Hungary (2016)

    Google Scholar 

  44. Ger, R.: Fischer-Muszély additivity – a half century story. In: Brzdek, J., Ciepliński, K., Rassias, Th.M. (eds.) Developments in Functional Equations and Related Topics. Springer Optimization and Its Applications, pp. 69–102. Springer, Basel (2017)

    Google Scholar 

  45. Ger, R., Koclega, B.: Isometries and a generalized Cauchy equation. Aequationes Math. 60, 72–79 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  46. Ger, R., Koclega, B.: An interplay between Jensen’s and Pexider’s functional equations on semigroups. Ann. Univ. Sci. Budapest Sect Comp. 35, 107–124 (2011)

    MathSciNet  MATH  Google Scholar 

  47. Gilányi, A.: Eine zur Parallelogrammgleichung äquivalente Ungleichung. Aequationes Math. 62, 303–309 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  48. Gilányi, A.: On a problem by K. Nikodem. Math. Ineq. Appl. 5, 707–710 (2002)

    MATH  Google Scholar 

  49. Gilányi, A., Troczka-Pawelec, K.: Regularity of weakly subquadratic functions. J. Math. Anal. Appl. 382, 814–821 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  50. Gilányi, A., Kézi, Cs., Troczka-Pawelec, K.: On two different concepts of subquadraticity. Inequalities and Applications 2010. International Series of Numerical Mathematics, vol. 161, pp. 209–215. Birkhäuser, Basel (2012)

    Google Scholar 

  51. Giles, J.R.: Classes of semi-inner-product spaces. Trans. Am. Math. Soc. 129, 436–446 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  52. Glavosits, T., Száz, Á.: Divisible and cancellable subsets of groupoids. Ann. Math. Inform. 43, 67–91 (2014)

    MathSciNet  MATH  Google Scholar 

  53. Golab, S., Światak, H.: Note on inner products in vector spaces. Aequationes Math. 8, 74–75 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  54. Haruki, H.: On the functional equations | f (x + iy)| = | f (x) + f (i y)| and | f (x + iy)| = | f (x) − f (i y)| and on Ivory’s theorem. Canad. Math. Bull. 9, 473–480 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  55. Haruki, H.: On the equivalence of Hille’s and Robinson’s functional equations. Ann. Polon. Math. 28, 261–264 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  56. Hosszú, M.: On an alternative functional equation (Hungarian). Mat. Lapok 14, 98–102 (1963)

    MathSciNet  MATH  Google Scholar 

  57. Hosszú, M.: A remark on the square norm. Aequationes Math. 2, 190–193 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  58. Istrǎtescu, V.I.: Inner Product Structures, Theory and Applications. Reidel, Dordrecht (1987)

    Google Scholar 

  59. Joichi, J.T.: Normed linear spaces equivalent to inner product spaces. Proc. Am. Math. Soc. 17, 423–426 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  60. Jordan, P., von Neumann, J.: On inner products in linear metric spaces. Ann. Math. 36, 719–723 (1935)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  63. Kannappan, Pl.: Functional Equations and Inequalities with Applications. Springer, Dordrecht (2009)

    Book  MATH  Google Scholar 

  64. Kominek, Z.: Stability of a quadratic functional equation on semigroups. Publ. Math. Debrecen 75, 173–178 (2009)

    MathSciNet  MATH  Google Scholar 

  65. Kominek, Z.: On a Dygras inequality. Tatra Mt. Math. Publ. 52, 65–70 (2012)

    MathSciNet  MATH  Google Scholar 

  66. Kominek, Z., Troczka, K.: Some remarks on subquadratic functions. Demonstr. Math. 39, 751–758 (2006)

    MathSciNet  MATH  Google Scholar 

  67. Kominek, Z., Troczka, K.: Continuity of real valued subquaratic functions. Comment. Math. 51, 71–75 (2011)

    MathSciNet  MATH  Google Scholar 

  68. Kuczma, M.: On some alternative functional equations. Aequationes Math. 17, 182–198 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  69. Kuczma, M.: An Introduction to the Theory of Functional Equations and Inequalities. Państwowe Wydawnictwo Naukowe, Warszawa (1985)

    Google Scholar 

  70. Kurepa, S.: The Cauchy functional equation and scalar product in vector spaces. Glasnik Mat. 19, 23–35 (1964)

    MathSciNet  MATH  Google Scholar 

  71. Kurepa, S.: Quadratic and sesquilinear functionals. Glasnik Mat. 20, 79–91 (1965)

    MathSciNet  MATH  Google Scholar 

  72. Kurepa, S.: On bimorphisms and quadratic forms on groups. Aequationes Math. 9, 30–45 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  73. Kurepa, S.: On the definition of a quadratic form. Publ. Ints. Math (Beograd) 42, 35–41 (1987)

    MathSciNet  MATH  Google Scholar 

  74. Kurepa, S.: On P. Volkmann’s paper. Glasnik Mat. 22, 371–374 (1987)

    MathSciNet  Google Scholar 

  75. Kwon, Y.H., Lee, H.M., Sim, J.S., Yand, J., Park, C.: Jordan-von Neumann type functional inequalities. J. Chungcheong Mah. Soc. 20, 269–277 (2007)

    Google Scholar 

  76. Lee, Y.-H., Jung, S.-M.: A general uniqueness theorem concerning the stability of additive and quadratic equations. J. Funct. Spaces 2015, 8 (2015). Art. ID 643969

    Google Scholar 

  77. Lee, J.R., Park, C., Shin, D.Y.: On the stability of generalized additive functional inequalities in Banach spaces. J. Ineq. Appl. 2008, 13 (2008). Art. ID 210626

    Google Scholar 

  78. Lumer, G.: Semi-inner-product spaces. Trans. Am. Math. Soc. 100, 29–43 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  79. Makai, I.: Über invertierbare Lösungen der additive Cauchy-Functionalgleichung. Publ. Math. Debrecen 16, 239–243 (1969)

    MathSciNet  MATH  Google Scholar 

  80. Maksa, Gy.: A remark of symmetric biadditive functions having nonnegative diagonalization. Glasnik Mat. 15, 279–282 (1980)

    Google Scholar 

  81. Maksa, Gy.: Remark on the talk of P. Volkmann. In: Proceedings of the Twenty-third International Symposium on Functional Equations. Gargano, Italy (1985)

    Google Scholar 

  82. Maksa, Gy., Volkmann, P.: Characterizations of group homomorphisms having values in an inner product space. Publ. Math. Debrecen 56, 197–200 (2000)

    Google Scholar 

  83. Manar, Y., Elqorachi, E.: On functional inequalities associated with Drygas functional equation. Tbilisi Math. J. 7, 73–78 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  84. Marinescu, D.S., Monea, M., Opincariu, M., Stroe, M.: Some equivalent characterizations of inner product spaces and their consequences. Filomat 29, 1587–1599 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  85. Mitrinović, D.S., Pečarić, J.E., Fink, A.M.: Classical and New Inequalities in Analysis. Kluwer, Dordrecht (1993)

    Book  MATH  Google Scholar 

  86. Multarziński, P.: Semi-inner product structures for groupoids (2013). arXiv:1301.0764v1 [math.GR]

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  88. Nakmahachalasint, P.: On the generalized Ulam-Gavruta-Rassias stability of mixed-type linear and Euler-Lagrange-Rassias functional equations. Int. J. Math. Math. Sci. 2007, 10 (2007). At. ID 63239

    Google Scholar 

  89. Nakmahachalasint, P.: Hyers-Ulam-Rassias and Ulam-Gavruta-Rassias stabilities of an additive functional equation in several variables. Int. J. Math. Math. Sci. 2007, 6 (2007). At. ID 13437

    Google Scholar 

  90. Ng, C.T., Zhao, H.Y.: Kernel of the second order Cauchy difference on groups. Aequationes Math. 86, 155–170 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  91. Nikodem, K.: 7. Problem. Aequationes Math. 61, 301 (2001)

    Google Scholar 

  92. Oikhberg, T., Rosenthal, H.: A metric characterization of normed linear spaces. Rocky Mont. J. Math. 37, 597–608 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  93. Oubbi, L.: Ulam-Hyers-Rassias stability problem for several kinds of mappings. Afr. Mat. 24, 525–542 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  94. Park, Ch., Cho, Y.S., Han, M.-H. Functional inequalities associated with Jordan–von Neumann type additive functional equations. J. Ineq. Appl. 2007, 13 (2007). Art. ID 41820

    Google Scholar 

  95. Park, C., Lee, J.R., Rassias, Th.M.: Functional inequalities in Banach spaces and fuzzy Banach spaces. In: Rassias, Th.M., Pardalos, P.M. (eds.) Essays in Mathematics and Its Applications, pp. 263–310. Springer, Cham (2016)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  97. Piejko, K.: Note on Robinson’s Functional equation. Demonstratio Math. 32, 713–715 (1999)

    MathSciNet  MATH  Google Scholar 

  98. Popoviciu, T.: Sur certaines inéqualités qui caractérisentes fonctions convexes. An. Stiint. Univ. Al. I. Cuza Iasi 11, 155–164 (1965)

    Google Scholar 

  99. Rassias, J.M.: On the Ulam stability of mixed type mappings on restricted domains. J. Math. Anal. Appl. 276, 747–762 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  100. Rätz, J.: On inequalities associated with the Jordan-von Neumann functional equation. Aequationes Math. 66, 191–200 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  101. Robinson, R.M.: A curious trigonometric identity. Am. Math. Monthly 64, 83–85 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  102. Roh, J., Chang, I.-S.: Functional inequalities associated with additive mappings. Abst. Appl. Anal. 2008, 11 (2008). Art. ID 136592

    Google Scholar 

  103. Röhmer, J.: Ein Charakterisierung quadratischer Formen durch eine Functionalgleichung. Aequationes Math. 15, 163–168 (1977)

    Article  MathSciNet  Google Scholar 

  104. Rosenbaum, R.A.: Sub-additive functions. Duke Math. J. 17, 227–247 (1950)

    Article  MathSciNet  MATH  Google Scholar 

  105. Schöpf, P.: Solutions of ∥ f(ξ + η) ∥ = ∥ f (ξ) + f (η) ∥. Math. Pannon. 8, 117–127 (1997)

    MathSciNet  Google Scholar 

  106. Šemrl, P.: A characterization of normed spaces. J. Math. Anal. Appl. 343, 1047–1051 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  107. Šemrl, P.: A characterization of normed spaces among metric spaces. Rocky Mt. J. Math. 41, 293–298 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  108. Sikorska, J.: Stability of the preservation of the equality of distance. J. Math. Anal. Appl. 311, 209–217 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  109. Sinopoulos, P.: Functional equations on semigroups. Aequationes Math. 59, 255–261 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  110. Skof, F.: On the functional equation ∥ f (x + y) − f (x) ∥ = ∥ f (y) ∥. Atti Acad. Sci. Torino Cl. Sci. Fis. Mat. Natur. 127, 229–237 (1993)

    MathSciNet  MATH  Google Scholar 

  111. Skof, F.: On two conditional forms of the equation ∥ f (x + y) ∥ = ∥ f (x) + f(y) ∥. Aequationes Math. 45, 167–178 (1993)

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  113. Smajdor, W.: On a Jensen type functional equation. J. Appl. Anal. 13, 19–31 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  114. Stetkaer, H.: Functional Equations on Groups. World Scientific, New Jersey (2013)

    Book  MATH  Google Scholar 

  115. Światak, H.: On the functional equation f (x + y)2 = [ f (x) + f (y) ]2. Publ. Techn. Univ. Miskolc 30, 307–308 (1970)

    Google Scholar 

  116. Światak, H., Hosszú, M.: Remarks on the functional equation e (x, y) f (x + y) = f (x) + f (y). Publ. Techn. Univ. Miskolc 30, 323–325 (1970)

    Google Scholar 

  117. Szabó, Gy.: A conditional Cauchy equation on normed spaces. Publ. Math. Debrecen 42, 265–271 (1993)

    Google Scholar 

  118. Száz, Á.: Preseminormed spaces. Publ. Math. Debrecen 30, 217–224 (1983)

    MathSciNet  MATH  Google Scholar 

  119. Száz, Á.: An instructive treatment of convergence, closure and orthogonality in semi-inner product spaces, Technical Report, 2006/2, 29 pp. Institute of Mathematics, University of Debrecen, Debrecen (2006)

    Google Scholar 

  120. Száz, Á.: Applications of fat and dense sets in the theory of additive functions, Technical Report, 2007/3, 29 pp. Institute of Mathematics, University of Debrecen, Debrecen (2007)

    Google Scholar 

  121. Száz, Á.: A generalization of a theorem of Maksa and Volkmann on additive functions, Technical Report, 2016/5, 6 pp. Institute of Mathematics, University of Debrecen (2016)

    Google Scholar 

  122. Száz, Á.: Remarks and Problems at the Conference on Inequalities and Applications, Hajdúszoboszló, Hungary, 2016, Technical Report, 2016/9, 34 pp. Institute of Mathematics, University of Debrecen, Debrecen (2016)

    Google Scholar 

  123. Száz, Á.: A natural Galois connection between generalized norms and metrics. Acta Univ. Sapientiae Math. 9, 360–373 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  124. Száz, Á.: Generalizations of an asymptotic stability theorem of Bahyrycz, Páles and Piszczek on Cauchy differences to generalized cocycles. Stud. Univ. Babes-Bolyai Math. 63, 109–124 (2018)

    Article  MATH  Google Scholar 

  125. Tabor, J.: Stability of the Fisher–Muszély functional equation. Publ. Math. Debrecen 62, 205–211 (2003)

    MathSciNet  MATH  Google Scholar 

  126. Tabor, J., Tabor, J.: 19. Remark (Solution of the 7. Problem posed by K. Nikodem.). Aequationes Math. 61, 307–309 (2001)

    Google Scholar 

  127. Toborg, I., Volkmann, P.: On stability of the Cauchy functional equation in groupoids. Ann. Math. Sil. 31, 155–164 (2017)

    MathSciNet  MATH  Google Scholar 

  128. Trif, T.: Hyers-Ulam-Rassias stability of a Jensen type functional equation. J. Math. Anal. Appl. 250, 579–588 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  129. Trif, T.: Popoviciu’s and related functional equations: a survey. In: Rassias, Th.M., Andrica, D. (eds.) Inequalities and Applications, pp. 273–286. Cluj-University Press, Cluj-Napoca (2008)

    Google Scholar 

  130. Troczka-Pawelec, K.: Some inequalities connected with a quadratic functional equation. Pr. Nauk. Akad. Jana Dlugosza Czest. Mat. 13, 73–79 (2008)

    MathSciNet  Google Scholar 

  131. Vincze, E.: On solutions of alternative functional equations (Humgarian). Mat. Lapok 15, 179–195 (1964)

    MathSciNet  Google Scholar 

  132. Vincze, E.: Beitrag zur Theorie der Cauchyschen Functionalgleichungen. Arch. Math. 15, 132–135 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  133. Vincze, E.: Über eine Verallgemeinerung der Cauchysche functionalgleichung. Funcialaj Ekvacioj 6, 55–62 (1964)

    MATH  Google Scholar 

  134. Volkmann, P.: Pour une fonction réelle f l’inéquation | f (x) + f (y)|≤|f (x + y)| et l’équation de Cauchy sont équivalentes. In: Proceedings of the Twenty-third International Symposium on Functional Equations, Gargano, Italy (1985)

    Google Scholar 

  135. Vrbová, P.: Quadratic functionals and bilinear forms. Časopis Pěst. Mat. 98, 159–161 (1973)

    MathSciNet  MATH  Google Scholar 

  136. Wilanski, A.: Modern Methods in Topological Vector Spaces, McGraw-Hill, New York (1978)

    Google Scholar 

  137. Youssef, M., Elhoucien, E.: On functional inequalities associated with Drygas functional equation, 5 pp (2014). arXiv: 1405.7942

    Google Scholar 

Download references

Acknowledgements

The work of the author has been supported by the Hungarian Scientific Research Fund (OTKA) Grant K-111651.

Moreover, the author is greatly indebted to Zoltán Boros, Gyula Maksa, Attila Gilányi and Jens Schwaiger for some inspiring conversations.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Debrecen, Debrecen, Hungary

    Árpád Száz

Authors
  1. Árpád Száz
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Árpád Száz .

Editor information

Editors and Affiliations

  1. Department of Applied Mathematics, AGH University of Science and Technology, Krakow, Poland

    Janusz Brzdęk

  2. Department of Mathematics, Technical University of Cluj-Napoca, Cluj-Napoca, Romania

    Dorian Popa

  3. Department of Mathematics, National Technical University of Athens, Athens, Greece

    Themistocles M. Rassias

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Száz, Á. (2019). Semi-Inner Products and Parapreseminorms on Groups and a Generalization of a Theorem of Maksa and Volkmann on Additive Functions. In: Brzdęk, J., Popa, D., Rassias, T. (eds) Ulam Type Stability . Springer, Cham. https://doi.org/10.1007/978-3-030-28972-0_18

Download citation

Publish with us

Policies and ethics

Search

Navigation