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The Hyers–Ulam and Hahn–Banach Theorems and Some Elementary Operations on Relations Motivated by Their Set-Valued Generalizations

Part of the Springer Optimization and Its Applications book series (SOIA, volume 68)

Abstract

In the first part of this paper, we provide several historical facts on the famous Hyers–Ulam stability theorems, Hahn–Banach extension theorems, and their set-valued generalizations with numerous references.

These generalizations will clearly show that the essence of the above mentioned theorems is nothing but the statement of the existence of a certain homogeneous, additive, or linear selection function of a particular relation.

In the second part of this paper, motivated by the above generalizations, we briefly review the most basic additivity and homogeneity properties of relations and investigate, in greater detail, some elementary operations on relations.

More concretely, for any relation F on one group X to another Y, we define two relations −F and \(\check{F}\) on X to Y such that \(\check{F}(x)=F(-x)\) and (−F)(x)=−F(x) for all xX. Moreover, we also define \(\hat{F}=-\check{F}\) and \(F^{\vartriangle}= F\cap\hat{F}\).

Furthermore, if in particular Y is a vector space over ℚ, then for any k∈ℤ, with k≠0, we also define a relation F k on X to Y such that F k (x)=k −1 F(kx) for all xX. Moreover, we also define \(F^{\star}=\bigcap_{ n=1}^{\infty} F_{n}\) and \(F^{\ast}=F^{\vartriangle\star}\).

The above operations and the intersection convolutions of relations, which can only be sketched here, will certainly allow of instructive treatments of some hoped-for common relational generalizations of the Hyers–Ulam and Hahn–Banach theorems.

Key words

Relations on groups Partial and global negatives Hyers transforms Intersection convolutions 

Mathematics Subject Classification

03E20 26E25 39B82 46A22 

Notes

Acknowledgements

The author is indebted to J. Horváth and Th.M. Rassias for several valuable pieces of advice.

Moreover, the author would also like to thank R. Ger, M. Sablik, Zs. Páles, and G. Horváth for some helpful discussions.

References1

  1. 1.
    Abreu, J., Etcheberry, A.: Hahn–Banach and Banach–Steinhaus theorems for convex processes. Period. Math. Hung. 20, 289–297 (1989) MathSciNetMATHGoogle Scholar
  2. 2.
    Aczél, J., Dhombres, J.: Functional Equations in Several Variables. Cambridge University Press, Cambridge York (1989) MATHGoogle Scholar
  3. 3.
    Adasch, N.: Der satz über offene lineare relationen in topologischen vectoräumen. Note Mat. 11, 1–5 (1991) MathSciNetMATHGoogle Scholar
  4. 4.
    Álvarez, T.: On the Browder essential sprectum of a linear relation. Publ. Math. (Debr.) 73, 145–154 (2008) MATHGoogle Scholar
  5. 5.
    Aoki, T.: On the stability of the linear transformation in Banach spaces. J. Math. Soc. Jpn. 2, 491–495 (1950) Google Scholar
  6. 6.
    Arens, R.: Operational calculus of linear relations. Pac. J. Math. 11, 9–23 (1961) MathSciNetMATHGoogle Scholar
  7. 7.
    Bahyrycz, A.: Forti’s example of an unstable homomorphism equation. Aequ. Math. 74, 310–313 (2007) MathSciNetMATHGoogle Scholar
  8. 8.
    Badora, R.: On some generalized invariant means and their application to the stability of the Hyers–Ulam type. Ann. Pol. Math. 58, 111–126 (1993) MathSciNetGoogle Scholar
  9. 9.
    Badora, R.: On approximately additive functions. Ann. Math. Sil. 8, 111–126 (1994) MathSciNetGoogle Scholar
  10. 10.
    Badora, R.: On the Hahn–Banach theorem for groups. Arch. Math. 86, 517–528 (2006) MathSciNetMATHGoogle Scholar
  11. 11.
    Badora, R., Ger, R., Páles, Zs.: Additive selections and the stability of the Cauchy functional equation. ANZIAM J. 44, 323–337 (2003) MathSciNetMATHGoogle Scholar
  12. 12.
    Baker, J.A.: On some mathematical characters. Glas. Mat. 25, 319–328 (1990) Google Scholar
  13. 13.
    Baker, J.A.: The stability of certain functional equations. Proc. Am. Math. Soc. 112, 729–732 (1991) MATHGoogle Scholar
  14. 14.
    Banach, S.: Sur les fonctionelles linéaires I–II. Stud. Math. 1, 211–216, 223–239 (1929) MATHGoogle Scholar
  15. 15.
    Banach, S.: Théorie des Opérations Linéaires. Druk M. Garasiński, Warsawa (1932) Google Scholar
  16. 16.
    Banach, S., Mazur, S.: Zur theorie der linearen dimension. Stud. Math. 4, 100–112 (1933) Google Scholar
  17. 17.
    Baron, K.: Functions with differences in subspaces. In: Proceedings of the 18th International Symposium on Functional Equations, Faculty of Mathematics, pp. 8–9. University of Waterloo (1980) Google Scholar
  18. 18.
    Baron, K.: A remark on the stability of the Cauchy equation. Rocz. Nauk.-Dydakt. Pr. Mat. 11, 7–12 (1985) Google Scholar
  19. 19.
    Baron, K., Kannappan, Pl.: On the Pexider difference. Fundam. Math. 134, 247–254 (1990) MathSciNetMATHGoogle Scholar
  20. 20.
    Baron, K., Volkmann, P.: On functions close to homomorphisms between square symmetric structures. Seminar LV 14, 1–12 (2002). http://www.mathematik.uni-karlsruhe.de/~semlv Google Scholar
  21. 21.
    Baron, K., Simon, A., Volkmann, P.: On functions having Cauchy differences in some prescribed sets. Aequ. Math. 52, 254–259 (1996) MathSciNetMATHGoogle Scholar
  22. 22.
    Baron, K., Sablik, M., Volkmann, P.: On decent solutions of a functional congruence. Rocz. Nauk.-Dydakt. Pr. Mat. 17, 27–40 (2000) MathSciNetMATHGoogle Scholar
  23. 23.
    Beg, I.: Fuzzy multivalued functions. Bull. Allahabad Math. Soc. 21, 41–104 (2006) MathSciNetMATHGoogle Scholar
  24. 24.
    Berge, C.: Topological Spaces Including a Treatment of Multi-Valued Functions, Vector Spaces and Convexity. Oliver and Boyd, London (1963) MATHGoogle Scholar
  25. 25.
    Berz, E.: Sublinear functions on ℝ. Aequ. Math. 12, 200–206 (1975) MathSciNetMATHGoogle Scholar
  26. 26.
    Boccuto, A., Candeloro, D.: Sandwich theorems and applications to invariant measures. Atti Sem. Math. Fis. Univ. Modena 38, 511–524 (1990) MathSciNetMATHGoogle Scholar
  27. 27.
    Bohnenblust, H.F., Sobczyk, A.: Extensions of functionals on complex linear spaces. Bull. Am. Math. Soc. 44, 91–93 (1938) Google Scholar
  28. 28.
    Boros, Z.: Stability of the Cauchy equation in ordered fields. Math. Pannon. 11, 191–197 (2000) MathSciNetMATHGoogle Scholar
  29. 29.
    Boros, Z., Száz, Á.: Reflexivity, transitivity, symmetry, and antisymmetry of the intersection convolution of relations. Rostock. Math. Kolloq. 63, 55–62 (2008) MATHGoogle Scholar
  30. 30.
    Bourgin, D.G.: Clases of transformations and bordering transformations. Bull. Am. Math. Soc. 57, 223–237 (1951) MathSciNetMATHGoogle Scholar
  31. 31.
    Brzdek, J.: On the Cauchy difference. Glas. Mat. 27, 263–269 (1992) MathSciNetMATHGoogle Scholar
  32. 32.
    Brzdek, J.: On a method of proving the Hyers–Ulam stability of functional equations on restricted domains. Aust. J. Math. Anal. Appl. 6, 1–10 (2009). Article 4 MathSciNetGoogle Scholar
  33. 33.
    Brzdek, J., Tabor, J.: Stability of the Cauchy congruence on restricted domain. Arch. Math. 82, 546–550 (2004) MathSciNetMATHGoogle Scholar
  34. 34.
    Brzdek, J., Popa, D., Xu, B.: Selections of set-valued maps satisfying a linear inclusion in a single variable. Nonlinear Anal. 74, 324–330 (2011) MathSciNetMATHGoogle Scholar
  35. 35.
    Burai, P., Száz, Á.: Relationships between homogeneity, subadditivity and convexity properties. Publ. Elektroteh. Fak. Univ. Beogr., Mat. 16, 77–87 (2005) Google Scholar
  36. 36.
    Buskes, G.: The Hahn–Banach theorem surveyed. Diss. Math. 327, 1–49 (1993) MathSciNetGoogle Scholar
  37. 37.
    Chu, H.-Y., Kang, D.S., Rassias, Th.M.: On the stability of a mixed n-dimensional quadratic functional equation. Bull. Belg. Math. Soc. 15, 9–24 (2008) MathSciNetMATHGoogle Scholar
  38. 38.
    Castillo, E., Ruiz-Cobo, M.R.: Functional Equations and Modelling in Science and Engineering. Dekker, New York (1992) MATHGoogle Scholar
  39. 39.
    Coddington, E.A.: Extension theory of formally normal and symmetric subspaces. Mem. Am. Math. Soc. 134, 5–7 (1973) MathSciNetGoogle Scholar
  40. 40.
    Coddington, E.A., Dijksma, A.: Adjoint subspaces in Banach spaces with applications to ordinary differential subspaces. Ann. Mat. Pura Appl. 118, 1–118 (1978) MathSciNetMATHGoogle Scholar
  41. 41.
    Cross, R.: Multivalued Linear Operators. Dekker, New York (1998) MATHGoogle Scholar
  42. 42.
    Czerwik, S.: Functional Equations and Inequalities in Several Variables. World Scientific, London (2002) MATHGoogle Scholar
  43. 43.
    Czerwik, S. (ed.): Stability of Functional Equations of Ulam–Hyers–Rassias Type. Hadronic Press, Palm Harbor (2003) Google Scholar
  44. 44.
    Czerwik, S., Król, K.: Ulam stability of functional equations. Aust. J. Math. Anal. Appl. 6, 1–15 (2009). Article 6 MathSciNetGoogle Scholar
  45. 45.
    Dacić, R.: On multi-valued functions. Publ. Inst. Math. (Belgr.) 9, 5–7 (1969) Google Scholar
  46. 46.
    Dascǎl, J., Száz, Á.: Inclusion properties of the intersection convolution of relations. Ann. Math. Inform. 36, 47–60 (2009) MathSciNetGoogle Scholar
  47. 47.
    Dascǎl, J., Száz, Á.: A necessary condition for the extensions of subadditive partial selection relations. Tech. Rep., Inst. Math., Univ. Debrecen 1, 1–13 (2009) Google Scholar
  48. 48.
    Faziev, V.A., Rassias, Th.M.: The space of (ψ,ν)-pseudocharacters on semigroups. Nonlinear Funct. Anal. Appl. 5, 107–126 (2000) Google Scholar
  49. 49.
    Faziev, V.A., Rassias, Th.M., Sahoo, P.K.: The space of (ψ,ν)-additive mappings onsemigroups. Trans. Am. Math. Soc. 354, 4455–4472 (2002) Google Scholar
  50. 50.
    Farkas, T., Száz, Á.: Minkowski functionals of summative sequences of absorbing and balanced sets. Bul. Ştiinţ. Univ. Baia Mare, Ser. B, Fasc. Mat.-Inform. 16, 323–334 (2000) MATHGoogle Scholar
  51. 51.
    Fechner, W.: Separation theorems for conditional functional equations. Ann. Math. Sil. 21, 31–40 (2007) MathSciNetMATHGoogle Scholar
  52. 52.
    Fechner, W.: On an abstract version of a functional inequality. Math. Inequal. Appl. 11, 381–392 (2008) MathSciNetMATHGoogle Scholar
  53. 53.
    Figula, A., Száz, Á.: Graphical relationships between the infimum and the intersection convolutions. Math. Pannon. 21, 23–35 (2010) MathSciNetGoogle Scholar
  54. 54.
    Findlay, G.D.: Reflexive holomorphic relations. Can. Math. Bull. 3, 131–132 (1960) MathSciNetMATHGoogle Scholar
  55. 55.
    Forti, G.L.: An existence and stability theorem for a class of functional equations. Stohastica 4, 23–30 (1980) MathSciNetMATHGoogle Scholar
  56. 56.
    Forti, G.L.: Remark 11. Aequ. Math. 29, 90–91 (1985) MathSciNetGoogle Scholar
  57. 57.
    Forti, G.L.: The stability of homomorphisms and amenability, with applications to functional equations. Abh. Math. Sem. Univ. Hamburg 57, 215–226 (1987) MathSciNetMATHGoogle Scholar
  58. 58.
    Forti, G.L.: Hyers–Ulam stability of functional equations in several variables. Aequ. Math. 50, 143–190 (1995) MathSciNetMATHGoogle Scholar
  59. 59.
    Forti, G.L.: Comments on the core of the direct method for proving Hyers–Ulam stability of functional equations. J. Math. Anal. Appl. Aequ. Math. 295, 127–133 (2004) MathSciNetMATHGoogle Scholar
  60. 60.
    Forti, G.L., Schwaiger, J.: Stability of homomorphisms and completeness. C. R. Math. Rep. Acad. Sci. Can. 11, 215–220 (1989) MathSciNetMATHGoogle Scholar
  61. 61.
    Fuchssteiner, B.: Sandwich theorems and lattice semigroups. J. Funct. Anal. 16, 1–14 (1974) MathSciNetMATHGoogle Scholar
  62. 62.
    Fuchssteiner, B., Horváth, J.: Die Bedeutung der Schnitteigenschaften beim Hahn–Banachschen Satz, Jahrbuch Überblicke Math, pp. 107–121. Bibliograph. Inst., Mannheim (1979). (The publication of an expanded English version of this paper in the Publ. Math. Debrecen was prevented in 1997 by the Editorial Board headed by L. Tamássy) Google Scholar
  63. 63.
    Fuchssteiner, B., Lusky, W.: Convex Cones. North-Holland, New York (1981) MATHGoogle Scholar
  64. 64.
    Fullerton, R.E.: An intersection property for cones in a linear space. Proc. Am. Math. Soc. 9, 558–561 (1958) MathSciNetMATHGoogle Scholar
  65. 65.
    Gajda, Z.: On stability of the Cauchy equation on semigroups. Aequ. Math. 36, 76–79 (1988) MathSciNetMATHGoogle Scholar
  66. 66.
    Gajda, Z.: On stability of additive mappings. Int. J. Math. Sci. 14, 431–434 (1991) MathSciNetMATHGoogle Scholar
  67. 67.
    Gajda, Z.: Invariant means and representations of semigroups in the theory of functional equations. Pr. Nauk. Uniw. Ślask. Katowic. 1273, 1–81 (1992) MathSciNetGoogle Scholar
  68. 68.
    Gajda, Z.: Sandwich theorems and amenable semigroups of transformations. Grazer Math. Ber. 316, 43–58 (1992) MathSciNetMATHGoogle Scholar
  69. 69.
    Gajda, Z., Ger, R.: Subadditive multifunctions and Hyers–Ulam stability. In: Walter, W. (ed.) General Inequalities 5. Internat. Ser. Numer. Math., vol. 80, pp. 281–291. Birkhäuser, Basel (1987) Google Scholar
  70. 70.
    Gajda, Z., Kominek, Z.: On separation theorems for subadditive and superadditive functionals. Stud. Math. 100, 25–38 (1991) MathSciNetMATHGoogle Scholar
  71. 71.
    Gajda, Z., Smajdor, A., Smajdor, W.: A theorem of the Hahn–Banach type and its applications. Ann. Pol. Math. 57, 243–252 (1992) MathSciNetMATHGoogle Scholar
  72. 72.
    Gǎvruţǎ, P.: A generalization of the Hyers–Ulam–Rassias stability of approximately additive mappings. J. Math. Anal. Appl. 184, 431–436 (1994) MathSciNetGoogle Scholar
  73. 73.
    Gǎvruţǎ, P.: On a problem of G. Isac and Th.M. Rassias concerning the stability of mappings. J. Math. Anal. Appl. 261, 543–553 (2001) MathSciNetGoogle Scholar
  74. 74.
    Gǎvruta, P., Gǎvruta, L.: A new method for the generalized Hyers–Ulam–Rassias stability. Int. J. Nonlinear Anal. Appl. 1, 11–18 (2010) Google Scholar
  75. 75.
    Gǎvruta, P., Hossu, M., Popescu, D., Cǎprǎu, C.: On the stability of mappings and an answer to a problem of Th.M. Rassias. Ann. Math. Blaise Pascal 2, 55–60 (1995) MathSciNetGoogle Scholar
  76. 76.
    Ger, R.: The singular case in the stability behaviour of linear mappings. Grazer Math. Ber. 316, 59–70 (1992) MathSciNetMATHGoogle Scholar
  77. 77.
    Ger, R.: On a factorization of mappings with a prescribed behavior of the Cauchy difference. Ann. Math. Sil. 8, 147–155 (1994) MathSciNetGoogle Scholar
  78. 78.
    Ger, R.: A survey of recent results on stability of functional equations. In: Proceedings of the 4th International Conference on Functional Equations and Inequalities, pp. 5–36. Pedagogical University of Krakow (1994) Google Scholar
  79. 79.
    Ger, R., Volkmann, P.: On sums of linear and bounded mappings. Abh. Math. Semin. Univ. Hamb. 68, 103–108 (1998) MathSciNetMATHGoogle Scholar
  80. 80.
    Gilányi, A., Kaiser, Z., Páles, Zs.: Estimates to the stability of functional equations. Aequ. Math. 73, 125–143 (2007) MATHGoogle Scholar
  81. 81.
    Glavosits, T., Kézi, Cs.: On the domain of oddness of an infimal convolution. Miskolc Math. Notes 12, 31–40 (2011) MathSciNetMATHGoogle Scholar
  82. 82.
    Glavosits, T., Száz, Á.: Pointwise and global sums and negatives of binary relations. An. St., Univ. Ovidius Constanta 11, 87–94 (2003) Google Scholar
  83. 83.
    Glavosits, T., Száz, Á.: Pointwise and global sums and negatives of translation relations. An. St., Univ. Ovidius Constanta 12, 27–44 (2004) MATHGoogle Scholar
  84. 84.
    Glavosits, T., Száz, Á.: On the existence of odd selections. Adv. Stud. Contemp. Math. (Kyungshang) 8, 155–164 (2004) MathSciNetMATHGoogle Scholar
  85. 85.
    Glavosits, T., Száz, Á.: General conditions for the subadditivity and superadditivity of relations. Scientia, Ser. A, Math. Sci. 11, 31–43 (2005) MATHGoogle Scholar
  86. 86.
    Glavosits, T., Száz, Á.: Constructions and extensions of free and controlled additive relations. Tech. Rep., Inst. Math., Univ. Debrecen 1, 1–49 (2010) Google Scholar
  87. 87.
    Glavosits, T., Száz, Á.: The infimal convolution can be used to easily prove the classical Hahn–Banach theorem. Rostock. Math. Kolloq. 65, 71–83 (2010) MATHGoogle Scholar
  88. 88.
    Glavosits, T., Száz, Á.: The generalized infimal convolution can be used naturally prove some dominated monotone additive extension theorems. Tech. Rep., Inst. Math., Univ. Debrecen 4, 1–26 (2010) Google Scholar
  89. 89.
    Glavosits, T., Száz, Á.: A Hahn–Banach type generalization of the Hyers–Ulam theorem. An. St., Univ. Ovidius Constanta 19, 139–144 (2011) MATHGoogle Scholar
  90. 90.
    Godini, G.: Set-valued Cauchy functional equation. Rev. Roum. Math. Pures Appl. 20, 1113–1121 (1975) MathSciNetMATHGoogle Scholar
  91. 91.
    Godini, G.: An approach to generalizing Banach spaces: normed almost linear spaces. Rend. Circ. Mat. Palermo, Suppl. 5, 33–50 (1984) MathSciNetMATHGoogle Scholar
  92. 92.
    Grabiec, A.: The generalized Hyers–Ulam stability of a class of functional equations. Publ. Math. (Debr.) 48, 217–235 (1996) MathSciNetGoogle Scholar
  93. 93.
    Gselmann, E., Száz, Á.: An instructive treatment of a generalization of Gǎvruţǎ’s stability theorem. Sarajevo J. Math. 6, 3–21 (2010) MathSciNetGoogle Scholar
  94. 94.
    Hahn, H.: Über lineare Gleichungsysteme in linearen Räumen. J. Reine Angew. Math. 157, 214–229 (1927) MATHGoogle Scholar
  95. 95.
    Harrop, R., Weston, J.D.: An intersection property in locally convex spaces. Proc. Am. Math. Soc. 7, 535–538 (1956) MathSciNetMATHGoogle Scholar
  96. 96.
    Harte, R.E.: A generalization of the Hahn–Banach theorem. J. Lond. Math. Soc. 40, 283–287 (1965) MathSciNetMATHGoogle Scholar
  97. 97.
    Hassi, S., Sebestyén, Z., De Snoo, H.S.V., Szafraniec, F.H.: A canonical decomposition for linear operators and linear relations. Acta Math. Hung. 115, 281–307 (2007) MATHGoogle Scholar
  98. 98.
    Hasumi, M.: The extension property of complex Banach spaces. Tohoku Math. J. 10, 135–142 (1958) MathSciNetMATHGoogle Scholar
  99. 99.
    Helly, E.: Über lineare Functionaloperationen. Sitzungsber. Acad. Wiss. Wien 121, 265–297 (1912) MATHGoogle Scholar
  100. 100.
    Henney, D.: The structure of set-valued additive functions. Port. Math. 26, 463–471 (1967) MathSciNetMATHGoogle Scholar
  101. 101.
    Henney, D.: Properties of set-valued additive functions. Am. Math. Mon. 75, 203–206 (1968) MathSciNetGoogle Scholar
  102. 102.
    Henney, D.: Representations of set-valued additive functions. Aequ. Math. 3, 230–235 (1969) MathSciNetMATHGoogle Scholar
  103. 103.
    Hochstadt, H.: Eduard Helly, father of the Hahn–Banach theorem. Math. Intell. 2, 123–125 (1980) MathSciNetMATHGoogle Scholar
  104. 104.
    Holá, L’.: Some properties of almost continuous linear relations. Acta Math. Univ. Comen. 50–51, 61–69 (1987) Google Scholar
  105. 105.
    Holá, L’., Kupka, I.: Closed graph and open mapping theorems for linear relations. Acta Math. Univ. Comen. 46–47, 157–162 (1985) Google Scholar
  106. 106.
    Holá, L’., Maličký, P.: Continuous linear selectors of linear relations. Acta Math. Univ. Comen. 48–49, 153–157 (1986) Google Scholar
  107. 107.
    Holbrook, J.A.R.: Concerning the Hahn–Banach theorem. Proc. Am. Math. Soc. 50, 322–327 (1975) MathSciNetMATHGoogle Scholar
  108. 108.
    Horváth, J.: Some selected results of Professor Baltasar Rodríguez-Salinas. Rev. Mat. Univ. Complut. Madr. 9, 23–72 (1996) MATHGoogle Scholar
  109. 109.
    Huang, J., Li, Y.: The Hahn–Banach theorem on arbitrary groups. Kyungpook Math. J. 49, 245–254 (2009) MathSciNetMATHGoogle Scholar
  110. 110.
    Hustad, O.: A note on complex Open image in new window spaces. Isr. J. Math. 16, 117–119 (1973) MathSciNetMATHGoogle Scholar
  111. 111.
    Hyers, D.H.: On the stability of the linear functional equation. Proc. Natl. Acad. Sci. USA 27, 222–224 (1941) MathSciNetGoogle Scholar
  112. 112.
    Hyers, D.H.: The stability of homomorphisms and related topics. In: Rassias, Th.M. (ed.) Global Analysis on manifolds. Teubner-Texte Math., vol. 57, pp. 140–153, Leipzig (1983) Google Scholar
  113. 113.
    Hyers, D.H., Rassias, Th.M.: Approximate homomorphisms. Aequ. Math. 44, 125–153 (1992) MathSciNetMATHGoogle Scholar
  114. 114.
    Hyers, D.H., Isac, G., Rassias, Th.M.: Topics in Nonlinear Analysis and Applications. World Scientific, London (1997) MATHGoogle Scholar
  115. 115.
    Hyers, D.H., Isac, G., Rassias, Th.M.: Stability of Functional Equations in Several Variables. Birkhäuser, Boston (1998) MATHGoogle Scholar
  116. 116.
    Ingleton, A.W.: The Hahn–Banach theorem for non-Archimedean-valued fields. Proc. Camb. Philol. Soc. 48, 41–45 (1952) MathSciNetMATHGoogle Scholar
  117. 117.
    Ioffe, A.D.: A new proof of the equivalence of the Hahn–Banach extension and the least upper bound properties. Proc. Am. Math. Soc. 82, 385–389 (1981) MathSciNetMATHGoogle Scholar
  118. 118.
    Ioffe, A.D.: Nonsmooth analysis: differential calculus of nondifferentiable mappings. Trans. Am. Math. Soc. 266, 1–56 (1981) MathSciNetMATHGoogle Scholar
  119. 119.
    Isac, G., Rassias, Th.M.: On the Hyers–Ulam stability of ψ-additive mappings. J. Approx. Theory 72, 131–137 (1993) MathSciNetMATHGoogle Scholar
  120. 120.
    Isac, G., Rassias, Th.M.: Functional inequalities for approximately additive mappings. In: Rassias, Th.M., Tabor, J. (eds.) Stability of Mappings of Hyers–Ulam Type, pp. 117–125. Hadronic Press, Palm Harbor (1994) Google Scholar
  121. 121.
    Isac, G., Rassias, Th.M.: Stability of ψ-additive mappings: applications to non-linear analysis. Int. J. Math. Math. Sci. 19, 219–228 (1996) MathSciNetMATHGoogle Scholar
  122. 122.
    Jameson, G.: Ordered Linear Spaces. Lect. Notes Math., vol. 141. Springer, Berlin (1970) MATHGoogle Scholar
  123. 123.
    Jun, K.-W., Shin, D.-S., Kim, B.-D.: On Hyers–Ulam–Rassias stability of the Pexider equation. J. Math. Anal. Appl. 239, 20–29 (1999) MathSciNetMATHGoogle Scholar
  124. 124.
    Jung, S.-M.: Hyers–Ulam–Rassias stability of functional equations. Dyn. Syst. Appl. 6, 541–565 (1997) MATHGoogle Scholar
  125. 125.
    Jung, S.-M.: Hyers–Ulam–Rassias Stability of Functional Equations in Mathematical Analysis. Hadronic Press, Palm Harbor (2001) MATHGoogle Scholar
  126. 126.
    Jung, S.-M.: Survey on the Hyers–Ulam–Rassias stability of functional equations. In: Czerwik, S. (ed.) Stability of Functional Equations of Ulam–Hyers–Rassias Type, pp. 91–117. Hadronic Press, Palm Harbor (2003) Google Scholar
  127. 127.
    Kaiser, Z., Páles, Zs.: An example of a stable functional equation when the Hyers method does not work. J. Inequal. Pure Appl. Math. 6, 1–11 (2005) Google Scholar
  128. 128.
    Kaufman, R.: Extension of functionals and inequalities on an Abelian semi-group. Proc. Am. Math. Soc. 17, 83–85 (1966) MathSciNetMATHGoogle Scholar
  129. 129.
    Kaufman, R.: Interpolation of additive functionals. Stud. Math. 27, 269–272 (1966) MathSciNetMATHGoogle Scholar
  130. 130.
    Kazhdan, D.: On ε-representations. Isr. J. Math. 43, 315–323 (1982) MathSciNetMATHGoogle Scholar
  131. 131.
    Kelley, J.L.: Banach spaces with the extension property. Trans. Am. Math. Soc. 72, 323–326 (1952) MathSciNetMATHGoogle Scholar
  132. 132.
    Kelley, J.L., Namioka, I.: Linear Topological Spaces. Van Nostrand, New York (1963) MATHGoogle Scholar
  133. 133.
    Khodaei, H., Rassias, Th.M.: Approximately generalized additive functions in several variables. Int. J. Nonlinear Anal. Appl. 1, 22–41 (2010) Google Scholar
  134. 134.
    Kim, G.K.: On the stability of functional equations with square-symmetric operations. Math. Inequal. Appl. 4, 257–266 (2001) MathSciNetMATHGoogle Scholar
  135. 135.
    Kominek, Z.: On Hyers–Ulam stability of the Pexider equation. Demonstr. Math. 37, 373–376 (2004) MathSciNetMATHGoogle Scholar
  136. 136.
    Kotarski, W.: A remark on the Hahn–Banach theorem. Rad. Mat. 3, 105–109 (1987) MathSciNetMATHGoogle Scholar
  137. 137.
    Kranz, P.: Additive functionals on abelian semigroups. Ann. Soc. Math. Pol. 16, 239–246 (1972) MathSciNetMATHGoogle Scholar
  138. 138.
    Kranz, P.: Extensions of additive functionals and semicharacters on commutative semigroups. Semigroup Forum 18, 293–305 (1979) MathSciNetMATHGoogle Scholar
  139. 139.
    Kuczma, M.: An Introduction to the Theory of Functional Equations and Inequalities. Polish Sci. Publ. and Univ. Ślaski, Warszawa (1985) MATHGoogle Scholar
  140. 140.
    Lambek, J.: Goursat theorem and the Zassenhaus lemma. Can. J. Math. 10, 45–56 (1958) MathSciNetMATHGoogle Scholar
  141. 141.
    Larsen, R.: An Introduction to the Theory of Multipliers. Sringer, Berlin (1971) MATHGoogle Scholar
  142. 142.
    Lee, Y.-H., Jun, K.-W.: On the stability of approximately additive mappings. Proc. Am. Math. Soc. 128, 1361–1369 (1999) MathSciNetGoogle Scholar
  143. 143.
    Lee, S.J., Nashed, M.Z.: Algebraic and topological selections of multi-valued linear relations. Ann. Scuola Norm. Sup. Pisa 17, 111–126 (1990) MathSciNetGoogle Scholar
  144. 144.
    Lee, S.J., Nashed, M.Z.: Normed linear relations: domain decomposability, adjoint subspaces, and selections. Linear Algebra Appl. 153, 135–159 (1991) MathSciNetMATHGoogle Scholar
  145. 145.
    Lorenzen, P.: Über die Korrespondenzen Einer. Struct. Mat. Zeitshr. 60, 61–65 (1954) MathSciNetMATHGoogle Scholar
  146. 146.
    Lu, G., Park, Ch.: Hyers–Ulam stability of additive set-valued functional equations. Appl. Math. Lett. (2011). doi: 10.1016/j.aml.2011.02.024 MathSciNetGoogle Scholar
  147. 147.
    MacLane, S.: An algebra of additive relations. Proc. Natl. Acad. Sci. USA 47, 1043–1051 (1961) MathSciNetMATHGoogle Scholar
  148. 148.
    MacLane, S.: Homology. Springer, Berlin (1963) MATHGoogle Scholar
  149. 149.
    Maligranda, L.: A result of Tosio Aoki about a generalization of Hyers–Ulam stability of additive functions – a question of priority. Aequ. Math. 75, 289–296 (2008) MathSciNetMATHGoogle Scholar
  150. 150.
    Mihet, D.: The Hyers–Ulam stability for two functional equation in a single variable. Banach J. Math. Anal. 2, 48–52 (2008) MathSciNetMATHGoogle Scholar
  151. 151.
    Moreau, J.J.: Inf-convolution, sous-additivité, convexité des fonctions numériques. J. Math. Pures Appl. 49, 109–154 (1970) MathSciNetMATHGoogle Scholar
  152. 152.
    Moslehian, M.S., Rassias, Th.M.: Stability of functional equations in non-Archimedean spaces. Appl. Anal. Discrete Math. 1, 325–334 (2007) MathSciNetMATHGoogle Scholar
  153. 153.
    Moszner, Z.: On the stability of functional equations. Aequ. Math. 77, 33–88 (2009) MathSciNetMATHGoogle Scholar
  154. 154.
    Murray, F.J.: Linear transformations in L p (p>1). Trans. Am. Math. Soc. 39, 83–100 (1936) Google Scholar
  155. 155.
    Nachbin, L.: A theorem of the Hahn–Banach type for linear transformations. Trans. Am. Math. Soc. 68, 28–46 (1950) MathSciNetMATHGoogle Scholar
  156. 156.
    Najati, A., Rassias, Th.M.: Stability of homomorphisms and (θ,φ)-derivations. Appl. Anal. Discrete Math. 3, 264–281 (2009) MathSciNetMATHGoogle Scholar
  157. 157.
    Najati, A., Rassias, Th.M.: Stability of a mixed functional equation in several variables on Banach modules. Nonlinear Anal. 72, 1755–1767 (2010) MathSciNetMATHGoogle Scholar
  158. 158.
    Narici, L.: On the Hahn–Banach theorem. In: Advanced Courses of Mathematical Analysis II, pp. 87–122. World Sci., Hackensack (2007) Google Scholar
  159. 159.
    Narici, L., Beckenstein, E.: The Hahn–Banach theorem: the life and times. Topol. Appl. 77, 193–211 (1997) MathSciNetMATHGoogle Scholar
  160. 160.
    Narici, L., Beckenstein, E.: The Hahn–Banach theorem and the sad life of E. Helly. In: Advanced Courses of Mathematical Analysis III, pp. 97–110. World Sci., Hackensack (2008) Google Scholar
  161. 161.
    Von Neumann, J.: Über adjungierte Functional-operatoren. Ann. Math. 33, 294–310 (1932) Google Scholar
  162. 162.
    Von Neumann, J.: Functional Operators: The Geometry of Orthogonal Spaces. Ann. Math. Stud., vol. 22. Princeton University Press, Princeton (1950) MATHGoogle Scholar
  163. 163.
    Nikodem, K.: Additive selections of additive set-valued functions. Univ. u Novom Sadu, Zb. Rad. Prirod.-Mat. Fak., Ser. Mat. 18, 143–148 (1988) MathSciNetMATHGoogle Scholar
  164. 164.
    Nikodem, K.: K-convex and K-concave set-valued functions. Zeszyty Nauk. Politech. Lódz. Mat. 559, 1–75 (1989) Google Scholar
  165. 165.
    Nikodem, K.: The stability of the Pexider equation. Ann. Math. Sil. 5, 91–93 (1991) MathSciNetGoogle Scholar
  166. 166.
    Nikodem, K.: Remarks on additive injective set-valued functions. An. Univ. Oradea, Fasc. Mat. 11, 175–180 (2004) MathSciNetMATHGoogle Scholar
  167. 167.
    Nikodem, K., Popa, D.: On single-valuedness of set-valued maps satisfying linear inclusions. Banach J. Math. Anal 3, 44–51 (2009) MathSciNetGoogle Scholar
  168. 168.
    Nikodem, K., Popa, D.: On selections of general linear inclusions. Publ. Math. (Debr.) 75, 239–249 (2009) MathSciNetMATHGoogle Scholar
  169. 169.
    Nikodem, K., Wasowicz, Sz.: A sandwich theorem and Hyers–Ulam stability of affine functions. Aequ. Math. Sil. 49, 160–164 (1995) MathSciNetMATHGoogle Scholar
  170. 170.
    Nikodem, K., Páles, Zs., Wasowicz, Sz.: Abstract separation theorems of Rodé type and their applications. Ann. Pol. Math. 72, 207–217 (1999) MATHGoogle Scholar
  171. 171.
    Nikodem, K., Sadowska, E., Wasowicz, Sz.: Note on separation by subadditive and sublinear functions. Ann. Math. Sil. 14, 7–21 (2000) MathSciNetGoogle Scholar
  172. 172.
    Paganoni, L.: Soluzione di una equazione funzionale su dominio ristretto. Boll. Unione Mat. Ital. 17-B, 979–993 (1980) MathSciNetGoogle Scholar
  173. 173.
    Páles, Zs.: Linear selections for set-valued functions and extension of bilinear forms. Arch. Math. (Basel) 62, 427–432 (1994) MathSciNetMATHGoogle Scholar
  174. 174.
    Páles, Zs.: Separation with symmetric bilinear forms and symmetric selections of set-valued functions. Publ. Math. (Debr.) 46, 321–331 (1995) MATHGoogle Scholar
  175. 175.
    Páles, Zs.: Separation by semidefinite bilinear forms. Int. Ser. Numer. Math. 123, 259–267 (1997) Google Scholar
  176. 176.
    Páles, Zs.: Generalized stability of the Cauchy functional equations. Aequ. Math. 56, 222–232 (1998) MATHGoogle Scholar
  177. 177.
    Páles, Zs.: Hyers–Ulam stability of the Cauchy functional equation on square-symmetric groupoids. Publ. Math. (Debr.) 58, 651–666 (2001) MATHGoogle Scholar
  178. 178.
    Páles, Zs., Székelyhidi, L.: On approximate sandwich and decomposition theorems. Ann. Univ. Sci. Budapest Sect. Comput. 23, 59–70 (2004) MathSciNetMATHGoogle Scholar
  179. 179.
    Park, Ch., Rassias, Th.M.: Fixed points and generalized Hyers–Ulam stability of quadratic functional equations. J. Math. Inequal. 1, 515–528 (2007) MathSciNetMATHGoogle Scholar
  180. 180.
    Park, Ch., Rassias, Th.M.: Fixed points and stability of the Cauchy functional equation. Aust. J. Math. Anal. Appl. 6, 14 (2009). pp. 1–9 MathSciNetGoogle Scholar
  181. 181.
    Pataki, G.: On a generalized infimal convolution of set functions. Manuscript Google Scholar
  182. 182.
    Peng, J., Lee, H.W.J., Rong, W., Yang, X.: A generalization of Hahn–Banach extension theorem. J. Math. Anal. Appl. 302, 441–449 (2005) MathSciNetMATHGoogle Scholar
  183. 183.
    Pérez-Garcia, C.: The Hahn–Banach extension property in p-adic analysis. In: P-adic Functional Analysis. Lect. Notes in Pure and Appl. Math., vol. 137, pp. 127–140. Dekker, New York (1992) Google Scholar
  184. 184.
    Piao, Y.J.: The existence and uniqueness of additive selections for (α,β)–(β,α) type subadditive set-valued maps. J. Northeast Norm. University 41, 33–40 (2009) MathSciNetGoogle Scholar
  185. 185.
    Plappert, P.: Sandwich theorem for monotone additive functions. Semigroup Forum 51, 347–355 (1995) MathSciNetMATHGoogle Scholar
  186. 186.
    Plewnia, J.: A generalization of the Hahn–Banach theorem. Ann. Pol. Math. 58, 47–51 (1993) MathSciNetMATHGoogle Scholar
  187. 187.
    Pólya, Gy., Szegő, G.: Aufgaben und Lehrsätze aus der Analysis I. Springer, Berlin (1925) MATHGoogle Scholar
  188. 188.
    Popa, D.: Additive selections of (α,β)-subadditive set valued maps. Glas. Mat. 36, 11–16 (2001) MATHGoogle Scholar
  189. 189.
    Popa, D.: Functional inclusions on square-symmetric grupoids and Hyers–Ulam stability. Math. Inequal. Appl. 7, 419–428 (2004) MathSciNetMATHGoogle Scholar
  190. 190.
    Popa, D., Vornicescu, N.: Locally compact set-valued solutions for the general linear equation. Aequ. Math. 67, 205–215 (2004) MathSciNetMATHGoogle Scholar
  191. 191.
    Przeslawski, K.: Linear and lipschitz continuous selectors for the family of convex sets in Euclidean vector spaces. Bull. Acad. Pol. Sci. Ser. Sci. Math. 33, 31–33 (1985) MathSciNetMATHGoogle Scholar
  192. 192.
    Rådström, H.: One-parameter semigroups of subsets of a real linear space. Ark. Mat. 4, 87–97 (1960) MathSciNetMATHGoogle Scholar
  193. 193.
    Radu, V.: The fixed point alternative and the stability of functional equations. Fixed Point Theory 4, 91–96 (2003) MathSciNetMATHGoogle Scholar
  194. 194.
    Rassias, J.M.: On approximation of approximately linear mappings by linear mappings. J. Funct. Anal. 46, 126–130 (1982) MathSciNetMATHGoogle Scholar
  195. 195.
    Rassias, J.M.: On approximation of approximately linear mappings by linear mappings. Bull. Sci. Math. 108, 445–446 (1984) MathSciNetMATHGoogle Scholar
  196. 196.
    Rassias, J.M.: Solution of a problem of Ulam. J. Approx. Theory 57, 268–273 (1989) MathSciNetMATHGoogle Scholar
  197. 197.
    Rassias, Th.M.: On the stability of the linear mapping in Banach spaces. Proc. Am. Math. Soc. 72, 279–300 (1978) Google Scholar
  198. 198.
    Rassias, Th.M.: On a modified Hyers–Ulam sequence. J. Math. Anal. Appl. 158, 106–113 (1991) MathSciNetMATHGoogle Scholar
  199. 199.
    Rassias, Th.M.: On the stability of the quadratic functional equation and its applications. Stud. Univ. Babeş-Bolyai, Math. 43, 89–124 (1998) MATHGoogle Scholar
  200. 200.
    Rassias, Th.M.: Stability and set-valued functions. In: Analysis and Topology, pp. 585–614. World Sci., River Edge (1998) Google Scholar
  201. 201.
    Rassias, Th.M.: On the stability of functional equations and a problem of Ulam. Acta Appl. Math. 62, 23–130 (2000) MathSciNetMATHGoogle Scholar
  202. 202.
    Rassias, Th.M.: The problem of S.M. Ulam for approximately multiplicative mappings. J. Math. Anal. Appl. 246, 352–378 (2000) MathSciNetMATHGoogle Scholar
  203. 203.
    Rassias, Th.M.: On the stability of functional equations in Banach spaces. J. Math. Anal. Appl. 251, 264–284 (2000) MathSciNetMATHGoogle Scholar
  204. 204.
    Rassias, Th.M.: On the stability of functional equations originated by a problem of Ulam. Mathematica (Cluj) 44, 39–75 (2002) Google Scholar
  205. 205.
    Rassias, Th.M., Šemrl, P.: On the behavior of mappings which do not satisfy Hyers–Ulam stability. Proc. Am. Math. Soc. 114, 989–993 (1992) MATHGoogle Scholar
  206. 206.
    Rassias, Th.M., Šemrl, P.: On the Hyers–Ulam stability of linear mappings. J. Math. Anal. Appl. 173, 325–338 (1993) MathSciNetMATHGoogle Scholar
  207. 207.
    Rassias, Th.M., Tabor, J.: What is left of Hyers–Ulam stability? J. Nat. Geom. 1, 65–69 (1992) MathSciNetMATHGoogle Scholar
  208. 208.
    Rassias, Th.M., Tabor, J. (eds.): Stability of Mappings of Hyers–Ulam Type. Hadronic Press, Palm Harbor (1994) MATHGoogle Scholar
  209. 209.
    Rätz, J.: On approximately additive mappings. In: Beckenbach, E.F. (ed.) General Inequalities 2, Oberwolfach, 1978. Int. Ser. Num. Math., vol. 47, pp. 233–251. Birkhäuser, Basel (1980) Google Scholar
  210. 210.
    Revenko, A.V.: On the extension of linear functionals. Ukr. Math. Visn. 6, 113–125 (2009) (Russian) MathSciNetGoogle Scholar
  211. 211.
    Riesz, F.: Sur les systémes orthogonaux de fonctions. C. R. Acad. Sci. Paris 144, 615–619 (1907) MATHGoogle Scholar
  212. 212.
    Riesz, F.: Untersuchungen über Systeme integrierbare Funktionen. Math. Ann. 69, 449–497 (1910) MathSciNetMATHGoogle Scholar
  213. 213.
    Robinson, S.M.: Normed convex processes. Trans. Am. Math. Soc. 174, 127–140 (1992) Google Scholar
  214. 214.
    Rockafellar, R.T.: Monotone processes of convex and concave type. Mem. Amer. Math. Soc. 77 (1967) Google Scholar
  215. 215.
    Rodríguez-Salinas, B.: Generalización sobre módulos del teorema de Hahn–Banach y sus aplicaciones. Collect. Math. 14, 105–151 (1962) MathSciNetGoogle Scholar
  216. 216.
    Rodríguez-Salinas, B.: Algunos problemas y teoremas de extension de aplicaciones lineales. Rev. R. Acad. Cienc. Exactas Fís. Nat. Madr. 65, 677–704 (1971) Google Scholar
  217. 217.
    Rodríguez-Salinas, B., Bou, L.: A Hahn–Banach theorem for arbitrary vector spaces. Boll. Unione Mat. Ital. 10, 390–393 (1974) MATHGoogle Scholar
  218. 218.
    Roth, W.: Hahn–Banach type theorems for locally convex cones. J. Aust. Math. Soc. 68, 104–125 (2000) MATHGoogle Scholar
  219. 219.
    Sablik, M.: A functional congruence revisited. Grazer Math. Ber. 316, 181–200 (1992) MathSciNetMATHGoogle Scholar
  220. 220.
    Sánchez, F.C., Castillo, J.M.F.: Banach space techniques underpinning a theory for nearly additive mappings. Diss. Math. 404, 1–73 (2002) Google Scholar
  221. 221.
    Saccoman, J.J.: Extension theorems by Helly and Riesz revisited. Riv. Mat. Univ. Parma 16, 223–230 (1990) MathSciNetMATHGoogle Scholar
  222. 222.
    Saccoman, J.J.: On the extension of linear operators. Int. J. Math. Math. Sci. 28, 621–623 (2001). http://ijmms.hindawi.com MathSciNetMATHGoogle Scholar
  223. 223.
    Sandovici, A., de Snoo, H., Winkler, H.: Ascent, descent, nullity, defect, and related notions for linear relations in linear spaces. Linear Algebra Appl. 423, 456–497 (2007) MathSciNetMATHGoogle Scholar
  224. 224.
    Saveliev, P.: Lomonosov’s invariant subspace theorem for multivalued linear operators. Proc. Am. Math. Soc. 131, 825–834 (2002) MathSciNetGoogle Scholar
  225. 225.
    Schwaiger, J.: Remark 12. Aequ. Math. 35, 120–121 (1988) Google Scholar
  226. 226.
    Simons, S.: Extended and sandwich versions of the Hahn–Banach theorem. J. Math. Anal. Appl. 21, 112–122 (1968) MathSciNetMATHGoogle Scholar
  227. 227.
    Simons, S.: From Hahn–Banach to Monotonicity. Springer, Berlin (2008) MATHGoogle Scholar
  228. 228.
    Smajdor, A.: Additive selections of superadditive set-valued functions. Aequ. Math. 39, 121–128 (1990) MathSciNetMATHGoogle Scholar
  229. 229.
    Smajdor, A.: The stability of the Pexider equation for set-valued functions. Rocz. Nauk.-Dydakt. Pr. Mat. 13, 277–286 (1993) MathSciNetMATHGoogle Scholar
  230. 230.
    Smajdor, A., Smajdor, W.: Affine selections of convex set-valued functions. Aequ. Math. 51, 12–20 (1996) MathSciNetMATHGoogle Scholar
  231. 231.
    Smajdor, W.: Subadditive set-valued functions. Glas. Mat. 21, 343–348 (1986) MathSciNetGoogle Scholar
  232. 232.
    Smajdor, W.: Superadditive set-valued functions and Banach–Steinhaus theorem. Rad. Mat. 3, 203–214 (1987) MathSciNetMATHGoogle Scholar
  233. 233.
    Smajdor, W.: Subadditive and subquadratic set-valued functions. Pr. Nauk. Univ. Ślask. Katowic. 889, 1–73 (1987) MathSciNetGoogle Scholar
  234. 234.
    Smajdor, W., Szczawińska, J.: A theorem of the Hahn–Banach type. Demonstr. Math. 28, 155–160 (1995) MATHGoogle Scholar
  235. 235.
    Soukhomlinov, G.A.: On the extension of linear functionals in complex and quaternion linear spaces. Mat. Sb. 3, 353–358 (1938) Google Scholar
  236. 236.
    Špakula, J., Zlatoš, P.: Almost homomorphisms of compact groups. Ill. J. Math. 48, 1183–1189 (2004) MATHGoogle Scholar
  237. 237.
    Strömberg, T.: The operation of infimal convolution. Diss. Math. 352, 1–58 (1996) Google Scholar
  238. 238.
    Száz, Á.: Convolution multipliers and distributions. Pac. J. Math. 60, 267–275 (1975) MATHGoogle Scholar
  239. 239.
    P185R1: Aequ. Math. 22, 308–309 (1981) Google Scholar
  240. 240.
    Száz, Á.: Generalized preseminormed spaces of linear manifolds and bounded linear relations. Univ. u Novom Sadu Zb. Rad. Prirod. Mat. Fak. Ser. Mat. 14, 49–78 (1984) MathSciNetMATHGoogle Scholar
  241. 241.
    Száz, Á.: Projective generation of preseminormed spaces by linear relations. Studia Sci. Math. Hung. 23, 297–313 (1988) MATHGoogle Scholar
  242. 242.
    Száz, Á.: Pointwise limits of nets of multilinear maps. Acta Sci. Math. (Szeged) 55, 103–117 (1991) MathSciNetMATHGoogle Scholar
  243. 243.
    Száz, Á.: The intersection convolution of relations and the Hahn–Banach type theorems. Ann. Pol. Math. 69, 235–249 (1998) MATHGoogle Scholar
  244. 244.
    Száz, Á.: An extension of Banach’s closed graph theorem to linear relations. Leaflets Math. Pécs, 144–145 (1998) Google Scholar
  245. 245.
    Száz, Á.: Translation relations, the building blocks of compatible relators. Math. Montisnigri 12, 135–156 (2000) MathSciNetMATHGoogle Scholar
  246. 246.
    Száz, Á.: Preseminorm generating relations. Publ. Elektroteh. Fak. Univ. Beogr., Mat. 12, 16–34 (2001) MATHGoogle Scholar
  247. 247.
    Száz, Á.: Partial multipliers on partially ordered sets. Novi Sad J. Math. 32, 25–45 (2002) MathSciNetGoogle Scholar
  248. 248.
    Száz, Á.: Relationships between translation and additive relations. Acta Acad. Paed. Agriensis, Sect. Math. (N.S.) 30, 179–190 (2003) MATHGoogle Scholar
  249. 249.
    Száz, Á.: Linear extensions of relations between vector spaces. Comment. Math. Univ. Carol. 44, 367–385 (2003) MATHGoogle Scholar
  250. 250.
    Száz, Á.: Lower semicontinuity properties of relations in relator spaces. Tech. Rep., Inst. Math., Univ. Debrecen 4, 1–52 (2006) Google Scholar
  251. 251.
    Száz, Á.: An extension of an additive selection theorem of Z. Gajda and R. Ger. Tech. Rep., Inst. Math., Univ. Debrecen 8, 1–24 (2006) Google Scholar
  252. 252.
    Száz, Á.: Applications of fat and dense sets in the theory of additive functions. Tech. Rep., Inst. Math., Univ. Debrecen 3, 1–29 (2007) Google Scholar
  253. 253.
    Száz, Á.: An instructive treatment of a generalization of Hyers’s stability theorem. In: Rassias, Th.M., Andrica, D. (eds.) Inequalities and Applications, pp. 245–271. Cluj University Press, Romania (2008) Google Scholar
  254. 254.
    Száz, Á.: Relationships between the intersection convolution and other important operations on relations. Math. Pannon. 20, 99–107 (2009) MathSciNetMATHGoogle Scholar
  255. 255.
    Száz, Á.: Applications of relations and relators in the extensions of stability theorems for homogeneous and additive functions. Aust. J. Math. Anal. Appl. 6, 16 (2009), pp. 66 MathSciNetGoogle Scholar
  256. 256.
    Száz, Á.: A reduction theorem for a generalized infimal convolution. Tech. Rep., Inst. Math., Univ. Debrecen 11, 1–4 (2009) Google Scholar
  257. 257.
    Száz, Á.: Foundations of the theory of vector relators. Adv. Stud. Contemp. Math. (Kyungshang), 20, 139–195 (2010) MathSciNetMATHGoogle Scholar
  258. 258.
    Száz, Á.: The intersection convolution of relations. Creative Math. Inf. 19, 209–217 (2010) MATHGoogle Scholar
  259. 259.
    Száz, Á.: The infimal convolution can be used to derive extension theorems from the sandwich ones. Acta Sci. Math. (Szeged) 76, 489–499 (2010) MathSciNetMATHGoogle Scholar
  260. 260.
    Száz, Á.: Relation theoretic operations on box and totalization relations. Tech. Rep., Inst. Math., Univ. Debrecen 13, 1–22 (2010) Google Scholar
  261. 261.
    Száz, Á., Száz, G.: Additive relations. Publ. Math. (Debr.) 20, 172–259 (1973) Google Scholar
  262. 262.
    Száz, Á., Száz, G.: Linear relations. Publ. Math. (Debr.) 27, 219–227 (1980) MATHGoogle Scholar
  263. 263.
    Száz, Á., Száz, G.: Absolutely linear relations. Aequ. Math. 21, 8–15 (1980) MATHGoogle Scholar
  264. 264.
    Száz, Á., Száz, G.: Quotient spaces defined by linear relations. Czechoslov. Math. J. 32, 227–232 (1982) Google Scholar
  265. 265.
    Száz, Á., Száz, G.: Multilinear relations. Publ. Math. (Debr.) 31, 163–164 (1984) MATHGoogle Scholar
  266. 266.
    Száz, Á., Túri, J.: Pointwise and global sums and negatives of odd and additive relations. Octogon 11, 114–125 (2003) Google Scholar
  267. 267.
    Szczawińska, J.: Selections of biadditive set-valued functions. Ann. Math. Sil. 8, 227–240 (1994) Google Scholar
  268. 268.
    Székelyhidi, L.: Remark 17. Aequ. Math. 29, 95–96 (1985) Google Scholar
  269. 269.
    Székelyhidi, L.: Note on Hyers’s theorem. C. R. Math. Rep. Acad. Sci. Can. 8, 127–129 (1986) MATHGoogle Scholar
  270. 270.
    Székelyhidi, L.: Ulam’s problem, Hyers’s solution—and to where they led. In: Rassias, Th.M. (ed.) Functional Equations and Inequalities. Math. Appl., vol. 518, pp. 259–285. Kluwer Acad., Dordrecht (2000) Google Scholar
  271. 271.
    Tabor, J.: Remark 18. Aequ. Math. 29, 96 (1985) Google Scholar
  272. 272.
    Tabor, J. Jr.: Ideally convex sets and hyers theorem. Funkc. Ekvacioj 43, 121–125 (2000) MathSciNetMATHGoogle Scholar
  273. 273.
    Tabor, J. Jr.: Hyers theorem and the cocycle property. In: Daróczy, Z., Páles, Zs. (eds.) Funtional Equations—Results and Advances, pp. 275–291. Kluwer, Boston (2002) Google Scholar
  274. 274.
    Tabor, J., Tabor, J.: Restricted stability and shadowing. Publ. Math. (Debr.) 73, 49–58 (2008) MathSciNetMATHGoogle Scholar
  275. 275.
    Tabor, J., Tabor, J.: Stability of the Cauchy functional equation in metric groupoids. Aequ. Math. 76, 92–104 (2008) MathSciNetMATHGoogle Scholar
  276. 276.
    Takahasi, S.-E., Miura, T., Takagi, H.: On a Hyers–Ulam–Aoki–Rassias type stability and a fixed point theorem. J. Nonlinear Convex Anal. 11, 423–439 (2010) MathSciNetMATHGoogle Scholar
  277. 277.
    Ulam, S.M.: Collection of Mathematical Problems. Interscience, New York (1960) MATHGoogle Scholar
  278. 278.
    Ursescu, C.: Multifunctions with convex closed graph. Czechoslov. Math. J. 25, 438–441 (1975) MathSciNetGoogle Scholar
  279. 279.
    Volkmann, P.: On the stability of the Cauchy equation. Leaflets Math. Pécs, 150–151 (1998) Google Scholar
  280. 280.
    Volkmann, P.: Zur Rolle der ideal konvexen Mengen bei der Stabilität der Cauchyschen Funktionalgleichung. Seminar LV 6, 1–6 (1999). http://www.mathematik.uni-karlsruhe.de/~semlv Google Scholar
  281. 281.
    Volkmann, P.: Instabilität einer zu f(x+y)=f(x)+f(y) äquivalenten Functionalgleichung. Seminar LV 23, 1 (2006). http://www.mathematik.uni-karlsruhe.de/~semlv MathSciNetGoogle Scholar
  282. 282.
    Whitehead, W.: Elements of Homotopy Theory. Springer, Berlin (1978) MATHGoogle Scholar
  283. 283.
    Zǎlinescu, C.: Hahn–Banach extension theorems for multifunctions. Math. Methods Oper. Res. 68, 493–508 (2008) MathSciNetGoogle Scholar

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  1. 1.Institute of MathematicsUniversity of DebrecenDebrecen, Pf. 12Hungary

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