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In the following 2 theorems, we prove the modified Hyers–Ulam stability of the quadratic functional equation for the cases p < 2 and p > 2 separately

# On the stability of functional equations and a problem of Ulam

Acta Applicandae Mathematica, no. 1 (2001): 23-130

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 ar...更多

• Šemrl [RS1] introduced a simple counterexample to Theorem 1.3 for p = 1 as follows: The continuous real-valued function defined by f (x) = x log2(x + 1) for x 0, x log2 |x − 1| for x < 0 satisfies the inequality (1.7) with θ = 1 and |f (x) − cx|/|x| → ∞, as x → ∞, for any real number c.
• It can be proved that if G is a Banach space and if f is continuous in t for every fixed x ∈ G A is a linear function in the same way as in Theorem 1.3.

• Ulam [Ul1], for very general functional equations, one can ask the following question: When is it true that the solution of an equation differing slightly from a given one, must of necessity be close to the solution of the given equation? if we replace a given functional equation by a functional inequality, when can one assert that the solutions of the inequality lie near the solutions of the strict equation?
• Ulam gave a wide ranging talk before a Mathematical Colloquium at the University of Wisconsin in which he discussed a number of important unsolved problems. Among those was the following question concerning the stability of homomorphisms: Let G1 be a group and let G2 be a metric group with a metric d(·, ·)
• Ulam [Ul1] raised the following question concerning the stability of homomorphisms: Let G1 be a group and let G2 be a metric group with a metric d(·, ·)
• In the following 2 theorems, we prove the modified Hyers–Ulam stability of the quadratic functional equation (3.1) for the cases p < 2 and p > 2 separately
• We introduce a Hyers–Ulam stability result which was presented by J

• The authors have seen that the condition that a function f satisfies one of the inequalities (1.2), (1.5), (1.10), (1.12), (1.15) and (1.18) on the whole space, assures them of the existence of a unique additive function which approximates f within a given distance.
• F. Skof [Sk2] proved the following theorem and applied the result to the study of an asymptotic behavior of additive functions.
• According to Theorem 1.1, there is a unique additive function A: R → E such that the inequality (1.23) holds for any x in R.
• According to Theorem 1.15, there exists a unique additive function An: E1 → E2 satisfying g(x) − An(x) 9δn (b) for all x in E1.
• The authors look at the so-called Hammerstein equation: x = λAf (x) subject to the following conditions: (H1) (E, K) and (F, P ) are real Banach spaces ordered by the cones K and P , respectively, and the cone K is normal and solid; (H2) The function f : E → F is continuous and the operator A: F → E is linear and compact; (H3) A is strongly positive, that is y − x ∈ P \{0} implies that A(y) − A(x) ∈ int(K).

• Let conditions (H1), (H2), (H3) hold and suppose that the followings are true: [1] f (K) ⊂ P ; [2] f is ψ-additive; [3] limn→∞ 2−nf (2nx) > 0 for all x ∈ K \{0}.
• Assume that the mapping E is RN+-asymptotically equivalent to a ψ-additive mapping H : RN → RN and the following hypotheses are satisfied: [1] ψ satisfies assumptions (i), (ii), (iii); [2] H is off-diagonal negative with respect to RN+; [3] the operator T (x) = limn→∞ 2−nH (2nx) is nonsingular on RN+ (i.e., T (x) =

• Šemrl [RS1] introduced a simple counterexample to Theorem 1.3 for p = 1 as follows: The continuous real-valued function defined by f (x) = x log2(x + 1) for x 0, x log2 |x − 1| for x < 0 satisfies the inequality (1.7) with θ = 1 and |f (x) − cx|/|x| → ∞, as x → ∞, for any real number c.
• It can be proved that if G is a Banach space and if f is continuous in t for every fixed x ∈ G A is a linear function in the same way as in Theorem 1.3.
• The authors have seen that the condition that a function f satisfies one of the inequalities (1.2), (1.5), (1.10), (1.12), (1.15) and (1.18) on the whole space, assures them of the existence of a unique additive function which approximates f within a given distance.
• F. Skof [Sk2] proved the following theorem and applied the result to the study of an asymptotic behavior of additive functions.
• According to Theorem 1.1, there is a unique additive function A: R → E such that the inequality (1.23) holds for any x in R.
• According to Theorem 1.15, there exists a unique additive function An: E1 → E2 satisfying g(x) − An(x) 9δn (b) for all x in E1.
• The authors look at the so-called Hammerstein equation: x = λAf (x) subject to the following conditions: (H1) (E, K) and (F, P ) are real Banach spaces ordered by the cones K and P , respectively, and the cone K is normal and solid; (H2) The function f : E → F is continuous and the operator A: F → E is linear and compact; (H3) A is strongly positive, that is y − x ∈ P \{0} implies that A(y) − A(x) ∈ int(K).
• Let conditions (H1), (H2), (H3) hold and suppose that the followings are true: [1] f (K) ⊂ P ; [2] f is ψ-additive; [3] limn→∞ 2−nf (2nx) > 0 for all x ∈ K \{0}.
• Assume that the mapping E is RN+-asymptotically equivalent to a ψ-additive mapping H : RN → RN and the following hypotheses are satisfied: [1] ψ satisfies assumptions (i), (ii), (iii); [2] H is off-diagonal negative with respect to RN+; [3] the operator T (x) = limn→∞ 2−nH (2nx) is nonsingular on RN+ (i.e., T (x) =

• Aczél, J.: Lectures on Functional Equations and Their Applications, Academic Press, New York, 1966.
• Aczél, J.: A Short Course on Functional Equations, D. Reidel, Dordrecht, 1987.
• Aczél, J. and Dhombres, J.: Functional Equations in Several Variables, Cambridge Univ. Press, Cambridge, 1989.
• Amann, H.: Fixed points of asymptotically linear maps in ordered Banach spaces, J. Funct. Anal. 14 (1973), 162–171.
• 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.
• Baker, J.: The stability of the cosine equation, Proc. Amer. Math. Soc. 80 (1980), 411– 416.
• 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.
• 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.
• 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.
• Borelli, C.: On Hyers–Ulam stability of Hosszú’s functional equation, Results Math. 26 (1994), 221–224.
• Borelli, C. and Forti, G. L.: On a general Hyers–Ulam stability result, Internat. J. Math. Math. Sci. 18 (1995), 229–236.
• Bourgin, D. G.: Classes of transformations and bordering transformations, Bull. Amer. Math. Soc. 57 (1951), 223–237.
• 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.
• Brzdek, 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.
• Brzdek, J.: On functionals which are orthogonally additive modulo Z, Results Math. 30 (1996), 25–38.
• Brzdek, J.: The Cauchy and Jensen differences on semigroups, Publ. Math. Debrecen 48 (1996), 117–136.
• Brzdek, J.: On the Cauchy difference on normed spaces, Abh. Math. Sem. Univ. Hamburg 66 (1996), 143–150.
• Brzdek, J.: On orthogonally exponential and orthogonally additive mappings, Proc. Amer. Math. Soc. 125 (1997), 2127–2132.
• Câc, N. P. and Gatica, J. A.: Fixed point theorems for mappings in ordered Banach spaces, J. Math. Anal. Appl. 71 (1979), 547–557.
• Castillo, E. and Ruiz-Cobo, M. R.: Functional Equations and Modelling in Science and Engineering, Dekker, New York, 1992.
• Cauchy, A. L.: Cours d’analyse de l’École Polytechnique, Vol. I, Analyse algébrique, Debure, Paris, 1821.
• Cholewa, P. W.: The stability of the sine equation, Proc. Amer. Math. Soc. 88 (1983), 631–634.
• Cholewa, P. W.: Remarks on the stability of functional equations, Aequationes Math. 27 (1984), 76–86. van der Corput, J. G.: Goniometrische functies gekarakteriseerd door een functionaal betrekking, Euclides 17 (1940), 55–75.
• Cottle, R. W., Pang, J. S. and Stone, R. E.: The Linear Complementarity Problem, Academic Press, 1992.
• Czerwik, S.: On the stability of the quadratic mapping in normed spaces, Abh. Math. Sem. Univ. Hamburg 62 (1992), 59–64.
• Czerwik, S.: On the stability of the homogeneous mapping, C.R. Math. Rep. Acad. Sci. Canada 14 (1992), 268–272.
• 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.
• Davison, T. M. K.: The complete solution of Hosszú’s functional equation over a field, Aequationes Math. 11 (1974), 114–115.
• Day, M. M.: Normed Linear Spaces, Ergeb. Math. Grenzgeb., Springer, Berlin, 1958.
• Forti, G. L.: An existence and stability theorem for a class of functional equations, Stochastica 4 (1980), 23–30.
• Forti, G. L.: The stability of homomorphisms and amenability, with applications to functional equations, Abh. Math. Sem. Univ. Hamburg 57 (1987), 215–226.
• Forti, G. L.: Hyers–Ulam stability of functional equations in several variables, Aequationes Math. 50 (1995), 143–190.
• 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.
• 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.
• Gajda, Z.: On stability of the Cauchy equation on semigroups, Aequationes Math. 36 (1988), 76–79.
• Gajda, Z.: On stability of additive mappings, Internat. J. Math. Math. Sci. 14 (1991), 431–434.
• Gavruta, P.: A generalization of the Hyers–Ulam–Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), 431–436.
• Gavruta, 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.
• Gavruta, P., Hossu, M., Popescu, D. and Caprau, C.: On the stability of mappings, Bull. Appl. Math. Tech. Univ. Budapest 73 (1994), 169–176.
• Gavruta, P., Hossu, M., Popescu, D. and Caprau, C.: On the stability of mappings and an answer to a problem of Th. M. Rássias, Ann. Math. Blaise Pascal 2 (1995), 55–60.
• Ger, R.: Stability of addition formulae for trigonometric mappings, Zeszyty Nauk. Politech. Slasiej, Ser. Mat. Fiz. 64 (1990), 75–84.
• Ger, R.: Superstability is not natural, Rocznik Naukowo-Dydaktyczny WSP w Krakowie, Prace Mat. 159 (1993), 109–123.
• Ger, R. and Šemrl, P.: The stability of the exponential equation, Proc. Amer. Math. Soc. 124 (1996), 779–787.
• Greenleaf, F. P.: Invariant Means on Topological Groups, Van Nostrand Math. Stud. 16, New York, 1969.
• Haruki, H. and Rassias, Th. M.: New generalizations of Jensen’s functional equation, Proc. Amer. Math. Soc. 123 (1995), 495–503.
• 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.
• 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.
• Hyers, D. H.: On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A. 27 (1941), 222–224.
• 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.
• Hyers, D. H., Isac, G. and Rassias, Th. M.: Stability of Functional Equations in Several Variables, Birkhäuser, Boston, 1998.
• Hyers, D. H. and Rassias, Th. M.: Approximate homomorphisms, Aequationes Math. 44 (1992), 125–153.
• Isac, G.: Opérateurs asymptotiquement linéaires sur des espaces locallement convexes, Colloq. Math. 46 (1982), 67–72.
• Isac, G.: Complementarity Problems, Lecture Notes in Math. 1528, Springer, 1992.
• Isac, G.: The fold complementarity problem and the order complementarity problem, In: Topological Methods in Nonlinear Analysis, 1997.
• Isac, G. and Rassias, Th. M.: On the Hyers–Ulam stability of ψ-additive mappings, J. Approx. Theory 72 (1993), 131–137.
• 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.
• Isac, G. and Rassias, Th. M.: Stability of ψ-additive mappings: Applications to nonlinear analysis, Internat. J. Math. Math. Sci. 19 (1996), 219–228.
• Jarosz, K.: Perturbations of Banach Algebras, Lecture Notes in Math. 1120, Springer, Berlin, 1985.
• Johnson, B. E.: Approximately multiplicative functionals, J. London Math. Soc. (2) 34 (1986), 489–510.
• Johnson, B. E.: Approximately multiplicative maps between Banach algebras, J. London Math. Soc. (2) 37 (1988), 294–316.
• Jung, S.-M.: On the Hyers–Ulam–Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 204 (1996), 221–226.
• 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.
• Jung, S.-M.: On the superstability of the functional equation f (xy ) = yf (x), Abh. Math. Sem. Univ. Hamburg 67 (1997), 315–322.
• Jung, S.-M.: Hyers–Ulam–Rassias stability of functional equations, Dynamic Systems Appl. 6 (1997), 541–566.
• Jung, S.-M.: On the modified Hyers–Ulam–Rassias stability of the functional equation for gamma function, Mathematica 39 (62) (1997), 235–239.
• Jung, S.-M.: On a modified Hyers–Ulam stability of homogeneous equation, Internat. J. Math. Math. Sci. 21 (1998), 475–478.
• Jung, S.-M.: On modified Hyers–Ulam–Rassias stability of a generalized Cauchy functional equation, Nonlinear Studies 5 (1998), 59–67.
• Jung, S.-M.: Hyers–Ulam–Rassias stability of Jensen’s equation and its application, Proc. Amer. Math. Soc. 126 (1998), 3137–3143.
• Jung, S.-M.: Superstability of homogeneous functional equation, Kyungpook Math. J. 38 (1998), 251–257.
• 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.
• 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.
• 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.
• Kairies, H. H.: Die Gammafunktion als stetige Lösung eines Systems von GaussFunktionalgleichungen, Results Math. 26 (1994), 306–315.
• Kannappan, Pl.: Quadratic functional equation and inner product spaces, Results Math. 27 (1995), 368–372.
• Kominek, Z.: On a local stability of the Jensen functional equation, Demonstratio Math. 22 (1989), 499–507.
• Krasnoselskii, M. A.: Positive Solutions of Operator Equations, Nordhoff, Groningen, 1964.
• Krasnoselskii, M. A. and Zabreiko, P. P.: Geometrical Methods of Nonlinear Analysis, Springer, Berlin, 1984.
• Kravvaritis, D.: Nonlinear random operators of monotone type in Banach spaces, J. Math. Anal. Appl. 78 (1980), 488–496.
• Kuczma, M.: An Introduction to the Theory of Functional Equations and Inequalities, Panstwowe Wydawnictwo Naukowe-Uniwersytet Slaski, Warszawa, 1985.
• Kurepa, S.: On the quadratic functional, Publ. Inst. Math. Acad. Serbe Sci. Beograd 13 (1959), 57–72.
• Lajkó, K.: Applications of extensions of additive functions, Aequationes Math. 11 (1974), 68–76.
• Losonczi, L.: On the stability of Hosszú’s functional equation, Results Math. 29 (1996), 305–310.
• Maksa, G.: Problems 18, In: Report on the 34th ISFE, Aequationes Math. 53, 1997, pp. 194.
• Mininni, M.: Coincidence degree and solvability of some nonlinear functional equations in normed spaces: A spectral approach, Nonlinear Anal. 1 (1977), 105–122.
• Ng, C. T.: Jensen’s functional equation on groups, Aequationes Math. 39 (1990), 85–90.
• Opoitsev, V. I.: Nonlinear Statical Systems, Economics–Mathematics Library, Nauka, Moscow (Russian), 1986.
• Páles, Z.: Remark 27, In: Report on the 34th ISFE, Aequationes Math. 53, 1997, pp. 200–201.
• Parnami, J. C. and Vasudeva, H. L.: On Jensen’s functional equation, Aequationes Math. 43 (1992), 211–218.
• Rassias, J. M.: On approximation of approximately linear mappings, J. Funct. Anal. 46 (1982), 126–130.
• Rassias, J. M.: On a new approximation of approximately linear mappings by linear mappings, Discuss. Math 7 (1985), 193–196.
• Rassias, Th. M.: On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300.
• Rassias, Th. M.: The stability of linear mappings and some problems on isometries, In: S. M. Mazhar, A. Hamoui and N. S. Faour (eds), Mathematical Analysis and Its Applications, Pergamon Press, New York, 1985.
• Rassias, Th. M.: On the stability of mappings, Rend. Sem. Mat. Fis. Milano 58 (1988), 91–99.
• Rassias, Th. M.: Seven problems in mathematical analysis, In: Th. M. Rassias (ed.), Topics in Mathematical Analysis, World Sci. Publ., Singapore, 1989.
• Rassias, Th. M.: The stability of mappings and related topics, In: Report on the 27th ISFE, Aequationes Math. 39, 1990, pp. 292–293.
• Rassias, Th. M.: On a modified Hyers–Ulam sequence, J. Math. Anal. Appl. 158 (1991), 106–113.
• Rassias, Th. M.: Problem 18, In: Report on the 31st ISFE, Aequationes Math. 47, 1994, pp. 312–313.
• Rassias, Th. M.: Remark and problem 19, In: Report on the 31st ISFE, Aequationes Math. 47, 1994, pp. 313–314.
• 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.
• Rassias, Th. M.: Inner Product Spaces and Applications, Longman, 1997.
• Rassias, Th. M.: On the stability of the quadratic functional equation and its applications, to appear. Rassias, Th. M.: On the stability of the quadratic functional equation, to appear. Rassias, Th. M.: On the stability of functional equations originated by a problem of Ulam, Studia Univ. Babes-Bolyai, to appear. 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.
• Rassias, Th. M. and Šemrl, P.: On the Hyers–Ulam stability of linear mappings, J. Math. Anal. Appl. 173 (1993), 325–338.
• Rassias, Th. M. and Tabor, J.: What is left of Hyers–Ulam stability?, J. Natural Geom. 1 (1992), 65–69.
• 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.
• Rätz, J.: On approximately additive mappings, In: E. F. Beckenbach (ed.), General Inequalities 2, Birkhäuser, Basel, 1980, pp. 233–251.
• Rudin, W.: Functional Analysis, McGraw-Hill, New York, 1991.
• Schwaiger, J.: Remark 10, In: Report on the 30th ISFE, Aequationes Math. 46, 1993, pp. 289.
• 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.
• Šemrl, P.: The functional equation of multiplicative derivation is superstable on standard operator algebras, Integral Equations Operator Theory 18 (1994), 118–122.
• Skof, F.: Sull’approssimazione delle applicazioni localmente δ-additive, Atti Accad. Sci. Torino 117 (1983), 377–389.
• Skof, F.: Proprietá locali e approssimazione di operatori, Rend. Sem. Mat. Fis. Milano 53 (1983), 113–129.
• Skof, F.: Approssimazione di funzioni δ-quadratiche su dominio ristretto, Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur. 118 (1984), 58–70.
• Skof, F.: On two conditional forms of the equation f (x + y) = f (x) + f (y), Aequationes Math. 45 (1993), 167–178.
• Skof, F.: On the stability on functional equations on a restricted domain and a related topic, In: Th. M. Rassias and J. Tabor (eds), Stability of Mappings of Hyers–Ulam Type, Hadronic Press, Florida, 1994, pp. 141–151.
• 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.
• 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.
• Székelyhidi, L.: On a theorem of Baker, Lawrence and Zorzitto, Proc. Amer. Math. Soc. 84 (1982), 95–96.
• Székelyhidi, L.: Remark 17, In: Report on the 22nd ISFE, Aequationes Math. 29, 1985, pp. 95–96.
• Székelyhidi, L.: Note on Hyers’s theorem, C.R. Math. Rep. Acad. Sci. Canada 8 (1986), 127–129.
• Székelyhidi, L.: Fréchet equation and Hyers’s theorem on noncommutative semigroups, Ann. Polon. Math. 48 (1988), 183–189.
• Székelyhidi, L.: The stability of the sine and cosine functional equations, Proc. Amer. Math. Soc. 110 (1990), 109–115.
• Tabor, J.: On approximate by linear mappings, Manuscript. 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.
• Tabor, J.: Hosszú’s functional equation on the unit interval is not stable, Publ. Math. Debrecen 49 (1996), 335–340.
• Tabor, J.: Remark 20, In: Report on the 34th ISFE, Aequationes Math. 53, 1997, pp. 194–196.
• Tabor, J. and Tabor, J.: Homogeneity is superstable, Publ. Math. Debrecen 45 (1994), 123–130.
• Ulam, S. M.: A Collection of Mathematical Problems, Interscience Publ., New York, 1960.
• Villar, A.: Operator Theorems with Applications to Distributive Problems and Equilibrium Models, Lecture Notes in Econom. and Math. Systems 377, Springer, 1992.
• Zeidler, E.: Nonlinear Functional Analysis and Its Applications I (Fixed Point Theorems), Springer, New York, 1986.

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