About this Journal Submit a Manuscript Table of Contents
Journal of Function Spaces and Applications
Volume 2013 (2013), Article ID 912718, 4 pages
http://dx.doi.org/10.1155/2013/912718
Research Article

Hyperstability of the Drygas Functional Equation

Institute of Mathematics, Pedagogical University, Podchorążych 2, 30-084 Kraków, Poland

Received 16 May 2013; Accepted 7 July 2013

Academic Editor: Alberto Fiorenza

Copyright © 2013 Magdalena Piszczek and Joanna Szczawińska. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We use the fixed point theorem for functional spaces to obtain the hyperstability result for the Drygas functional equation on a restricted domain. Namely, we show that a function satisfying the Drygas equation approximately must be exactly the solution of it.

1. Introduction

We say that a function satisfies the Drygas equation if The above equation was introduced in [1] in order to obtain a characterization of the quasi-inner-product spaces. The general solution of (1), obtained by Ebanks et al. in [2] (see also [3]), is of the following form: where is an additive function and is a quadratic function; that is, satisfies the quadratic functional equation A set-valued version of (1) was considered by Smajdor in [4].

The stability in the Hyers-Ulam sense of the Drygas equation has been investigated, for example, in [58]. For more information and numerous references on the stability of functional equations in general, we refer to [9, 10].

The term hyperstability was used for the first time probably in [11]; however, it seems that the first hyperstability result was published in [12] and concerned the ring homomorphisms. The hyperstability results for the Cauchy equation were investigated by Brzdȩk in [13, 14]. Gselmann in [15] studied the hyperstability of the parametric fundamental equation of information.

In the paper, we prove the hyperstability of the Drygas equation on a restricted domain. Let be a nonempty subset of an Abelian group and a semigroup. We say that a function satisfies the Drygas functional equation on if for all such that . We will show that (4) is hyperstable; that is, each function (under some additional assumptions on ) satisfying the inequality for all such that with must satisfy the Drygas equation (4).

The method of the proof of the main theorem is motivated by an idea used by Brzdȩk in [13] and further by Piszczek in [16]. It is based on a fixed point theorem for functional spaces obtained by Brzdęk et al. (see [17, Theorem  1]). Some generalizations of their result were proved by Cădariu et al. in [18]. The case of fixed point theorem for non-Archimedean metric spaces was also studied by Brzdęk and Ciepliński in [19]. It is worth mentioning that using fixed point theorems is now one of the most popular methods of investigating the stability of functional equations in single as well as in several variables. Let us recall a few recent approaches of Jung in [20], Lee and Jung in [21], or Cădariu and Radu in [22, 23]. More information on the application of the fixed point method was collected by Ciepliński in [24].

First, we take the following three hypotheses (all notations come from [17]).(H1) is a nonempty set, is a Banach space, and and are given.(H2) is an operator satisfying the inequality for all , . (H3) is defined by

The mentioned fixed point theorem is stated as follows.

Theorem 1. Let hypotheses (H1)–(H3) be valid and functions and let fulfil the following two conditions: Then, there exists a unique fixed point of with Moreover,

Throughout the paper, , , and denote the set of all positive integers, the set of all nonnegative integers, and the set of all integers greater than or equal to , respectively.

2. The Main Result

Theorem 2. Assume that is a nonempty, symmetric with respect to subset of a normed space such that and there exists with for , . Let be a Banach space, , and . If satisfies for all such that , then satisfies the Drygas equation on .

Proof. First observe that there exists such that Let us fix . Replacing with and with in (11), we get Further put Inequality (13) now takes the form The operator has the form described in (H3) with and , , , , , and for . Moreover, for every ,  , As , we have Thus, according to Theorem 1 there exists a unique solution of the equation such that Moreover,
To prove that satisfies the Drygas equation on , observe that for every ,  , and .
Indeed, if , then (22) is simply (11). So, fix and suppose that (22) holds for and such that . Then By induction, we have shown that (22) holds for all ,  , and . Letting in (22), we obtain Thus, we have proved that for every there exists a function such that is a solution of the Drygas equation on and Since , the sequence tends to zero. Consequently, satisfies the Drygas equation on as the pointwise limit of .

Remark 3. If is a normed space and satisfies (11) for , with , then by Theorem 2 we know that satisfies the Drygas equation on . It is easy to see that if , then satisfies the Drygas equation on the whole .

We end with two examples. First notice that the assumption of unboundedness of the set is necessary.

Example 4. Let and let be defined by , . Then for all such that , with , but does not satisfy the Drygas equation on .

In the case , the considered Drygas equation is not hyperstable.

Example 5. Let and let be a constant function , for some . Then satisfies the inequality for all such that , with , but is not a solution of the Drygas equation on .

References

  1. H. Drygas, “Quasi-inner products and their applications,” in Advances in Multivariate Statistical Analysis, K. Gupta, Ed., pp. 13–30, Reidel, 1987. View at Zentralblatt MATH · View at MathSciNet
  2. B. R. Ebanks, Pl. Kannappan, and P. K. Sahoo, “A common generalization of functional equations characterizing normed and quasi-inner-product spaces,” Canadian Mathematical Bulletin, vol. 35, no. 3, pp. 321–327, 1992. View at Publisher · View at Google Scholar · View at MathSciNet
  3. P. K. Sahoo and P. Kannappan, Introduction to Functional Equations, CRC Press, Boca Raton, Fla, USA, 2011. View at MathSciNet
  4. W. Smajdor, “On set-valued solutions of a functional equation of Drygas,” Aequationes Mathematicae, vol. 77, no. 1-2, pp. 89–97, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. G. L. Forti and J. Sikorska, “Variations on the Drygas equation and its stability,” Nonlinear Analysis, vol. 74, no. 2, pp. 343–350, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. S.-M. Jung and P. K. Sahoo, “Stability of a functional equation of Drygas,” Aequationes Mathematicae, vol. 64, no. 3, pp. 263–273, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. J. Sikorska, “On a direct method for proving the Hyers-Ulam stability of functional equations,” Journal of Mathematical Analysis and Applications, vol. 372, no. 1, pp. 99–109, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. D. Yang, “Remarks on the stability of Drygas' equation and the Pexider-quadratic equation,” Aequationes Mathematicae, vol. 68, no. 1-2, pp. 108–116, 2004. View at Zentralblatt MATH · View at MathSciNet
  9. N. Brillouët-Belluot, J. Brzdęk, and K. Ciepliński, “On some recent developments in Ulam's type stability,” Abstract and Applied Analysis, vol. 2012, Article ID 716936, 41 pages, 2012. View at Publisher · View at Google Scholar
  10. S.-M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis, vol. 48 of Springer Optimization and Its Applications, Springer, New York, NY, USA, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  11. G. Maksa and Z. Páles, “Hyperstability of a class of linear functional equations,” Acta Mathematica, vol. 17, no. 2, pp. 107–112, 2001. View at Zentralblatt MATH · View at MathSciNet
  12. D. G. Bourgin, “Approximately isometric and multiplicative transformations on continuous function rings,” Duke Mathematical Journal, vol. 16, pp. 385–397, 1949. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. J. Brzdȩk, “Hyperstability of the Cauchy equation on restricted domains,” Acta Mathematica Hungarica, 2013. View at Publisher · View at Google Scholar
  14. J. Brzdȩk, “Remarks on hyperstability of the Cauchy functional equation,” Aequationes Mathematicae, 2012. View at Publisher · View at Google Scholar
  15. E. Gselmann, “Hyperstability of a functional equation,” Acta Mathematica Hungarica, vol. 124, no. 1-2, pp. 179–188, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. M. Piszczek, “Remark on hyperstability of the general linear equation,” Aequationes Mathematicae, 2013. View at Publisher · View at Google Scholar
  17. J. Brzdęk, J. Chudziak, and Z. Páles, “A fixed point approach to stability of functional equations,” Nonlinear Analysis, vol. 74, no. 17, pp. 6728–6732, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. L. Cădariu, L. Găvruţa, and P. Găvruţa, “Fixed points and generalized Hyers-Ulam stability,” Abstract and Applied Analysis, vol. 2012, Article ID 712743, 10 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. J. Brzdęk and K. Ciepliński, “A fixed point approach to the stability of functional equations in non-Archimedean metric spaces,” Nonlinear Analysis, vol. 74, no. 18, pp. 6861–6867, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. S.-M. Jung, “A fixed point approach to the stability of differential equations y=F(x,y),” Bulletin of the Malaysian Mathematical Sciences Society, vol. 33, no. 1, pp. 47–56, 2010. View at Zentralblatt MATH · View at MathSciNet
  21. Y. H. Lee and S.-M. Jung, “A fixed point approach to the stability of an n-dimensional mixed-type additive and quadratic functional equation,” Abstract and Applied Analysis, vol. 2012, Article ID 482936, 14 pages, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  22. L. C. Cădariu and V. Radu, “A general fixed point method for the stability of Cauchy functional equation,” in Functional Equations in Mathematical Analysis, M. Th. Rassias and J. Brzdęk, Eds., vol. 52 of Springer Optimization and Its Applications, pp. 19–32, Springer, New York, NY, USA, 2011. View at Publisher · View at Google Scholar
  23. L. Cădariu and V. Radu, “A general fixed point method for the stability of the monomial functional equation,” Carpathian Journal of Mathematics, vol. 28, no. 1, pp. 25–36, 2012. View at Zentralblatt MATH · View at MathSciNet
  24. K. Ciepliński, “Applications of fixed point theorems to the Hyers-Ulam stability of functional equations—a survey,” Annals of Functional Analysis, vol. 3, no. 1, pp. 151–164, 2012. View at Zentralblatt MATH · View at MathSciNet