/ / Article
Special Issue

## Ulam’s Type Stability and Fixed Points Methods 2016

View this Special Issue

Research Article | Open Access

Volume 2016 |Article ID 1804206 | 7 pages | https://doi.org/10.1155/2016/1804206

# Stability of a Quartic Functional Equation in Restricted Domains

Revised11 May 2016
Accepted16 May 2016
Published26 Jul 2016

#### Abstract

Let be a real normed space and a Banach space and . We prove the Ulam-Hyers stability theorem for the quartic functional equation in restricted domains. As a consequence we consider a measure zero stability problem of the above inequality when .

#### 1. Introduction

Let , , and be the set of real numbers, a real normed space, and a real Banach space, respectively. A mapping is called a quartic mapping if satisfies the functional equationfor all . It is known in [1, Theorem 2.1] that the general solutions of (1) are given by for all , where is a symmetric function which is additive for each variable when the other three variables are fixed. The following is a modified version of [1, Theorem 3.1]. We refer the reader to [2, 3] for the stability of generalized quartic mappings.

Theorem 1. Let be fixed. Suppose that satisfies the cubic functional inequalityfor all . Then there exists a unique quartic mapping such thatfor all .

It is a very natural subject to study functional equations or inequalities satisfied on restricted domains or satisfied under restricted conditions . Among the results, Jung  and Rassias  proved the Ulam-Hyers stability of the quadratic functional equations in a restricted domain. As a refined version of the results in [14, 17] we state the result in .

Theorem 2. Let . Suppose that satisfies the inequalityfor all . Then there exists a unique mapping satisfyingfor all such thatfor all .

Also, Chung-Ju-Rassias  proved the following Ulam-Hyers stability of cubic functional equation in restricted domains.

Theorem 3. Let . Suppose that satisfies the inequalityfor all . Then there exists a unique mapping satisfyingfor all such thatfor all .

In this paper, we consider the Ulam-Hyers stability of quartic functional equation (1) in some restricted domains . First, imposing condition (C) on (see Section 2) we prove that if satisfies inequality (2) for all , then there exists a unique quartic mapping such thatfor all . Since satisfies condition (C), we obtain the parallel result for quartic functional equation as Theorem 2 for a quadratic functional equation and Theorem 3 for a cubic functional equation.

Secondly, constructing a subset of measure 0 satisfying condition (C) we consider a measure zero stability problem of quartic functional equation (1) for all , where and has 2-dimensional Lebesgue measure 0.

As an application we consider an asymptotic behavior of satisfying the weak conditionas only for , where has 2-dimensional Lebesgue measure 0.

#### 2. Stability of the Quartic Functional Equation in Restricted Domain

Throughout this section we assume that satisfies the following condition:For given there exists such that(C), , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , .

Theorem 4. Let be fixed. Suppose that satisfies inequality (2) for all . Then there exists a unique quartic mapping satisfying (10) for all .

Proof. Letfor all . ThenThus, we get the functional identitySince satisfies condition (C), for given , there exists such thatfor all . Thus, dividing (14) by 6 and using the triangle inequality and (15) we havefor all . Let for all . Then from (16) we havefor all . Using Theorem 1 with (17), we get (10). This completes the proof.

Remark 5. Letting in (10) and dividing the result by 232 we have . Thus, inequality (10) impliesfor all .

Let . It is easy to see that satisfies condition (C). Indeed, for given if we choose , then . Thus, as a direct consequence of Theorem 4 we obtain the following result (see [14, 16, 17] for similar results).

Corollary 6. Let be fixed. Suppose that satisfies inequality (2) for all such that . Then there exists a unique quartic mapping satisfying (10) for all .

In particular, if , we have the following.

Corollary 7. Suppose that (1) for all . Then, (1) holds for all .

#### 3. Stability Problem in a Set of Lebesgue Measure Zero

In this section, we show that even a set of Lebesgue measure zero can satisfy condition (C) when . From now on, we identify with .

Lemma 8. Let , where with for all . Then there exists such that satisfies for all .

Proof. The coefficients and are given by for all . Now, the equation has only a finite number of zeros in . Thus, we can choose such that . This completes the proof.

Lemma 9. One can find a set of Lebesgue measure 0 such that, for any countable subsets , and , there exists satisfying

Proof. It is shown in [22, Theorem 1.6] that there exists a set of Lebesgue measure 0 such that is of the first Baire category; that is, is a countable union of nowhere dense subsets of . Let and and . Then, since is of the first Baire category, are also of the first Baire category for all . Since each consists of a countable union of nowhere dense subsets, by the Baire category theorem, countably many of them cannot cover ; that is, Thus, there exists such that for all . This means that for all . This completes the proof.

Theorem 10. Let be fixed. Then there exists a set of 2-dimensional Lebesgue measure 0 which satisfies condition (C).

Proof. Let be the set in condition (C). Then by Lemma 8 we can choose such that satisfies for all . Let be the set in Lemma 9. Then has 2-dimensional Lebesgue measure 0. Now, we show that satisfies condition (C); that is, for given , there exists satisfying the conditionsLet , , and . Then we haveNow, by Lemma 9, for given there exists such thatFrom (24) and (25) we haveSince , using the triangle inequality we have for all . Thus, satisfies (C). This completes the proof.

Now, as a direct consequence of Theorems 4 and 10 we have the following.

Corollary 11. Let be fixed. Suppose that satisfies inequality (2) for all . Then there exists a unique quartic mapping satisfying (10) for all .

As a consequence of Corollary 11 we obtain an asymptotic behavior of satisfying condition (11) as only for .

Corollary 12. Suppose that satisfies condition (11). Then is a quartic mapping.

Proof. Condition (11) implies that, for each , there exists such thatfor all . By Corollary 11, there exists a unique quartic mapping such thatfor all . Replacing by in (28) and using the triangle inequality we havefor all . Let for all . Then by (29), is a bounded quartic mapping. Thus, we have and hence for all . Letting in (28) we have and hence for all . This completes the proof.

Remark 13. If we define as an appropriate rotation of -product of , then has -dimensional measure 0 and satisfies condition (C). Consequently, we obtain the following.

Theorem 14. Suppose that satisfies inequality (2) for all . Then there exists a unique quartic mapping satisfying (10) for all .

#### Competing Interests

The authors declare that there are no competing interests regarding the publication of this paper.

#### Acknowledgments

This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (no. 2015R1D1A3A01019573).

1. S. H. Lee, S. M. Im, and I. S. Hwang, “Quartic functional equations,” Journal of Mathematical Analysis and Applications, vol. 307, no. 2, pp. 387–394, 2005. View at: Publisher Site | Google Scholar | MathSciNet
2. A. Bodaghi, “Stability of a quartic functional equation,” The Scientific World Journal, vol. 2014, Article ID 752146, 9 pages, 2014. View at: Publisher Site | Google Scholar
3. D. Kang, “On the stability of generalized quartic mappings in quasi-β-normed spaces,” Journal of Inequalities and Applications, vol. 2010, Article ID 198098, 11 pages, 2010. View at: Publisher Site | Google Scholar
4. C. Alsina and J.-L. Garcia-Roig, “On a conditional Cauchy equation on rhombuses,” in Functional Analysis, Approximation theory and Numerical Analysis, J. M. Rassias, Ed., pp. 5–7, World Scientific, River Edge, NJ, USA, 1994. View at: Google Scholar | MathSciNet
5. B. Batko, “Stability of an alternative functional equation,” Journal of Mathematical Analysis and Applications, vol. 339, no. 1, pp. 303–311, 2008.
6. B. Batko, “On approximation of approximate solutions of Dhombres' equation,” Journal of Mathematical Analysis and Applications, vol. 340, no. 1, pp. 424–432, 2008.
7. J. Brzdęk, “On the quotient stability of a family of functional equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 10, pp. 4396–4404, 2009. View at: Publisher Site | Google Scholar | MathSciNet
8. J. Brzdęk, “On a method of proving the Hyers-Ulam stability of functional equations on restricted domains,” The Australian Journal of Mathematical Analycis and Applications, vol. 6, no. 1, pp. 1–10, 2009. View at: Google Scholar | MathSciNet
9. A. Bahyrycz and J. Brzdęk, “On solutions of the d'Alembert equation on a restricted domain,” Aequationes Mathematicae, vol. 85, no. 1-2, pp. 169–183, 2013. View at: Publisher Site | Google Scholar | MathSciNet
10. J. Brzdęk and J. Sikorska, “A conditional exponential functional equation and its stability,” Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 6, pp. 2923–2934, 2010. View at: Publisher Site | Google Scholar | MathSciNet
11. J.-Y. Chung, “Stability of functional equations on restricted domains in a group and their asymptotic behaviors,” Computers and Mathematics with Applications, vol. 60, no. 9, pp. 2653–2665, 2010. View at: Publisher Site | Google Scholar | MathSciNet
12. R. Ger and J. Sikorska, “On the Cauchy equation on spheres,” Annales Mathematicae Silesianae, vol. 11, pp. 89–99, 1997. View at: Google Scholar | MathSciNet
13. S.-M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analisis, vol. 48, Springer, New York, NY, USA, 2011. View at: Publisher Site | MathSciNet
14. S.-M. Jung, “On the Hyers-Ulam stability of the functional equations that have the quadratic property,” Journal of Mathematical Analysis and Applications, vol. 222, no. 1, pp. 126–137, 1998.
15. M. Kuczma, “Functional equations on restricted domains,” Aequationes Mathematicae, vol. 18, no. 1-2, pp. 1–34, 1978.
16. J. M. Rassias and M. J. Rassias, “On the Ulam stability of Jensen and Jensen type mappings on restricted domains,” Journal of Mathematical Analysis and Applications, vol. 281, no. 2, pp. 516–524, 2003. View at: Publisher Site | Google Scholar | MathSciNet
17. J. M. Rassias, “On the Ulam stability of mixed type mappings on restricted domains,” Journal of Mathematical Analysis and Applications, vol. 276, no. 2, pp. 747–762, 2002.
18. J. Sikorska, “On two conditional Pexider functinal equations and their stabilities,” Nonlinear Analysis. Theory, Methods & Applications A: Theory and Methods, vol. 70, no. 7, pp. 2673–2684, 2009. View at: Publisher Site | Google Scholar | MathSciNet
19. F. Skof, “Sull'approssimazione delle applicazioni localmente $\delta$-dadditive,” Atti dell'Accademia delle Scienze di Torino, vol. 117, pp. 377–389, 1983. View at: Google Scholar
20. J. Chung and J. M. Rassias, “On a measure zero stability problem of a cyclic equation,” Bulletin of the Australian Mathematical Society, vol. 93, no. 2, pp. 272–282, 2016.
21. J. Chung, Y. M. Ju, and J. M. Rassias, “Cubic functional equation on restricted domains of Lebesgue measure zero,” Canadian Mathematical Bulletin, In press. View at: Google Scholar
22. J. C. Oxtoby, Measure and Category, vol. 2, Springer, New York, NY, USA, 2nd edition, 1980. View at: MathSciNet

#### More related articles

We are committed to sharing findings related to COVID-19 as quickly and safely as possible. Any author submitting a COVID-19 paper should notify us at help@hindawi.com to ensure their research is fast-tracked and made available on a preprint server as soon as possible. We will be providing unlimited waivers of publication charges for accepted articles related to COVID-19. Sign up here as a reviewer to help fast-track new submissions.