The Scientific World Journal

Volume 2014 (2014), Article ID 752146, 9 pages

http://dx.doi.org/10.1155/2014/752146

Research Article

Stability of a Quartic Functional Equation

Abasalt Bodaghi

Department of Mathematics, Garmsar Branch, Islamic Azad University, Garmsar, Iran

Received 8 August 2013; Accepted 20 November 2013; Published 23 January 2014

Academic Editors: A. Ibeas and B. Meng

Copyright © 2014 Abasalt Bodaghi. 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 obtain the general solution of the generalized quartic functional equation + for a fixed positive integer . We prove the Hyers-Ulam stability for this quartic functional equation by the directed method and the fixed point method on real Banach spaces. We also investigate the Hyers-Ulam stability for the mentioned quartic functional equation in non-Archimedean spaces.

1. Introduction

We say a functional equation is stable if any function satisfying the equation approximately is near to exact solution of . Moreover, a functional equation is hyperstable if any function satisfying the equation   approximately is a true solution of .

The study of stability problems for functional equations is related to a question of Ulam [1] concerning the stability of group homomorphisms, affirmatively answered for Banach spaces by Hyers [2]. Subsequently, the result of Hyers was generalized by a number of authors. For example, Bodaghi et al. investigated the Hyers-Ulam stability of Jordan -derivation pairs for the Cauchy additive functional equation and the Cauchy additive functional inequality in [3]. For some results on the stability of various functional equations, see also [4–9].

The oldest quartic functional equation was introduced by Rassias in [10] and then was employed by other authors. Rassias [10] investigated stability properties of the following quartic functional equation:

Since , we get

In [11], Chung and Sahoo determined the general solution of (2) without assuming any regularity conditions on the unknown function. Indeed, they proved that the function is a solution of (2) if and only if where the function is symmetric and additive in each variable. The fact that every solution of (2) is even implies that it can be written as follows:

Lee et al. [12] obtained the general solution of (3) and proved the Hyers-Ulam stability of this equation. Also Park [13] investigated the stability problem of (3) in the orthogonality normed space. Lee and Chung [14] considered the following quartic functional equation, which is a generalization of (3):

for fixed integer with . They obtained the general solution of (4) and proved its Hyers-Ulam stability.

Bodaghi et al. [15] applied the fixed point alternative theorem (Theorem 8 of the current paper) to establish Hyers-Ulam stability of (3). They also showed that the functional equation (3) can be hyperstable under some conditions. This method which is different from the “direct method,” initiated by Hyers in 1941, had been applied by Cădariu and Radu for the first time in [16]. In other words, they employed this fixed point method to the investigation of the Cauchy functional equation [17] and of the quadratic functional equation [16] (for more applications of this method, see [18–20]).

In this paper, we consider the following functional equation which is somewhat different from (2), (3), and (4):

for a fixed positive integer . In case , then (5) is the celebrated Jordan-von Neumann equation. Then we find out the general solution of (5). We also prove the Hyers-Ulam stability problem and the hyperstability for (5) by the directed method and the fixed point method.

2. General Solution of (5)

To achieve our aim in this section, we need the following lemma.

Lemma 1. Let and be real vector spaces. If a function satisfies the functional equation (5) for all integers , then satisfies for all integers .

Proof. Letting in (5), we get . Once more, by putting in (5), we obtain

In the case that , by replacing , by , in (5), respectively, we have

The above equality and (6) imply that . Now, assume that, for every , we have . If , then . Since , we have , and thus . Let . Then, by substituting , by , in (5), respectively, we have

Since , we have and . Replacing these equalities in (8) and using (6), we get . This completes the proof.

Remark 2. It is shown in [21, Lemma 2.1] that a mapping satisfies the functional equation (1) if and only if satisfies

There is a gap in its proof. In fact, in the proof, the author only showed that the functional equation (1) implies (9) but the converse is not proved. Theorem 3 resolved this problem. Indeed, we solve the equation of (5).

Theorem 3. Let and be real vector spaces. Then a mapping satisfies the functional equation (2) if and only if it satisfies (5) where . Therefore, every solution of the functional equation (5) is also a quartic mapping.

Proof. Suppose that satisfies the functional equation (2). Putting in (2), we get . Let in (2) to get for all . Setting in (2) and using the fact that , we have . Letting in (2), we have for all . By induction, we obtain for all positive integers . Replacing by and in (2), respectively, we have

In Similar way to the above, we get

Using the above method, we can deduce that

for which

Solving the above recurrence equations is routine, and so we get

for all and each positive integer .

Conversely, assume that satisfies the functional equation for each , in particular, for . Hence for each , we have

By Lemma 1, we have and so

On the other hand,

Using (16) and (17), we get

A calculation shows that

Thus, if satisfies the functional equation (5) for all , then it satisfies (5) for . In particular, satisfies (2).

3. Hyers-Ulam Stability of (5)

Let be an integer with . We use the abbreviation for the given mapping as follows:

Throughout this section, we assume that is a normed real linear space with norm and is a real Banach space with norm . We are going to prove the stability of the quartic functional equation (5).

Theorem 4. Let be a real number and let be a mapping for which there exists a function such that

for all , where is an integer with . Then there exists a unique quartic mapping such that

for all .

Proof. Putting in (22), we have

for all . Thus

for all . Replacing by in (25) and continuing this method, we get

On the other hand, we can use induction to obtain

for all , and . Thus the sequence is Cauchy by (21) and (27). Completeness of allows us to assume that there exists a map so that

Taking the limit as in (26) and applying (28), we can see that inequality (23) holds. Now, we replace , by ,, respectively, in (22); then

Letting the limit as , we obtain for all positive integers  and all . Hence, by Theorem 3, it indicates that is a quartic mapping. Now, let be another quartic mapping satisfying (23). Then we have

for all . Taking in the preceding inequality, we immediately find the uniqueness of . This completes the proof.

Corollary 5. Let , , , , and be nonnegative real numbers such that and . Suppose that is a mapping fulfilling

for all , where is an integer with . Then there exists a unique quartic mapping such that

for alland all if .

Proof. Setting in Theorem 12, we have

It follows from (31) that

By these statements we can get the result.

We have the following result which is analogous to Theorem 12 for the quartic functional equation (5). We include its proof.

Theorem 6. Suppose that is a mapping for which there exists a function such that

for all , where is an integer with . Then there exists a unique quartic mapping such that

for all .

Proof. It follows from (35) that . Thus from (36) we have . Putting in (36), we get

for all . If we replace by in the above inequality and divide both sides by , we have

Using triangular inequality and proceeding this way, we obtain

for all . If we show that the sequence is Cauchy, then it will be convergent by the completeness of . For this, if we replace by in (40) and then multiply both sides by , then we get

for all , and . Thus the mentioned sequence is convergent to the mapping ; that is,

Now, in a similar way to the proof of Theorem 12, we can complete the rest of the proof.

Corollary 7. Let , , , and be nonnegative real numbers such that . Suppose that is a mapping fulfilling

for all , where is an integer with . Then there exists a unique quartic mapping such that

for all .

Proof. First, we note that if we put in (43), we have . Taking in Theorem 14, we can obtain the desired result.

We are going to investigate the hyperstability of the given quartic functional equation (5) by using the fixed point method. First, we bring the next theorem which was proved in [22]. This result plays a fundamental role to achieve our goal.

Theorem 8 (the fixed point alternative theorem). Let be a complete generalized metric space and let be a mapping with Lipschitz constant . Then, for each element , either for all or there exists a natural number such that

(i) for all ;(ii)the sequence is convergent to a fixed point of ;(iii) is the unique fixed point of in the set ;(iv) for all .

Theorem 9. Let be a mapping with and let be a function such that

for all , where is an integer with . If there exists a constant , such that

for all , then there exists a unique quartic mapping such that

for all .

Proof. We wish to make the conditions of Theorem 8. We consider the set

and define the mapping on as follows:

if there exists such constant , and , otherwise. In a similar way to the proof of [15, Theorem 2.2], we can show that is a generalized metric on and the metric space is complete. Here, we define the mapping by

If such that , by definitions of and , we have

for all . Using (46), we get

for all . The above inequality shows that for all . Hence, is a strictly contractive mapping on with a Lipschitz constant . We now show that . Putting in (45), we obtain

for all . We conclude from the last inequality that

Theorem 8 shows that for all , and thus in this theorem we have . Consequently, the parts (iii) and (iv) of Theorem 8 hold on the whole . Hence there exists a unique mapping such that is a fixed point of and that as . Thus

for all , and so

The above inequalities show that (47) is true for all . Now, it follows from (46) that

Substituting and by and , respectively, in (45), we get

Taking the limit as , we obtain for all integersand all. It follows from Theorem 3 that is a quartic mapping which is unique.

Corollary 10. Let , , and be nonnegative real numbers with and let be a mapping such that

for all . Then there exists a unique quartic mapping satisfying

for all .

Proof. Note that inequality (59) implies that . If we put in Theorem 9, we obtain the desired result.

In the next result, we prove the hyperstability of quartic functional equations under some conditions.

Corollary 11. Let , , and be nonnegative real numbers with and let be a mapping such that

for all . Then is a quartic mapping on .

Proof. Putting in (61), we get . Again, if we put in (61), then we have for all . It is easy to check that , and so for all and . Now, it follows from Theorem 9 that is a quartic mapping when .

4. Stability of (5) in Non-Archimedean Spaces

We recall some basic facts concerning non-Archimedean spaces and some preliminary results. By a non-Archimedean field we mean a field equipped with a function (valuation) from into such that if and only if , , and max for all . Clearly and for all .

Let be a vector space over a scalar field with a non-Archimedean nontrivial valuation . A function is a non-Archimedean norm (valuation) if it satisfies the following conditions:

(i) if and only if ;(ii), ;(iii)the strong triangle inequality (ultrametric); namely,

Then is called a non-Archimedean space. Due to the fact that

a sequence is Cauchy if and only if converges to zero in a non-Archimedean normed space . By a complete non-Archimedean normed space we mean one in which every Cauchy sequence is convergent.

In [23], Hensel discovered the -adic numbers as a number theoretical analogue of power series in complex analysis. The most interesting example of non-Archimedean spaces is -adic numbers. A key property of -adic numbers is that they do not satisfy the Archimedean axiom: for all , there exists an integer such that .

Let be a prime number. For any nonzero rational number in which and are coprime to the prime number . Consider the -adic absolute value on . It is easy to check that is a non-Archimedean norm on . The completion of with respect to which is denoted by is said to be the -adic number field. One should remember that if , then for all integers . In [24], the stability of some functional equations in non-Archimedean normed spaces is investigated (see also [25]).

Here and subsequently, we assume that is a normed space and is a complete non-Archimedean space unless otherwise stated explicitly. In the upcoming theorem, we prove the stability of the functional equation (5).

Theorem 12. Let such that

for all . Suppose that is a mapping satisfying the equality

for all , where is an integer with . Then there exists a unique quartic mapping such that

for all where .

Proof. Putting in (65), we get

for all . Thus we have

for all . Replacing by in (68) and then dividing both sides by , we have

for all and all nonnegative integers . Thus the sequence is Cauchy by (64) and (69). Due to the completeness of as a non-Archimedean space, there exists a mapping so that

For each and non-negative integers , we have

Taking in (71) and applying (70), we can see that the inequality (66) holds when . It follows from (64), (65), and (70) that, for all ,

Hence, the mapping satisfies (5). Now, let be another quartic mapping satisfying (66). Then we have

for all . This shows the uniqueness of .

Corollary 13. Let , be a non-Archimedean space and let be a function satisfying for all for which . Suppose that is a mapping satisfying the inequality

for all , where is an integer with . Then there exists a unique quartic mapping such that

for all .

Proof. Defining by , we have

for all . We also have

for all . Now, Theorem 12 implies the desired result.

We have the following result which is analogous to Theorem 12 for the functional equation (5).

Theorem 14. Let such that

for all . Suppose that is a mapping satisfying the inequality

for all , where is an integer with . Then there exists a unique quartic mapping such that

for all where .

Proof. In a similar way to the proof of Theorem 12, we have

for all . If we replace by in the above inequality and multiply both sides of (81) to, we get

for all and all non-negative integers . Thus, we conclude from (78) and (82) that the sequence is Cauchy. Since the non-Archimedean space is complete, this sequence converges in to the mapping . Indeed,

Using induction and (81), one can show that

for all and non-negative integers . Since the right hand side of inequality (84) goes to as , by applying (83), we deduce inequality (80). Now, in a similar way to the proof of Theorem 12, we can complete the rest of the proof.

Corollary 15. Let , be a non-Archimedean space and let be a function satisfying for all for which . Suppose that is a mapping satisfying the inequality

for all , where is an integer with . Then there exists a unique quartic mapping such that

for all .

Proof. The proof is a direct consequence of Theorem 14 and similar to the proof of Corollary 13.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

References

S. M. Ulam, Problems in Modern Mathematics, chapter 6, John Wiley & Sons, New York, NY, USA, 1964.

D. H. Hyers, “On the stability of the linear functional equation,” Proceedings of the National Academy of Sciences of the United States of America, vol. 27, pp. 222–224, 1941.

A. Bodaghi, S. Y. Jang, and C. Park, “On the Stability of Jordan *-Derivation Pairs,” Results in Mathematics, vol. 64, no. 3-4, pp. 289–303, 2013.

T. Aoki, “On the stability of the linear transformation in Banach spaces,” Journal of the Mathematical Society of Japan, vol. 2, pp. 64–66, 1950.

A. Bodaghi and I. A. Alias, “Approximate ternary quadratic derivations on ternary Banach algebras and -ternary rings,” Advances in Difference Equations, vol. 2012, article 11, 2012.

M. Eshaghi Gordji and A. Bodaghi, “On the stability of quadratic double centralizers on Banach algebras,” Journal of Computational Analysis and Applications, vol. 13, no. 4, pp. 724–729, 2011.

D. H. Hyers, G. Isac, and T. M. Rassias, Stability of functional equations in several variables, Birkhäuser Boston, Boston, Mass, USA, 1998.

C. Park and A. Bodaghi, “On the stability of -derivations on Banach -algebras,” Advances in Difference Equations, vol. 2012, article 138, 2012.

S. Y. Yang, A. Bodaghi, and K. A. M. Atan, “Approximate cubic *-derivations on Banach *-algebras,” Abstract and Applied Analysis, Article ID 684179, 12 pages, 2012.

J. M. Rassias, “Solution of the Ulam stability problem for quartic mappings,” Glasnik Matematički, vol. 34, no. 2, pp. 243–252, 1999.

J. K. Chung and P. K. Sahoo, “On the general solution of a quartic functional equation,” Bulletin of the Korean Mathematical Society, vol. 40, no. 4, pp. 565–576, 2003.

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.

C.-G. Park, “On the stability of the orthogonally quartic functional equation,” Bulletin of the Iranian Mathematical Society, vol. 31, no. 1, pp. 63–70, 2005.

Y.-S. Lee and S.-Y. Chung, “Stability of quartic functional equations in the spaces of generalized functions,” Advances in Difference Equations, vol. 2009, Article ID 838347, 2009.

A. Bodaghi, I. A. Alias, and M. H. Ghahramani, “Ulam stability of a quartic functional equation,” Abstract and Applied Analysis, Article ID 232630, 9 pages, 2012.

L. Cǎdariu and V. Radu, “Fixed points and the stability of quadratic functional equations,” Analele Universitatii de Vest din Timisoara, vol. 41, no. 1, pp. 25–48, 2003.

L. Cădariu and V. Radu, “On the stability of the Cauchy functional equation: a fixed point approach,” Grazer Mathematische Berichte, vol. 346, pp. 43–52, 2004.

A. Bodaghi, M. Eshaghi Gordji, and K. Paykan, “Approximate multipliers and approximate double centralizers: a fixed point approach,” Analele Universitatii “Ovidius” Constanta, vol. 20, no. 3, pp. 21–32, 2012.

M. Eshaghi Gordji, A. Bodaghi, and C. Park, “A fixed point approach to the stability of double Jordan centralizers and Jordan multipliers on Banach algebras,” Scientific Bulletin-University Politehnica of Bucharest—Scientific Bulletin A, vol. 73, no. 2, pp. 65–74, 2011.

C. Park, “Fixed points and Hyers-Ulam-Rassias stability of Cauchy-Jensen functional equations in Banach algebras,” Fixed Point Theory and Applications, vol. 2007, Article ID 050175, 2007.

D. S. Kang, “On the stability of generalized quartic mappings in quasi--normed spaces,” Journal of Inequalities and Applications, Article ID 198098, 11 pages, 2010.

J. B. Diaz and B. Margolis, “A fixed point theorem of the alternative, for contractions on a generalized complete metric space,” Bulletin of the American Mathematical Society, vol. 74, pp. 305–309, 1968.

K. Hensel, “Uber eine neue Begrndung der Theorie der algebraischen Zahlen,” Deutsche Mathematiker-Vereinigung, vol. 6, pp. 83–88, 1897.

M. S. Moslehian and T. M. Rassias, “Stability of functional equations in non-Archimedean spaces,” Applicable Analysis and Discrete Mathematics, vol. 1, no. 2, pp. 325–334, 2007.

A. K. Mirmostafaee, “Approximately additive mappings in non-Archimedean normed spaces,” Bulletin of the Korean Mathematical Society, vol. 46, no. 2, pp. 387–400, 2009.