Stability of a Quartic Functional Equation

Bodaghi, Abasalt

doi:https://doi.org/10.1155/2014/752146

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Research Article | Open Access

Volume 2014 | Article ID 752146 | https://doi.org/10.1155/2014/752146

Stability of a Quartic Functional Equation

Abasalt Bodaghi¹

Academic Editor: A. Ibeas, B. Meng

Received08 Aug 2013

Accepted20 Nov 2013

Published23 Jan 2014

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 havefor 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.

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Copyright

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.

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