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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 232630, 9 pages
Ulam Stability of a Quartic Functional Equation
1Department of Mathematics, Garmsar Branch, Islamic Azad University, Garmsar, Iran
2Institute for Mathematical Research, University Putra Malaysia, 43400 UPM, Serdang, Selangor Darul Ehsan, Malaysia
Received 11 January 2012; Revised 9 February 2012; Accepted 13 February 2012
Academic Editor: Nicole Brillouet-Belluot
Copyright © 2012 Abasalt Bodaghi et al. 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.
The oldest quartic functional equation was introduced by J. M. Rassias in (1999), and then was employed by other authors. The functional equation is called a quartic functional equation, all of its solution is said to be a quartic function. In the current paper, the Hyers-Ulam stability and the superstability for quartic functional equations are established by using the fixed-point alternative theorem.
We say a functional equation is stable if any function satisfying the equation approximately is near to true solution of . Moreover, a functional equation is superstable if any function satisfying the equation approximately is a true solution of (see  for another notion of the superstability which may be called superstability modulo the bounded functions).
The stability problem for functional equations originated from a question by Ulam  in 1940, concerning the stability of group homomorphisms: let be a group, and let be a metric group with the metric . Given , does there exist such that, if a mapping satisfies the inequality for all , then there exists a homomorphism with for all In other words, under what condition a functional equation is stable? In the following year, Hyers  gave a partial affirmative answer to the question of Ulam for Banach spaces. In 1978, the generalized Hyers’ theorem was independently rediscovered by Th. M. Rassias  by obtaining a unique linear mapping under certain continuity assumption.
The functional equations are called quadratic and cubic functional equations, respectively. During the last decades, several stability problems for functional equations especially the quadratic and cubic and their generalized have been extensively investigated by many mathematicians (for instances, [5–9]).
In , Lee et al. considered the following quartic functional equation: It is easy to check that for every , the function is a solution of the above functional equation. They solved (1.2) and in fact showed that a function whenever and are real vector spaces is quadratic if and only if there exists a symmetric biquadratic function such that for all . They also proved the stability of (1.2). Zhou Xu et al. in  used the fixed-point alternative (Theorem 2.1 of the current paper) to establish Hyers-Ulam-Rassias stability of the general mixed additive-cubic functional equation, where functions map a linear space into a complete quasifuzzy -normed space. The generalized Hyers-Ulam stability of a general mixed AQCQ-functional in multi-Banach spaces is also proved by using the mentioned theorem in .
Recently, Bodaghi et al. in [13, 14] investigated the stability and the superstability of quadratic and cubic functional equations by a fixed-point method and applied this method to prove the stability of (quadratic, cubic) multipliers on Banach algebras.
2. Main Results
Throughout this paper, assume that is a normed vector space and is a Banach space. For a given mapping , we consider
for all .
To achieve our aim, we need the following known fixed-point theorem which has been proved in .
Theorem 2.1. Suppose that is a complete generalized metric space, and let be a strictly contractive 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 2.2. Assume that is a function satisfying for all . Let a mapping satisfy . If there exists such that for all , then there exists a unique quartic mapping such that for all .
Proof. By recurrence method, we can conclude from (2.4) that for all . Passing to the limit, we get
for all . Here, we intend to build the conditions of Theorem 2.1 and so consider the set and the mapping defined on by
if there exists such constant , and otherwise. It is easy to see that and , for all . For each , we have
Hence, . Now if , then for every fixed , we have , for all . This implies . Let be a -Cauchy sequence in , then , and thus , for all . Since is complete, then there exists such that in . Therefore, is a generalized metric on , and the metric space is complete. Now, we define the mapping by
Fix a and take such that . The definitions of and show that
for all . By using (2.4), we have for all . It follows from the above inequality that , for all . Hence, is a strictly contractive mapping on with a Lipschitz constant . Putting in (2.3) and dividing both sides of the resulting inequality by 32, we have for all . Thus, . Note that by Theorem 2.1, , for all . Thus, we get in this theorem, so (iii) and (iv) of Theorem 2.1 are true on the whole . However, the sequence converges to a unique fixed-point in the set , that is, for all . By the part (iv) of Theorem 2.1, we have
From (2.14), we observe that the inequality (2.5) holds for all . Substituting by in (2.3), respectively, and applying (2.6) and (2.13), we have for all . Therefore, is a quartic mapping which is unique by part (iii) of Theorem 2.1.
Corollary 2.3. Let be nonnegative real numbers such that , and let be a mapping (with when ) satisfying for all , then there exists a unique quartic mapping such that for all .
Proof. The result follows from Theorem 2.2 by using .
Now, we establish the superstability of quartic mapping on Banach spaces under some conditions.
Corollary 2.4. Let be nonnegative real numbers such that . Suppose that a mapping satisfies for all , then is a quartic mapping on .
Proof. Letting in Theorem 2.2, we have which shows (2.6) holds for . Putting in (2.18), we get . Furthermore, if we put in (2.18), then we have , for all . It is easy to see that by induction, we have , and so , for all and . Now, it follows from Theorem 2.2 that is a quartic mapping.
Let and be positive real numbers. Suppose that a mapping satisfies for all , then by considering in Theorem 2.2, the mapping is again a quartic mapping on .
The following result is proved in [16, Theorem 1].
Theorem 2.5. Let be a linear space, and let be a Banach space. Let be a mapping for which there exists a function such that for all , where , then there exists a unique quartic mapping such that for all .
One should note that in the above theorem, is not necessarily zero, but in the following result, we assume that and also consider the case . By these hypotheses and by applying Theorem 2.1, we obtain the specific result which is a way to prove the superstability of a quartic functional equation.
Theorem 2.6. Let be a mapping with , and let be a function satisfying for all . If there exists such that for all , then there exists a unique quartic mapping such that for all .
Proof. We take the set and consider the generalized metric on ,
if there exists such a constant , and otherwise. It follows from the proof of Theorem 2.2 that the metric space is complete (see the proof of Theorem 2.2). We will show that the mapping defined by is strictly contractive. Fix a and take such that , then we have
for all . By using (2.25), we obtain
for all . It follows from the last inequality that , for all . Hence, is a strictly contractive mapping on with a Lipschitz constant . By putting , replacing by in (2.24) and using (2.25), and then dividing both sides of the resulting inequality by 2, we have
for all . Hence, . By applying the fixed-point alternative Theorem 2.1, there exists a unique mapping in the set such that
for all . Again Theorem 2.1 shows that
Hence, inequality (2.32) implies (2.26). Replacing by in (2.24), respectively, and using (2.23) and (2.31), we conclude that for all . Therefore, is a quartic mapping.
Corollary 2.7. Let and be nonnegative real numbers such that . Suppose that is a mapping satisfying for all , then there exists a unique quartic mapping such that for all .
Proof. It is enough to let in Theorem 2.6.
Corollary 2.8. Let be nonnegative real numbers such that . Suppose that a mapping satisfies for all . Then is a quartic mapping on .
Proof. Putting in Theorem 2.6, we have
and thus, (2.6) holds. If we put in (2.36), then we get . Again, letting in (2.36), we conclude that , and thus, , for all and . Now, we can obtain the desired result by Theorem 2.6.
From Corollaries 2.4 and 2.8 we deduce the following result.
Corollary 2.9. Let be nonnegative real numbers such that and . Suppose that a mapping satisfies (2.36), for all then is a quartic mapping on .
This paper was prepared while the first author was attending as a Postdoctoral Researcher in University Putra Malaysia. He is pleased to thank the staff of the Institute for Mathematical Research for warm hospitality, and he wishes to express his gratitude to Professor Dato’ Dr. Hj. Kamel Ariffin Mohd Atan. The authors would like to thank the referees for careful reading and giving some useful comments in the first draft of the paper.
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