Abstract
In this article, we present generalized Hyers–Ulam stability results of a cubic functional equation associated with an approximate cubic Lie derivations on convex modular algebras with -condition on the convex modular functional
1. Introduction
In 1940, S. M. Ulam [1] raised the question 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 ? D. H. Hyers [2] has solved the problem of Ulam for the case of additive mappings in 1941. The result was generalized by T. Aoki [3] in 1950, by Th.M. Rassias [4] in 1978, by J. M. Rassias [5] in 1992, and by P. Gǎvruta [6] in 1994. Over the past few decades, many mathematicians have investigated the generalized Hyers–Ulam stability theorems of various functional equations [7–12].
Now, we recall some basic definitions and remarks of modular spaces with modular functions, which are primitive notions corresponding to norms or metrics, as in the following [13–15].
Definition 1. Let be a linear space.(a) A function is called a modular if, for arbitrary ,(1) if and only if ,(2) for every scalar with ,(3) for any scalars , , where and ;(b)alternatively, if (3) is replaced by(3)’ for every scalars , , where and , then we say that is a convex modular.
It is well known that a modular defines a corresponding modular space, i.e., the linear space given by Let be a convex modular. Then, we remark the modular space can be a Banach space equipped with a norm called the Luxemburg norm, defined byIf is a modular on , we note that is an increasing function in for each fixed ; that is, , whenever . In addition, if is a convex modular on , then for all and . Moreover, we see that for all and .
Remark. (a) In general, we note that for all and whenever [14].
(b) Consequently, we lead to for all and .
Definition 2. Let be a modular space and let be a sequence in . Then, (1) is -convergent to and we write if as ;(2) is called -Cauchy in if as ;(3)a subset of is called -complete if and only if any -Cauchy sequence in is -convergent to an element in .
They say that the modular has the Fatou property if and only if whenever the sequence is -convergent to . A modular function is said to satisfy the -condition if there exists such that for all
In 2014, G. Sadeghi [16] has demonstrated generalized Hyers–Ulam stability via the fixed point method of a generalized Jensen functional equation in convex modular spaces with the Fatou property satisfying the -condition with In [15], the authors have proved the generalized Hyers–Ulam stability of quadratic functional equations via the extensive studies of fixed point theory in the framework of modular spaces whose modulars are convex and lower semicontinuous but do not satisfy any relatives of -conditions (see also [17, 18]). Recently, the authors [14, 19, 20] have investigated stability theorems of functional equations in modular spaces without using the Fatou property and -condition. In 2001, J. M. Rassias [21] has introduced to study Hyers–Ulam stability of the following cubic functional equation: which is equivalent to whose general solution is characterized as where is symmetric and additive for each fixed one variable [22]. For this reason, every solution of the cubic functional equation is said to be a cubic mapping.
Now, we say that is called a (convex) modular algebra if the fundamental space is an algebra over or with (convex) modular subject to for all A subset of a convex modular algebra is called -complete if and only if any -Cauchy sequence in is -convergent to an element in . Throughout the paper, will be a -complete convex modular algebra and the symbol will denote the commutator We say that a mapping is cubic homogeneous if for all vectors and all scalars , and a cubic homogeneous mapping is called a cubic Lie derivation if for all vectors [23, 24].
In this article, we first investigate generalized Hyers–Ulam stability of the equationin -complete convex modular algebras without using the Fatou property and -condition and then present alternatively generalized Hyers–Ulam stability of (6) using necessarily -condition without the Fatou property in -complete convex modular algebras.
2. Generalized Hyers–Ulam Stability of (6)
First of all, we remark that (6) is equivalent to the original cubic functional equation, and so every solution of (6) is a cubic mapping.
For notational convenience, we let the difference operators of cubic equation (6) and of cubic derivation be as follows: for all in a linear space and In the following, we present a generalized Hyers–Ulam stability via direct method of the system and in -complete convex modular algebras without using both the Fatou property and -condition.
Theorem 3. Suppose that a mapping satisfiesand , are mappings such thatfor all and . If for each the mapping from to is continuous, then there exists a unique cubic Lie derivation which satisfies equation (6) andfor all .
Proof. Putting and in (8), we obtainwhich yieldsfor all Since , we prove the following functional inequality:for all by using the property of convex modular .
Now, replacing by in (13), we havewhich converges to zero as by assumption (9). Thus the above inequality implies that the sequence is -Cauchy for all and so it is convergent in since the space is -complete. Thus, we may define a mapping as for all .
Claim 1. is a cubic mapping satisfying approximation (10). In fact, if we put in (8) and then divide the resulting inequality by , one obtainsfor all , where is a fixed positive real. Thus we figure out by use of the first remarkfor all , and all positive integers . Taking the limit as , one obtains , and so for all . Hence, taking in , it follows that satisfies (6) and so it is cubic. On the other hand, since for all , it follows from (12) and the first remark that without applying the Fatou property of the modular for all and all , from which we obtain the approximation of by the cubic mapping as follows: for all by taking in the last inequality.
Claim 2. is cubic homogeneous. By (17), we have , which yields for all and . From the assumption that for each the mapping from to is continuous, it follows that for all and by the same argument as in the paper [4, 25]. Thus, for any nonzero for all and , which concludes that is cubic homogeneous.
Claim 3. is a cubic Lie derivation. From the second inequality in (9) and the second condition in (8), we arrive at for all , which tends to zero as tends to Therefore, one obtains , and so is a cubic Lie derivation.
Claim 4. is a unique cubic Lie derivation. To show the uniqueness of , let us assume that there exists a cubic Lie derivation which satisfies the inequality for all , but suppose for some . Then there exists a positive constant such that . For such given , it follows from (9) that there is a positive integer such that . Since and are cubic mappings, we see from the equality and that which leads a contradiction. Hence the mapping is a unique cubic Lie derivation near satisfying approximation (10) on the modular algebra .
As a corollary of Theorem 3, we obtain the following stability result of cubic equation (6) associated with cubic Lie derivation on the Banach algebra , which may be considered as endowed with modular .
Corollary 4. Suppose is a Banach algebra with norm For given real numbers , , and , , suppose that a mapping satisfies for all and , where whenever and that for each the mapping from to is continuous. Then there exists a unique cubic Lie derivation such that for all , where whenever
We observe that if the modular satisfies the -condition, then for nontrivial modular , and for nontrivial convex modular . See [13–16]. In the following theorem, we prove generalized Hyers–Ulam stability of the system and using necessarily -condition, which permits the existence of -Cauchy sequence in
Theorem 5. Let be a -complete convex modular space with -condition. Suppose there exist two functions for which a mapping satisfiesfor all and . If for each the mapping from to is continuous, then there exists a unique cubic Lie derivation satisfying (6) andfor all .
Proof. First, we remark that since and , we lead to , and so Thus, it follows from (12) that for all . Thus, one obtains the following inequality by the convexity of the modular and -condition: for all . Then using the repeated process for any , we prove the following functional inequality:for all . In fact, it is true for . Assume that inequality (31) holds true for . Thus, using the convexity of the modular , we deduce which proves (31) for . Now, replacing by in (31), we have which converges to zero as by assumption (27). Thus, the sequence is -Cauchy for all and so it is -convergent in since the space is -complete. Thus, we may define a mapping as for all .
Claim 1. is a cubic mapping with estimation (28) near . By -condition without using the Fatou property, we can see the following inequality: by taking , which yields approximation (28).
Now, setting in (26) and multiplying the resulting inequality by , we get which tends to zero as for all . Thus, it follows from the first remark that for all , , and all positive integers , where is a fixed real number. Taking the limit as , one obtains , and thus for all . Hence satisfies (6), and so it is cubic.
Claim 2. is a cubic Lie derivation. By the same proof of Theorem 3, the mapping is a cubic homogeneous mapping. From the last inequality in (27) and the last condition in (26), one obtains that for all , from which by taking and so is a cubic Lie derivation.
Claim 3. is unique. To show the uniqueness of , let us assume that there exists a cubic Lie derivation which satisfies the approximation (28). Since and are cubic mappings, we see from the equalities and that which tends to zero as for all . Hence the mapping is a unique cubic Lie derivation satisfying (28).
Remark. In Theorem 5 if is a Banach algebra with norm , and so , , then we see from (26) and (27) that there exists a unique cubic Lie derivation , defined as , , which satisfies (6) and for all .
As a corollary of Theorem 5, we obtain the following stability result of (6), which generalizes stability result on Banach algebras.
Corollary 6. Suppose is a Banach algebra with norm and . For given real numbers , , , and , if a mapping satisfies for all and , then there exists a unique cubic Lie derivation such that for all .
3. Conclusion
We introduce modular algebras with modular over and obtain stability results of a cubic equation associated with cubic derivations on -complete modular algebras, which generalizes stability results on Banach algebras.
Data Availability
Previously reported data were used to support this study and are available at https://journalofinequalitiesandapplications.springeropen.com/articles/10.1186/s13660-017-1422-z and https://www.hindawi.com/journals/jfs/2015/461719/. These prior studies (and datasets) are cited at relevant places within the text as [13–17].
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Acknowledgments
This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2016R1D1A3B03930971).