Research Article | Open Access
Approximate Cubic Lie Derivations on -Complete Convex Modular Algebras
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
In 1940, S. M. Ulam  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  has solved the problem of Ulam for the case of additive mappings in 1941. The result was generalized by T. Aoki  in 1950, by Th.M. Rassias  in 1978, by J. M. Rassias  in 1992, and by P. Gǎvruta  in 1994. Over the past few decades, many mathematicians have investigated the generalized Hyers–Ulam stability theorems of various functional equations [7–12].
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 .
(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  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 , 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  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 . 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)
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).
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 .
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.
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.
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
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).
- S. M. Ulam, Problems in Modern Mathematics, Science Editions, Chapter 6, Wiley, New York, NY, USA, 1960.
- D. H. Hyers, “On the stability of the linear functional equation,” Proceedings of the National Acadamy of Sciences of the United States of America, vol. 27, pp. 222–224, 1941.
- 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.
- Th. M. Rassias, “On the stability of the linear mapping in Banach spaces,” Proceedings of the American Mathematical Society, vol. 72, no. 2, pp. 297–300, 1978.
- J. M. Rassias, “On the stability of the Euler-Lagrange functional equation,” Chinese Journal of Mathematics, vol. 20, no. 2, pp. 185–190, 1992.
- P. Găvruţa, “A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings,” Journal of Mathematical Analysis and Applications, vol. 184, no. 3, pp. 431–436, 1994.
- C. Borelli and G. L. Forti, “On a general HyersUlam stability result,” International Journal of Mathematics and Mathematical Sciences, vol. 18, pp. 229–236, 1995.
- J. Brzdek, D. Popa, I. Rasa, and B. Xu, Ulam Stability of Operators, Mathematical Analysis and its Applications 1, Academic Press, Elsevier, 2018.
- S. Czerwik, “On the stability of the quadratic mapping in normed spaces,” Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, vol. 62, pp. 59–64, 1992.
- S.-M. Jung and S. Min, “Stability of the Wave Equation with a Source,” Journal of Function Spaces, vol. 2018, Article ID 8274159, 4 pages, 2018.
- A. Khan, K. Shah, Y. Li, and T. S. Khan, “Ulam Type Stability for a Coupled System of Boundary Value Problems of Nonlinear Fractional Differential Equations,” Journal of Function Spaces, vol. 2017, Article ID 3046013, 8 pages, 2017.
- F. Skof, “Local properties and approximation of operators,” Rendiconti del Seminario Matematico e Fisico di Milano, vol. 53, pp. 113–129, 1983.
- H.-M. Kim and Y. S. Hong, “Approximate Cauchy-Jensen type mappings in modular spaces,” Far East Journal of Mathematical Sciences, vol. 102, no. 7, pp. 1319–1336, 2017.
- H. Kim and Y. Hong, “Approximate quadratic mappings in modular spaces,” International Journal of Pure and Apllied Mathematics, vol. 116, no. 1, 2017.
- K. Wongkum, P. Chaipunya, and P. Kumam, “On the generalized UlamHyersRassias stability of quadratic mappings in modular spaces without Δ2-conditions,” Journal of Function Spaces, vol. 2015, Article ID 461719, 6 pages, 2015.
- G. Sadeghi, “A fixed point approach to stability of functional equations in modular spaces,” Bulletin of the Malaysian Mathematical Sciences Society. Second Series, vol. 37, no. 2, pp. 333–344, 2014.
- M. A. Khamsi, “Quasicontraction mappings in modular spaces without Δ2-condition,” Fixed Point Theory and Applications, Article ID 916187, 6 pages, 2008.
- C. I. Kim and S. W. Park, “A fixed point approach to stability of additive functional inequalities in fuzzy normed spaces,” Journal of the Chungcheong Mathematical Society, vol. 29, no. 3, pp. 453–464, 2016.
- H. Kim and H. Shin, “Approximation of almost cubic mappings by cubic mappings via modular functional,” Far East Journal of Mathematical Sciences (FJMS), vol. 104, no. 1, pp. 133–148, 2018.
- H.-M. Kim and H.-Y. Shin, “Refined stability of additive and quadratic functional equations in modular spaces,” Journal of Inequalities and Applications, Paper No. 146, 13 pages, 2017.
- J. M. Rassias, “Solution of the Ulam stability problem for cubic mappings,” Glasnik Matematicki Serija III, vol. 36(56), no. 1, pp. 63–72, 2001.
- K. Jun and H. Kim, “The generalized Hyers-Ulam-Rassias stability of a cubic functional equation,” Journal of Mathematical Analysis and Applications, vol. 274, no. 2, pp. 867–878, 2002.
- Ajda Fošner and Maja Fošner, “Approximate Cubic Lie Derivations,” Abstract and Applied Analysis, vol. 2013, Article ID 425784, 5 pages, 2013.
- D. Kang, “On the stability of cubic Lie *-derivations,” Journal of Computational Analysis and Applications, vol. 21, no. 3, pp. 587–596, 2016.
- K.-W. Jun and H.-M. Kim, “On the stability of Euler-Lagrange type cubic mappings in quasi-Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 332, no. 2, pp. 1335–1350, 2007.
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