Characterization and Stability of Multimixed Additive-Quartic Mappings: A Fixed Point Application

Bodaghi, Abasalt; Alias, Idham Arif; Mousavi, Lida; Hosseini, Sedigheh

doi:https://doi.org/10.1155/2021/9943199

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Volume 2021 | Article ID 9943199 | https://doi.org/10.1155/2021/9943199

Characterization and Stability of Multimixed Additive-Quartic Mappings: A Fixed Point Application

Abasalt Bodaghi,¹Idham Arif Alias,²Lida Mousavi,³and Sedigheh Hosseini⁴

Academic Editor: Zoran Mitrovic

Received25 Mar 2021

Accepted01 Nov 2021

Published12 Nov 2021

Abstract

In this article, we introduce the multi-additive-quartic and the multimixed additive-quartic mappings. We also describe and characterize the structure of such mappings. In other words, we unify the system of functional equations defining a multi-additive-quartic or a multimixed additive-quartic mapping to a single equation. We also show that under what conditions, a multimixed additive-quartic mapping can be multiadditive, multiquartic, and multi-additive-quartic. Moreover, by using a fixed point technique, we prove the Hyers-Ulam stability of multimixed additive-quartic functional equations thus generalizing some known results.

1. Introduction

Let be a commutative group, be a linear space over rational numbers, and be an integer with . A mapping is called (i)Multiadditive if it satisfies the Cauchy’s functional equation in each variable [1](ii)Multiquadratic if it fulfills quadratic functional equation in each variable [2, 3](iii)Multicubic if it satisfies the cubic equation in each variable [4, 5](iv)Multiquartic if it satisfies the quartic equationin each variable [6, 7].

We have the following observations about a several variables mapping . (i) is multiadditive [1] if and only if it satisfies(ii) is multiquadratic [8] if and only if it satisfies

where with . More information about the structure of multiadditive and multiquadratic mappings, we refer for instance to [9, 10].

Bodaghi et al. [4] (resp., [6]) provided a characterization of multicubic (resp., multiquartic) mappings, and they showed that every multicubic (resp., multiquartic) mapping can be shown a single functional equation and vice versa.

Lee et al. [11] introduced and obtained the general solution of the quartic functional equation which somewhat different from (1) as follows:

For the generalized forms of the quartic functional, equations (1) and (4) refer to [12, 13]. Recently, in [14] and motivated by (4), a new form of multiquartic mappings was introduced, and the structure of such mappings was described.

Speaking of the stability of a functional equation, we follow the question raised in 1940 by Ulam [15] for group homomorphisms. Hyers [16] presented a partial solution to the problem of Ulam. Later, Hyers’ theorem was extended and generalized in various forms by many mathematicians such as Aoki [17] and Rassias [18]. Recall that a functional equation is said to be stable if any mapping fulfilling approximately; then, it is near to an exact solution of . Next, several stability problems of various functional equations and mappings have been investigated by many mathematicians which can be found in literatures.

In the last two decades, the stability problem for several variable mappings such as multiadditive, multi-Jensen, multiquadratic, multicubic, and multiquartic mappings by applying direct and fixed point methods has been studied by a number of authors which are available for example in [1, 2, 4, 8, 9, 19–26].

In [27], Eshaghi Gordji introduced and obtained the general solution of the following mixed type additive and quartic functional equation

He also established the Hyers-Ulam Rassias stability of the above functional equation in real normed spaces. The stability of (5) in non-Archimedean orthogonality spaces is studied in [28]. A different and equivalent form of mixed type additive and quartic functional equation from (5) was introduced by the first author in [29] as follows:

It is easily verified that the function is a solution of equations (5) and (6); the generalized version of equation (6) can be found in [30].

This paper is organized as follows: In the second section, we firstly define multi-additive-quartic mappings and include a characterization of such mappings. In fact, we prove that every multi-additive-quartic mapping can be shown a single functional equation and vice versa (under some extra conditions). Section 3 is devoted to the study of stricture of multimixed additive-quartic mappings. In other words, motivated by equation (6), we introduce the multimixed additive-quartic mappings and reduce the system of equations defining the multimixed additive-quartic mappings to a single equation, namely, the multimixed additive-quartic functional equation. In Section 4, we prove the Hyers-Ulam stability for the multi-additive-quartic and the multimixed additive-quartic mappings in the setting of Banach spaces by applying a fixed point method [31]. As an application of this result, we establish the stability of multi-additive-quartic mappings. Finally, we show that under some mild conditions every multiadditive and multiquartic functional equations are -stable for a small positive number .

2. Characterization of Multi-Additive-Quartic Mappings

Throughout this paper, and stand for the set of all positive integers and the rational numbers, respectively, . For any , , and , we write and , where stands, as usual, for the th power of an element of the commutative group .

Let and be linear spaces, and . A mapping is called -additive and -quartic (briefly, multi-additive-quartic) if is additive in each of some variables and satisfies (4) in each of the other variables. In what follows, for simplicity, it is assume that is additive in each of the first variables. Moreover, for (), the above definition leads to the so-called multiadditive (multiquartic) mappings.

In the sequel, we assume that and are vector spaces over . Moreover, we identify with , where and . Let with and , where . Throughout, we shall denote by if there is no risk of mistake. Put also and . For and with and , set , where . Consider the subset of as follows:

To achieve our aims, for the multi-additive-quartic mappings, we use the oncoming notations:

For each , we consider the equation

for all and where .

It is shown in Proposition 2.2 in [14] that if a mapping is multiquartic, then it satisfies the equation

The next proposition shows that the system of equations defining a multi-additive-quartic mapping can be reduced to (10).

Proposition 1. Let and . Suppose that a mapping is -additive and -quartic (multi-additive-quartic) mapping. Then, fulfills equation (10).

Proof. For , the result follows from Proposition 2.2 in [14] and Theorem 2 in [1], and so we prove the assertion for the case that . For any , consider the mapping defined by for . The assumption shows that is -additive, and thus, we can obtain from Theorem 2 in [1] that The above equality implies that for all and . Repeat the above method, and for any , define the mapping via . This mapping is -quartic, and hence, by Proposition 2.2 from [14], we have for all . On the other hand, by the definition of , relation (14) converts to for all and . It now follows between (13) and (15) that for all and . This finishes the proof.

By Proposition 6, it is easily verified that the mapping satisfies (10), and so this equation is said to be multi-additive-quartic functional equation.

Definition 2. Let . Consider a mapping . We say (i)Satisfies (has) the -power condition in the th variable iffor all . Sometimes -power condition is called quartic condition. (ii)Has zero condition if for any with at least one component which is equal to zero

We remember that the binomial coefficient for all with is defined and denoted by .

We wish to show that if a mapping satisfies equation (10), then it is multi-additive-quartic. For doing it, we need the upcoming lemma. The method of the proof of Lemma 3 is similar to the proof of ([14], Lemma 2.5) and so we include lemma without the proof.

Lemma 3. Suppose that a mapping satisfies equation (10). Under one of the following assumptions, satisfying zero condition. (i) satisfies the quartic condition in the last variables(ii) is even in the last variables

Theorem 4. Suppose that a mapping fulfilling equation (10). Under one of the hypothesis of Lemma 3, is multi-additive-quartic.

Proof. It follows from Lemma 3; satisfies zero condition. Putting in the left side of (10) and applying the hypothesis, we obtain On the other hand, by using Lemma 3, the right side of (10) converts to Now, relations (18) and (19) necessitate that for all and . In light of Theorem 2 in [1], we see that is additive in each of the first variables. In addition, by considering in (10) and applying again Lemma 3, we have for all and , and thus, by Theorem 2.6 in [14], is quartic in each of the last variables. The proof of second part is similar.

3. Characterization of Multimixed Additive-Quartic Mappings

In this section, we introduce the multimixed additive-quartic mappings and then characterize them as an equation. We start this section with the definition of such mappings.

Definition 5. Let and be vector spaces over , . A mapping is called -multimixed additive-quartic or briefly multimixed additive-quartic if satisfies mixed additive-quartic equation (6) in each variable.

Let with and , where . For and with , put

Consider the subset of as follows:

Hereafter, for the multimixed additive-quartic mappings, we use the following notations:

Next, we reduce the system of equations defining the multimixed additive-quartic mapping to obtain a single functional equation.

Proposition 6. If a mapping is multimixed additive-quartic, it satisfies the equation where and are defined in (24) and (8), respectively.

Proof. The proof is based on induction for . For , it is obvious that satisfies (6). Assume that (26) holds for some positive integer . Then The assertion is now proved.

Since the mapping is multimixed additive-quartic, it satisfies (26) by proposition above, and so this equation is called multimixed additive-quartic functional equation.

Here, we bring an elementary lemma without proof.

Lemma 7. Let , such that , where . Then

Similar to Lemma 2.1 from [6], we need the following lemma in obtaining our goal in this section. The proof is similar, but we include some parts for the sake of completeness.

Lemma 8. If a mapping satisfies equation (26), then it has zero condition.

Proof. Putting in (26), we have Using Lemma 7 for and , the right side of (29) will be as follows: On the other hand, by a simple computation, the left side of (29) is It follows from relations (29), (30), and (31) that . One can continue this method to show that has zero condition.

Definition 9. A mapping is (iii)Odd in the th variable if(iv)Even in the th variable if

Proposition 10. Suppose that a mapping satisfies equation (26). Then, it is multimixed additive-quartic. Moreover, (i)If is odd in a variable, then it is additive in the same variable(ii)If is even in a variable, then it is quartic in the same variable

Proof. Let be arbitrary and fixed. Set Putting for all in (26) and using Lemma 8, we get The above equalities show that In other words, (6) is true for . Since is arbitrary, is a multimixed additive-quartic mapping. (i)Repeating the proof of Lemma 2.1 (i) from [29] for , we see that . This means that is additive in the th variable(ii)Similar to the previous part, it follows from the proof of part (ii) of Lemma 2.1 in [29] thatTherefore, is quartic in the th variable.

Corollary 11. Suppose a mapping satisfies equation (26). (i)If is odd in each variable, then it is multiadditive. Moreover, it satisfies (2)(ii)If is even in each variable, then it is multiquartic. In particular, it fulfills (11)(iii)If is odd in each of some variables and is even in each of the other variables, then it is multi-additive-quartic. In addition, (10) is valid for

4. Various Stability Results

In this section, we prove some Hyers-Ulam stability results by a fixed point method in the setting of Banach spaces. In what follows, we denote the set of all mappings from to by . We remember the following theorem which is an essential result in fixed point theory ([23], Theorem 1). This achievement is a key tool in obtaining our aim in this section.

Theorem 12. Let the hypotheses
(A1) is a Banach space, is a nonempty set, , , and
(A2) is an operator satisfying the inequality (A3) is an operator defined through hold, and a function and a mapping fulfill the following two conditions: Then, there exists a unique fixed point of such that Moreover, for all .

For the rest of this paper and for each mapping , we consider the difference operator defined via

where and are defined in (24) and (8), respectively. In the sequel, all mappings are assumed that satisfy zero condition. With this assumption, we have the next stability result for functional equation (26) in the odd case.

Theorem 13. Let be fixed, be a linear space, and be a Banach space. Suppose that is a mapping satisfying for all and for all . Assume also is a mapping fulfilling the inequality for all . If is odd in each variable, then there exists a unique multiadditive mapping such that for all .

Proof. Replacing by in (45) and using the assumptions, we have for all (here and the rest of the proof) and so Set Then, relation (48) can be modified as Define for all . It is seen that has the form (A3) of Theorem 12 for which , , and . Furthermore, for each , we get The above relation portrays that the hypothesis (A2) holds. By induction on , one can check that for any , we have Now, relations (44) and (52) necessitate that all assumptions of Theorem 12 are satisfied. Hence, there exists a unique mapping such that and (46) holds. In continuation, we prove that for all and . We argue by induction on . Clearly, inequality (54) is valid for by (45). Assume that (54) is true for an . Then for all . Letting in (54) and applying (43), we arrive at for all . This means that the mapping satisfies (26), and so it is multiadditive by Corollary 11. This finishes the proof.

Here, in analogy with Theorem 13, we bring the next stability result for functional equation (26) in the even case.

Theorem 14. Let be fixed, be a linear space, and be a Banach space. Suppose that is a mapping satisfying for all and for all . Assume also is a mapping fulfilling the inequality for all . If is even in each variable, then there exists a unique solution of (26) such that for all . In particular, if is even mapping in each variable, then it is multiquartic.

Proof. Replacing by in (58) and applying the hypotheses, we obtain where (here and the rest of the proof). On the other hand By the relations above (60) will be and so One can rewrite (63) as where Similar to the proof of Theorem 13, consider for all , and hence, satisfies (A3) of Theorem 12 with , , and . Moreover, for each , we obtain The last relation implies that the hypothesis (A2) is true. It is easily checked by induction on that for any and It now follows between (57) and (67) that all assumptions of Theorem 12 hold, and thus, there exists a unique mapping such that and (46) is valid. Similar to the proof of Theorem 13, one can show that for all and . Letting in (69) and applying (56), we arrive at for all , and therefore, the mapping satisfies (26). The last part follows from part (ii) of Corollary 11.

Here and subsequently, it is assumed that is a normed space and is a Banach space unless otherwise stated explicitly. In the following corollary, we show that the multiadditive and multiquartic mappings are stable. Since the proof is routine, we include it without proof.

Corollary 15. Given . Suppose that is a mapping satisfying the inequality for all . (i)If and is odd in each variable, then there exists a unique multiadditive mapping such that(ii)If and is even in each variable, then there exists a unique solution of (26) such thatfor all . Moreover, if is even mapping in each variable, then it is multiquartic.

The upcoming corollaries are direct consequences of Theorems 13 and 14 when the functional equation (26) is controlled by a small positive number .

Corollary 16. Let and be a mapping satisfying the inequality for all . (i)If is odd in each variable, then there exists a unique multiadditive mapping such thatfor all (ii)If is even in each variable, then there exists a unique solution of (26) such thatfor all . Furthemore, if is even mapping in each variable, then it is multiquartic.

Proof. Letting the constant function for all and using Theorem 13 and Theorem 14 in the case , one can obtain the desired result.

Given the mapping , we define the operator through for all and where and are defined in (8).

In the next result, we show that the functional equation (10) can be stable.

Theorem 17. Let be fixed, be a linear space, and be a Banach space. Suppose that is a mapping satisfying the inequality for all . Assume also is a mapping satisfying the inequality for all . Then, there exists a unique solution of (10) such that for all , where

Proof. Putting and in (79), we have in which . A computation shows that (81) can be rewritten as follows: and so Set and where . Then, relation (83) can be modified as Define for all . The rest of the proof is similar to the proof of Theorem 13.

Corollary 18. Let . If is a mapping satisfying the inequality for all , then there exists a unique solution of (10) such that for all .

Proof. Setting the constant function for all and applying Theorem 17 in the case , the result can be found.

5. Conclusion

In the present paper, we introduced the multi-additive-quartic and multimixed additive-quartic mappings. Indeed, we characterized the mentioned mappings and then unified the system of functional equations defining a multi-additive-quartic or a multimixed additive-quartic mapping to a single equation. We also showed that under which conditions a multimixed additive-quartic mapping is multiadditive, multiquartic, and multi-additive-quartic. Finally, we applied a fixed point theorem to establish the Hyers-Ulam stability of multi-additive-quartic mappings and multimixed additive-quartic functional equations.

Data Availability

All results are obtained without any software and found by manual computations.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors conceived of the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the final manuscript.

Acknowledgments

The present research was fully supported by the Journal Publication Fund of Universiti Putra Malaysia, Serdang, Selangor, Malaysia.

References

K. Ciepliński, “Generalized stability of multi-additive mappings,” Applied Mathematics Letters, vol. 23, no. 10, pp. 1291–1294, 2010.
View at: Publisher Site | Google Scholar
K. Ciepliński, “On the generalized Hyers-Ulam stability of multi-quadratic mappings,” Computers & Mathematcs with Applications, vol. 62, no. 9, pp. 3418–3426, 2011.
View at: Publisher Site | Google Scholar
F. Skof, “Proprieta locali e approssimazione di operatori,” Rendiconti del Seminario Matematico e Fisico di Milano, vol. 53, no. 1, pp. 113–129, 1983.
View at: Publisher Site | Google Scholar
A. Bodaghi and B. Shojaee, “On an equation characterizing multi-cubic mappings and its stability and hyperstability,” Fixed Point Theory, vol. 22, no. 1, pp. 83–92, 2021.
View at: Publisher Site | Google Scholar
K. W. Jun and H. M. 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.
View at: Publisher Site | Google Scholar
A. Bodaghi, C. Park, and O. T. Mewomo, “Multiquartic functional equations,” Advances in Difference Equations, vol. 2019, no. 1, 2019.
View at: Publisher Site | Google Scholar
J. M. Rassias, “Solution of the Ulam stability problem for quartic mappings,” Glasnik Matematicki Series III, vol. 34, no. 2, pp. 243–252, 1999.
View at: Google Scholar
X. Zhao, X. Yang, and C.-T. Pang, “Solution and stability of the multiquadratic functional equation,” Abstract and Applied Analysis, vol. 2013, Article ID 415053, 2013.
View at: Publisher Site | Google Scholar
A. Bodaghi, “Functional inequalities for generalized multi-quadratic mappings,” Journal of Inequalities and Applications, vol. 2021, no. 1, 2021.
View at: Publisher Site | Google Scholar
M. Kuczma, “An introduction to the theory of functional equations and inequalities,” in Cauchy’s Equation and Jensen’s Inequality, Birkhauser, Verlag, Basel, 2009.
View at: Publisher Site | Google Scholar
S. Lee, S. Im, and I. Hwang, “Quartic functional equations,” Journal of Mathematical Analysis and Applications, vol. 307, no. 2, pp. 387–394, 2005.
View at: Publisher Site | Google Scholar
A. Bodaghi, “Intuitionistic fuzzy stability of the generalized forms of cubic and quartic functional equations,” Journal of Intelligent & Fuzzy Systems, vol. 30, no. 4, pp. 2309–2317, 2016.
View at: Publisher Site | Google Scholar
D. Kang, “On the stability of generalized quartic mappings in quasi-β-normed spaces,” Journal of Inequalities and Applications, vol. 2010, Article ID 198098, 2010.
View at: Publisher Site | Google Scholar
Z. Abbasbeygi, A. Bodaghi, and A. Gharibkhajeh, “On an equation characterizing multi-quartic mappings and its stability,” International Journal of Nonlinear Analysis and Applications, vol. 13, no. 1, pp. 991–1002, 2022.
View at: Google Scholar
S. M. Ulam, Problems in Modern Mathematics, Chapter VI, Science Ed., Wiley, New York, 1940.
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, no. 4, pp. 222–224, 1941.
View at: Publisher Site | Google Scholar
T. Aoki, “On the stability of the linear transformation in Banach spaces,” Journal of the Mathematical Society of Japan, vol. 2, no. 1-2, pp. 64–66, 1950.
View at: Publisher Site | Google Scholar
T. 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.
View at: Publisher Site | Google Scholar
A. Bahyrycz, K. Cieplinski, and J. Olko, “On an equation characterizing multi-Cauchy-Jensen mappings and its Hyers-Ulam stability,” Acta Mathematica Scientia. Series B. English Edition, vol. 35, no. 6, pp. 1349–1358, 2015.
View at: Publisher Site | Google Scholar
A. Bahyrycz, K. Ciepliński, and J. Olko, “On an equation characterizing multi-additive-quadratic mappings and its Hyers- Ulam stability,” Applied Mathematics and Computation, vol. 265, pp. 448–455, 2015.
View at: Publisher Site | Google Scholar
A. Bahyrycz and J. Olko, “On stability and hyperstability of an equation characterizing multi-Cauchy-Jensen mappings,” Results in Mathematics, vol. 73, no. 2, 2018.
View at: Publisher Site | Google Scholar
A. Bodaghi, “Approximation on the multi--Jensen-quadratic mappings and a fixed point approach,” Mathematica Slovaca, vol. 71, no. 1, pp. 117–128, 2021.
View at: Publisher Site | Google Scholar
J. Brzdȩk, J. Chudziak, and Z. Páles, “A fixed point approach to stability of functional equations,” Nonlinear Anal., vol. 74, no. 17, pp. 6728–6732, 2011.
View at: Publisher Site | Google Scholar
M. B. Ghaemi, M. Majani, and M. E. Gordji, “General system of cubic functional equations in non-Archimedean spaces,” Tamsui Oxford Journal of Information and Mathematical Sciences, vol. 28, no. 4, pp. 407–423, 2012.
View at: Google Scholar
C.-G. Park, “Multi-quadratic mappings in Banach spaces,” Proceedings of the American Mathematical Society, vol. 131, no. 8, pp. 2501–2504, 2003.
View at: Publisher Site | Google Scholar
T. Z. Xu, “Stability of multi-Jensen mappings in non-Archimedean normed spaces,” Journal of Mathematical Physics, vol. 53, no. 2, article 023507, 2012.
View at: Publisher Site | Google Scholar
M. Eshaghi Gordji, “Stability of a functional equation deriving from quartic and additive functions,” Bulletin of the Korean Mathematical Society, vol. 47, no. 3, pp. 491–502, 2010.
View at: Publisher Site | Google Scholar
H. Lee, S. W. Kim, B. J. Son, D. H. Lee, and S. Y. Kang, “Additive-quartic functional equation in non-Archimedean orthogonality spaces,” Korean Journal of Mathematics, vol. 20, no. 1, pp. 33–46, 2012.
View at: Publisher Site | Google Scholar
A. Bodaghi, “Stability of a mixed type additive and quartic functional equation,” Filomat, vol. 28, no. 8, pp. 1629–1640, 2014.
View at: Publisher Site | Google Scholar
A. Bodaghi, “Approximate mixed type additive and quartic functional equation,” Boletim da Sociedade Paranaense de Matemática, vol. 35, no. 1, pp. 43–56, 2017.
View at: Publisher Site | Google Scholar
J. Brzdek, “Stability of the equation of the p-Wright affine functions,” Aequationes mathematicae, vol. 85, no. 3, pp. 497–503, 2013.
View at: Publisher Site | Google Scholar

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Copyright © 2021 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.

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