Abstract
In this research paper, the authors present a new mixed Euler-Lagrange -cubic-quartic functional equation. For this introduced mixed type functional equation, the authors obtain general solution and investigate the various stabilities related to the Ulam problem in Felbin’s type of fuzzy normed linear space (f-NLS) with suitable counterexamples. This approach leads us to approximate the Euler-Lagrange -cubic-quartic functional equation with better estimation.
1. Introduction
One of the famous questions concerning the stability of homomorphisms was raised by Ulam [1] in 1940. The author Hyers [2] provided a partial answer to Ulam’s question in 1941, and then, a generalized solution to Ulam’s question was given by Rassias [3] in 1978, which is called Hyers-Ulam-Rassias stability or generalized Hyers-Ulam stability. The generalization of Hyers stability result by Rassias [4] is called Ulam-Gavruta-Rassias stability. Later, Ravi et al. [5] investigated the stability using mixed powers of norms which is called Rassias stability.
Definition 1 (see [6]). A fuzzy subset on is said to be a fuzzy real number when it satisfies two axioms:
(N1)There exists such that
(N2)For each , where
Note that is -level set. We show the set of all fuzzy real numbers by . Also, is said to be a nonnegative fuzzy real number when and for . We show the set of all nonnegative fuzzy real numbers by .
We define as
Definition 2 (see [6]). We define onas(i), (ii), (iii), (iv), are additive and multiplicative identities, respectively. We also define as ; so, .
Definition 3 (see [6]). For , the notation shows fuzzy scalar multiplication and defied as and.
Definition 4 (see [7]). Consider the vector space and the left and right norms which are symmetric and nondecreasing functions satisfying , . So, is said to be a fuzzy norm and is a fuzzy normed linear space (in short f-NLS) if
(N1) if and only if
(N2) for all and
(N3)for all :
(N3L)if and , then
(N3R)if and , then ,where for and .
Lemma 5 (see [8]). Consider f-NLS , and let
(R1)
(R2) in which for every
(R3)
So, . The converse is not true.
Lemma 6 (see [8]). Consider f-NLS . Then, (1)if , then for all , for all (2) implies that, for every , there exists such that for every (3), implies that for every , there exists such that for every
Lemma 7. Consider f-NLS and let
(L1)
(L2) such that for all
(L3)
So, , but not conversely.
Lemma 8. Consider f-NLS , then (1), implying that , for every (2) implies that for every , there exists such that for every (3), implying that for every , there exists such that for every
Lemma 9 (see [7]). Consider f-NLS . Then, (1) and for all , implying that for all , then (N3R)(2) and for all , implying that for all , so (N3L)
Definition 10 (see [7]). Consider f-NLS and let. A sequence converges to , denoted by , if for every , and is called a Cauchy sequence if for every . A subset is said to be complete if every Cauchy sequence in converges in . A f-NLS is called a fuzzy Banach space (f-BS) if it is complete.
Lemma 11. Consider f-NLS which satisfies . Then,
(1) is continuous from into at for every (2)For every and , we have Recently, the stability problems of several functional equations (FEs) have been extensively investigated by a number of authors [4, 9–20] in Felbin type f-NLS. Our method helps to solve some new problems of stability and approximation of functional equations [21–28] in Felbin type f-NLS.
Motivated from the above historical developments in the field of FEs, the authors introduce a new mixed Euler-Lagrange -cubic-quartic functional equation (FE)
where . For this mixed type FE, authors obtain the general solution and investigate the various stabilities related to Ulam problem [1] in Felbin’s type f-NLS with suitable counterexamples.
2. General Solution of Euler-Lagrange -Cubic-Quartic FE
Theorem 12. Consider satisfies (2) and odd that is , then a mapping is cubic.
Proof. Assume satisfies (2). Putting in (2), we get . Setting by in (2), we obtain for all , and by assuming in (3) which leads Thus, is cubic.
Theorem 13. If satisfies (2) and even that is , then a mapping is quartic.
Proof. Assume holds (2). Putting in (2), we get . Setting by in (2), we arrive Allowing in (5), we arrive . Using and in (5), we get for all . Thus, is quartic.
3. Generalized Hyers-Ulam-Rassias Stability of a Euler-Lagrange -Cubic-Quartic FE
Consider the following abbreviation and the integer .
Theorem 14. Consider the odd mapping for which we can find for a linear space and a fuzzy Banach space (f-BS) where So, we can find a unique cubic function such that for all , where
Proof. Putting in (9) implies that Multiply both sides of equation (12) by , so we get Again multiplying (13) by and replacing by , we obtain and it leads to with , nonnegative integers. Now, (8) and (15) imply that the sequence is fuzzy Cauchy in . So, the sequence converges, which let us to define the mapping by Considering and allowing in (15), we obtain and it gives (10). Using (8) and (9), we have which implies that is cubic. Suppose that is a cubic mapping satisfying (10) and implies , which shows the uniqueness of .
Theorem 15. Consider and let there exist a function such that for a linear space and a fuzzy Banach space (f-BS) . So, we can find a unique cubic mapping , such that where The following corollary gives the Hyers-Ulam, Hyers-Ulam-Rassias, and Rassias stabilities of (2).
Corollary 16. Consider and let there be real numbers and such that then there is a unique cubic mapping such that In the next example, we consider the unstability of FE (2) for in Corollary 16.
Example 17. Define the mapping as in which is a fuzzy real number. Define as So, As a result, there does not exist a cubic mapping and a constant such that
Proof. The below inequality
showing the boundedness of . Now, we show that satisfies (27).
Let , then (27) is trivial. If , then the left-hand side of (27) is less than . If . So, we can find a positive integer such that
so that
and therefore, for each , we have
for . From (26) and (30), we have
Thus, satisfies (27) for all with . Corollary 16 shows for any in , and so,
But we can choose a positive integer with . If , then for all . For this , we have
which contradicts (34). Therefore, the functional equation (2) is not stable in the sense of Ulam, Hyers, and Rassias if .
Theorem 18. Consider the even mapping for which we can find such that and all . So, we can find a unique quartic mapping and , such that where
Proof. Putting in (37), we get Multiply (40) by , we obtain Replacing by and multiplying (41) by , we obtain Therefore, for all , there is such that with . From (36) and (43) and because is a f-BS, we have the sequence which is a fuzzy Cauchy in and converges . Now, we define by Assuming and allowing the limit as in (43), we have Therefore, we obtain (38). From (36) and (37), we have and hence, the mapping is quartic. Letting be a quartic mapping fulfills (38), and we have for all , and hence, is unique.
Theorem 19. Consider for which we can find a mapping such that and all . So, we can find a unique quartic mapping and , such that where
Corollary 20. Consider and let there be real numbers and such that so we can find a unique quartic mapping satisfying In the next example, we show that the FE (2) is not stable for in Corollary 20.
Example 21. Letting be a mapping defined by where is a fuzzy real number and is defined by a linear space and a fuzzy Banach space (f-BS) . Then fulfills the functional inequality So, we cannot find a quartic mapping and a constant such that
4. Conclusion
In our work, we have obtained the general solution of a new generalized mixed Euler-Lagrange -cubic-quartic functional equation and studied its generalized Hyers-Ulam-Rassias, Hyers-Ulam, Hyers-Ulam-Rassias, and Rassias stabilities in fuzzy normed linear space using Felbin’s concept. Moreover, some counterexamples show both stability and unstability of FE (2) in f-BS.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no competing interests.
Authors’ Contributions
All authors conceived 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 second author was funded by the Tamil Nadu State Council for Higher Education (TANCHE), Chennai, 600 005, Tamil Nadu, India, through the Minor Research Project 2017-2018 (Grant No. D.O.Rc.No.744/2017A dated 26.12.2017). The authors are grateful to the Basque Government for the support of this work through Grant IT1207-19.