Abstract
We propose a novel generalized quartic functional equation and investigate its Hyers–Ulam stability in modular spaces using a fixed point technique and the Fatou property in this paper.
1. Introduction
The idea of functional equation stability occurs when a functional equation is replaced by an inequality that acts as a perturbation on the equation. Take into account that the topic of functional equation stability was prompted by a query posed by Ulam in 1940 [1], and Hyers response was published in [2]. By considering an unbounded Cauchy difference, Aoki [3] and Rassias [4] extended Hyers’ theorem for additive mappings and linear mappings, respectively.
In [5–8], the authors examined the Hyers–Ulam-Rassias stability findings for functional equations involving many variables. They discussed the approximate solution of the septic functional equation in [9]. They established that this equation is stable in quasi--Banach spaces and in -Banach spaces. Additionally, they established the instability of the preceding radical functional equation in a pertinent example. Khamsi investigated the notion of quasicontraction mappings in modular function spaces in the absence of the -condition, establishing the presence of fixed points and thoroughly analysing their uniqueness in [10].
Kim and Tamilvanan [11] introduced a novel class of quartic functional equations and examined their Hyers–Ulam stability in fuzzy normed spaces using both the direct and fixed point methods. They demonstrated how sums and products of powers of norms may be used to manage the stability of this quartic functional equation. Wongkum et al. [12] studied the extended Ulam–Hyers–Rassias stability of quadratic functional equations extensively using fixed point theory. Their conclusions are achieved in the context of modular spaces whose modulars are lower semicontinuous but do not meet any -related requirements.
Nakano established the theory of modulars on linear spaces and the accompanying theory of modular linear spaces, which was further refined by Koshi and Shimogaki [13] and Yamamuro [14] and others. Luxemburg [15], Musielak [16], and Turpin [17] and their collaborators contributed the most comprehensive elaboration of these ideas. At the moment, modulars and modular spaces theory is widely employed, most notably in the study of different Orlicz spaces [18] and interpolation theory [19, 20], both of which have several applications [16]. The significance for applications stems from the complex structure of modular function spaces, which, in addition to being Banach spaces (or -spaces in a more generic setting), contain modular equivalents of norm or metric concepts.
Notably, Nakano pioneered modular spaces with relation to order spaces in 1950 [21]. The spaces were produced using the Orlicz spaces theory, which substitutes a particular, integrated nonlinear functionality for the abstract functional that regulates the expansion of space members. They established several features of modular space in [22] and shown that every uniformly limited composition operator mapping this space onto itself must satisfy the so-called Matkowski’s requirements. In 2014, Sadeghi [23] explored the stability of functional equations in modular space using a fixed point technique.
In [24], the existence and Hyers-Ulam stability of the nearly periodic solution to the fractional differential equation with impulse and fractional Brownian motion were studied under nonlocal circumstances. The research was conducted using the semigroups of operators and Mönch fixed point technique, as well as the basic theory of Hyers-Ulam stability. In [25], the authors examined at the existence and Hyers-Ulam stability of random impulsive stochastic functional differential equations with finite delays. They demonstrated that moderate solutions to the equations exist using Krasnoselskii’s fixed point approach. They then examined the Hyers-Ulam stability results under the Lipschitz condition on a bounded and closed interval. Finally, they demonstrated their findings with an example. The reader is recommended to see [26–28] and the references therein for further information on fixed point theory in modular spaces.
Definition 1 (see [29]). Let be a vector space over . A generalized functional is called a modular if for arbitrary , satisfies
(a) if and only if (b) for every scalar with (c), whenever and If we replace (c) by.
(c’) , whenever and , then, the modular is called convex. A modular defines a corresponding modular space, i.e., the vector space given by:
Definition 2 (see [29]). If is a modular space and the sequence in , then (i) if as (ii) is known as -Cauchy if as (iii)A subset is known as -complete iff every -Cauchy sequence is -convergent in
Definition 3 (see [29]). Let be a modular space and a non-empty subset . The mapping is referred to as a quasicontraction, if there is satisfies for any .
Definition 4 (see [29]). Let be a modular space, a nonempty subset , and a function , the orbit around a point is the quantity is then related to and is referred to as the orbital diameter of at . If , in particular, one says that has an orbit of that is limited to .
Fatou property: the -modular will have the Fatou property iff whenever . A modular function is stated to fulfil the conditions if there is which satisfies , for every .
In this work, we introduce new generalized quartic functional equation where , and investigate Hyers-Ulam stability of this quartic functional equation in modular space by using the fixed point method with the help of Fatou property.
2. Solution
We denote and as two real vector spaces.
Theorem 5. If an even mapping satisfies the functional equation (5) for all , then, the function is quartic.
Proof. In the view of evenness, we obtain for all in . Now, setting in equation (5), we have . Replacing by in equation (5), we get for all . Replacing by in equation (6), we have for all . Again, replacing by in equation (7), we obtain for all . For any nonnegative integer , we can generalize the result that for all . Similarly, we have for all . Next, replacing by , we obtain for all . Hence, the function is quartic.
3. Hyers-Ulam Stability
We assume that is a convex modular on -complete modular spaces with the Fatou property that meets the -condition with in this section. Let be a linear space as well. For convenience, we call a function by the shorthand : for all .
Theorem 6. Let be a function such that for all , with . If an even mapping with and such that for all , then, there exists a unique quartic mapping satisfying for all .
Proof. We consider the set
and define the function on as follows:
Now, we show that is a convex modular on . It is easy to verify that satisfies the axioms (a) and (b) of a modular. Next, we will show that is convex, and hence, is satisfied. Let be given, then, there exist real constants and such that
Also
for all . If and , then, we get
so we get
This concludes that is convex modular on . Now, we show that is -complete.
Let is a -Cauchy sequence in and let . Then, there exists a positive integer such that
for all . Then
for all and all . Therefore, is a -Cauchy sequence in . Since is -complete, so is convergent in , for each . Hence, we can define a function by
for all . Since satisfies the Fatou property, it follows from (25) that
so
for all . Thus, is -converges. Hence, is -complete.
Next, we show that satisfies the Fatou property. Suppose that is a sequence in which is -convergent to an element .
Let be given. For each , let be a real constant such that
So
for all . Since satisfies the Fatou property, we get
Thus, we obtain
Hence, satisfies the Fatou property. Consider the function by
for all and all . Let and let be an arbitrary constant with . From the definition of , we obtain
for all . By inequality (15) and the above inequality, we get
for all . Hence,
i.e., is a -contraction. Next, we show that has a bounded orbit at . Replacing by in (16), we get
for all . Replacing with in (37), we get
for all . By using (37) and (38), we get
for all . By induction, we can easily see that
for all . It follows from inequality (40) that
for all and all . By the definition of , we conclude that
which implies the boundedness of an orbit of at . It follows from Theorem 1.5 [29] that the sequence -converges to . Now, by the -contractivity of , we have
Passing to the limit and applying the Fatou property of , we obtain that
Therefore, is a fixed point of . Replacing by in (16), we get
for all . Therefore
Employing the limit , we get
for all . It follows from Theorem 2 that is quartic. By using (40), we get (17).
To prove the uniqueness of , let be another quartic mapping satisfying (17). Then, is a fixed point of .
which implies that . This proves that . Therefore, the function is unique. This completes the proof.
Corollary 7. Let a mapping such that for all with . Suppose that a mapping with and such that for all , then, there exists a unique quartic mapping satisfying for all .
Proof. It is known that every normed space is modular space with the modular and satisfies the -condition with .
Remark 8. If we replace by and letting in Corollary 7, we obtain the stability results for the sum of norms that for all , where and are constants with .
Remark 9. If we replace by and letting in Corollary 7, we obtain the stability results for the sum of product of norms that for all , where and are constants with .
Theorem 10. Let be a function such that for all with . Suppose that with and satisfies (16), then, there exists a unique quartic mapping satisfying for all .
Proof. We consider the set
and define the function on as follows:
Similar to the proof of Theorem 6, we have
(1) is a convex modular on (2) is -complete(3) satisfies the Fatou propertyNow, we consider the function defined by
for all and all . Let and let be an arbitrary constant with . From the definition of , we have
for all . By the assumption and the last inequality, we get
Hence,
i.e., is a -contraction.
Next, we prove then that has a bounded orbit at . Replacing by in (16), we get
for all . Replacing with in (64), we get
for all . Replacing with in (65), we get
for all . By using (64), (65), and (66), we get
for all . By induction, we can easily see that
for all . It follows from inequality (68) that
for all and all . By the definition of , we conclude that
which implies the boundedness of an orbit of at . It follows from Theorem 1.5 [29] that the sequence -converges to .
Now, by the -contractivity of , we have
Employing the limit and applying the Fatou property of , we obtain that
Therefore, is a fixed point of . Replacing by in (16), we get
for all . Therefore
Passing to the limit , we get
for all . It follows from Theorem 2 that is quartic. By using (68), we get (57).
In order to prove the uniqueness of , consider another quartic solution that satisfy the inequality (17). Then, is a fixed point of .
which implies that or . Hence, the proof is now completed.
Corollary 11. Let a mapping such that for all , with . Suppose that with and satisfies (51), then there exists a unique quartic mapping satisfying for all .
Proof. It is known that every normed space is modular space with the modular and satisfies the -condition with .
Remark 12. If we replace by and letting in Corollary 11, we obtain the stability results for the sum of norms that for all , where and are constants with .
Remark 13. If we replace by and letting in Corollary 11, we obtain the stability results for the sum of product of norms that for all , where and are constants with .
4. Counterexample
We present a counterexample to show instability of a particular condition of the equality (5) using modified example of Gajda [7].
Remark 14. If a function satisfies the functional equation (5), then, the following assertions hold: (1) and (2) if the function is continuous
Example 15. Let a mapping be defined as follows: where then, the mapping satisfies for all , but a quartic mapping does not exist satisfies for all , where and are a constant.
Proof. It is easy to show that is bounded by on . If or 0, then
Thus, (84) is valid. Next, suppose that
then, there exists an integer satisfies
So that and
Also, for ,
Next, by inequality (88), we obtain that
It follows from (88) that
Thus, the function satisfies the inequality (84). Assume on a contrary that there exist a quartic solution satisfying (85). For every in , since is continuous and bounded, is limited to an open interval of origin and continuous origin.
In the view of Remark 14, must be . So we obtain
Suppose, we can choose with . If , then, for all , we obtain
which contradicts.
Data Availability
No data were used to support this study.
Additional Points
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Conflicts of Interest
The authors declare that they have no competing interests.
Authors’ Contributions
The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.
Acknowledgments
The authors like to express their gratitude to referees for their suggestions. The authors are thankful to the Deanship of Scientific Research at Prince Sattam bin Abdulaziz University, Al-Kharj, Kingdom of Saudi Arabia, for supporting this research.