Characterization and Stability of Multi-Euler-Lagrange Quadratic Functional Equations

Bodaghi, Abasalt; Moshtagh, Hossein; Mousivand, Amir

doi:https://doi.org/10.1155/2022/3021457

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Unique and Non-Unique Fixed Points and their Applications 2022

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Volume 2022 | Article ID 3021457 | https://doi.org/10.1155/2022/3021457

Characterization and Stability of Multi-Euler-Lagrange Quadratic Functional Equations

Abasalt Bodaghi,¹Hossein Moshtagh,²and Amir Mousivand³

Academic Editor: Cristian Chifu

Received22 Mar 2022

Revised22 Aug 2022

Accepted22 Sept 2022

Published10 Oct 2022

Abstract

The aim of the current article is to characterize and to prove the stability of multi-Euler-Lagrange quadratic mappings. In other words, it reduces a system of equations defining the multi-Euler-Lagrange quadratic mappings to an equation, say, the multi-Euler-Lagrange quadratic functional equation. Moreover, some results corresponding to known stability (Hyers, Rassias, and Gӑvruta) outcomes regarding the multi-Euler-Lagrange quadratic functional equation are presented in quasi--normed and Banach spaces by using the fixed point methods. Lastly, an example for the nonstable multi-Euler-Lagrange quadratic functional equation is indicated.

1. Introduction

The celebrated Ulam challenge [1] arises from this question that how we can find an exact solution near to an approximate solution of an equation. This phenomenon of mathematics is called the stability of functional equations which has many applications in nonlinear analysis. The mentioned question has been partially solved by Hyers [2], Aoki [3], and Rassias [4] for the linear, additive, and linear (unbounded Cauchy difference) mappings, respectively. Next, many Hyers-Ulam stability problems for miscellaneous functional equations were studied by authors in the spirit of Rassias approach (see for instance [5–14] and other resources).

During the last two decades, stability problems for multivariable mappings were studied and extended by a number of authors. One of the mappings is the multiquadratic mapping, studied, for example, in [15–17]. Recall that a multivariable mapping is said to be multiquadratic [11] if it fulfills the famous quadratic equation in each component. Note that equation (1) is a suitable tool for obtaining some characterizations in the setting of inner product spaces and in fact plays a prominent role. In other words, any square norm on an inner product space fulfills which is called the parallelogram equality. However, some functional equations have been applied to characterize inner product spaces and are available in [18, 19] and references therein. In addition, the quadratic functional equation was used to characterize inner product spaces in [20, 21].

A lot of information about equation (1) and some equations which are equivalent to it (in particular, about their solutions and stability) and more applications can be found for instance in [22–24]. Park was the first author who studied the stability of multiquadratic in the setting of Banach algebras [16]. After that, some authors introduced various versions of multiquadratic mappings and investigated the Hyers-Ulam stability of such mappings in Banach spaces and non-Archimedean spaces; these results are available for instance in [15, 25–29]. As for an unification of the multiquadratic mappings, Zhao et al. [17] were the first authors who described the structure of multiquadratic mappings, and in fact, they showed that is a multiquadratic mapping if and only if the equation holds, where and .

Rassias [30] introduced the following notion of a generalized Euler-Lagrange-type quadratic mapping and investigated its generalized stability.

Definition 1. Suppose that and are linear spaces. A nonlinear mapping satisfying the functional equation is called 2-dimensional quadratic, where and are the fixed reals with .

It is easily seen that the Euler-Lagrange equality is valid for , defined in Definition 1 with any fixed reals , and hence, (4) is also called Euler-Lagrange quadratic functional equation; we refer to [31] for Euler-Lagrange type cubic functional equation and its stability. Note that equation (4) is a general form of (1) in the case that , and so the function satisfies (4). Next, Xu [32] extended the definition above to several variable mappings and presented the next definition.

Definition 2. Let and be vector spaces. A mapping is said to be the -Euler-Lagrange quadratic or multi-Euler-Lagrange quadratic if the mapping satisfies (4), for all and all .

In this article, we include a characterization of multi-Euler-Lagrange quadratic mappings and show that every multi-Euler-Lagrange quadratic mapping can be described as an equation (namely, the multi-Euler-Lagrange quadratic equation). Under the quadratic condition (2-power condition) in each variable, every multivariable mappings satisfying the mentioned earlier equation is multi-Euler-Lagrange quadratic (Theorem 5). Furthermore, we bring two Hyers-Ulam stability results for multi-Euler-Lagrange quadratic functional equations in quasi--normed and Banach spaces which their proof is based according to some known fixed point methods; see [33, 34] for more stability results in quasi--Banach spaces setting. Finally, we indicate an example to show that the multi-Euler-Lagrange quadratic functional equation is nonstable in the case of singularity.

2. Characterization of Multi-Euler-Lagrange Quadratic Mappings

Throughout, we consider the following known notations: (i)=the set of all natural numbers(ii)= the set of all integer numbers(iii)= the set of all rational numbers(iv)(v)

Let be a linear space over . Given , , and . We write and which belong to . Here and subsequently, is linear space over and , in which . Furthermore, for given the fixed elements such that , where and (here and the rest of the paper). We will write and simply and , respectively, when no confusion can arise.

For and , set

In continuation, we show that the equation is a general form of (4) for the multivariable case. In other words, we prove that every multi-Euler-Lagrange quadratic mapping fulfills (1) and vice versa. For doing this, we need some definitions and the upcoming lemma.

Definition 3. Let and be vector spaces over and be a multivariable mapping. (i)We say satisfies (has) the 2-power (quadratic) condition in the th component if for all , where for all (ii)If when the fixed is zero, then we say that has zero functional equation in the th variable. Moreover, if for any with at least one is zero, we say has zero functional equation

We consider two hypotheses as follows:

(H1) has the quadratic condition in all variables.

(H2) has zero functional equation.

Remark 4. It is clear that if a mapping satisfies the quadratic condition in the th variable, then it has zero functional equation in the same variable. Therefore, if fulfills (H1), then it satisfies (H2).

Theorem 5. For a mapping , the following assertions are equivalent: (i) is multi-Euler-Lagrange quadratic(ii) fulfills (8) and H1

Proof. ⇒ In view of [30], one can show that satisfies H1. By induction on , we now proceed the rest of this implication so that satisfies equation (8). Obviously, satisfies equation (4) for . The induction hypothesis is Then (ii) ⇒ (i) Let be arbitrary and fixed. Taking for all in (8) and applying Remark 4, the left side will be as follows: Once again, the same replacements convert the right side of (8) to It follows from (12) and (13) that is Euler-Lagrange -quadratic in the th component, and this completes the proof.

We should note that Theorem 5 necessitates that the mapping defined through fulfills equation (8). Hence, this equation can be called the multi-Euler-Lagrange quadratic functional equation.

3. Stability and Nonstability Results

The goals of this section are to prove miscellaneous result stability of multi-Euler-Lagrange quadratic equation (14) such as Hyers and Găvruta stability. Here, we mention a special case of equation (8) in which and , and so (8) converts to in which and (used here and from now on) for all .

For a set , a function is said to be a generalized metric on provided that fulfills the statements below, for all . (i) if and only if (ii)(iii)

The next theorem from [35] is one of fundamental results in fixed point theory and useful to achieve our first purpose in this section.

Theorem 6. Suppose that is a complete generalized metric space and is a mapping such that its Lipschitz constant is . Then, for each element , one of following cases can be happen: (i)(ii)There is an such that for all , and the sequence is convergent to a fixed point of which belongs to the set . Moreover, for all In the sequel, for any mapping , we define the operator via for the fixed nonzero integers where and are defined in (15) for all .
In the incoming stability result for equation (14), is controlled by a small positive number . We recall that for , we consider .

Theorem 7. Given . Let and be a linear space and a complete normed space, respectively. Suppose that a mapping fulfilling H2 and for all . Then, there exists a unique solution of (14) such that for all . In addition, for all .

Proof. Putting in (17) and using the assumption H2, we have for all , where Set for simply and for the rest of the proof, all the equations and inequalities are valid for all . Once more, by replacing instead of in (17), we get Multiplying both sides of (20) by and plugging to (22), we obtain and thus Let . For each , we define the function on as follows: Similar to the proof of ([36], Theorem 2.2), it is seen that is a complete generalized metric space. Define through for all . Take and with . Then, , and hence Therefore, . This shows that and in fact is a strictly contractive operator such that its Lipschitz is . It concludes from (24) that Hence, An application of Theorem 6 for the space , the operator , , and , shows that the sequence is convergent in and its limit; is a fixed point of . Indeed, and In other words, by induction on , it is easily verified that for each , we have and (19) follows. Note that clearly , and hence, part (iii) of Theorem 6 and (29) necessitate that which proves (18). In addition, for all . The last relation shows that for all and means that fulfills (14). Let us finally suppose that is another solution of equation (14) satisfies H2 such that inequality (18) holds. Then, satisfies (30), and so it is a fixed point of . Furthermore, by (18), we get and consequently, . It now follows from part (ii) of Theorem 6 that . This finishes the proof.

Remark 8. In the proof of Theorem 7, if we put , we can not reach to (20) unless it is assumed that is even in each component. Recall from [33] that is even in the th component if In other words, this condition is redundant, and we do not need it.

Hereafter, we concentrate our mind on the quasi--normed spaces.

Definition 9. Let be a fix real number with and denote either or . Suppose that is a vector space over . A quasi--norm is a real-valued function on fulfilling the next conditions for all and . (i) and moreover (ii)(iii)There exists a real number such that

When , the norm above is a quasinorm. Recall that is the modulus of concavity of the norm . Moreover, if is a quasi--norm on , the pair is said to be a quasi--normed space. Similar to normed spaces, a complete quasi--normed space is called a quasi--Banach space. For , if , for all , then the quasi--norm is called a -norm. In this case, every quasi--Banach space is said to be a -Banach space. A result of the Aoki-Rolewicz theorem [37] shows that every quasinorm can be equivalent to a -norm, for some .

A main tool of this section is the upcoming fixed point lemma which has been proved in ([38], Lemma 3.1).

Lemma 10. Given the fixed and with . Suppose that is a linear space and is a -Banach space with -norm . If is a function such that there exists an with for all and is a mapping satisfying for all , then there exists a uniquely determined mapping such that and Furthermore, for each , we have .

In the next theorem, we prove the Găvruta stability of (14) in quasi--normed spaces.

Theorem 11. Given . Let be a vector space over and be a -Banach space. Assume that is a function such that for all , where . If a mapping satisfying H2 and then there is a unique solution of (14) so that where whereas is the modulus of concavity of the norm .

Proof. Setting in (38) and applying H2, we have for all , where is defined in (21). Interchanging into in (38), we obtain for all . Multiplying both sides of (41) by , we get for all . It follows from (42), (43), and part (iii) of Definition 9 that for all , where is defined in (40). By Lemma 10, there exists a mapping which is unique such that and Lastly, we show that fulfilling (14). Note that Lemma 10 implies that for each , . For each and , by (38), we find Taking in the last relation, we observe that for all , and therefore, fulfills (14).

The following corollary is a consequence of Theorem 11 when the norm of is controlled by sum of variable norms of and with positive powers.

Corollary 12. Let be a quasi--normed space with quasi--norm , and be a -Banach space with -norm . Let and be positive numbers with . If a mapping satisfying for all , then there exists a unique solution of (14) such that for all , where .

Proof. Taking , the result concludes from Theorem 11.

We bring an elementary lemma without proof as follows.

Lemma 13. If a function is a continuous and satisfies (1), then it has the form , for all , where .

It is easily seen that when in (14), then this equation and (3) are the same. In the upcoming result, we extend Lemma 13 for multivariable functions. In fact, we use it to make a counterexample.

Proposition 14. Suppose that is a continuous which satisfies (3). Then, has the form where is a constant in .

Proof. We first recall from Theorem 2 in [17] that is a -quadratic mapping. By induction on , we proceed the proof. For , (49) holds by Lemma 13. Assume that (49) is valid for a , and is a continuous -quadratic function. Fix the variables in . Then, the function is quadratic and continuous, and hence, by Lemma 13, has the form where is a constant in . One should note that depends on , and hence Letting in (50) and applying (51), we have It is known that is -quadratic and is an -quadratic function. Therefore, by the induction assumption, there exists a real number so that It now follows from (50) and (53) that (49) holds for .
Here, we present a nonstable example for the multiquadratic mappings on (see [8]). Indeed, for the case , we show that the assumption can not be eliminated in Corollary 12.

Example 1. Given and . Set . The function is defined via Consider as a function defined by Obviously, is a nonnegative function and moreover is an even function in all components. Additionally, is bounded by and continuous. Since is a uniformly convergent series of continuous functions, it is continuous and bounded. In other words, we get for all . For , take . We shall prove that for all . Clearly, (56) holds for . Let with Inequality (57) necessitates that there is such that and so . It follows the last relation that for all and . Hence, . Let . Then It is known that is multiquadratic function on , and hence, for all . Now, the last equality and relation (58) imply that for all . Hence, (56) is valid for case (57). If , then Therefore, satisfies in (56) for all . Assume that there exists a number and a multiquadratic function for which the inequality is valid for all . An application of Proposition 14 shows that there is a constant such that , and hence Furthermore, choose such that . Take in which for all , then for all . Therefore which is a contradiction with (61).

We close the paper by an alternative stability result for equation (14) as follows.

Corollary 15. Let be a quasi--normed space with quasi--norm and be a -Banach space with -norm . Suppose for and with , where and . If a mapping fulfilling the inequality for all , then there exists a unique solution of (14) so that for all , where .

Proof. Setting in Theorem 11, one can obtain the desired results.

4. Conclusion

In this paper, by using Euler-Lagrange type quadratic functional equations, we have defined the multi-Euler-Lagrange quadratic mappings and have studied the structure of such mappings. Indeed, we have described the multi-Euler-Lagrange quadratic mapping as an equation. In continuation, we have shown that some fixed point theorems can be applied to prove the Hyers-Ulam stability version of multi-Euler-Lagrange quadratic functional equations in the setting of quasi--normed and Banach spaces. In the last part, we have brought an example which shows that such functional equations can be nonstable in the some cases.

The current work provides guidelines for further research and proposals for new directions and open problems with relevant discussions. Here, we give some questions and information on the connections between the fixed point theory and the Hyers-Ulam stability. (1)Which equation can describe the multi-Euler-Lagrange cubic mappings defined in [31]? Are these mappings stable on various Banach spaces? Can the known fixed point methods be useful to prove their Hyers-Ulam stability?(2)Definition of the multiadditive-quartic mappings by using [14] as a system of functional equations. The characterization of such mappings and discussion about their stability via a fixed point approach(3)Applying the functional equations indicated in [5, 12, 13, 34], we can generalize such mappings and equations to multiple variables

Data Availability

All results are obtained without any software and found by manual computations. In other words, the manuscript is in the pure mathematics (mathematical analysis) category.

Conflicts of Interest

There do not exist any competing interests regarding this article.

Authors’ Contributions

A.B proposed the topic. H.M and A.M prepared the first draft. Lastly, A.B edited and finalized the manuscript.

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Copyright © 2022 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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