Equivalence for Various Forms of Quadratic Functional Equations and Their Generalized Hyers–Ulam–Rassias Stability

Sharma, Ravinder Kumar; Chandok, Sumit

doi:https://doi.org/10.1155/2023/1721273

International Journal of Mathematics and Mathematical Sciences

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Abstract Introduction Preliminaries Conclusion Data Availability Conflicts of Interest References Copyright Related Articles

Research Article | Open Access

Volume 2023 | Article ID 1721273 | https://doi.org/10.1155/2023/1721273

Equivalence for Various Forms of Quadratic Functional Equations and Their Generalized Hyers–Ulam–Rassias Stability

Ravinder Kumar Sharma¹and Sumit Chandok¹

Academic Editor: Bikash Koli Dey

Received01 Jul 2022

Revised06 Nov 2022

Accepted18 Nov 2022

Published30 Mar 2023

Abstract

In this manuscript, we study the properties and equivalence results for different forms of quadratic functional equations. Using the Brzdȩk and Ciepliński fixed-point approach, we investigate the generalized Hyers–Ulam–Rassias stability for the 3-variables quadratic functional equation in the setting of 2-Banach space. Also, we obtain some hyperstability results for the 3-variables quadratic functional equation. The results obtained in this paper extend several known results of the literature to the setting of 2-Banach space.

1. Introduction

The theory of functional equations is a broad field of nonlinear analysis that is difficult to comprehend. Economic theory, game theory, statistics, geometry, measure theory, dynamics, and a variety of other subjects all use functional equations. The study of functional equation solutions and stability results is a popular area in analysis research. In nonlinear analysis, notably in fixed-point theory, the stability results of functional equations are used. The asymptotic properties of additive mappings are studied using the stability results.

One can find an efficient instrument to analyse errors in the theory of Ulam’s stability, which is to investigate the existence of an exact solution to the perturbed functional equation that is close to the given function. Ulam’s [1] address at the University of Wisconsin in 1940 provided the main impetus for the study of the stability of functional equations. In Banach space, Hyers [2] supplied a partial affirmative response to Ulam’s question. Following that, Aoki [3] generalized Hyers theorem for additive mapping, while Rassias [4] generalized Hyers theorem for linear mapping by taking into account an unbounded Cauchy difference. Găvruta [5] generalized Rassias and Aoki theorem and discussed the stability of additive Cauchy linear functional equations.

A functional equation is hyperstable if a function satisfying the functional equation approximately is a true solution of it. In 1949, Bourgin [6] gave the first hyperstability result and concerned the ring homomorphisms. In recent years, many authors have investigated the Hyers–Ulam stability and hyperstability results of various functional equations in the literature. Some following are given below.

In 2016, Bahyrycz et al. [7] proved some general stability and hyperstability results for a generalization of the well-known Fréchet equation. In 2017, Aiemsomboon and Sintunavarat [8] proved new generalized hyperstability results for the general linear equation. In 2021, Rana et al. [9] investigated the stability of complex functional equation in 2-Banach space. In 2022, Sharma and Chandok [10] investigated the Ulam-type stability of quartic functional equation in -Banach space and non-Archimedean -normed space.

Hyers–Ulam stability concept has a wide range of applications in economics, optimization, biology, numerical analysis, and other allied areas. In 2019, Ali et al. [11] investigated the stability of solutions to a class of nonlinear impulsive boundary value problems of fractional order differential equations. In the same year, Li et al. [12] investigated the existence of solutions to random impulsive stochastic functional differential equations with delays by using Krasnoselskii’s fixed point. They also obtain some results on the Hyers–Ulam stability of random impulsive stochastic functional differential equations with delays under Lipschitz conditions. In 2021, Guo et al. [13] study Hyers–Ulam stability of the almost periodic solution to the fractional differential equation with impulse and fractional Brownian motion under nonlocal condition. In 2022, Jamil et al. [14] established necessary and sufficient conditions for the existence Hyers–Ulam and generalized Hyers-Ulam stability for the fractional order hybrid sequential integro-differential equations. In 2020, Mohanapriya et al. [15] studied the Hyers–Ulam stability of the second-order differential equation using Fourier transform. Unyong et al. [16] study the Hyers–Ulam stability with respect to the linear differential condition of the fourth order. In 2021, Govindan et al. [17] investigated the general solution of -variable quadratic functional equation and discussed its Ulam stability by using the direct method and fixed point method. Sharma and Chandok [18] investigated the hyperstability results for the general linear equation in 2022.

Quadratic functional equation is one of the famous functional equations of the following form:where is unknown function and . The quadratic function is a solution of the functional equation (1).

Quadratic functions are more than just mathematical puzzles; they are widely used in science, business, and engineering. A parabola’s U-shape can describe the paths of water jets in a fountain and a bouncing ball, or it can be incorporated into structures such as the parabolic reflectors that form the base of satellite dishes and car headlights. Quadratic functions aid in the prediction of business profit and loss, the tracking of moving objects, and the determination of minimum and maximum values. Most everyday objects, from cars to clocks, would not exist if quadratic functions had not been used in their design by someone somewhere. When gravity is involved, such as the path of a ball or the shape of cables in a suspension bridge, quadratic equations are used. A quadratic equation is the characteristic equation of a second-order linear difference equation or differential equation. There are numerous types of quadratic equations.

As a result, it is known as the quadratic functional equation or the Rassias–Euler–Lagrange functional equation introduced by Rassias [19]. Skof developed a Hyers–Ulam stability theorem for the equation (1) for the function where is a normed space and is a Banach space [20]. The Hyers–Ulam–Rassias stability of the quadratic functional equation (1) was demonstrated by Czerwik [21], and this result was generalized by several mathematicians (see [22–24]). Various other forms of quadratic functional equation (1) are discussed in [25–27].

We consider the following functional equations:

In 1995, Kannappan [28] solved the functional equation (2). In 1998, Jung [26] explored the Hyers–Ulam stability of the functional equation (2) on a restricted (unbounded) domain and applied the conclusion to the research of asymptotic behaviour of the quadratic functions. Thereafter, in 2000, Bae [29] investigated that functional equation (3) is equivalent to functional equation (1). He also demonstrated the Hyers–Ulam–Rassias stability of the quadratic functional equation (3) and demonstrated the Hyers–Ulam stability of the functional equation in restricted (unbounded) domains. In 2001, Bae and Jun [30] showed the generalized Hyers–Ulam–Rassias stability of the quadratic functional equation (3) and Hyers–Ulam–Rassias stability of the functional equation in bounded domains. In 2002, Bae and Jung [31] showed the Hyers–Ulam stability of the quadratic functional equation (3) on Abelian group. The hyperstability findings for a functional quadratic equation (3) in the class of functions from the Abelian group into a Banach space using a fixed point method were demonstrated by El-Fassi and Kim [32] in 2016.

The concept of an approximation solution and the concept of the nearness of two functions, on the other hand, can be construed in a variety of nonstandard ways depending on the situation. One of these nonclassical distance measures can be introduced using the concept of a 2-norm. Remember that Gahler introduced the concept of linear 2-normed space in [33], and it appears that [34] is the first work on the Hyers–Ulam stability of functional equations in 2-Banach spaces. In 2018, Brzdek and Ciepliński [35] proved a new fixed point theorem in 2-Banach space and showed its applications to the Ulam stability of some single-variable equations and the most important functional equation in several variables, namely, the Cauchy equation. In addition, Brzdęk and El-hady [36] provided an extension of an earlier stability result that has been motivated by a problem of Rassias for the functions taking values in 2-Banach spaces. During the past few years, many authors have written extensively on the subject of 2-normed space in a wide range of subjects (see [9, 37, 38] and references cited therein).

In this paper, we study the properties and equivalence of the functional equations and (1)–(3). Using the Brzdȩk and Ciepliński fixed-point approach, we investigated the generalized Hyers–Ulam–Rassias stability for the 3-variables quadratic functional equation in the setting of 2-Banach space. Also, we obtain some hyperstability results for 3-variables quadratic functional equation. The results of El-Fassi and Kim [32] are supplemented by our findings.

2. Preliminaries

Throughout this paper, stands for the set of natural numbers, stands for the set of integers, and stands for the set of reals. Let be the set of nonnegative real numbers and denote the family of all mappings from a nonempty set into a nonempty set .

In this section, we present some definitions, properties, and results of 2-normed space, which will be used in the next section.

In 1965, Gähler et al. [33] introduced 2-normed space.

Definition 1. Suppose that is a real linear space of dimension greater than 1. A 2-norm is a real valued function on satisfying the following axioms, for all and . , = if and only if and are linearly dependent in = = The pair is a called a 2-normed space if is 2-norm on . Then, function is called a 2-norm on and the pair is a called a 2-normed space.

Example 1. Let and be defined byfor all . Then, is a 2-norm on .

Example 2. Let and consider the following 2-norm on:where and . Then, is a 2-normed space.
For Cauchy sequence, convergent sequence, and completeness of -normed space see [33, 39].
Now, we state the following results as a lemma (see [38] for the details).

Lemma 1. Let be a 2-normed space. Then,(1) for all .(2); then, , where and are linearly independent (LI).(3)For a convergent sequence ,

In 2018, Brzdȩk and Ciepliński [35] proved the following fixed point theorem in 2-Banach space.

Theorem 1. (1)Let be a nonempty set, be a 2-Banach space, be a subset of containing two LI vectors, , , and be given mapping for .(2)Suppose that is an operator satisfies the following inequality: for all , , and .(3)Assume that the function and a mapping satisfies the following conditions: for every , and where is a linear operator defined by for , , and .

Then, there exists a unique fixed point of for which

Moreover, for every , the limit is

3. Equivalence of Functional Equations (1)–(3)

In this section, we investigate some properties and equivalence of functional equations (1)–(3). Throughout this section, and are vector spaces.

Lemma 2. Suppose that a function satisfiesfor all . Then, satisfiesfor every and is scalar.

Proof. Letting in (13), we have . Thus, we can say that equation satisfies (14) for when .
By taking in (13) and using , we have . Thus, we can say that equation satisfies (14) for when .
Letting in (13) and using , we have . Thus, we can say that equation satisfies (14) for when .
Letting in (13) and using , we have . Thus, we can say that equation satisfies (14) for when .
Suppose thatholds for , where is a positive integer.
If we take , we haveSo, we have . Now, suppose that (15) is true for , where is a positive integer. Furthermore, we take :So, we have . Therefore, is quadratic; then, , for positive integer .
Since is even function , so is true for all integers .

Remark 1. Suppose that a function satisfiesfor all . Case A1: If in (18), we have . Case A2: If in (18) and using case A1, we have So, we have . Case A3: If in (18) and using case A1, we have Taking , we have . Case A4: If , in (18), we have Taking and using cases A1 and A3, we have So, we have .

Proposition 1. Suppose that a function satisfiesfor all ; then, is quadratic, i.e.,

If a mapping is quadratic, then the mapping satisfies (23).

Proof. Taking in (23), we haveUsing Remark 1, we haveConversely, let a mapping is quadratic; then,Taking , we haveSince is quadratic and using Lemma 2, so we have

Remark 2. Suppose that a function satisfies (3), Case B1: If in (3), we have . Case B2: If in (3) and using case B1, we have So, we have . Case B3: If in (3) and using case B1, we have So, we have . Case B4: If , in (3) and using case B1, we have So, we have . Case B5: If in (3) and using case B1, we have So, we have . Case B6: If in (3), we have Using cases B1, B3, and B4, we have So, .On the similar lines, we can show and so on.

Proposition 2. Suppose that a function satisfiesfor all ; then, is quadratic, i.e.,

If a mapping is quadratic, then the mapping satisfies (36).

Proof. Taking in (36), we haveUsing Remark 2, we haveConversely, let a mapping is quadratic; then,for all . Taking , we haveUsing Lemma 2, we haveSince is quadratic, so we haveUsing Propositions 1 and 2, we get the following result.

Proposition 3. Suppose that a function satisfiesfor all ; then, is a 3-variables quadratic functional equationfor all .

If a mapping is 3-variables quadratic functional equation (45), then the mapping satisfies (44).

Proof. Let and be vector spaces and be a mapping satisfying (44)Using Proposition 1, we haveSo, we haveConversely, let be a mapping satisfying (45)Using Proposition 2, we haveSo, we have

4. Generalized HUR Stability

In this section, using the Brzdȩk and Ciepliński fixed point approach (Theorem 1), we investigate the generalized Hyers–Ulam–Rassias stability of 3-variables quadratic functional equation (3). Consider to be an abelian group, to be a 2-Banach space, and to be a subset of containing two LI vectors. stands for the family of all automorphisms of . will be used to represent the identity function on .

Theorem 2. Let be a mapping such thatwhere , ,, and .

We assume thatis a nonempty set, where.

Then, there exists a unique function satisfying (3) andfor all , where .

Proof. fix . Replacing by in (52), we haveWe define the operators and byfor all , , and . Then, (57) becomesfor all . The operator has the form given by (10) with and , , , , , and for all . Furthermore, using Definition 1, we havefor all and . Using the definition of and for all . Using induction on , we have for all , where . For , we haveAs the operator is linear, we obtainfor all .
Therefore by Theorem 1, there exists a unique solution of the equationfor all , which is a fixed point of such thatfor all . Moreover,Now, to prove that satisfies the functional equation (3) on , we have to prove the following inequality:for all .
Indeed, if , then (66) is simply (52). So, we suppose that (66) holds for , , and . Then, from (58) and Definition 1, we haveSo, we haveLetting in (68), using Lemma 1 and equation (65), we obtainfor all . Thus, we have proven that, for every , there exists a function which is the solution of functional equation (3) on and satisfies , for all .
Now, we prove that for all . Fix and note that satisfies (64) with replaced by . Hence, by replacing with in (69), we get , for . Using Definition 1, we havefor every . Using the linearity of and (61), we obtainwhere for every , , and . Letting ; we get . Thus, we havefor all . Therefore, we get (56). Using (69), it is obvious that is a solution of (3). Now to demonstrate the uniqueness of the mapping , suppose that there exists a mapping satisfying (3) and inequalityThen,Furthermore, for every . As a result, with a fixed , we obtainfor all and . Taking we get . The proof of the theorem is complete.

Theorem 3. Let and be functions, and conditions (53)–(55) hold. We assume thatfor all . Then, satisfies (3) on .

Proof. Suppose thatTherefore, from Theorem 2, we obtain . Then, satisfies (3) on .

Remark 3. In Theorem 2, if(this is the case when, i.e., ), then (53) holds andNow, we prove that a function satisfying the 3-dimensional quadratic functional equation on fulfills it on the whole space .

Proposition 4. If satisfiesfor all , then .

Proof. Note that it is enough to show that .
Putting in (80), we havePutting in (80), we havePutting in (80), we haveFrom (82) and (83)Putting in (80), we haveUsing (84), we haveUsing (81), we havePutting and in (80), we haveUsing (84), we haveUsing (87), we haveThus, we get .
Now, we are in a position to prove the following.

Theorem 4. Let be a mapping such thatwhere , .

We assume thatis a nonempty set, wherefor .

Then, there exists a unique function satisfying (3) andfor all , , where .

5. Hyperstability

In this section, we will look at how Theorem 2 can be used in the following two situations:, and andwhere for and .

Throughout this section, is an abelian group, is a 2-Banach space, is a subspace of containing two LI vectors, and is a nonempty subgroup of . Also, we obtain that is a solution of (3) on if (3) holds for every with .

Corollary 1. Let satisfyfor all , with , , , and . Then, is a solution of (3) on .

Proof. The proof follows from Theorem 2 by takingfor all with some real number , . For each , we define by . Then,for all and . Hence,for all and . Then (76) is valid with for , and there exists with such that andTherefore, we can say that (53) is satisfied with . Hence, by Theorem 3, every function satisfies (3) on .

Corollary 2. Let satisfyfor all , with ,, , and . Then is a solution of (3) on , such that .

Proof. The proof follows from Theorem 2 by takingfor all with some real number , . For each , we define by . Then,for all and . Hence,for all and . Then, (76) is valid with for , and there exists with such that andTherefore, we can say that (53) is satisfied with . Hence by the Theorem 3, every function , satisfies (3) on .

Corollary 3. Let be an abelian group, be a 2-Banach space, be a subspace of containing two LI vectors, and conditions (54), (55), and (76) be valid. If is a mapping such that for some andfor all , then the function equationfor all , has no solution in the class of function .

Proof. Suppose that is a solution of (108). Then, (52) holds, and consequently, according to Theorem 3, satisfies (3) on , which means that . This is a contradiction.

6. Conclusion

In this work, by discussing some of the properties of the quadratic functional equations, we obtain their equivalence. We primarily obtain the generalized Hyers–Ulam–Rassias stability of a 3-variables quadratic functional equation using the fixed point approach. Also, we obtain some hyperstability results for 3-variables quadratic functional equation. Since quadratic equation is the characteristic equation of a second-order linear difference equation or differential equation, and there are numerous types of quadratic equations. So, one can investigate the stability and hyperstability of several types of second-order linear difference equations or differential equations.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest.

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Copyright © 2023 Ravinder Kumar Sharma and Sumit Chandok. 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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