General Solution and Stability of Additive-Quadratic Functional Equation in IRN-Space

Tamilvanan, K.; Alessa, Nazek; Loganathan, K.; Balasubramanian, G.; Namgyel, Ngawang

doi:https://doi.org/10.1155/2021/8019135

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Volume 2021 | Article ID 8019135 | https://doi.org/10.1155/2021/8019135

General Solution and Stability of Additive-Quadratic Functional Equation in IRN-Space

K. Tamilvanan,¹Nazek Alessa,²K. Loganathan,³G. Balasubramanian,¹and Ngawang Namgyel⁴

Academic Editor: Liliana Guran

Received20 Jun 2021

Revised06 Jul 2021

Accepted16 Jul 2021

Published04 Aug 2021

Abstract

The investigation of the stabilities of various types of equations is an interesting and evolving research area in the field of mathematical analysis. Recently, there are many research papers published on this topic, especially additive, quadratic, cubic, and mixed type functional equations. We propose a new functional equation in this study which is quite different from the functional equations already dealt in the literature. The main feature of the equation dealt in this study is that it has three different solutions, namely, additive, quadratic, and mixed type functions. We also prove that the stability results hold good for this equation in intuitionistic random normed space (briefly, IRN-space).

1. Introduction

The theory of random normed spaces (RN-spaces) is important as a generalization of the deterministic result of linear normed spaces and also in the study of random operator equations. The RN-spaces may also provide us with the tools to study the geometry of nuclear physics and have important applications in quantum particle physics.

The concept of stability of a functional equation arises when one replaces a functional equation by an inequality which acts as a perturbation of the equation. The first stability problem concerning group homomorphisms was raised by Ulam [1] in 1940 and affirmatively solved by Hyers [2]. Aoki generalized the result of Hyers [3] for approximate additive mappings and by Rassias [4] for approximate linear mappings by allowing the difference Cauchy equation to be controlled by . In 1994, a generalization of the Th.M. Rassias’ theorem was got by Gavruta [5], who replaced by a general control function . For additional information regarding the outcomes about such issues, the related background in [6–12] can be examined. Absorbing new outcomes concerning mixed-type functional equations has as of late been acquired by Najati et al. [13–15], Jun and Kim [16, 17], and Park [18–22].

The functional equations and are called the additive and quadratic functional equations, respectively. Every solution of the additive and quadratic functional equations is said to be additive mapping and quadratic mapping, respectively.

As of late, Zhang [23] examined the cubic functional equation in intuitionistic random space. The stability of various equations in RN-spaces has been as of late concentrated in Alsina [24], Eshaghi Gordji et al. [25, 26], Mihet and Radu [27–29], and Saadati et al. [30]. Xu et al. [31–33] presented the various mixed types of functional equations investigated in Intuitionistic fuzzy normed spaces, quasi Banach spaces, and random normed spaces. Also, Shu et al. [33–35] discussed various differential equations to study the Hyers-Ulam stability, which provides a wide view of this stability problem.

In this present work, we introduce a new mixed type additive-quadratic functional equation where is a fixed integer and and investigate the Ulam-Hyers stability results of this mixed type additive-quadratic functional equation in an intuitionistic random normed space.

So far various forms of additive and quadratic functional equations are considered in this research field to obtain their stability results through different methods. For the first time, a new mixed additive-quadratic functional equation is proposed in this paper, and its stability results are proved in an intuitionistic random normed space.

This type of functional equation can be of use in solving many physical problems and also has significant relevance in various scientific fields of research and study. In particular, additive-quadratic functional equations have applications in electric circuit theory, physics, and relations connecting the harmonic mean and arithmetic mean of several values. Providing a wealth of essential insights and new concepts in the field of functional equations.

2. Preliminaries

We recall the following ideas and conceptions of IRN-spaces in [36–41].

Definition 1 (see [42]). A mapping is said to be a measure distribution function, if is left continuous on , non-decreasing, , and .

Definition 2 (see [42]). A mapping is said to be a non-measure distribution function, if is right continuous on , non-increasing, , and .

Lemma 3 (see [43, 44]). Let be a set with an operator is defined by Then, the pair is a complete lattice.

We denote its units by and . Typically, a triangular norm (t-norm) on is defined as an increasing, commutative, associative mapping satisfying for every , and a triangular conorm (t-conorm) is defined as an increasing, commutative, associative mapping satisfying for all .

By using the lattice , these definitions can be straightforwardly extended.

Definition 4 (see [44]). A triangular norm (t-norm) onis a mappingsatisfying the following conditions:(i)Boundary conditioni.e., , (ii)Commutativityi.e., , (iii)Associativityi.e., , (iv)Monotonicityi.e., and for all
If is an Abelian topological monoid with unit , then is called a continuous t-norm.

Definition 5 (see [42]). A negator onis any decreasing mappingfromtosatisfyingand. Iffor all, thenis called an involutive negator. A negator onis a decreasing mappingsatisfyingand.
denotes the standard negator on defined by for all .

Definition 6 (see [23]). Let and be measure and nonmeasure distribution functions from to such that

The triple is said to be an intuitionistic random normed space if a vector space , continuous t-representable , and a mapping holds the following conditions: for all and

Thus, is called an intuitionistic random norm. Hence,

Example 1 (see [42]). Letbe a normed space. Letfor all, and letbe measure and non-measure distribution functions defined byThen, is an IRN-space.

Definition 7 (see [42]). Letbe an IRN-space.(i)A sequence in is known as a Cauchy sequence if, for some and , there is an satisfies(ii)The sequence is convergent to any point if as for all (iii)An intuitionistic random normed space is known as complete if every Cauchy sequence in is convergent to a point

3. Solution of the Functional Equation (3)

In this section, let us consider and are two real vector spaces.

Theorem 8. If an odd mappingsatisfies the functional equation (3) for all, then the functionis additive.

Proof. In the view of the oddness of , we have for all . Using the oddness property, the functional equation (3) reduces as for all . Now, replacing by in (11), we get . Interchanging with in (11), we get Again interchanging with in (12), we have for all . Replacing by in (13), we obtain From the equalities (12)–(14), we can generalize the results for any nonnegative integer as Similarly, we have Replacing by in (11), we have Hence, the function is additive.☐

Theorem 9. If an even mappingsatisfies the functional equation (3) for all, then the functionis quadratic.

Proof. Since, in the view of evenness of , we have for all . Now, the functional equation (3) reduces as for all . Now, replacing by in (18), we obtain . Interchanging with in (18), we obtain Replacing by in (19), we reach Switching by in (20), we get From (19)–(21), we can generalize the results for any nonnegative integer as Similarly, we have Replacing by in (18), we obtain Hence, the function is quadratic.☐

Theorem 10. If a mappingsatisfiesand satisfies the functional equation (3) for allif and only if there exists a symmetric biadditive mappingand a additive mappingsatisfiesfor all.

Proof. Let a mapping with satisfies the functional equation (3). We divide the function into the odd part and even part as respectively. Clearly, for all .☐

It is easy to prove that and satisfies the functional equation (3). By Theorems 8 and 9, we conclude that and are additive and quadratic, respectively. Then, there exist a symmetric biadditive mapping which satisfies and an additive mapping which satisfies for all . Hence, for all .

Conversely, suppose that there exists a symmetric biadditive mapping and an additive mapping and satisfies for all . It is easy to prove that the mappings and satisfy the functional equation (3). Hence, the mapping satisfies the functional equation (3).

For our notational convenience, we can define a mapping by for all .

In the following sections, we consider is a linear space, is an intuitionistic random normed space and is a complete intuitionistic random normed space.

4. Stability Results for Even Case

Theorem 11. Let, whereis denoted by, is denoted byandis denoted by, be a mapping such thatfor all and all , and for all and all . If an even mapping with satisfies for all and all , then there exists a unique quadratic mapping such that for all and all .

Proof. Replacing by in (29), we have for all and all . From inequality (31), we get Interchanging with in (32), we obtain Replacing by and divide by in (33), we conclude that for all and all . Thus, for all and all . To prove the convergence of the sequence , replacing by in (35), we obtain for all and all and all . Since the R.H.S of the inequality (36) tends to as , the sequence is a Cauchy sequence in . Since is a complete IRN-space, this sequence converges to some point . So one can define the mapping by for all . Letting in (36), we obtain for all and all . Taking the limit in (38), we get for all and all .

Next, we prove that the function is quadratic. Replacing by in (29), we obtain for all and all . Taking the limit as , we find that for all and all , which implies . Thus, the function satisfies the functional equation (3). Hence, is a quadratic mapping. Passing to the limit as in (35), we have (30).

Finally, to show the uniqueness of subject to (30), consider that there exists an another quadratic function which satisfies the inequality (30). Clearly, and for all and , from (30) and (28) that for all and all . By taking in (41), we show the uniqueness of . This ends the proof of the uniqueness, as desired.☐

Corollary 12. If an even mappingsatisfiesfor all and all , and for all and all , then there exists a unique quadratic mapping such that for all and all .

Proof. By taking in Theorem 11, we obtain our desired result.☐

5. Stability Results for Odd Case

Theorem 13. Let, whereis denoted by, is denoted byandis denoted by, be a mapping such thatfor all and all , and for all and all . If an odd mapping with satisfies for all and all , then there exist a unique additive mapping such that for all and all .

Proof. Replacing by in (47), we obtain for all and all . From inequality (49), we get for all and all . Replacing by in the above inequality (50), we have for all and all . Replacing by in (51), we conclude that for all and all . Thus, for all and all . To prove the convergence of the sequence , replacing by in (53), we obtain for all and all and all . Since the R.H.S of the inequality (54) tends to as , the sequence is a Cauchy sequence in . Since is a complete IRN-space, this sequence converges to some point . So one can define the mapping by for all . Letting in (54), we obtain for all and all . Taking the limit as in (56), we get for all and all .

Next, we want to prove that the function is additive. Replacing by in (47), we obtain for all and all . Taking the limit as , we find that for all and all , which implies . Thus, satisfies the functional equation (3). Hence, the function is additive. Passing to the limit as in (53), we have (48).

Finally, to show the uniqueness of the additive function subject to (48), consider that there exists another additive function which satisfies the inequality (48). Evidently, and for all and , from (48) and (46) that for all and all . By taking the limit in (59), we show the uniqueness of .☐

Corollary 14. If an odd mappingsatisfiesfor all and all , and for all and all . Then, there exists a unique additive mapping such that for all and all .

Proof. By taking in Theorem 13, we obtain our desired result.☐

6. Stability Results for Mixed Case

Theorem 15. Letbe mappings satisfying (27), (28), (45), and (46) for alland all. If a mappingwithsatisfies (29) for alland all, then there exist a unique quadratic mappingand a unique additive mappingsatisfying (3) andfor all .

Proof. Let for all . Thus, , and for all and all , By Theorem 11, there exists a quadratic mapping such that for all and all .☐

On the other hand, let for all . Then , . By Theorem 13, there exists a additive mapping satisfies for all and all . From inequalities (65) and (66), we obtain our desired result (64).

Corollary 16. If a mappingsatisfiesfor all and all , and for all and all . Then, there exists a unique quadratic mapping and a unique additive mapping such that for all and all .

7. Conclusion

In this paper, we introduced a new mixed type of additive-quadratic functional equation, and we applied the Hyers direct technique to investigate the Hyers-Ulam stability of the mixed type of additive-quadratic functional equation. Moreover, we have derived its general solution. The main objective of this work has been discussed: In Section 4, we have proved its Ulam-Hyers stability for the even case; in Section 5, examined Ulam-Hyers stability for odd case, and in Section 6, investigated Ulam-Hyers stability for the mixed cases, respectively, in intuitionistic random normed space.

Data Availability

No data were used in this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This research was funded by the Deanship of Scientific Research at Princess Nourah Bint Abdulrahman University through the Fast-track Research Funding Program.

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Copyright © 2021 K. Tamilvanan 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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