General Solution and Stability of Additive-Quadratic Functional Equation in IRN-Space
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 ‖f(x + y) − f(x) − f(y)‖ to be controlled by ε(∥x∥p+∥y∥p). In 1994, a generalization of the Th.M. Rassias’ theorem was got by Gavruta [5], who replaced ε(∥x∥p+∥y∥p) by a general control function φ(x, y). 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].
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.
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 μ : ℝ⟶[0, 1] is said to be a measure distribution function, if μ is left continuous on ℝ, non-decreasing, inft∈ℝμ(t) = 0, and supt∈ℝμ(t) = 1.
Definition 2 (see [42].)A mapping ν : ℝ⟶[0, 1] is said to be a non-measure distribution function, if ν is right continuous on ℝ, non-increasing, supt∈ℝν(t) = 1, and inft∈ℝν(t) = 0.
Lemma 3 (see [43], [44].)Let L∗ 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) ∗ = Φ on [0, 1] is defined as an increasing, commutative, associative mapping Φ : [0, 1]2⟶[0, 1] satisfying Φ(1, p) = 1∗p = p for every p ∈ [0, 1], and a triangular conorm (t-conorm) Y = ⋄ is defined as an increasing, commutative, associative mapping Y : [0, 1]2⟶[0, 1] satisfying Y(0, p) = 0⋄p = p for all p ∈ [0, 1].
By using the lattice , these definitions can be straightforwardly extended.
Definition 4 (see [44].)A triangular norm (t-norm) on L∗ is a mapping satisfying the following conditions:
- (i)
Boundary condition
i.e., , ∀p ∈ L∗;
- (ii)
Commutativity
i.e., Φ(p, q) = Φ(q, p),
- (iii)
Associativity
i.e., Φ(p, Φ(q, r)) = Φ(Φ(p, q), r),
- (iv)
Monotonicity
i.e., and for all
If is an Abelian topological monoid with unit , then Φ is called a continuous t-norm.
Definition 5 (see [42].)A negator on L∗ is any decreasing mapping N from L∗ to L∗ satisfying and . If N(N(p)) = p for all p ∈ L∗, then N is called an involutive negator. A negator on [0, 1] is a decreasing mapping N : [0, 1]⟶[0, 1] satisfying N(0) = 1 and N(1) = 0.
Ns denotes the standard negator on [0, 1] defined by
Definition 6 (see [23].)Let μ and ν be measure and nonmeasure distribution functions from V × (0, +∞) to [0, 1] such that
Example 1 (see [42].)Let (V, ‖·‖) be a normed space. Let Φ(p, q) = (p1q1, min(p2 + q2, 1)) for all p = (p1, p2), q = (q1, q2) ∈ L∗ and let μ, ν be measure and non-measure distribution functions defined by
Then, (V, Iμ,ν, Φ) is an IRN-space.
Definition 7 (see [42].)Let (V, Iμ,ν, Φ) be an IRN-space.
- (i)
A sequence {pm} in (V, Iμ,ν, Φ) is known as a Cauchy sequence if, for some δ > 0 and t > 0, there is an m0 ∈ ℕ satisfies
- (ii)
The sequence {pm} is convergent to any point p ∈ V if as m⟶∞ for all t > 0
- (iii)
An intuitionistic random normed space (V, Iμ,ν, Φ) is known as complete if every Cauchy sequence in V is convergent to a point p ∈ V
3. Solution of the Functional Equation (3)
In this section, let us consider V and W are two real vector spaces.
Theorem 8. If an odd mapping φ : V⟶W satisfies the functional equation (3) for all v1, v2, ⋯, vm ∈ V, then the function φ is additive.
Proof. In the view of the oddness of φ, we have φ(−v) = −φ(v) for all v ∈ V. Using the oddness property, the functional equation (3) reduces as
Again interchanging v with av in (12), we have
From the equalities (12)–(14), we can generalize the results for any nonnegative integer m as
Similarly, we have
Replacing (v1, v2, ⋯, vm) by ((x/a), (y/a2), 0, ⋯, 0) in (11), we have
Hence, the function φ is additive.☐
Theorem 9. If an even mapping φ : V⟶W satisfies the functional equation (3) for all v1, v2, ⋯, vm ∈ V, then the function φ is quadratic.
Proof. Since, in the view of evenness of φ, we have φ(−v) = φ(v) for all v ∈ V. Now, the functional equation (3) reduces as
Replacing v by av in (19), we reach
Switching v by av in (20), we get
From (19)–(21), we can generalize the results for any nonnegative integer m as
Similarly, we have
Replacing (v1, v2, ⋯, vm) by ((x/a), (y/a2), 0, ⋯, 0) in (18), we obtain
Hence, the function φ is quadratic.☐
Theorem 10. If a mapping φ : V⟶W satisfies φ(0) = 0 and satisfies the functional equation (3) for all v1, v2, ⋯, vm ∈ V if and only if there exists a symmetric biadditive mapping Q : V × V⟶W and a additive mapping A : V⟶W satisfies φ(v) = Q(v, v) + A(v) for all v ∈ V.
Proof. Let a mapping φ : V⟶W with φ(0) = 0 satisfies the functional equation (3). We divide the function φ into the odd part and even part as
It is easy to prove that φo and φe satisfies the functional equation (3). By Theorems 8 and 9, we conclude that φo and φe are additive and quadratic, respectively. Then, there exist a symmetric biadditive mapping Q : V × V⟶W which satisfies φe(v) = Q(v, v) and an additive mapping A : V⟶W which satisfies φo(v) = A(v) for all v ∈ V. Hence, φ(v) = Q(v, v) + A(v) for all v ∈ V.
Conversely, suppose that there exists a symmetric biadditive mapping Q : V × V⟶W and an additive mapping A : V⟶W and satisfies φ(v) = Q(v, v) + A(v) for all v ∈ V. It is easy to prove that the mappings v ↦ Q(v, v) and A : V⟶W satisfy the functional equation (3). Hence, the mapping φ : V⟶W satisfies the functional equation (3).
In the following sections, we consider V is a linear space, is an intuitionistic random normed space and (W, Iμ,ν, Y) is a complete intuitionistic random normed space.
4. Stability Results for Even Case
Theorem 11. Let α, β : Vm⟶D+, where α(v1, v2, ⋯, vm) is denoted by , β(v1, v2, ⋯, vm) is denoted by and is denoted by Θα,β(v1, v2, ⋯, vm, ε), be a mapping such that
Proof. Replacing (v1, v2, ⋯, vm) by (v, 0, ⋯, 0) in (29), we have
Interchanging v with av in (32), we obtain
Replacing v by al−1v and divide by a2l in (33), we conclude that
Corollary 12. If an even mapping φ : V⟶W satisfies
Proof. By taking in Theorem 11, we obtain our desired result.☐
5. Stability Results for Odd Case
Theorem 13. Let α, β : Vm⟶D+, where α(v1, v2, ⋯, vm) is denoted by , β(v1, v2, ⋯, vm) is denoted by and is denoted by Θα,β(v1, v2, ⋯, vm, ε), be a mapping such that
Proof. Replacing (v1, v2, ⋯, vm) by (v, 0, ⋯, 0) in (47), we obtain
Corollary 14. If an odd mapping φ : V⟶W satisfies
Proof. By taking in Theorem 13, we obtain our desired result.☐
6. Stability Results for Mixed Case
Theorem 15. Let α, β : Vm⟶D+ be mappings satisfying (27), (28), (45), and (46) for all v1, v2, ⋯, vm, v ∈ V and all ε > 0. If a mapping φ : V⟶W with φ(0) = 0 satisfies (29) for all v1, v2, ⋯, vm ∈ V and all ε > 0, then there exist a unique quadratic mapping Q2 : V⟶W and a unique additive mapping A1 : V⟶W satisfying (3) and
Proof. Let φe(v) = φ(v) + φ(−v)/2 for all v ∈ V. Thus, φe(0) = 0, φe(−v) = φe(v) and for all v1, v2, ⋯, vm ∈ V and all ε > 0,
By Theorem 11, there exists a quadratic mapping Q2 : V⟶W such that
Corollary 16. If a mapping φ : V⟶W satisfies
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.
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.
Open Research
Data Availability
No data were used in this study.