Volume 2021, Issue 1 8019135
Research Article
Open Access

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

K. Tamilvanan

Corresponding Author

K. Tamilvanan

Department of Mathematics, Government Arts College for Men, Krishnagiri, 635 001 Tamil Nadu, India gacmenkrishnagiri.org

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Nazek Alessa

Nazek Alessa

Department of Mathematical Sciences, Faculty of Science, Princess Nourah Bint Abdulrahman University, Saudi Arabia pnu.edu.sa

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K. Loganathan

K. Loganathan

Research and Development Wing, Live4Research, Tiruppur, 638106 Tamil Nadu, India smvdu.ac.in

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G. Balasubramanian

G. Balasubramanian

Department of Mathematics, Government Arts College for Men, Krishnagiri, 635 001 Tamil Nadu, India gacmenkrishnagiri.org

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Ngawang Namgyel

Corresponding Author

Ngawang Namgyel

Department of Humanities and Management, Jigme Namgyel Engineering College, Royal University of Bhutan, Dewathang, Bhutan rub.edu.bt

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First published: 04 August 2021
Citations: 3
Academic Editor: Liliana Guran

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(∥xp+∥yp). In 1994, a generalization of the Th.M. Rassias’ theorem was got by Gavruta [5], who replaced ε(∥xp+∥yp) by a general control function φ(x, y). For additional information regarding the outcomes about such issues, the related background in [612] can be examined. Absorbing new outcomes concerning mixed-type functional equations has as of late been acquired by Najati et al. [1315], Jun and Kim [16, 17], and Park [1822].

The functional equations
(1)
and
(2)
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 [2729], and Saadati et al. [30]. Xu et al. [3133] 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. [3335] 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
(3)
where a is a fixed integer and m ≥ 5 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 [3641].

Definition 1 (see [42].)A mapping μ : ⟶[0, 1] is said to be a measure distribution function, if μ is left continuous on , non-decreasing, inftμ(t) = 0, and suptμ(t) = 1.

Definition 2 (see [42].)A mapping ν : ⟶[0, 1] is said to be a non-measure distribution function, if ν is right continuous on , non-increasing, suptν(t) = 1, and inftν(t) = 0.

Lemma 3 (see [43], [44].)Let L be a set with an operator is defined by

(4)

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., , ∀pL;

  • (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 pL, 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

(5)
for all p ∈ [0, 1].

Definition 6 (see [23].)Let μ and ν be measure and nonmeasure distribution functions from V × (0, +∞) to [0, 1] such that

(6)

The triple (V, Iμ,ν, Φ) is said to be an intuitionistic random normed space if a vector space V, continuous t-representable Φ, and a mapping Iμ,ν : V × (0, +∞)⟶L holds the following conditions: for all p, qV and t1, t2 > 0
(7)
Thus, Iμ,ν is called an intuitionistic random norm. Hence,
(8)

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

(9)

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 satisfies

(10)
  • (ii)

    The sequence {pm} is convergent to any point pV 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 pV

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 : VW satisfies the functional equation (3) for all v1, v2, ⋯, vmV, then the function φ is additive.

Proof. In the view of the oddness of φ, we have φ(−v) = −φ(v) for all vV. Using the oddness property, the functional equation (3) reduces as

(11)
for all v1, v2, ⋯, vmV. Now, replacing (v1, v2, ⋯, vm) by (0, 0⋯, 0) in (11), we get φ(0) = 0. Interchanging (v1, v2, ⋯, vm) with (v, 0, 0, ⋯, 0) in (11), we get
(12)

Again interchanging v with av in (12), we have

(13)
for all vV. Replacing v by av in (13), we obtain
(14)

From the equalities (12)–(14), we can generalize the results for any nonnegative integer m as

(15)

Similarly, we have

(16)

Replacing (v1, v2, ⋯, vm) by ((x/a), (y/a2), 0, ⋯, 0) in (11), we have

(17)

Hence, the function φ is additive.☐

Theorem 9. If an even mapping φ : VW satisfies the functional equation (3) for all v1, v2, ⋯, vmV, then the function φ is quadratic.

Proof. Since, in the view of evenness of φ, we have φ(−v) = φ(v) for all vV. Now, the functional equation (3) reduces as

(18)
for all v1, v2, ⋯, vmV. Now, replacing (v1, v2, ⋯, vm) by (0, 0, ⋯, 0) in (18), we obtain φ(0) = 0. Interchanging (v1, v2, ⋯, vm) with (v, 0, 0, ⋯, 0) in (18), we obtain
(19)

Replacing v by av in (19), we reach

(20)

Switching v by av in (20), we get

(21)

From (19)–(21), we can generalize the results for any nonnegative integer m as

(22)

Similarly, we have

(23)

Replacing (v1, v2, ⋯, vm) by ((x/a), (y/a2), 0, ⋯, 0) in (18), we obtain

(24)

Hence, the function φ is quadratic.☐

Theorem 10. If a mapping φ : VW satisfies φ(0) = 0 and satisfies the functional equation (3) for all v1, v2, ⋯, vmV if and only if there exists a symmetric biadditive mapping Q : V × VW and a additive mapping A : VW satisfies φ(v) = Q(v, v) + A(v) for all vV.

Proof. Let a mapping φ : VW with φ(0) = 0 satisfies the functional equation (3). We divide the function φ into the odd part and even part as

(25)
respectively. Clearly, φ(v) = φe(v) + φo(v) for all vV.☐

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 × VW which satisfies φe(v) = Q(v, v) and an additive mapping A : VW which satisfies φo(v) = A(v) for all vV. Hence, φ(v) = Q(v, v) + A(v) for all vV.

Conversely, suppose that there exists a symmetric biadditive mapping Q : V × VW and an additive mapping A : VW and satisfies φ(v) = Q(v, v) + A(v) for all vV. It is easy to prove that the mappings vQ(v, v) and A : VW satisfy the functional equation (3). Hence, the mapping φ : VW satisfies the functional equation (3).

For our notational convenience, we can define a mapping φ : VW by
(26)
for all v1, v2, ⋯, vmV.

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 : VmD+, 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

(27)
for all v1, v2, ⋯, vmV and all ε > 0, and
(28)
for all vV and all ε > 0. If an even mapping φ : VW with φ(0) = 0 satisfies
(29)
for all v1, v2, ⋯, vmV and all ε > 0, then there exists a unique quadratic mapping Q2 : VW such that
(30)
for all vV and all ε > 0.

Proof. Replacing (v1, v2, ⋯, vm) by (v, 0, ⋯, 0) in (29), we have

(31)
for all vV and all ε > 0. From inequality (31), we get
(32)

Interchanging v with av in (32), we obtain

(33)

Replacing v by al−1v and divide by a2l in (33), we conclude that

(34)
for all vV and all ε > 0. Thus,
(35)
for all vV and all ε > 0. To prove the convergence of the sequence {φ(awv)/a2w}, replacing v by akv in (35), we obtain
(36)
for all vV and all ε > 0 and all k, w ≥ 0. Since the R.H.S of the inequality (36) tends to as w, k⟶∞, the sequence {φ(awv)/a2w} is a Cauchy sequence in (W, Iμ,ν, Y). Since (W, Iμ,ν, Y) is a complete IRN-space, this sequence converges to some point Q2(v) ∈ W. So one can define the mapping Q2 : VW by
(37)
for all vV. Letting k = 0 in (36), we obtain
(38)
for all vV and all ε > 0. Taking the limit w⟶∞ in (38), we get
(39)
for all vV and all ε > 0.

Next, we prove that the function Q2 is quadratic. Replacing (v1, v2, ⋯, vm) by (awv1, awv2, ⋯, awvm) in (29), we obtain
(40)
for all v1, v2, ⋯, vmV and all ε > 0. Taking the limit as w⟶∞, we find that for all v1, v2, ⋯, vmV and all ε > 0, which implies DQ2(v1, v2, ⋯, vm) = 0. Thus, the function Q2 satisfies the functional equation (3). Hence, Q2 : VW is a quadratic mapping. Passing to the limit as w⟶∞ in (35), we have (30).
Finally, to show the uniqueness of Q2 subject to (30), consider that there exists an another quadratic function which satisfies the inequality (30). Clearly, Q2(awv) = a2wQ2(v) and for all vV and w, from (30) and (28) that
(41)
for all vV and all ε > 0. By taking w⟶∞ in (41), we show the uniqueness of Q2. This ends the proof of the uniqueness, as desired.☐

Corollary 12. If an even mapping φ : VW satisfies

(42)
for all v1, v2, ⋯, vmV and all ε > 0, and
(43)
for all vV and all ε > 0, then there exists a unique quadratic mapping Q2 : VW such that
(44)
for all vV and all ε > 0.

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

5. Stability Results for Odd Case

Theorem 13. Let α, β : VmD+, 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

(45)
for all v1, v2, ⋯, vmV and all ε > 0, and
(46)
for all vV and all ε > 0. If an odd mapping φ : VW with φ(0) = 0 satisfies
(47)
for all v1, v2, ⋯, vmV and all ε > 0, then there exist a unique additive mapping A1 : VW such that
(48)
for all vV and all ε > 0.

Proof. Replacing (v1, v2, ⋯, vm) by (v, 0, ⋯, 0) in (47), we obtain

(49)
for all vV and all ε > 0. From inequality (49), we get
(50)
for all vV and all ε > 0. Replacing v by av in the above inequality (50), we have
(51)
for all vV and all ε > 0. Replacing v by al−1v in (51), we conclude that
(52)
for all vV and all ε > 0. Thus,
(53)
for all vV and all ε > 0. To prove the convergence of the sequence {φ(awv)/aw}, replacing v by akv in (53), we obtain
(54)
for all vV and all ε > 0 and all k, w ≥ 0. Since the R.H.S of the inequality (54) tends to as w, k⟶∞, the sequence {φ(awv)/aw} is a Cauchy sequence in (W, Iμ,ν, Y). Since (W, Iμ,ν, Y) is a complete IRN-space, this sequence converges to some point A1(v) ∈ W. So one can define the mapping A1 : VW by
(55)
for all vV. Letting k = 0 in (54), we obtain
(56)
for all vV and all ε > 0. Taking the limit as w⟶∞ in (56), we get
(57)
for all vV and all ε > 0.

Next, we want to prove that the function A1 is additive. Replacing (v1, v2, ⋯, vm) by (awv1, awv2, ⋯, awvm) in (47), we obtain
(58)
for all v1, v2, ⋯, vmV and all ε > 0. Taking the limit as w⟶∞, we find that for all v1, v2, ⋯, vmV and all ε > 0, which implies DA1(v1, v2, ⋯, vm) = 0. Thus, A1 satisfies the functional equation (3). Hence, the function A1 : VW is additive. Passing to the limit as w⟶∞ in (53), we have (48).
Finally, to show the uniqueness of the additive function A1 subject to (48), consider that there exists another additive function which satisfies the inequality (48). Evidently, A1(awv) = awA1(v) and for all vV and w, from (48) and (46) that
(59)
for all vV and all ε > 0. By taking the limit w⟶∞ in (59), we show the uniqueness of A1.☐

Corollary 14. If an odd mapping φ : VW satisfies

(60)
for all v1, v2, ⋯, vmV and all ε > 0, and
(61)
for all vV and all ε > 0. Then, there exists a unique additive mapping A1 : VW such that
(62)
for all vV and all ε > 0.

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

6. Stability Results for Mixed Case

Theorem 15. Let α, β : VmD+ be mappings satisfying (27), (28), (45), and (46) for all v1, v2, ⋯, vm, vV and all ε > 0. If a mapping φ : VW with φ(0) = 0 satisfies (29) for all v1, v2, ⋯, vmV and all ε > 0, then there exist a unique quadratic mapping Q2 : VW and a unique additive mapping A1 : VW satisfying (3) and

(63)
for all vV.

Proof. Let φe(v) = φ(v) + φ(−v)/2 for all vV. Thus, φe(0) = 0, φe(−v) = φe(v) and for all v1, v2, ⋯, vmV and all ε > 0,

(64)

By Theorem 11, there exists a quadratic mapping Q2 : VW such that

(65)
for all vV and all ε > 0.☐

On the other hand, let φo(v) = φ(v) − φ(−v)/2 for all vV. Then φo(0) = 0, φo(−v) = −φo(v). By Theorem 13, there exists a additive mapping A1 : VW satisfies
(66)
for all vV and all ε > 0. From inequalities (65) and (66), we obtain our desired result (64).

Corollary 16. If a mapping φ : VW satisfies

(67)
for all v1, v2, ⋯, vmV and all ε > 0, and
(68)
for all vV and all ε > 0. Then, there exists a unique quadratic mapping Q2 : VW and a unique additive mapping A1 : VW such that
(69)
for all vV and all ε > 0.

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.

    Data Availability

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