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

Authors: K. Tamilvanan; Nazek Alessa; K. Loganathan; G. Balasubramanian; Ngawang Namgyel
Journal: Journal of Function Spaces
 (2021)

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## 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).


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## Body

## 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 fx+y−fx−fy 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 φx,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].The functional equations(1)fx+y=fx+fy,and
(2)fx+y+fx−y=2fx+2fy,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(3)φ∑i=1maivi+∑i=1mφ−aivi+∑j=1;i≠jmajvj=m−3∑i=1mφaivi+ajvj−m2−5m+2∑i=1ma2iφvi+φ−vi2−m2−5m+4∑i=1maiφvi−φ−vi2,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 [36–41].Definition 1 (see [42]).
A mappingμ:ℝ⟶0,1 is said to be a measure distribution function, if μ is left continuous on ℝ, non-decreasing, inft∈ℝμt=0, and supt∈ℝμt=1.Definition 2 (see [42]).
A mappingν:ℝ⟶0,1 is said to be a non-measure distribution function, if ν is right continuous on ℝ, non-increasing, supt∈ℝνt=1, and inft∈ℝνt=0.Lemma 3 (see [43, 44]).
LetL∗ be a set with an operator ≤L∗ is defined by
(4)L∗=v1,v2:v1,v2∈0,12andv1+v2≤1,v1,v2≤L∗w1,w2⇔v1≤w1,v2≥w2,v1,v2,w1,w2∈L∗.
Then, the pairL∗,≤L∗ is a complete lattice.We denote its units by0L∗=0,1 and 1L∗=1,0. Typically, a triangular norm (t-norm) ∗=Φ on 0,1 is defined as an increasing, commutative, associative mapping Φ:0,12⟶0,1 satisfying Φ1,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:0,12⟶0,1 satisfying Y0,p=0⋄p=p for all p∈0,1.By using the latticeL∗,≤L∗, these definitions can be straightforwardly extended.Definition 4 (see [44]).
A triangular norm (t-norm) onL∗is a mappingΦ:L∗2⟶L∗satisfying the following conditions:(i)
Boundary condition
i.e.,Φp,1L∗=p, ∀p∈L∗;(ii)
Commutativity
i.e.,Φp,q=Φq,p, ∀p,q∈L∗2;(iii)
Associativity
i.e.,Φp,Φq,r=ΦΦp,q,r, ∀p,q,r∈L∗3;(iv)
Monotonicity
i.e.,p≤L∗p′ and q≤L∗q′⇒Φp,q≤L∗Φp′,q′ for all p,p′,q,q′∈L∗4.
IfL∗,≤L∗,Φ is an Abelian topological monoid with unit 1L∗, then Φ is called a continuous t-norm.Definition 5 (see [42]).
A negator onL∗is any decreasing mappingNfromL∗toL∗satisfyingN1L∗=0L∗andN1L∗=0L∗. IfNNp=pfor allp∈L∗, thenNis called an involutive negator. A negator on0,1is a decreasing mappingN:0,1⟶0,1satisfyingN0=1andN1=0.
Ns denotes the standard negator on 0,1 defined by
(5)Nsp=1−p,for all p∈0,1.Definition 6 (see [23]).
Letμ and ν be measure and nonmeasure distribution functions from V×0,+∞ to 0,1 such that
(6)μpt+νpt≤1,p∈V,t>0.The tripleV,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,q∈V and t1,t2>0(7)Iμ,νp,0=0L∗,Iμ,νp,t1=1L∗⇔p=0,Iμ,ναp,t1=Iμ,νp,t1α,∀α≠0,Iμ,νp+q,t1+t2≥L∗ΦIμ,νp,t1,Iμ,νq,t2.Thus,Iμ,ν is called an intuitionistic random norm. Hence,
(8)Iμ,νp,t1=μpt1,νpt1.Example 1 (see [42]).
LetV,·be a normed space. LetΦp,q=p1q1,minp2+q2,1for allp=p1,p2, q=q1,q2∈L∗and letμ,νbe measure and non-measure distribution functions defined by(9)Iμ,νv,ε=μvε,νvε=εε+∥v∥,∥v∥ε+∥v∥,∀ε∈R+.
Then,V,Iμ,ν,Φ is an IRN-space.Definition 7 (see [42]).
LetV,Iμ,ν,Φbe an IRN-space.(i)
A sequencepm 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(10)Iμ,νpm−pn,t≥L∗Nsδ,δ,m,n≥m0.(ii)
The sequencepm is convergent to any point p∈V if Iμ,νpm−p,t⟶1L∗ as m⟶∞ for all t>0(iii)
An intuitionistic random normed spaceV,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 considerV and W are two real vector spaces.Theorem 8.
If an odd mappingφ:V⟶Wsatisfies the functional equation (3) for allv1,v2,⋯,vm∈V, then the functionφis additive.Proof.
In the view of the oddness ofφ, we have φ−v=−φv for all v∈V. Using the oddness property, the functional equation (3) reduces as
(11)φ∑i=1maivi+∑i=1mφ−aivi+∑j=1;i≠jmajvj=m−3∑i=1mφaivi+ajvj−m2−5m+4∑i=1maiφvi,for all v1,v2,⋯,vm∈V. Now, replacing v1,v2,⋯,vm by 0,0⋯,0 in (11), we get φ0=0. Interchanging v1,v2,⋯,vm with v,0,0,⋯,0 in (11), we get
(12)φav=aφv,v∈V.
Again interchangingv with av in (12), we have
(13)φa2v=a2φv,for all v∈V. Replacing v by av in (13), we obtain
(14)φa3v=a3φv,v∈V.
From the equalities (12)–(14), we can generalize the results for any nonnegative integer m as
(15)φamv=amφv,v∈V.
Similarly, we have(16)φvam=1amφv,v∈V.
Replacingv1,v2,⋯,vm by x/a,y/a2,0,⋯,0 in (11), we have
(17)φx+y=φx+φy,x,y∈V.
Hence, the functionφ is additive.☐Theorem 9.
If an even mappingφ:V⟶Wsatisfies the functional equation (3) for allv1,v2,⋯,vm∈V, then the functionφis quadratic.Proof.
Since, in the view of evenness ofφ, we have φ−v=φv for all v∈V. Now, the functional equation (3) reduces as
(18)φ∑i=1maivi+∑i=1mφ−aivi+∑j=1;i≠jmajvj=m−3∑i=1mφaivi+ajvj−m2−5m+2∑i=1ma2iφvi.for all v1,v2,⋯,vm∈V. Now, replacing v1,v2,⋯,vm by 0,0,⋯,0 in (18), we obtain φ0=0. Interchanging v1,v2,⋯,vm with v,0,0,⋯,0 in (18), we obtain
(19)φav=a2φv,v∈V.
Replacingv by av in (19), we reach
(20)φa2v=a4φv,v∈V.
Switchingv by av in (20), we get
(21)φa3v=a6φv,v∈V.
From (19)–(21), we can generalize the results for any nonnegative integer m as
(22)φamv=a2mφv,v∈V.
Similarly, we have(23)φvam=1a2mφv,v∈V.
Replacingv1,v2,⋯,vm by x/a,y/a2,0,⋯,0 in (18), we obtain
(24)φx+y+φx−y=2φx+2φy,x,y∈V.
Hence, the functionφ is quadratic.☐Theorem 10.
If a mappingφ:V⟶Wsatisfiesφ0=0and satisfies the functional equation (3) for allv1,v2,⋯,vm∈Vif and only if there exists a symmetric biadditive mappingQ:V×V⟶Wand a additive mappingA:V⟶Wsatisfiesφv=Qv,v+Avfor allv∈V.Proof.
Let a mappingφ:V⟶W with φ0=0 satisfies the functional equation (3). We divide the function φ into the odd part and even part as
(25)φov=φv−φ−v2,φev=φv+φ−v2,v∈V,respectively. Clearly, φv=φev+φov for all v∈V.☐It is easy to prove thatφo 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 φev=Qv,v and an additive mapping A:V⟶W which satisfies φov=Av for all v∈V. Hence, φv=Qv,v+Av for all v∈V.Conversely, suppose that there exists a symmetric biadditive mappingQ:V×V⟶W and an additive mapping A:V⟶W and satisfies φv=Qv,v+Av for all v∈V. It is easy to prove that the mappings v↦Qv,v and A:V⟶W satisfy the functional equation (3). Hence, the mapping φ:V⟶W satisfies the functional equation (3).For our notational convenience, we can define a mappingφ:V⟶W by
(26)Dφv1,v2,⋯,vm=φ∑i=1maivi+∑i=1mφ−aivi+∑j=1;i≠jmajvj−m−3∑i=1mφaivi+ajvj+m2−5m+2∑i=1ma2iφvi+φ−vi2+m2−5m+4∑i=1maiφvi−φ−vi2,for all v1,v2,⋯,vm∈V.In the following sections, we considerV is a linear space, V,Iμ′,ν′,Y 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.
Letα,β:Vm⟶D+, whereαv1,v2,⋯,vmis denoted byαv1,v2,⋯,vm, βv1,v2,⋯,vmis denoted byβv1,v2,⋯,vmandαv1,v2,⋯,vmε,βv1,v2,⋯,vmεis denoted byΘα,βv1,v2,⋯,vm,ε, be a mapping such that(27)limw⟶∞Θα,βawv1,awv2,⋯,awvm,a2wε=1L∗,for all v1,v2,⋯,vm∈V and all ε>0, and
(28)limw⟶∞Φi=1∞Θα,βaw+i−1v,0,⋯,0,a2w+iε=1L∗,for all v∈V and all ε>0. If an even mapping φ:V⟶W with φ0=0 satisfies
(29)Iμ,νDφv1,v2,⋯,vm,ε≥L∗Θα,βv1,v2,⋯,vm,ε,for all v1,v2,⋯,vm∈V and all ε>0, then there exists a unique quadratic mapping Q2:V⟶W such that
(30)Iμ,νφv−Q2v,ε≥L∗Φi=1∞Θα,βai−1v,0,⋯,0,aim2−5m+2ε,for all v∈V and all ε>0.Proof.
Replacingv1,v2,⋯,vm by v,0,⋯,0 in (29), we have
(31)Iμ,νm2−5m+2φav−m2−5m+2a2φv,ε≥L∗Θα,βv,0,⋯,0,ε,for all v∈V and all ε>0. From inequality (31), we get
(32)Iμ,νφava2−φv,ε≥L∗Θα,βv,0,⋯,0,a2m2−5m+2ε,v∈V,ε>0.
Interchangingv with av in (32), we obtain
(33)Iμ,νφa2va4−φava2,ε≥L∗Θα,βav,0,⋯,0,a22m2−5m+2ε,v∈V,ε>0.
Replacingv by al−1v and divide by a2l in (33), we conclude that
(34)Iμ,νφal+1va2l+1−φalva2l,ε≥L∗Θα,βalv,0,⋯,0,a2l+1m2−5m+2ε,for all v∈V and all ε>0. Thus,
(35)Iμ,νφawva2w−φv,ε≥L∗Φl=0w−1Iμ,νφal+1va2l+1−φalva2l,∑l=0w−1εal+1,for all v∈V and all ε>0. To prove the convergence of the sequence φawv/a2w, replacing v by akv in (35), we obtain
(36)Iμ,νφaw+kva2w+k−φakva2k,ε≥L∗Φl=kw+k−1Iμ,νφal+1va2l+1−φalva2l,∑l=kw+k−1εal+1,for all v∈V and all ε>0 and all k,w≥0. Since the R.H.S of the inequality (36) tends to 1L∗ as w,k⟶∞, the sequence φawv/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 Q2v∈W. So one can define the mapping Q2:V⟶W by
(37)Q2v=limw⟶∞φawva2w,for all v∈V. Letting k=0 in (36), we obtain
(38)Iμ,νφawva2w−φv,ε≥L∗Φl=0w−1Iμ,νφal+1va2l+1−φalva2l,∑l=0w−1εal+1,for all v∈V and all ε>0. Taking the limit w⟶∞ in (38), we get
(39)Iμ,νφv−Q2v,ε≥L∗Φl=1∞Θα,βal−1v,0,⋯,0,alm2−5m+2ε,for all v∈V and all ε>0.Next, we prove that the functionQ2 is quadratic. Replacing v1,v2,⋯,vm by awv1,awv2,⋯,awvm in (29), we obtain
(40)Iμ,ν1a2wDφawv1,awv2,⋯,awvm,ε≥L∗Θα,βawv1,awv2,⋯,awvm,a2wε,for all v1,v2,⋯,vm∈V and all ε>0. Taking the limit as w⟶∞, we find that Iμ,νDQ2v1,v2,⋯,vm,ε=1L∗ for all v1,v2,⋯,vm∈V and all ε>0, which implies DQ2v1,v2,⋯,vm=0. Thus, the function Q2 satisfies the functional equation (3). Hence, Q2:V⟶W is a quadratic mapping. Passing to the limit as w⟶∞ in (35), we have (30).Finally, to show the uniqueness ofQ2 subject to (30), consider that there exists an another quadratic function Q2′ which satisfies the inequality (30). Clearly, Q2awv=a2wQ2v and Q2′awv=a2wQ2′v for all v∈V and w∈ℕ, from (30) and (28) that
(41)Iμ,νQ2v−Q2′v,ε≥L∗Iμ,νQ2awv−Q2′awv,a2wε≥L∗ΦIμ,νQ2awv−φawv,a2w2ε,Iμ,νφawv−Q2′awv,a2w2ε≥L∗ΦΦi=1∞Θα,βaw+i−1v,0,⋯,0,a2w+im2−5m+2ε2,Φi=1∞Θα,βaw+i−1v,0,⋯,0,a2w+im2−5m+2ε2,for all v∈V 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 mappingφ:V⟶Wsatisfies(42)Iμ,νDφv1,v2,⋯,vm,ε≥L∗Iμ′,ν′∑i=1mvi,ε,for all v1,v2,⋯,vm∈V and all ε>0, and
(43)limw⟶∞Φi=1∞Iμ′,ν′aw+i−1v,a2w+im2−5m+2ε=1L∗,for all v∈V and all ε>0, then there exists a unique quadratic mapping Q2:V⟶W such that
(44)Iμ,νφv−Q2v,ε≥L∗Φi=1∞Iμ′,ν′ai−1v,aim2−5m+2ε,for all v∈V and all ε>0.Proof.
By takingΘα,βv1,v2,⋯,vm,ε=Iμ′,ν′∑i=1mvi,ε in Theorem 11, we obtain our desired result.☐


## 5. Stability Results for Odd Case

Theorem 13.
Letα,β:Vm⟶D+, whereαv1,v2,⋯,vmis denoted byαv1,v2,⋯,vm, βv1,v2,⋯,vmis denoted byβv1,v2,⋯,vmandαv1,v2,⋯,vmε,βv1,v2,⋯,vmεis denoted byΘα,βv1,v2,⋯,vm,ε, be a mapping such that(45)limw⟶∞Θα,βawv1,awv2,⋯,awvm,awε=1L∗,for all v1,v2,⋯,vm∈V and all ε>0, and
(46)limw⟶∞Φi=1∞Θα,βaw+i−1v,0,⋯,0,awε=1L∗,for all v∈V and all ε>0. If an odd mapping φ:V⟶W with φ0=0 satisfies
(47)Iμ,νDφv1,v2,⋯,vm,ε≥L∗Θα,βv1,v2,⋯,vm,ε,for all v1,v2,⋯,vm∈V and all ε>0, then there exist a unique additive mapping A1:V⟶W such that
(48)Iμ,νφv−A1v,ε≥L∗Φi=1∞Θα,βai−1v,0,⋯,0,m2−5m+4ε,for all v∈V and all ε>0.Proof.
Replacingv1,v2,⋯,vm by v,0,⋯,0 in (47), we obtain
(49)Iμ,νm2−5m+4φav−m2−5m+4aφv,ε≥L∗Θα,βv,0,⋯,0,ε,for all v∈V and all ε>0. From inequality (49), we get
(50)Iμ,νφava−φv,ε≥L∗Θα,βv,0,⋯,0,am2−5m+4ε,for all v∈V and all ε>0. Replacing v by av in the above inequality (50), we have
(51)Iμ,νφa2va2−φava,ε≥L∗Θα,βav,0,⋯,0,a2m2−5m+4ε,for all v∈V and all ε>0. Replacing v by al−1v in (51), we conclude that
(52)Iμ,νφal+1val+1−φalval,ε≥L∗Θα,βalv,0,⋯,0,al+1m2−5m+4ε,for all v∈V and all ε>0. Thus,
(53)Iμ,νφawvaw−φv,ε≥L∗Φl=0w−1Iμ,νφal+1val+1−φalval,∑l=0w−1εal+1,for all v∈V and all ε>0. To prove the convergence of the sequence φawv/aw, replacing v by akv in (53), we obtain
(54)Iμ,νφaw+kvaw+k−φakvak,ε≥L∗Φl=kw+k−1Iμ,νφal+1val+1−φalval,∑l=kw+k−1εal+1,for all v∈V and all ε>0 and all k,w≥0. Since the R.H.S of the inequality (54) tends to 1L∗ as w,k⟶∞, the sequence φawv/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 A1v∈W. So one can define the mapping A1:V⟶W by
(55)A1v=limw⟶∞φawvaw,for all v∈V. Letting k=0 in (54), we obtain
(56)Iμ,νφawvaw−φv,ε≥L∗Φl=0w−1Iμ,νφal+1val+1−φalval,∑l=0w−1εal+1,for all v∈V and all ε>0. Taking the limit as w⟶∞ in (56), we get
(57)Iμ,νφv−A1v,ε≥L∗Φl=1∞Θα,βal−1v,0,⋯,0,m2−5m+4ε,for all v∈V and all ε>0.Next, we want to prove that the functionA1 is additive. Replacing v1,v2,⋯,vm by awv1,awv2,⋯,awvm in (47), we obtain
(58)Iμ,ν1awDφawv1,awv2,⋯,awvm,ε≥L∗Θα,βawv1,awv2,⋯,awvm,awε,for all v1,v2,⋯,vm∈V and all ε>0. Taking the limit as w⟶∞, we find that Iμ,νDA1v1,v2,⋯,vm,ε=1L∗ for all v1,v2,⋯,vm∈V and all ε>0, which implies DA1v1,v2,⋯,vm=0. Thus, A1 satisfies the functional equation (3). Hence, the function A1:V⟶W is additive. Passing to the limit as w⟶∞ in (53), we have (48).Finally, to show the uniqueness of the additive functionA1 subject to (48), consider that there exists another additive function A1′ which satisfies the inequality (48). Evidently, A1awv=awA1v and A1′awv=awA1′v for all v∈V and w∈ℕ, from (48) and (46) that
(59)Iμ,νA1v−A1′v,ε≥L∗Iμ,νA1awv−A1′awv,awε≥L∗ΦIμ,νA1awv−φawv,aw2ε,Iμ,νφawv−A1′awv,aw2ε≥L∗ΦΦi=1∞Θα,βaw+i−1v,0,⋯,0,awm2−5m+4ε2,Φi=1∞Θα,βaw+i−1v,0,⋯,0,awm2−5m+4ε2,for all v∈V and all ε>0. By taking the limit w⟶∞ in (59), we show the uniqueness of A1.☐Corollary 14.
If an odd mappingφ:V⟶Wsatisfies(60)Iμ,νDφv1,v2,⋯,vm,ε≥L∗Iμ′,ν′∑i=1mvi,ε,for all v1,v2,⋯,vm∈V and all ε>0, and
(61)limw⟶∞Φi=1∞Iμ′,ν′aw+i−1v,awm2−5m+4ε=1L∗,for all v∈V and all ε>0. Then, there exists a unique additive mapping A1:V⟶W such that
(62)Iμ,νφv−A1v,ε≥L∗Φi=1∞Iμ′,ν′ai−1v,m2−5m+4ε,for all v∈V and all ε>0.Proof.
By takingΘα,βv1,v2,⋯,vm,ε=Iμ′,ν′∑i=1mvi,ε in Theorem 13, we obtain our desired result.☐


## 6. Stability Results for Mixed Case

Theorem 15.
Letα,β:Vm⟶D+be mappings satisfying (27), (28), (45), and (46) for allv1,v2,⋯,vm,v∈Vand allε>0. If a mappingφ:V⟶Wwithφ0=0satisfies (29) for allv1,v2,⋯,vm∈Vand allε>0, then there exist a unique quadratic mappingQ2:V⟶Wand a unique additive mappingA1:V⟶Wsatisfying (3) and(63)Iμ,νφv−Q2v−A1v,ε≥L∗ΦΦΦi=1∞ai−1v,0,⋯,0,aim2−5m+2ε,Φi=1∞−ai−1v,0,⋯,0,aim2−5m+2εΦΦi=1∞ai−1v,0,⋯,0,m2−5m+4ε,Φi=1∞−ai−1v,0,⋯,0,m2−5m+4ε,for all v∈V.Proof.
Letφev=φv+φ−v/2 for all v∈V. Thus, φe0=0, φe−v=φev and for all v1,v2,⋯,vm∈V and all ε>0,
(64)Iμ,νDφev1,v2,⋯,vm,ε=Iμ,νDφv1,v2,⋯,vm+Dφ−v1,−v2,⋯,−vm2,ε,≥L∗ΦIμ,νDφv1,v2,⋯,vm,ε,Iμ,νDφ−v1,−v2,⋯,−vm,ε,≥L∗ΦΘα,βv1,v2,⋯,vm,ε,Θα,β−v1,−v2,⋯,−vm,ε.,
By Theorem11, there exists a quadratic mapping Q2:V⟶W such that
(65)Iμ,νφev−Q2v,ε≥L∗ΦΦi=1∞ai−1v,0,⋯,0,aim2−5m+2ε,Φi=1∞−ai−1v,0,⋯,0,aim2−5m+2ε,for all v∈V and all ε>0.☐On the other hand, letφov=φv−φ−v/2 for all v∈V. Then φo0=0, φo−v=−φov. By Theorem 13, there exists a additive mapping A1:V⟶W satisfies
(66)Iμ,νφ0v−A1v,ε≥L∗ΦΦi=1∞ai−1v,0,⋯,0,m2−5m+4ε,Φi=1∞−ai−1v,0,⋯,0,m2−5m+4ε,for all v∈V and all ε>0. From inequalities (65) and (66), we obtain our desired result (64).Corollary 16.
If a mappingφ:V⟶Wsatisfies(67)Iμ,νDφv1,v2,⋯,vm,ε≥L∗Iμ′,ν′∑i=1mvi,ε,for all v1,v2,⋯,vm∈V and all ε>0, and
(68)limw⟶∞Φi=1∞Iμ′,ν′aw+i−1v,a2w+im2−5m+2ε=1L∗,limw⟶∞Φi=1∞Iμ′,ν′aw+i−1v,awm2−5m+4ε=1L∗,for all v∈V and all ε>0. Then, there exists a unique quadratic mapping Q2:V⟶W and a unique additive mapping A1:V⟶W such that
(69)Iμ,νφv−Q2v−A1v,ε≥L∗ΦΦΦi=1∞ai−1v,aim2−5m+2ε,Φi=1∞−ai−1v,aim2−5m+2εΦΦi=1∞ai−1v,m2−5m+4ε,Φi=1∞−ai−1v,m2−5m+4ε,for all v∈V 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 Section4, 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.



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