# Hyers–Ulam Stability of Additive Functional Equation Using Direct and Fixed-Point Methods

Authors: K. Tamilvanan; G. Balasubramanian; Nazek Alessa; K. Loganathan
Journal: Journal of Mathematics
 (2020)

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## Abstract

In this present work, we obtain the solution of the generalized additive functional equation and also establish Hyers–Ulam stability results by using alternative fixed point for a generalized additive functional equationχ∑g=1lvg=∑1≤g<h<i≤lχvg+vh+vi−∑1≤g<h≤lχvg+vh−l2−5l+2/2∑g=1lχvg−χ−vg/2. where l is a nonnegative integer with ℕ−0,1,2,3,4 in Banach spaces.


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## Body

## 1. Introduction

The problem of Ulam–Hyers stability concerns determining circumstances under which, given an approximate solution of a functional equation, one may locate an exact key that is closer to it in some sense. The investigation of stability problem for functional equations is identified to a question of Ulam [1] about the stability of group homomorphisms and affirmatively answered for Banach space by Hyers [2, 3]. It was further generalized and interesting results were obtained by a number of authors [4–6].Rassias investigated the Hyers–Ulam stability results for the various functional equations in [7–10] through different spaces. Czerwik [11, 12] examined the stability of the quadratic functional equation involving several variables in the normed spaces. Numerous mathematicians investigated the various stability results in [13–32].In 2019, Park et al. [33] introduced additive s-functional inequality. Using the fixed-point method and direct method, he established the Hyers–Ulam stability for the abovementioned one in complex Banach spaces. Also, he examined the Hyers–Ulam stability of homomorphism and derivations in complex Banach algebras. In 2018, Almahalebi [34] investigated the quadratic functional equation in Banach spaces. And, he established the hyperstability outcome of the same equation through the fixed-point approach.Radu [35] investigated various results about the stability problem by using the fixed-point alternative. He applied the fixed-point method to examine the stability of Cauchy functional equation and Jensen’s functional equations. After his work, numerous authors used the fixed-point method to investigate several functional equations [36–41]. The functional equation(1)fx+y=fx+fy,is called the Cauchy additive functional equation and it is the most famous functional equation. As fx=cx is the solution of (1), every solution of the additive equation is called an additive function.In this present work, we derive the solution of the generalized additive functional equation along with established Hyers–Ulam stability results by using direct and fixed-point methods for a generalized additive functional equation(2)χ∑g=1lvg=∑1≤g<h<i≤lχvg+vh+vi−∑1≤g<h≤lχvg+vh−l2−5l+22∑g=1lχvg−χ−vg2,where l≥5 is a nonnegative integer in Banach spaces.


## 2. General Solution of the Functional Equation (2)

In this section, we derive the general solution of the generalized additive functional equation (2).Here, we considerΦ and Ω be real vector spaces.Theorem 1.
If a mappingχ:Φ⟶Ω satisfies the functional equation (2) for all v1,v2,…,vn∈Φ, then the mapping χ:Φ⟶Ω satisfies the functional equation (1) for all x,y∈Φ.Proof.
Suppose a mappingχ:Φ⟶Ω satisfies the functional equation (2). Replacing v1,v2,v3,…,vl by 0,0,0,…,0 in the functional equation (2), we have χ0=0. Replacing v1,v2,v3,…,vl by v,0,0,…,0 in equation (2), we get χ−v=−χv for all v∈Φ. Therefore, the function χ is an odd function. Replacing v1,v2,v3,…,vl by v,v,0,0,…,0 in equation (2) and using the property of odd function, we have(3)χ2v=2χv,for all v∈Φ. Replacing v by 2v in (3), we obtain(4)χ22v=22χv,for all v∈Φ. Again, replacing v by 2v in (5) and using (3), we have(5)χ23v=23χv,for all v∈Φ. We can generalize for any nonnegative integer n and we get(6)χ2nv=2nχv,for all v∈Φ. Now, replacing v1,v2,v3,…,vl by x,y,0,…,0 in (2), we obtain our desired result of equation (1).Remark 1.
LetΩ be a linear space and a function χ:Φ⟶Ω satisfies the functional equation (2). Then, the following claims hold:(1)
χrkv=rkχv for all v∈ℝ, r∈ℚ, k integers(2)
χv=vχ1 for all v∈ℝ if χ is continuous
In Sections3 and 4, we take Φ be a normed space and Ω be a Banach space. For our convincing effortlessness, we describe a function Θ:Φ⟶Ω as(7)Θv1,v2,v3,…,vl=χ∑g=1lvg−∑1≤g<h<i≤lχvg+vh+vi−∑1≤g<h≤lχvg+vh−l2−5l+22∑g=1lχvg−χ−vg2,for every v1,v2,…,vl∈Φ.


## 3. Hyers–Ulam Stability of the Functional Equation (2): Direct Method

In this section, we investigated the Hyers–Ulam stability of the generalized additive functional equation (2) in Banach space by using the direct method.Theorem 2.
Letζ∈−1,1, ξ:Φl⟶0,∞ be a mapping such that(8)∑w=0∞ξ2wζv1,2wζv2,…,2wζvl2wζ,converges in ℝ with(9)limw⟶∞ξ2wζv1,2wζv2,…,2wζvl2wζ=0,for all v1,v2,…,vl∈Φ. If a mapping Θ:Φ⟶Ω satisfies the inequality(10)Θv1,v2,…,vl≤ξv1,v2,…,vl,for all v1,v2,…,vl∈Φ, then there exists a unique additive mapping Ψ:Φ⟶Ω satisfying equation (2) and(11)χv−Ψv≤12l−4∑w=1−ζ/2∞ξ2wζv,2wζv,0,…,02wζ,for all v∈Φ.Proof.
Assumeζ=1. Replacing v1,v2,…,vl by v,v,0,…,0 in (10), we have(12)l−4χ2v−2l−4χv≤ξv,v,0,…,0,for all v∈Φ. From equality (12), we get(13)χ2v2−χv≤ξv,v,0,…,02l−4,for all v∈Φ. Exchanging v through 2v in (13), we obtain(14)χ22v2−χ2v≤ξ2v,2v,0,…,02l−4,for all v∈Φ. From (14), we achieve(15)χ22v22−χ2v2≤12ξ2v,2v,0,…,02l−4,for all v∈Φ. Adding together (13) and (15), we get the following outcome:(16)χ22v22−χv≤12l−4ξv,v,0,…,0+ξ2v,2v,0,…,02,for all v∈Φ. It follows from (13), (15), and (16), and we can generalize that as follows:(17)χ2lv2l−χv≤12l−4∑w=0l−1ξ2wv,2wv,0,…,02w,∀v∈Φ,⟹χ2lv2l−χv≤12l−4∑w=0∞ξ2wv,2wv,0,…,02w,for all v∈Φ. In order to establish the convergence of the sequence χ2wv/2w, switch v through 2sv and also divide by 2s in (17). We conclude that, for some w,s>0,(18)χ2w+sv2w+s−χ2sv2s=12sχ2w+sv2w−χ2sv≤12l−4∑w=0l−1ξ2w+sv,2w+sv,0,…,02w+s≤12l−4∑w=0∞ξ2w+sv,2w+sv,0,…,02w+s⟶0,ass⟶∞,for all v∈Φ. Therefore, the sequence χ2wv/2w is a Cauchy. As Ω is complete, there exists Ψ:Φ⟶Ω so that Ψv=limw⟶∞χ2wv/2w for all v∈Φ. Taking the limit w⟶∞ in (17), we obtain that result (11) holds for all v∈Φ. To prove that the function Ψ satisfies equation (2), replacing v1,v2,…,vl by 2wv1,2wv2,…,2wvl and also dividing by 2w in (10), we get(19)12wΘ2wv1,2wv2,…,2wvl≤12wξ2wv1,2wv2,…,2wvl,for all v1,v2,…,vl∈Φ. Taking the limit w⟶∞ in the above inequality and using the definition of Ψv, we have Ψv1,v2,…,vl=0 for all v1,v2,…,vl∈Φ. Thus, the function Ψ satisfies equation (2). To prove that the function Ψ is unique, let φ:Φ⟶Ω be another additive mapping satisfying the functional equation (2) and (11). Hence,(20)Ψv−φv≤12sΨ2sv−χ2sv+χ2sv−φ2sv≤12l−4∑w=0∞ξ2w+sv,2w+sv,0,…,02w+s,∀v∈Φ⟶0ass⟶∞.
Hence,Ψ is unique. Now, replacing v through v/2 in (12), we have(21)l−4χv−2l−4χv2≤ξv2,v2,0,…,0,for all v∈Φ. The rest of the proof is similar to that when ζ=1. So for ζ=−1, we can prove the results by a similar manner. Hence, the proof is completed.Corollary 1.
Letϕ and ϑ be positive real numbers. If there exists a mapping Θ:Φ⟶Ω satisfying the inequality(22)Θv1,v2,…,vl≤ϕ,ϕ∑i=1lviϑ,ϕ∏i=1lviϑ+∑i=1lvilϑ,for all v1,v2,…,vl∈Φ, then there exists a unique additive mapping Ψ:Φ⟶Ω such that(23)χv−Ψv≤ϕl−4,2ϕvϑl−42−2ϑ;ϑ≠1,2ϕvlϑl−42−2lϑ;ϑ≠1l,for all v∈Φ.


## 4. Hyers–Ulam Stability of the Functional Equation (2): Fixed-Point Method

In this section, we examined the Hyers–Ulam stability of the generalized additive functional equation (2) in Banach space by using the fixed-point method.Theorem 3.
LetΨ:Φ⟶Ω be a mapping for which there exists a mapping ξ:Φl⟶0,∞ and(24)limw⟶∞ξηgwv1,ηgwv2,…,ηgwvlηgw=0,where ηg=2,ifg=0,1/2,ifg=1, and such that it satisfies the inequality(25)Ψv1,v2,…,vl≤ξv1,v2,…,vl,for all v1,v2,…,vl∈Φ. If there exists a Lipschitz constant L=Lg such that(26)v⟶λv=1l−4ξv2,v2,0,…,0,has the property(27)ληgvηg=Lλv,for all v∈Φ. Then, there exists a unique additive mapping Ψ:Φ⟶Ω satisfying equation (2) and(28)χv−Ψv≤L1−g1−Lλv,for all v∈Φ.Proof.
Consider a setψ=m/m:Φ⟶Ω,m0=0 and initiate the generalized metric on ψ, dm,n=infw∈0,∞:mv−nv≤wλv,v∈Ψ. It is easy to view that ψ,d is complete. Fix ν:ψ⟶ψ as νmv=1/ηgmηgv for all v∈Φ. For m,n∈ψ and v∈Φ, we have(29)dm,n=w⟹mv−nv≤wλv⟹mηgvηg−nηgvηg≤1ηgwληgv⟹νmv−νnv≤1ηgwληgv⟹dνmv,νnv≤wL.
That is,dνm,νn≤Ldm,n. Accordingly, ν is a strictly contractive mapping on ψ with Lipschitz constant L. From inequality (12), we have(30)l−4χ2v−2l−4χv≤ξv,v,0,…,0,for all v∈Φ. It follows from (30) that(31)χ2v2−χv≤ξv,v,0,…,02l−4,for all v∈Φ. Using inequality (27) when g=0, we get(32)χv−χ2v2≤12λv⟹χv−νmv≤Lλv,for all v∈Φ. So we obtain(33)dνχv,χv≤L=L1−g<∞,for all v∈Φ. Replacing v by v/2 in (31), we have(34)χv−2χv2≤ξv/2,v/2,0,…,0l−4,for all v∈Φ. Using inequality (27) when g=1, we have(35)2χv2−χv≤λv⟹νχv−χv≤λv,for all v∈Φ. Therefore, we obtain the result that(36)dχv,νχv≤1=L0=L1−g,for all v∈Φ. From inequalities (33) and (36), we conclude that(37)dχv,νχv≤L1−g<∞,for all v∈Φ. Next, using fixed-point alternative theorem [35], there exists a fixed point Ψ of ν in ψ such that(38)Ψv=limw⟶∞χηgwvηgw,for all v∈Φ. In order to establish Ψ:Φ⟶Ω satisfying equation (2), we use an argument similar to that in the proof of Theorem 2. As Ψ is a unique fixed point of ν in the set Δ=χ∈ψ/dχ,Ψ<∞, Ψ is a unique function such that(39)dχ,Ψ≤11−Ldχ,νχ⟹dχ,Ψ≤L1−g1−L⟹χv−Ψv≤L1−g1−Lλv,for all v∈Φ. This accomplished the proof.Corollary 2.
Letϕ and ϑ be positive real numbers. If a mapping Θ:Φ⟶Ω satisfies inequality (22) for all v1,v2,…,vl∈Φ, then there exists a unique additive mapping such that (23) for all v∈Φ.Proof.
Consider(40)ξv1,v2,…,vl≤ϕ,ϕ∑i=1lviϑ,ϕ∏i=1lviϑ+∑i=1lvilϑ,for all v1,v2,…,vl∈Φ. Now(41)ξηgwv1,ηgwv2,…,ηgwvlηgw=ϕηgw,ϕηgw∑i=1lηgviϑ,ϕηgw∏i=1lηgviϑ+∑i=1lηgvilϑ,=⟶0,asw⟶∞,⟶0,asw⟶∞,⟶0,asw⟶∞.
That is, (24) holds. As we have(42)λv=1l−4ξv2,v2,0,…,0.
Also,(43)λv=1l−4ξv2,v2,0,…,0=ϕl−4,2ϕvϑl−42ϑ2ϕvlϑl−42lϑ.
And also,(44)1ηgληgv=ϕl−4ηg,2ηgϕvϑηgϑl−42ϑ2ηgϕvlϑηglϑl−42lϑ,=ηg−1λv,ηgϑ−1λv,ηglϑ−1λv,for all v∈Φ. Inequity (2) holds for the following cases:(i)
L=2−1 if g=0 and L=21 if g=1(ii)
L=2ϑ−1 for ϑ<1 if g=0 and L=21−ϑ for ϑ>1 if g=1(iii)
L=2lϑ−1 for ϑ<1/l if g=0 and L=21−lϑ for ϑ>1/l if g=1
Now, from (28), we verify the following cases:Case 1:L=2−1 if g=0:(45)χv−Ψv≤L1−g1−Lλv=2−1ϕl−41−2−1=ϕl−4.Case 2:L=2 if g=1:(46)χv−Ψv≤L1−g1−Lλv=ϕl−41−2=−ϕl−4.Case 3:L=2ϑ−1 for ϑ<1 if g=0:(47)χv−Ψv≤L1−g1−Lλv=2ϑ−11−2ϑ−12ϕvϑl−42ϑ=2ϕvϑl−42−2ϑ.Case 4:L=21−ϑ for ϑ>1 if g=1:(48)χv−Ψv≤L1−g1−Lλv=11−21−ϑ2ϕvϑl−42ϑ=2ϕvϑl−42ϑ−2.Case 5:L=2lϑ−1 for ϑ<1/l if g=0:(49)χv−Ψv≤L1−g1−Lλv=2lϑ−11−2lϑ−12ϕvlϑl−42lϑ=2ϕvlϑl−42−2lϑ.Case 6:L=21−lϑ for ϑ>1/l if g=1:(50)χv−Ψv≤L1−g1−Lλv=11−21−lϑ2ϕvlϑl−42lϑ=2ϕvlϑl−42lϑ−2.
Hence, the proof is accomplished.


## 5. Counterexample

In this section, we examine the following counterexample changed by the well-known counterexample of Gajda [42] to (2).Example 1.
Let a mappingΘ:Φ⟶Ω be defined by(51)Θv=∑l=0∞ϕ2lv2l,where(52)ϕv=μv,−1<v<1,μ,otherwise,where μ is a constant, and then, the mapping Θ:Φ⟶Ω satisfies the inequality(53)Θv1,v2,…,vl≤l2−5l+828μ∑i=1nvi,for all v1,v2,…,vl∈Φ, but there exists an additive function Ψ:Φ⟶Ω with a constant α such that(54)χv−Ψv≤αv,for all v∈Φ.Proof.
It is easy to show thatχ is bounded by 2μ on Φ. If ∑i=1lvi≥1/2 or 0, then the left side of (53) is less than l2−5l+8/22μ, and thus, (53) is true. Assume that 0<∑i=1lvi≥1/2. Then there exists an integer m such that(55)12m+2≤∑i=1lvi<12m+1.
So that2mv1<1/2,2mv2<1/2,…,2mvl<1/2 and 2lv1,2lv2,…,2lvl∈−1,1 for all l=0,1,2,…,m−1. So, for l=0,1,…,m−1(56)ϕ∑g=1l2lvg−∑1≤g<h<i≤lϕ2lvg+vh+vi+∑1≤g<h≤lϕ2lvg+vh+l2−5l+22∑g=1lϕ2lvg+ϕ2l−vg2=0.
From the definition ofχ, we have(57)Θv1,v2,…,vl≤∑i=m∞12iϕ2iv1,2iv2,…,2ivl≤∑i=m∞12il2−5l+82μ≤l2−5l+8221−mμ.
It follows from (55) that(58)Θv1,v2,…,vl≤l2−5l+828μ∑i=1nvi,for all v1,v2,…,vl∈Φ. Thus, χ satisfies (53) for all v1,v2,…,vl∈Φ.
Assume on the contrary that there exists a mappingΨ:Φ⟶Ω additive with a constant α>0 satisfying (54). As χ is bounded and continuous for all v∈Φ, Ψ is bounded on any open interval containing the origin and continuous at the origin.
By Remark1, Ψ defined by Ψv=av for all v∈Φ. Hence(59)χv≤α+av,for all v∈Φ. However, we can select a nonnegative integer m and mμ>α+a. If v∈0,1/2m−1, then 2lv∈0,1 for all l=0,1,…,m−1, and for this v, we get(60)χv=∑l=0∞ϕ2lv2l≥∑l=0m−1μ2lv2l=mμv>α+av,which contradicts.


## 6. Conclusion

In this work, we introduced the generalized finite-dimensional additive functional equation (2) and obtain its general solution in Section 2. In Section 3, we investigated the Hyers–Ulam stability results in Banach space by using the direct method, and in Section 4, we examined the Hyers–Ulam stability results in Banach space by using the fixed-point method. In Section 5, we proved the counterexample changed by the well-known counterexample of Gajda [42] to show the nonstability of the generalized additive functional equation (2).



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