# Nearly General Septic Functional Equation

Authors: Ick-Soon Chang; Yang-Hi Lee; Jaiok Roh
Journal: Journal of Function Spaces
 (2021)

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## Abstract

If a mapping can be expressed by sum of a septic mapping, a sextic mapping, a quintic mapping, a quartic mapping, a cubic mapping, a quadratic mapping, an additive mapping, and a constant mapping, we say that it is a general septic mapping. A functional equation is said to be a general septic functional equation provided that each solution of that equation is a general septic mapping. In fact, there are a lot of ways to show the stability of functional equations, but by using the method of Ga˘vruta, we examine the stability of general septic functional equation ∑i=08C8i−18−ifx+i−4y=0 which considered. The method of Ga˘vruta as just mentioned was given in the reference Gavruta (1994).


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## Body

## 1. Introduction

The concept of stability for a functional equation arising when replacing the functional equation by an inequality which acts as a perturbation of the equation. Ulam [1] posed the question concerning the stability of group homomorphisms. Hyers [2] gave the first partial affirmative answer to the question of Ulam, which states that if δ>0 and f:X⟶Y is a mapping with X a normed space, Y a Banach space such that
(1)fx+y−fx−fy≤δforallx,y∈X,then there exists a unique additive mapping T:X⟶Y such that
(2)fx−Tx≤δforallx∈X.Thereafter, many authors have generalized the Hyers’ result (refer to [3–8]). On the other hand, a generalization of the Hyers result was also obtained by Ga˘vruta [9]. That is, let G,+ be an abelian group and let Y be a Banach space. Suppose that a function φ:G2⟶0,∞ satisfies the assumption
(3)φ~x,y≔∑k=0∞2−kφx,y<∞,for all x,y∈G, and that f:G⟶Y is a mapping such that the inequality
(4)fx+y−fx−fy≤φx,y,holds for all x,y∈G. Then, there exists a unique additive mapping T:G⟶Y with
(5)fx−Tx≤12φ~x,xforallx∈G.We, meanwhile, call the functional equation(6)∑i=0nCni−1n−ifx+iy−n!fy=0,the n-monomial functional equation. The 1-monomial, the 2-monomial, the 3-monomial, the 4-monomial, the 5-monomial, the 6-monomial, and the 7-monomial functional equation are called additive, quadratic, cubic, quartic, quintic, sextic, and septic functional equation, respectively. In this case, we say that each solution of the previous equation is additive, quadratic, cubic, quartic, quintic, sextic, and septic mapping, respectively. The function f:R⟶R defined by fx≔axn is a particular solution of the n-monomial functional equation. Quite recently, Lee [6] showed the stability of the n-monomial functional equation in the sense of Ga˘vruta.The following functional equation is called(7)∑i=0nCni−1n−ifx+iy=0,as Jensen, general quadratic, general cubic, general quartic, general quintic, general sextic, and general septic functional equation, respectively, for n=2,3,4,5,6,7,8. The solution of the general septic functional equation is said to be a general septic mapping. The function f:R⟶R given by fx≔∑i=07aixi is a particular solution of the general septic mapping. More detailed term for the concept of a general septic mapping can be found in Baker’s paper [10] by the term generalized polynomial mapping of degree at most 7.For a number of years now, many interesting results of the stability problems to several functional equations (or involving the range from additive functional equation to sextic functional equation) have been investigated; see, e.g., [11–26].Our principal purpose is to consider the following general septic functional equation(8)∑i=08C8i−18−ifx+i−4y=0,and then we are going to obtain the stability theorems of the functional equation (8) in the spirit of Ga˘vruta approach.


## 2. Stability of the General Septic Functional Equation (8)

In this section, we letY and V be a real Banach space and a real vector space, respectively. For a given mapping f:V⟶Yand all x,y∈V, we use the following abbreviations
(9)fox=fx−f−x2,fex=fx+f−x2,Dfx,y=∑i=08C8i−18−ifx+i−4y,Γx=Dfo8x,2x+8Dfo6x,2x+36Dfo4x,2x+120Dfo2x,2x+160Dfo4x,x+1280Dfo3x,x+4032Dfo2x,x+5376Dfox,x,Δfx=Dfe4x,x+8Dfe3x,x+36Dfe2x,x+120Dfex,x+123Dfe0,x.On the other hand, iff~ is a mapping defined by f~x=fx−f0, we know that
(10)Df~x,y=Dfx,yandf~0=0.In addition, through tedious computation, we then get the following expressions(11)Δfx=fe8x−84fe4x+1344fe2x−4096fex,(12)Γfx=fo16x−170fo8x+5712fo4x−43520fo2x+65536fox,for all x∈V.Lemma 1.
Letf:V⟶Y be a mapping with f0=0. Suppose that Jnf:V⟶Y and J′nf:V⟶Y are mappings given by
(13)Jnfx≔4n−20·16n+64·64n45fex2n−804n−17·16n+16·64n45fex2n+1+10244n−5·16n+4·64n45fex2n+2−2n−84·8n+1344·32n−4096·128n2835fox2n+1682n−81·8n+1104·32n−1024·128n2835fox2n+1−53762n−69·8n+324·32n−256·128n2835fox2n+2+327682n−21·8n+84·32n−64·128n2835fox2n+3,for all x∈V and all integers n≥0 and
(14)Jn′fx≔−1128n−8432n+13448n−40962n512fo2nx512·2835+1128n−8132n+11048n−10242n336fo2n+1x512·2835−1128n−6932n+3248n−2562n42fo2n+2x512·2835+1128n−2132n+848n−642nfo2n+3x512·2835+164n−516n+44nfe2n+2x64·45−2064n−34016n+3204nfe2n+1x64·45+6464n−128016n+40964nfe2nx64·45,for all x∈V and all integers n≥0. Then,
(15)Jnfx−Jn+1fx=4n45−20·16n45+64·64n45Δfx2n+3−2n−84·8n+1344·32n−4096·128n2835Γfx2n+4,holds for all x∈V and all integers n≥0 and
(16)Jn′fx−Jn+1′fx=−44n+1−516n+1+164n+1Δf2nx2880−1128n+1−2132n+1+848n+1−642n+1Γf2nx512·2835,is fulfilled for all x∈V and all integers n≥0.Proof.
By using (11) and the definitions of Jnf and J′nf, we can obtain the result after tedious calculations. Therefore, the proof will be omitted here.Lemma 2.
Assume thatf:V⟶Y is a mapping with f0=0 subject to the equation
(17)Dfx,y=0,for all x,y∈V. Then, we have
(18)Jnfx=fxandJn′fx=fx,for all x∈V and all positive integers n.Proof.
We have by the definitions ofΔfx and Γfx that
(19)Δfx=0andΓfx=0forallx∈V.
So, we figure out(20)fx−Jnfx=∑i=0n−1Jifx−Ji+1fx=∑i=0n−14i45−20·16i45+64·64i45Δfx2i+3−2i−84·8i+1344·32i−4096·128i2835Γfx2i+4=0,which implies Jnfx=fx for all x∈V and all positive integers n. Similarly, we get the equality J′nfx=fx for all x∈V and all positive integers n.We are now in a position to prove the following theorem.Theorem 3.
Suppose that a functionφ:V2⟶0,∞ satisfies the condition
(21)∑n=0∞128nφx2n,y2n<∞,for all x,y∈V. Assume that f:V⟶Y is a mapping subject to the inequality
(22)Dfx,y≤φx,y,for all x,y∈V. Then, there exists a unique general septic mapping F with F0=0 such that
(23)fx−f0−Fx≤∑n=0∞4n45−20·16n45+64·64n45Φx2n+3+−2n+84·8n−1344·32n+4096·128n2835Φ′x2n+4,for all x∈V, where φe:V2⟶0,∞ and Φ,Φ′:V⟶0,∞ are functions defined by
(24)φex,y≔φx,y+φ−x,−y2,Φx≔φe4x,x+8φe3x,x+36φe2x,x+120φex,x+123φe0,x,Φ′x≔φe8x,2x+8φe6x,2x+36φe4x,2x+120φe2x,2x+160φe4x,x+1280φe3x,x+4032φe2x,x+5376φex,x.Proof.
Considering a mappingf~ defined by f~x=fx−f0, we see that f~ satisfies the properties
(25)Df~x,y=Dfx,yandf~0=0.
Then, by (11) and the definitions of Γf and Δf, we obtain that
(26)Γf~x=Dfo8x,2x+8Dfo6x,2x+36Dfo4x,2x+120Dfo2x,2x+160Dfo4x,x+1280Dfo3x,x+4032Dfo2x,x+5376Dfox,x≤Φ′x,(27)Δf~x=Dfe4x,x+8Dfe3x,x+36Dfe2x,x+120Dfex,x+123Dfe0,x≤Φx,hold for all x∈V. It follows from (15) and (26) that
(28)Jnf~x−Jn+1f~x=4n45−20·16n45+64·64n45Δf~x2n+3−2n−84·8n+1344·32n−4096·128n2835Γf~x2n+4≤4n45−20·16n45+64·64n45Φx2n+3+−2n+84·8n−1344·32n+4096·128n2835Φ′x2n+4,for all x∈V. This gives that
(29)Jnf~x−Jn+mf~x≤∑i=nn+m−14i45−20·16i45+64·64i45Φx2i+3+−2i+84·8i−1344·32i+4096·128i2835Φ′x2i+4,for all x∈V and all nonnegative integers n and m. By the definition of Φ and Φ′ together with (21) and (29), the sequence Jnf~x is Cauchy in Y. And since Y is complete, the sequence Jnf~x converges. Therefore, we can define a mapping F:V⟶Y by
(30)Fx≔limn⟶∞Jnf~xforallx∈V.
Note thatJ0f~x=fx−f0 and F0=0 follow from f~0=0. Furthermore, by letting n=0 and passing the limit as m⟶∞ in (29), we arrive at the inequality (23). On the other hand, from (21) and the definition of F, we yield the following inequality
(31)DFx,y=limn⟶∞DJnfx,y≤limn⟶∞64n+145φex2n,y2n+20·64n+145φex2n+1,y2n+1+64n+245φex2n+2,y2n+2+32·128n+12835φex2n,y2n+1344·128n+12835φex2n+1,y2n+1+42·128n+22835φex2n+2,y2n+2+128n+32835φex2n+3,y2n+3=0,for all x,y∈V.
In order to prove the uniqueness ofF, we suppose that F′:V⟶Y is another general septic mapping satisfying (23) and F′0=0. However, it is also possible to show uniqueness by replacing condition (23) with weaker condition. So, we want to prove that there is a unique mapping satisfying the weaker condition
(32)f~x−Fx≤∑i=0∞128i+1Φx2i+3+Φ′x2i+4,for all x∈V. According to Lemma 2, we get F′x=JnF′x for all positive integers n. Thus, by using the condition (32) and the definition of Jn, after tedious calculations, we have
(33)Jnf~x−F′x=Jnf~x−JnF′x≤2∑i=0∞128i+n+2Φx2i+n+3+Φ′x2i+n+4+2∑i=0∞128i+n+2Φx2i+n+4+Φ′x2i+n+5+2∑i=0∞128i+n+3Φx2i+n+5+Φ′x2i+n+6+∑i=0∞128i+n+3Φx2i+n+6+Φ′x2i+n+7≤∑i=n+3∞128iΦx2i+Φ′x2i,for all x∈V and all positive integer n. Taking the limit as n⟶∞ in the last inequality, we then have
(34)F′x=limn⟶∞Jnf~xforallx∈V.
This implies thatF=F′.Theorem 4.
Suppose that a functionφ:V2⟶0,∞ satisfies the condition
(35)∑n=0∞2−nφ2nx,2ny<∞,for all x,y∈V. Assume that f:V⟶Y is a mapping subject to the inequality
(36)Dfx,y≤φx,y,for all x,y∈V. Then, there exists a unique general septic mapping F with F0=0 such that
(37)fx−f0−Fx≤∑n=0∞44n+1−516n+1+164n+1Φ2nx2880+642n+1−848n+1+2132n+1−1128n+1Φ′2nx512·2835,for all x∈V, where φe,Φ and Φ′ are the functions in Theorem 3.Proof.
We first define a mappingf~ by f~x=fx−f0. Then, we find that
(38)Df~x,y=Dfx,yandf~0=0.
With help of the definitions ofΓf and Δf, the relations (16), (26), and (36) guarantee that
(39)J′nf~x−J′n+1f~x≤44n+1−516n+1+164n+1Φ2nx2880+642n+1−848n+1+2132n+1−1128n+1Φ′2nx512·2835,for all x∈V, which leads to the inequality
(40)Jn′f~x−Jn+m′f~x≤∑i=nn+m−144i+1−516i+1+164i+1Φ2ix2880+642i+1−848i+1+2132i+1−1128i+1Φ′2ix512·2835,for all x∈V and all nonnegative integers n and m. In view of (35) and (40), the sequence J′nf~x is Cauchy in x∈Y. By completeness of Y, the sequence J′nf~x converges. Thereby, we can define a mapping F:V⟶Y by
(41)Fx≔limn⟶∞J′nf~xforallx∈V.
Sincef~0=0, we have F0=0. Note that J′0f~x=fx−f0 for all x∈V. Letting n=0 and sending the limit in (40) as m⟶∞, we obtain the desired result (37).
But, we intend to claim thatDFx,y=0. The expression (35) and the definition of F provide that
(42)DFx,y=limn⟶∞DJn′fx,y≤limn⟶∞2φe2nx,2ny+φe2n+1x,2n+1y+φe2n+2x,2n+2y4n+2φe2nx,2ny+φe2n+1x,2n+1y+φe2n+2x,2n+2y+φe2n+3x,2n+3y2n=0,forallx,y∈V.
To show the aforementioned uniqueness, let us assume thatF′:V⟶Y is another general septic mapping with (37) and F′0=0. Then, this proof is analogue to that of Theorem 3. That is, instead of the condition (37), it suffices to prove that there is a unique mapping satisfying the weaker condition
(43)f~x−Fx≤∑i=0∞Φ2ix+Φ′2ix2i,for all x∈V. From Lemma 2, the equality F′x=J′nF′x holds for all positive integer n. So, we have by (43) that
(44)J′nf~x−F′x=J′nf~x−J′nF′x≤4∑i=0∞Φ2n+ix+Φ′2n+ix2n+i+2∑i=0∞Φ2n+i+1x+Φ′2n+i+1x2n+i+1+2∑i=0∞Φ2n+i+2x+Φ′2n+i+2x2n+i+2+2∑i=0∞Φ2n+i+3x+Φ′2n+i+3x2n+i+3≤10∑i=n∞Φ2ix+Φ′2ix2i,for all x∈V and all positive integer n. Taking the limit as n⟶∞ in the previous inequality, we get the relation
(45)F′x=limn⟶∞Jnf~xforallx∈V,which means that F=F′.


## 3. Discussion

In this paper, we investigated the stability of general septic functional equation by using the method of Ga˘vruta. In Theorem 3, we proved that if the function f:V⟶Y satisfies the inequality
(46)Dfx,y=∑i=08C8i−18−ifx+i−4y≤φx,y,where a function φ:V2⟶0,∞ satisfies the condition
(47)∑n=0∞128nφx2n,y2n<∞,for all x,y∈V, then there exists a unique general septic mapping F near the function f.And in Theorem4, we proved similar result when the function φ:V2⟶0,∞ satisfies the condition
(48)∑n=0∞2−nφ2nx,2ny<∞,forallx,y∈V.Lee [27] proved the stability of the general sextic functional equation for the mapping f such that
(49)Dfx,y=∑i=07C7i−17−ifx+iy≤θxp+yp.For the future work, we want to see if we can get similar results for the general septic functional equation. And by using our results, we want to know what we can say for the stability of the general octic functional equation.Also, by fixed point theorem, Roh et al. [24] showed the stability of the general sextic functional equation for the mapping f such that
(50)Dfx,y=∑i=07C7i−17−ifx+iy≤φx,y,where φ:V2⟶0,∞ is a function for which there exists a constant 0<L<1 such that
(51)φ2x,2y≤421cosθ−14Lφx,y,for all x,y∈V, and θ is a real constant such that 0<θ<π/4 and cos3θ=−17/2121. In fact, for the stability of the septic functional equation, we first tried this method, but we failed to get the good result. The calculation became complicated because it resulted in a quadratic equation problem. So, we leave this problem as an open problem.



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*Source: 5643145-2021-12-13.xml*