# The Stability of a General Sextic Functional Equation by Fixed Point Theory

Authors: Jaiok Roh; Yang-Hi Lee; Soon-Mo Jung
Journal: Journal of Function Spaces
 (2020)

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## Abstract

In this paper, we will consider the generalized sextic functional equation∑i=077Ci−17−ifx+iy=0. And by applying the fixed point theorem in the sense of Ca˘dariu and Radu, we will discuss the stability of the solutions for this functional equation.


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## Body

## 1. Introduction

In 1940, Ulam [1] remarked the problem concerning the stability of group homomorphisms. In 1941, Hyers [2] gave an answer to this question for additive mappings between Banach spaces. Subsequently, many mathematicians came to deal with this question (cf. [3–10]). Let V and W be real vector spaces, X be a real normed space, Y be a real Banach space, n∈ℕ (the set of natural numbers), and f:V⟶W be a given mapping. Consider the functional equation
(1)∑i=0nnCi−1n−ifix+y−n!fx=0,for everyx,y∈V, where nCi=n!/i!n−i!. The functional equation (1) is called an n-monomial functional equation, and every solution of the functional equation (1) is called to be a monomial mapping of degree n. The function f:ℝ⟶ℝ given by
(2)fx=axn,is a particular solution of the functional equation (1). In particular, the functional equation (1) is called a sextic functional equation for the case n=6, and every solution of the functional equation (1) is called to be a sextic mapping for the case n=6. Many mathematicians [11–17] have previously investigated the stability of the sextic functional equation, and many authors [18–26] have studied the stability of the n-monomial functional equation in various spaces.The solution of the functional equation(3)∑i=0nnCi−1n−ifx+iy=0,is called a generalized polynomial mapping of degreen∈ℕ (See Baker [27]). The function f:ℝ⟶ℝ given by fx=∑i=0naixi, is a particular solution of the functional equation (3). Some mathematicians [28–31] have previously investigated the stability of the functional equation (3) for the cases n=4,5,6,7. In particular, the functional equation
(4)∑i=077Ci−17−ifx+iy=0,is called a general sextic functional equation, and every solution of the functional equation (4) is said to be the general sextic mapping.In this paper, we will partially generalize the results in [31] for the stability of the general sextic functional equation. For the details, one can refer Corollary 4 and Corollary 7 which are special cases of main theorems. Specifically, in this paper, we will show that there is only one solution F of the general sextic functional equation (4) near the function f, which approximates the functional equation (4) by using fixed point theorem [32–35]. Moreover, the solution mapping F of the functional equation (4) can be explicitly constructed by the formula
(*)Fx=limn→∞∑i=0nnCi∑j=0iiCj−84n−i1344i4096nfe223n−i−jx+−42i−j336j512nfo23n−i−jx,or(**)Fx=limn→∞∑i=0nnCi∑j=0iiCj42j−336i−j512n−ifox23n−i−j+84j−1344i−j4096n−ifex23n−i−j,which approximates the mappingf.


## 2. Main Results

We first recall the following Margolis and Diaz fixed point theorem, which is necessary to obtain the main results of this paper.Proposition 1 (see [36]).
SupposeX,d is a complete generalized metric space, which means that the metric d may assume infinite values, and J:X⟶X is a strictly contractive mapping with the Lipschitz constant 0<L<1. Then, for each given element x∈X, either
(5)dJnx,Jn+1x=+∞foralln∈ℕ∪0,
or there exists an integerk≥0 such that:
(i)
dJnx,Jn+1x<+∞ for all n≥k(ii)
The sequenceJnx converge to a fixed point y∗ of J(iii)
y∗ is a unique fixed point of J in Y=y∈X:dJkx,y<+∞(iv)
dy,y∗≤1/1−Ldy,Jy for every y∈Y
In this paper, we letV and W be real vector spaces, X be a real normed space, and Y be a real Banach space. For a mapping g:V⟶W, we use the following abbreviations
gex=gx+g−x2,gox=gx−g−x2,Dgx,y=∑i=077Ci−17−igx+iy,
for everyx,y∈V.Now, we will see useful lemma for the proof of main theorem.Lemma 2.
Letθ be a real constant such that 0<θ<π/4 and cos3θ=−17/2121. Let φ:V2⟶0,∞ be a function for which there exists a constant 0<L<1 such that
(6)φ2x,2y≤421cosθ−14Lφx,y,
for allx,y∈V. Then, 1<421cosθ−14<2,
(7)421cosθ−143+42421cosθ−142+336421cosθ−14512=1,and the equality
(8)limn→∞1512n∑i=0nnCi∑j=0iiCj42i−j336jφe23n−i−jx,23n−i−jy=0,
holds for allx,y∈V.Proof.
When0<θ<π/4 and cos3θ=−17/2121, it is not difficult to see that 1.74837<3θ<1.7484 and 0.83493<cosθ<0.834925 in the trigonometric function table. So 1.3<421cosθ−14<1.4.
We can also obtain the equality (7) by the following calculation:
(9)336421cosθ−14512+42421cosθ−145122+421cosθ−145123=134421cosθ−4704512+14112cos2θ−470421cosθ+8232512+134421cos3θ−14112cos2θ+235221cosθ−2744512=336214cos3θ−3cosθ+784512=33621cos3θ+784512=33621×−17/2121+784512=1.
And, to obtain the equality (8), by (7), we obtain the following calculation:
(10)limn→∞1512n∑i=0nnCi∑j=0iiCj42i−j336jφe23n−i−jx,23n−i−jy≤limn→∞1512n∑i=0nnCi∑j=0iiCj42i−j336j421cosθ−142n−i−jL2n−i−jφe2nx,2ny≤limn→∞1512n∑i=0nnCi421cosθ−142n−2i∑j=0iiCj42i−j421cosθ−14i−j336j×φe2nx,2ny=limn→∞1512n∑i=0nnCi421cosθ−142n−2i42421cosθ−14+336iφe2nx,2ny≤limn→∞1512n421cosθ−142+42421cosθ−14+336nφe2nx,2ny=limn→∞Ln421cosθ−143+42421cosθ−142+336421cosθ−14512n×φex,y=limn→∞Lnφex,y=0,forallx,y∈V.In the following main theorem, we will prove the generalized Hyers-Ulam stability of the functional equation (4) by using the direct method.Theorem 3.
Letθ, L, and φ be as in Lemma 2. If f:V⟶Y is a mapping satisfying f0=0 and the inequality
(11)Dfx,y≤φx,y,forallx,y∈V,
then there exists the unique solution mappingF:V⟶Y of (4) such that
(12)fx−Fx≤Φx1−L,
for allx∈V, where
Φx=9φe−6x,2x+56φe−x,x+392φe−2x,x+1008φe−3x,x4096.
In particular,F is represented by
(13)Fx=limn→∞∑i=0n∑j=0inCiiCj−84i−j1344j4096nfe23n−i−jx+−42i−j336j512nfo23n−i−jx,
for allx∈V.Proof.
We let the setS be the set of the functions g:V⟶Y with g0=0. And we define a generalized metric on Sby
dg,h=infK∈R+gx−hx≤KΦxforallx∈V.
Then, it is not so difficult to show thatS,d is a complete generalized metric space (see ([34], Theorem 2.5) or the proof of ([37], Theorem 3.1)). Next, we see the mapping J:S⟶S, which is defined by
Jgx=4032g2x8192−1344g−2x8192−420g4x8192+252g−4x8192+9g8x8192−7g−8x8192,for all x∈V.
And, by using the oddness and the evenness ofgo and ge and nCi−1+nCi=n+1Ci, due to mathematical induction, we can get
Jngx=∑i=0nnCi∑j=0iiCj−84i−j1344j4096nge23n−i−jx+−42i−j336j512ngo23n−i−jx,holds for all n∈ℕ and x∈V.
Letg,h∈S and we choose K∈0,∞ as an arbitrary constant with dg,h≤K. Due to the definition of d and (7) in Lemma 2, we have
Jgx−Jhx≤40328192g2x−h2x+13448192g−2x−h−2x+4208192g4x−h4x+2528192g−4x−h−4x+98192g8x−h8x+78192g−8x−h−8x≤K336512Φ2x+42512Φ4x+1512Φ8x≤K336421cosθ−14L512+42421cosθ−142L2512+421cosθ−143L3512Φx≤K336421cosθ−14512+42421cosθ−142512+421cosθ−143512LΦx≤LKΦx,
for everyx∈V, which implies that
dJg,Jh≤Ldg,h,for all g,h∈S, where 0<L<1. That is, with the Lipschitz constant L, J is a strictly contractive self-mapping of S, where 0<L<1.
Now, after long and tedious calculation, we havefx−Jfx=14096Dfe−6x,2x+8Dfe−x,x+56Dfe−2x,x+112Dfe−3x,x+1512Dfo−6x,2x+6Dfo−x,x+42Dfo−2x,x+112Dfo−3x,x.
And, by (11) we obtain
fx−Jfx≤14096Dfe−6x,2x+8Dfe−x,x+56Dfe−2x,x+112Dfe−3x,x+1512Dfo−6x,2x+6Dfo−x,x+42Dfo−2x,x+112Dfo−3x,x≤9φe−6x,2x+56φe−x,x+392φe−2x,x+1008φe−3x,x4096=Φx,
for everyx∈V. It implies that df,Jf≤1<∞ from the definition of d and due to Proposition 1, the sequence Jnf converges to only one fixed point F:V⟶Y of J in the set T=g∈S:df,g<∞ which implies (13). Moreover, by Proposition 1, we have
df,F≤11−Ldf,Jf≤11−L,which implies (12).
Also, by the equality (8) in Lemma 2, since one has
limn→∞14096n∑i=0nnCi∑j=0iiCj−84i−j1344jDfe23n−i−jx,23n−i−jy≤limn→∞14096n∑i=0nnCi∑j=0iiCj84i−j1344jφe23n−i−jx,23n−i−jy≤limn→∞1512n∑i=0nnCi∑j=0iiCj42i−j336jφe23n−i−jx,23n−i−jy=0,and
limn→∞1512n∑i=0nnCi∑j=0iiCj−42i−j336jDfo23n−i−jx,23n−i−jy≤limn→∞1512n∑i=0nnCi∑j=0iiCj42i−j336jφe23n−i−jx,23n−i−jy=0,
for allx,y∈V, we obtain
DFx,y=limn→∞∑i=0nnCi∑j=0iiCj−84i−j1344j4096nDfe23n−i−jx+−42i−j336j512nDfo23n−i−jx,23n−i−jy=0,
for everyx,y∈V.
Therefore,F is the unique solution of the functional equation (4) with (12). Finally, we see that if F is a solution of the sextic functional equation (4) with F0=0, then we can derive that F is a fixed point of J from the equality
(14)Fx−JFx=14096DFe−6x,2x+8DFe−x,x+56DFe−2x,x+112DFe−3x,x1512DFo−6x,2x+6DFo−x,x+42DFo−2x,x+112DFo−3x,x.In next corollary, we will consider special functionφx,y=∥x∥p+∥y∥p in Theorem 3 to compare with the results in [31].Corollary 4.
LetX be a real normed space, θ be as in Lemma 2, and p be a fixed real number such that 0<p<log2421−14. If f:X⟶Y satisfies the equality f0=0 and the inequality
(15)Dfx,y≤xp+yp,
for allx,y∈X, then there exists a unique solution mapping F:V⟶Y of (4) satisfying the inequality
(16)fx−Fx≤1512+401·2p+1008·3p+9·6p221−7xp8192421−14−2p,
for allx∈X.Proof.
If we putφx,y=xp+yp and L=2p/421−14, then we have the equalities φ2x,2y=2pφx,y=421−14Lφx,y for all x,y∈X. So the condition (6) in Lemma 2 holds for all x,y∈X. According to Theorem 3, there exists a unique solution mapping F:V⟶Y of (4) satisfying the inequality (16) for all x,y∈X.Next, we will try to prove the stability of the sextic functional equation (4) from another point of view. For that, we first will introduce useful facts in the following lemma.Lemma 5.
Letθ be a real constant such that 0<θ<π/4 and cos3θ=637/7777. Let φ:V2⟶0,∞ be a function for which there exists a constant 0<L<1 such that
(17)Lφ2x,2y≥877cosθ+28φx,y,
for allx,y∈V. Then, we have 97<877cosθ+28<97.8,
(18)84877cosθ+28+1344877cosθ+282+4096877cosθ+283=1,and the equality
(19)limn→∞∑i=0nnCi∑j=0iiCj84j1344i−j4096n−iφex23n−i−j,y23n−i−j=0,
holds for allx,y∈V.Proof.
When0<θ<π/4 and cos3θ=637/7777, it is not difficult to see that 0<3θ<0.33997 and 97<877cosθ+28<97.8 in the trigonometric function table. Also, we obtain the equality (18) from the following calculation:
(20)4096+1344877cosθ+28+84877cosθ+282−877cosθ+283=4096+1075277cosθ+37632+413952cos2θ+3763277cosθ+65856−3942477cos3θ−413952cos2θ−1881677cosθ−21952=−9856774cos3θ−3cosθ+81536=−985677cos3θ+81536=0.
And by (17) and (18), we have
(21)∑i=0nnCi∑j=0iiCj84j1344i−j4096n−iφex23n−i−j,y23n−i−j≤∑i=0nnCi∑j=0iiCj84j1344L877cosθ+28i−j4096n−iφex23n−2i,y23n−2i=∑i=0nnCi84+1344L877cosθ+28i4096n−iφex23n−2i,y23n−2i≤∑i=0nnCi84+1344L877cosθ+28i4096L2877cosθ+282n−iφex2n,y2n≤84+1344L877cosθ+28+4096L2877cosθ+282nL877cosθ+28nφex,y≤84877cosθ+28+1344877cosθ+282+4096877cosθ+283nLnφex,y=Lnφex,y,
for allx,y∈V. Therefore, by taking the limit, we complete the proof of (19).In the following theorem, we will prove the stability of the solution for the sextic functional equation (4) with different types of functions compared to Theorem 3.Theorem 6.
Letθ, L, and φ be as in Lemma 5. If f:V⟶Y is a mapping such that the inequality (11) in Theorem 3 holds for every x,y∈V, then there exists only one solution F:V⟶Y of (4) satisfying the following inequality
(22)fx−Fx≤Ψx1−L,
for everyx∈V, where
Ψx=2φe−3x4,x4+14φe−x8,x8+98φe−x4,x8+224φe−3x8,x8.
In particular,F is represented by
(23)Fx=limn→∞∑i=0nnCi∑j=0iiCj42j−336i−j512n−ifox23n−i−j+84j−1344i−j4096n−ifex23n−i−j,
for anyx∈V.Proof.
Similar to Theorem3, we consider the set S which contains all functions g:V⟶Y with g0=0, and we define a generalized metric on S as
dg,h=infK∈R+gx−hx≤KΨxforallx∈V.
We now consider the mappingJ:S⟶S defined by
Jgx=63gx2+21g−x2−840gx4−504g−x4+2304gx8+1792g−x8,
for everyx∈V. Then, similar to Theorem 3, by mathematical induction, we obtain that
Jngx=∑i=0nnCi∑j=0iiCj42j−336i−j512n−igox23n−i−j33mm+84j−1344i−j4096n−igex23n−i−j,
holds for everyn∈ℕ and x∈V.
Letg,h∈S and we assume K∈0,∞ as an arbitrary constant with dg,h≤K. By the definition of d and (18) in Lemma 5, we have
Jgx−Jhx≤63gx2−hx2+21g−x2−h−x2+840gx4−hx4+504g−x4−h−x4+2304gx8−hx8+1792g−x8−h−x8≤84KΨx2+1344KΨx4+4096KΨx8≤84LKΨx877cosθ+28+1344L2KΨx877cosθ+282+4096L3KΨx877cosθ+283≤84877cosθ+28+1344877cosθ+282+4096877cosθ+283LKΨx≤LKΨx,
for everyx∈V, which implies that
dJg,Jh≤Ldg,h,
for allg,h∈S, where 0<L<1. So, with the Lipschitz constant L, J is a strictly contractive self-mapping of S, where 0<L<1.
Moreover, by the definition ofDfx,y, with long and tedious calculation, we have
fx−Jfx=Dfe−3x4,x4+8Dfe−x8,x8+56Dfe−x4,x8+112Dfe−3x8,x8+Dfo−3x4,x4+6Dfo−x8,x8+42Dfo−x4,x8+112Dfo−3x8,x8.
And, by (11) in assumption, we obtain
fx−Jfx≤Dfe−3x4,x4+8Dfe−x8,x8+56Dfe−x4,x8+112Dfe−3x8,x8+Dfo−3x4,x4+6Dfo−x8,x8+42Dfo−x4,x8+112Dfo−3x8,x8≤2φe−3x4,x4+14φe−x8,x8+98φe−x4,x8+224φe−3x8,x8=Ψx,
for everyx∈V.
It implies thatdf,Jf≤1<∞ by the definition of d. Therefore, according to Proposition 1, the sequence Jnf converges to only one fixed point F:V⟶Y of J in the set T=g∈S∣df,g<∞, which is represented by (23) for every x∈V.
We also due to Proposition1 obtain that
df,F≤11−Ldf,Jf≤11−L,which implies (22).
Now, by (19) in Lemma 5, since we have
limn→∞∑i=0nnCi∑j=0iiCj84j−1344i−j4096n−iDfex23n−i−j,y23n−i−j≤limn→∞∑i=0nnCi∑j=0iiCj84j1344i−j4096n−iφex23n−i−j,y23n−i−j=0,limn→∞∑i=0nnCi∑j=0iiCj42j−336i−j512n−iDfox23n−i−j,y23n−i−j≤limn→∞∑i=0nnCi∑j=0iiCj42j336i−j512n−iφex23n−i−j,y23n−i−j≤limn→∞∑i=0nnCi∑j=0iiCj84j1344i−j4096n−iφex23n−i−j,y23n−i−j=0,
for allx,y∈V, due to the equality (23), we obtain
DFx,y=limn→∞∑i=0nnCi∑j=0iiCj42j−336i−j512n−iDfox23n−i−j,y23n−i−j+limn→∞∑i=0nnCi∑j=0iiCj84j−1344i−j4096n−iDfex23n−i−j,y23n−i−j=0,
which conclude thatF is a solution of the sextic functional equation (4).
Finally, we see that ifF is a solution of the sextic functional equation (4), then the equality
Fx−JFx=DFe−3x4,x4+8DFe−x8,x8+56DFe−x4,x8+112DFe−3x8,x8+DFo−3x4,x4+6DFo−x8,x8+42DFo−x4,x8+112DFo−3x8,x8,
implies thatF is a fixed point of J.In next corollary, we will consider special functionφx,y=xp+yp in Theorem 6 to compare with the results in [31].Corollary 7.
LetX be a real normed space, θ be as in Lemma 5, and p be a fixed real number such that log2877cosθ+28<p. Iff:X⟶Y satisfies the equality f0=0 and the inequality (15) for all x,y∈X, then there exists a unique solution mapping F:V⟶Y of (4) such that
(24)fx−Fx≤175+50·2p+112·3p+6p∥x∥p4p2p−877cosθ+28,
for allx∈X.Proof.
If we putφx,y=xp+yp and L=877cosθ+28/2p, then we have the equalities Lφ2x,2y=2pLφx,y=877cosθ+28φx,y for all x,y∈X. So the condition (17) in Lemma 5 holds for all x,y∈X. According to Theorem 6, there exists a unique solution mapping F:V⟶Y of (4) satisfying the inequality (24) for all x,y∈X.



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*Source: 6497408-2020-12-24.xml*