# A New Approach to Hyers-Ulam Stability ofr-Variable Quadratic Functional Equations

Authors: Vediyappan Govindan; Porpattama Hammachukiattikul; Grienggrai Rajchakit; Nallappan Gunasekaran; R. Vadivel
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this paper, we investigate the general solution of a new quadratic functional equation of the form∑1≤i<j<k≤rϕli+lj+lk=r−2∑i=1,i≠jrϕli+lj+−r2+3r−2/2∑i=1rϕli. We prove that a function admits, in appropriate conditions, a unique quadratic mapping satisfying the corresponding functional equation. Finally, we discuss the Ulam stability of that functional equation by using the directed method and fixed-point method, respectively.


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## Body

## 1. Introduction

The stability problem of functional equations originated from a question of Ulam [1] concerning about the stability. Hyers [2] gave a first affirmative answer to the question of Ulam for Banach spaces. In addition, various generalizations of Ulam’s problem and Hyer’s theorem have been extensively studied and many elegant results have been obtained [3–9]. The theory of nonlinear analysis has become a fast-developing field during the past decades. Functional equations have substantially grown to become an important branch of this field. In [10], the authors deal with a comprehensive illustration of the stability of functional equations, and in [11], the authors studied functional equations and inequalities in several variables. Very recently, most classical results on the Hyers-Ulam-Rassias stability have been offered in an integrated and self-contained version in [12]. It is worth noting that among the stability problem of functional equations, the study of the Ulam stability of different types of quadratic functional equations is an important and interesting topic, and it has attracted many scholars [13–18]. In addition, very recently, authors studied various types of stability results and have been discussed with differential equation [19–29]. To the best of the author’s knowledge, a new approach to Hyers-Ulam stability of r-variable quadratic functional equations has not been studied so far, which motivates the present study.Consider the functional equation as follows:(1)ϕl+m+ϕl−m=2ϕl+ϕm,is called a quadratic functional equation. Every solution of the quadratic functional equation is a quadratic mapping. In this paper, we investigate the general solution of a new quadratic functional equation of the form
(2)∑1≤i<j<k≤rϕli+lj+lk=r−2∑i=1,i≠jrϕli+lj+−r2+3r−22∑i=1rϕli.Motivated by the above discussion, we prove that a function admits in appropriate conditions and a unique quadratic mapping satisfying the corresponding functional equation. Finally, we discuss the Ulam stability of that functional equation by using the directed method and fixed-point method, respectively.


## 2. Preliminaries

Definition 1.
Letlbe a real linear space. A functionΦ:L×I⟶0,1is said to be a fuzzy norm onLif for alll,m∈Land allu,v∈I
(N1)Φl,d=0 for d≤0
(N2)l=0 if and only if Φl,d=1 for all d>0
(N3)Φdl,v=Φl,v/∣d∣ if d≠0
(N4)Φl+m,u+v≥minΦl,u,Φv,m
(N5)Φl,. is a nondecreasing function on I and limv→∞Φl,u=1
(N6) for l≠0,Φl,. is continuous on I
The pairL,Φ is called fuzzy normed linear space one may regard Φl,u as the truth value of the statement; the norm of l is less than or equal to the real number u.Definition 2.
LetL,Φbe a fuzzy normed linear space. Letlrbe a sequence inL. Then,lris said to be convergent if there existsl∈Lsuch thatlimr→∞lr−l,u=1for allv>0. In that case,lis called the limit of the sequencelrand we denote it byΦ−limr→∞lr=l.Definition 3.
A sequencelrbe inlis called Cauchy if for eachε>0and eachv>0, there existsr0such that for allr≥r0and alln>0, we haveΦlr+n−lr,v>1−ε.Definition 4.
Every convergent sequence in fuzzy normed space is Cauchy. If each Cauchy sequence is convergent, then the fuzzy norm is said to be complete and the fuzzy normed space is called a fuzzy Banach space.Theorem 5.
(Banach’s contraction principle). LetL,c be a complete metric space and consider a mapping W:L⟶L which is strictly contractive mapping, that is,
(A1)cWl,Wm≤Lcl,m for some (Lipchitz constant) L<1, then
(i)
The mappingT has one and only fixed point l∗=Wl∗(ii)
The fixed point for each given elementl∗ is globally attractive that is
(A2)limr→∞Wrl=l∗, for any starting point l∈L(iii)
One has the following estimation inequalities:
(A3)cWrl,l∗≤1/1−LcWrl,Wr+1l, for all r≥0,l∈L
(A4)cl,l∗≤1/1−Lcl,l∗, with respect to l∈LTheorem 6.
(the alternative of fixed point). Suppose that for a complete generalized metric spaceL,c and a strictly contractive mapping W:L⟶M with Lipschitz constant L. Then, for each given element l∈L, either
(B1)cWrl,Wr+1l=∞,∀r≥0 or
(B2) there exists natural numberr0 such that:
(i)
cWrl,Wl+1r<∞, for all r≥r0(ii)
The sequenceWrl is convergent to a fixed point m∗ of W(iii)
m∗ is the unique fixed point of W in the set M=m∈L:cWrl,m∞(iv)
cm,m∗≤1/1−Lcv,Wm for all m∈L


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

In this sector, the authors obtain the general solution of the functional equation (2). All over this sector, let L and M be real vector space.Theorem 7.
LetLandMbe a real vector spaces. The mappingϕ:L⟶Msatisfies the functional equation ((2)) for alll1,l2,l3,⋯,ln∈L, thenϕ:L⟶Msatisfies the functional equation ((1)) for alll,m∈L.Proof.
We first assume that the mappingϕ:L⟶M satisfies (1). Setting l=m=0 in (1), we get ϕ0=0. Replacing l=0, m=l in (1), then
(3)ϕ−l=ϕl,
for alll∈L. Therefore, ϕ is even. If we choose l=l, m=l, and l=2l, m=l in (1), we get
(4)ϕ2l=4ϕl,ϕ3l=9ϕl,
for alll∈L. In general for any positive integer r such that
(5)ϕrl=r2ϕl,
for alll∈L. Conversely, replacing l1,⋯,lr by 0,0,⋯,0⏟3−times in (2), we get
(6)ϕ0=3r−2ϕ0+3−r2+3r−22ϕ0.
Replacingl1,⋯,lr by 0,0,⋯,0⏟4−times in (2), we have
(7)ϕ0=6r−2ϕ0+4−r2+3r−22ϕ0.
Settingl1=⋯=lr by 0,0,⋯,0⏟5−times in (2), we get
(8)ϕ0=10r−2ϕ0+5−r2+3r−22ϕ0.
Adding (6), (7), and (8) up to r-times attack, we get
(9)ϕ0=r−23+3r−3+r2−7r+122ϕ0+r−r2+3r−22ϕ0.
It follows from (9), and using evenness of ϕ, we get
(10)ϕ0=0.
Replacingl1=⋯=lr by l,−l,0,0,⋯,0⏟1−times in (2), we obtain
(11)ϕ0=r−2ϕ0+ϕl+ϕ−l+−r2+3r−22ϕl+ϕ−l,
for alll∈L. Switching l1=⋯=lr by l,−l,0,0,⋯,0⏟2−times in (2), we get
(12)2ϕ0+ϕl+ϕ−l=r−2ϕ0+2ϕl+2ϕ−l+−r2+3r−22ϕl+ϕ−l,
for alll∈L. Setting l1=⋯=lr by l,−l,0,0,⋯,0⏟3−times in (2) that
(13)3ϕ0+3ϕl+3ϕ−l=r−2ϕ0+3ϕl+3ϕ−l+−r2+3r−22ϕl+ϕ−l,
for alll∈L. Adding (11), (12), and (13), we obtain
(14)ϕl1+l2+l3=ϕl1+ϕl2+ϕl3r−2+r2−7r−122ϕl+r−2+r2−7r−122ϕ−l=r−2r−2ϕl+r−2r−2ϕ−l+−r2+3r−22ϕl+ϕ−l,
for alll∈L. It follows from (14); it reduces that
(15)ϕ−l=ϕl,
for alll∈L. Replacing l1=⋯=lr by l,−l,0,0,⋯,0⏟1−times in (2), we get
(16)ϕ2l=r−2ϕ2l+2ϕl+−r2+3r−222ϕl,
for alll∈L. Substituting l1=⋯=lr by l,l,0,0,⋯,0⏟2−times in (2), we arrive
(17)2ϕ2l+2ϕl=r−2ϕ2l+4ϕl+−r2+3r−222ϕl,
for alll∈L. Replacing l1=⋯=lr by l,l,0,0,⋯,0⏟3−times in (2), we get
(18)3ϕ2l+6ϕl=r−2ϕ2l+6ϕl+−r2+3r−222ϕl,
for alll∈L. Adding (16), (17), and (18) using evenness of ϕ, then we get
(19)r−2ϕ2l+2r−3+r2−7r+12ϕl=r−2ϕ2l+2r−4ϕl+−r2+3r−222ϕl,
for alll∈L. It follows from (19); we get
(20)ϕ2l=4ϕl,
for alll∈L. In general for any positive integer r, then can be written as
(21)ϕrl=r2ϕl,
for alll∈L. Replacing l1=⋯=lr by l,m,0,0,⋯,0⏟r−2−times, we arrive
(22)r−2ϕl+m+r−3+r2−7r+122ϕl+r−3+r2−7r+122ϕmr−2ϕl+m+r−2ϕl+r−2ϕm+−r2+3r−222ϕl+ϕm,
for alll,m∈L. Setting l1=⋯=lr by l,−l,m,0,0,⋯,0⏟r−3−times(23)ϕm=r−2ϕl+m+ϕ−l+m+−r2+3r−22ϕl+ϕ−l+ϕm,
for alll,m∈L. Replacing l1=⋯=lr by l,−l,m,0,0,⋯,0⏟r−3−times in (2), we get
(24)ϕm+ϕl+m+ϕl+m=l−2ϕl+m+ϕ−l+m+2ϕl+2ϕm+−r2+3r−22ϕl+ϕ−l+ϕm,
for alll,m∈L. Substituting l1=⋯=lr by l,−l,m,0,0,⋯,0⏟r−3−times in (2) that
(25)2ϕm+2ϕl+m+2ϕ−l+m+2ϕl=r−2ϕl+m+ϕ−l+m+4ϕl+2ϕm+−r2+3r−22ϕl+ϕ−l+ϕm,
for alll,m∈L. Adding (23), (24), and (25) and using evenness of ϕ, we get
(26)1+r2−7r+122ϕm+r2−7r+12ϕl+2ϕl+m+r−2ϕ−l+m+r−2ϕl=r−2ϕl+m+ϕ−l+m+r−22r−6ϕl+r−2r−3ϕm+−r2+3r−22ϕl+ϕ−l+ϕm,
for alll,m∈L. Replace m by −m in (26), we get
(27)1+r2−7r+122ϕ−m+r2−7r+12ϕl+2ϕl−m+r−2ϕ−l−m+r−2ϕl=r−2ϕl−m+ϕ−l−m+r−22r−6ϕl+r−2r−3ϕ−m+−l2+3l−22ϕl+ϕ−l+ϕ−m,
for alll,m∈L. Adding (26) and (27) and using evenness of ϕ, then
(28)ϕl+m+ϕl−m=2ϕl+2ϕm,
for alll,m∈L. So the mapping ϕ:L⟶M is quadratic.In Sections4 and 5, using L be a normed space and M be a Banach space. For notational handiness, we define a function Dϕ:L⟶M by
(29)Dϕl1,l2,⋯lr=∑1≤i<j<k≤rϕli+lj+lk−r−2∑i=1,i≠jrϕli+lj+−r2+3r−22∑i=1rϕli,for all l1,l2,⋯lr∈L.


## 4. Stability of the Functional Equation (2): Direct Method

In this section, we establish the stability of (2) in a fuzzy Banach space using a direct method.Theorem 8.
Letβ∈−1,1. Letχ:Lr⟶Nbe a mapping with(30)0<c32<1Φ′χ3βκl1,3βκl2,⋯,3βκlr,n≥Φ′cβχl,l,⋯,l,n,
for alll∈L and all n>0,c>0 and
(31)limκ→Φ′χ3βκl1,3βκl2,⋯,3βκlr,3βκn=1,
for alll1,l2,⋯,lr∈L and all n>0. Suppose that a function Dϕ:L⟶M satisfies the inequality
(32)ΦDϕl1,l2,⋯,lr,n≥Φ′χl1,l2,⋯,lr,n,
for alln>0 and l1,l2,⋯,lr∈L. Then, the limit
(33)Ql=Φ−limκ→∞ϕ3βκl3βκ
exists for alll∈L and the mapping Q:L⟶M is a unique quadratic mapping such that
(34)ϕϕl−Ql,n≥Φ′χl,l,⋯,l,3r3−3r2+2r2n∣32−c∣,
for alll∈L and for all n>0.Proof.
First, assume thatβ=1. Replacing l1,l2,⋯,lr by l,l,⋯,l, in (32), we have
(35)Φϕr3−3r2+2r6ϕ3l−3r3−3r2+2r2ϕl,n≥Φ′χl,l,⋯,l,n,
for alll∈L and for all n>0. Replacing l by 3κl in (35), we obtain
(36)Φϕ3κ+1l32−ϕ3κl,2n3r3−3r2+2r≥Φ′χ3κl,3κl,⋯,3κl,n,
for alll∈L and for all n>0. Using (30) and (N3) in (36), we have
(37)Φϕ3κ+1l32−ϕ3κl,2n3r3−3r2+2r≥Φ′χ3κl,3κl,⋯,3κl,ncκ,
for alll∈L and for all n>0; it is easy to verify from (37) that
(38)Φϕ3κ+1l32κ+1−ϕ3κl32κ,2n3r3−3r2+2r32l≥Φ′χ3κl,3κl,⋯,3κl,ncκ,
holds for alll∈L and for all n>0. Replacing n by cκn in (38), we get
(39)Φϕ3κ+1l32κ+1−ϕ3κl32κ,2vκn3r3−3r2+2r32κ≥Φ′χ3κl,3κl,⋯,3κl,n,
for alll∈L and for all n>0; it is easy to see that
(40)ϕ3κ+1l32κ+1−ϕl=∑i=0κ−1ϕ3i+1l32i+1−Φ3ix32i,
for alll∈L. From equations (39) and (40), we get
(41)Φϕ3κl32κ−ϕl,∑i=0κ−12cin3r3−3r2+2r32i≥min⋃i=1κ−1ϕ3i+1l32i+1−ϕ3il32i,2cin3r3−3r2+2r32i≥min⋃i=1κ−1Φ′χl,l,⋯,l,n≥Φ′χl,l,⋯,l,n,
for alll∈L and for all n>0. Replacing l by 3ml in (41) and using (30) and (N3), we obtain
(42)Φϕ3κ+ml32κ+m−ϕ3ml32m,∑i=0m+κ−12cin3r3−3r2+2r32i≥Φ′χl,l,⋯,l,ncm,
for alll∈L and for all n>0. And all m,κ≥0. Replacing n by cmn in (42), we get
(43)Φϕ3κ+ml32κ+m−ϕ3ml32m,∑i=mm+κ−12cin3r3−3r2+2r32i≥Φ′χl,l,⋯,l,n,
for alll∈L and for all n>0. And all m,κ≥0. Using (N3) in (42), we have
(44)Φϕ3κ+ml32κ+m−ϕ3ml32m,n≥Φ′χl,l,⋯,l,n∑i=mm+κ−12ci/3r3−3r2+2r32i,
for alll∈L and for all n>0. And all m,κ≥0. Since 0<c<32 and ∑i=0κc/32i<∞, the Cauchy criterion for convergence and (N5) implies that ϕ3κl/32κ is a Cauchy sequence in M,Φ′ is a fuzzy Banach space. This sequence converges to some point Ql∈M so one can define the mapping Q:L⟶M by
(45)Ql=Φ−limκ→∞ϕ3βκl32βκ,for all l∈L. Letting m=0 in (44), we receive
(46)Φϕ3κl32κ−ϕl,n≥Φ′χl,l,⋯,l,n∑i=0κ−1cin/3r3−3r2+2r32i,
for alll∈L. Letting κ⟶∞ in (46) and using (N6), we have
(47)ϕϕt−Ql,n≥Φ′χl,l,⋯,l,3r3−3r2+2r2n32−c,for all l∈L and for all n>0. To prove Q satisfies (2), replacing l1,l2,⋯,lr by 3κl1,3κl2,⋯,3κlr in (32), we get
(48)Φ13κDϕ3κl1,3κl2,⋯,3κlr,r≥Φ′χ3κl1,3κl2,⋯,3κlr,32κn,for all n>0 and all l1,l2,⋯,lr∈L, since limκ→∞Φ′χ3βκl1,3βκl2,⋯,3βκlr,32βκn=1.
Hence,Q satisfies the quadratic functional equation (2), in order to prove Ql is unique.
We letRl be another quadratic functional equation satisfying (2) and (34). Hence,
(49)ϕQl−Rl,n=ΦQ3κl32κ−R3κl32κ≥minΦQ3κl32κ−ϕ3κl32κ,n2,Φϕ3κl32κ−R3κl32κ,n2≥Φ′χ3κl,3κl,⋯,3κlr3−3r2+2r32κn32−c4≥Φχl,l,⋯,l,3r3−3r2+2r32κn32−c4cκ,
for alll∈L and for n>0. Since
(50)limκ→∞3r3−3r2+2r32κn32−c4cκ=∞,we obtain
(51)Φ′χ3κl,3κl,⋯,3κl,3r3−3r2+2r32κn32−c4cκ=1.
Thus,ΦQl−Rl,n=1 for all l∈L and for n>0. Hence, Ql=Rl. Therefore, Ql is unique. For β=−1, we can prove the result by a similar method. This completes the proof of the theorem.The following Corollary9 is an immediate consequence of Theorem 8 concerning the stability of (2).Corollary 9.
Suppose that the functionDϕ:L⟶Msatisfies the inequality(52)ϕDϕl1,l2,⋯,lr,n≥Φ′ε,n,Φ′ε∑i=1rlis,n,Φ′ε∏i=1nlis,r,Φ′ε∑i=1rlirs+∏i=1rlis,n,for all l1,l2,⋯,lr∈L and all n>0, where ε,s are constants. Then, there exists a unique quadratic mapping Q:L⟶M such that
(53)ϕϕt−Ql,n≥Φ′ε,3r3−3r2+2r2n8,Φ′rε∥l∥s,3r3−3r2+2r2m32−3s;s≠2,Φ′ε∥l∥rs,3r3−3r2+2r2n32−3rs;s≠2r,Φ′εr+1∥l∥rs,3r3−3r2+2r2n32−3rs;s≠2r,for all l∈L and for n>0.


## 5. Stability of the Functional Equation (2): Fixed-Point Method

In this section, the authors investigate the generalized Ulam-Hyers stability of the functional equation (2) in fuzzy normed space using the fixed-point method.To prove the stability result, we define the followingψi is a constant such that
(54)ψi=3ifi=0,13ifi=1,and Ω is the set such that ω=u\u:L⟶M,u0=0.Theorem 10.
LetDϕ:L⟶Mbe a mapping for which there exists a functionχ:Lr⟶Nwith condition(55)limκ→∞Φ′χψiκl1,ψiκl2,⋯.,ψiκlr,ψiκn=1,
for alll1,l2,⋯,lr∈L,n>0 and satisfying the inequality
(56)ϕDϕl1,l2,⋯,lr,n≥Φ′Dϕl1,l2,⋯,lr,n,
for alll1,l2,⋯,lr∈L and n>0.
If there existsL=Li such that the function
(57)l⟶ρl=2r3−3r2+2rχl3,l3,⋯,l3has the property
(58)Φ′Liψi2ρψil,n=Φ′ρl,n
for alll∈L and n>0. Then, there exists a unique quadratic function Q:L⟶M satisfying the functional equation (2) and
(59)ϕϕt−Ql,n≥Φ′L1−i1−Lρl,n,for all l∈L and n>0.Proof.
Letc be a general metric on Ω, such that
(60)cu,v=infκ∈0,∞ϕul−vl,n≥Φ′ρl,κn,l∈L,l>0.
It is easy to see thatΩ,c is complete.
DefineW:Ω⟶Ω by Wl=1/ψi2uψil,∀l∈L.
Foru,v∈Ω, we get
(61)cu,v≤κ⟹ϕul−vl,n≥Φ′ρl,κn⟹Φuψilψi2−vψilψi2,n≥Φ′ρψil,κψi2n⟹ϕWul−Wvl,n≥Φ′ρψil,κψi2n⟹ϕWul−Wvl,n≥Φ′ρl,κLn⟹cWul,Wvl,n≤κLn⟹cWu,Wv,n≤κcu,v,∀u,v∈Ω.
Therefore,W is strictly contractive mapping on Ω with Lipschitz constant L, replacing l1,l2,⋯,lr by l,l,⋯,l in (56), we get
(62)Φ3r3−3r2+2r6ϕ3l−3r3−3r2+2r2ϕl,n≥Φ′χl,l,⋯,l,n,
for alll∈L and n>0. Using (N3) in (62), we have
(63)Φϕ3l32−ϕt,n≥Φ′23r3−3r2+2rχl,l,⋯,l,n,
for alll∈L and n>0 with the help of (58), when i=0. It follows from (63) that
(64)⟹Φϕ3l32−ϕt,n≥Φ′Lρl,n⟹cWϕ,ϕ≤L1−i.
Replacingl by l/3 in (62), we get
(65)Φϕl−32ϕl3,n≥Φ′23r3−3r2+2rχl3,l3,⋯,l3,n,
for alll∈L and n>0 when i=1; it follows from (65); we arrive
(66)⟹Φϕl−32ϕl3,n≥Φ′ρl,n⟹cϕ,Wϕ≤L1−i.
Then from (64) and (66), we get
(67)⟹cϕ,Wϕ≤L1−i≤∞.
Now from the fixed-point alternative in both cases, it follows that there exists a fixed pointQ of W in Ω such that
(68)Qx=Φ−limκ→∞ϕψκlψ2κ,
for alll∈L and n>0. Replacing l1,l2,⋯,lr by ψiκl1,ψiκl2,⋯,ψiκlr in (56), we get
(69)Φ1ψi2κDϕψiκl1,ψiκl2,⋯,ψiκlr,n≥Φ′χψiκl1,ψiκl2,⋯,ψiκlr,ψi2κn,for all n>0 and all l1,l2,⋯,lr∈L. Utilizing the same procedure in Theorem 8, we can prove the function Q:L⟶M is quadratic and it satisfies the functional equation (2) by a fixed-point alternative, since Q is a unique fixed point of W in the set Δ=ϕ∈Ω/cϕ,Q<∞. Therefore, Q is a unique function such that
(70)ϕϕt−Ql,n≥Φ′ρl,κn,
for alll∈L and n>0. Again using the fixed-point alternative, we get
(71)cϕ,Q≤11−Lcϕ,Wϕ⟹cϕ,Q≤L1−i1−L⟹ϕϕt−Ql,n≥Φ′ρlL1−i1−L,n.
This completes the proof.The following Corollary11 is an immediate consequence of Theorem 10 concerning the stability of (2).Corollary 11.
Suppose that the functionDϕ:L⟶Msatisfies the inequality(72)ϕDϕl1,l2,⋯,lr,n≥Φ′ε,n,Φ′ε∑i=1rlis,n,Φ′ε∏i=1rlis,n,Φ′ε∑i=1rlins+∏i=1r∥li∥s,n,
for alll1,l2,⋯,lr∈L and all n>0, where ε,s are constants. Then, there exists a unique quadratic mapping A:L⟶M such that
(73)ϕϕl−Al,n≥Φ′ε,3r3−3r2+2r2n8,Φ′rεlis,3r3−3r2+2r2n32−3s;s≠2,Φ′εlirs,3r3−3r2+2r2n32−3rs;s≠2r,Φ′εr+1lirs,3r3−3r2+2r2n32−3rs;s≠2r,
for alll∈L and for n>0.Proof.
Setting(74)χl1,l2,⋯,lr≤ε,ε∑i=1rlis,ε∏i=1rlis,ε∑i=1rlirs+∏i=1rlis,for all l1,l2,⋯,lr∈L. Then,
(75)Φ′χψiκl1,ψiκl2,⋯,ψiκlr,ψi2κn=Φ′ε,ψiKn,Φ′ε∑i=1rlis,ψi2−sKn,Φ′ε∑i=1rlis,ψi2−rsKn,Φ′ε∑i=1rlirs+∏i=1rlis,ψi2−rsKn,=⟶1asκ⟶∞,⟶1asκ⟶∞,⟶1asκ⟶∞,⟶1asκ⟶∞,i.e., (55) holds. We have ρl=2/3r3−3r2+2rχl/3,l/3,⋯,l/3 that has the property Φ′L1/ψi2ρψil,n=Φ′ρl,n for all l∈L and n>0. Hence,
(76)Φ′ρl,n=Φ′χl3,l3,⋯,l3,3r3−3r2+2r2n=Φ′ε,3r3−3r2+2r2n,Φ′rεls,3r3−3r2+2r23sn,Φ′εlrs,3r3−3r2+2r23rsn,Φ′εr+1lrs,3r3−3r2+2r23rsn.
Now,(77)Φ′1ψi2ρψil,n=Φ′εψi2,3r3−3r2+2r2n,Φ′rεlsψisψi23s,3r3−3r2+2r2n,Φ′εlrsψirsψi23rs,3r3−3r2+2r2n,Φ′r+1εlrsψirsψi23rs,3r3−3r2+2r2n,=ψi−2ρl,ψis−2ρl,ψirs−2ρl,ψirs−2ρl,
for alll∈L. The following cases hold with the below conditions:
L=3−2 if i=0 and L=32 if i=1.
L=3s−2 for s>2 if i=0 and L=32−s if for s21 if i=1.
L=3rs−2 for s>2/r if i=0 and L=32−rs if for s<2/r if i=1.
L=3rs−2 for s>2/r if i=0 and L=32−rs if for s<2/r if i=1.Case 1.
L=3−2 if i=0(78)ϕϕt−Ql,n≤Φ′L1−i1−Lρl,n=Φ′3−21−3−22ε3r3−3r2+2r,n=Φ′ε,24r3−3r2+2r2n.Case 2.
L=32 if i=1(79)ϕϕt−Ql,n≤Φ′L1−i1−Lρl,n=Φ′11−332ε3r3−3r2+2r,n=Φ′ε,−24r3−3r2+2r2n.Case 3.
L=3s−2 for s>2 if i=0(80)ϕϕt−Ql,n≤Φ′L1−i1−Lρl,n=Φ′3s−21−3s−22rε∥l∥s3r3−3r2+2r3s,n=Φ′rε∥l∥s,3r3−3r2+2r232−3sn.Case 4.
L=32−s for s<2 if i=1(81)ϕϕt−Ql,n≤Φ′L1−i1−Lρl,n=Φ′11−32−s2rε∥l∥s3r3−3r2+2r3s,n=Φ′rε∥l∥s,3r3−3r2+2r232−3sn.Case 5.
L=3rs−2 for s>2/r if i=0(82)ϕϕt−Ql,n≤Φ′L1−i1−Lρl,n=Φ′3rs−21−3rs−22ε∥l∥rs3r3−3r2+2r3rs,n=Phi′ε∥l∥rs,3r3−3r2+2r232−3rsn.Case 6.
L=32−rs for s<2/r if i=1(83)ϕϕt−Ql,n≤Φ′L1−i1−Lρl,n=Φ′11−32−rs2ε∥l∥rs3r3−3r2+2r3rsn=Φ′ε∥l∥rs,3r3−3r2+2r3rs23rs−32n.Hence, the proof is completed.



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*Source: 6628733-2021-01-18.xml*