# Solution and Stability of Euler-Lagrange-Rassias Quartic Functional Equations in Various Quasinormed Spaces

Authors: Heejeong Koh; Dongseung Kang
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We obtain the general solution of Euler-Lagrange-Rassias quartic functional equation of the followingfax+by+fbx+ay+(1/2)ab(a-b)2f(x-y)=(a2-b2)2fx+fy+(1/2)ab(a+b)2f(x+y). We also prove the Hyers-Ulam-Rassias stability in various quasinormed spaces when b=1.


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## Body

## 1. Introduction

One of the interesting questions concerning the stability problems of functional equations is as follows: when is it true that a mapping satisfying a functional equation approximately must be close to the solution of the given functional equation? Such an idea was suggested in 1940 by Ulam [1] as follows. LetG1be a group and letG2be a metric group with the metricd(·,·). Givenε>0, does there exist aδ>0such that if a functionh:G1→G2satisfies the inequalityd(h(xy),h(x)h(y))<δfor allx,y∈G1then there is a homomorphismH:G1→G2withd(h(x),H(x))<εfor allx∈G1? In other words, we are looking for situations when the homomorphisms are stable; that is, if a mapping is almost a homomorphism, then there exists a true homomorphism near it. In 1941, Hyers [2] considered the case of approximately additive mappings in Banach spaces and satisfying the well-known weak Hyers inequality controlled by a positive constant. The famous Hyers stability result that appeared in [2] was generalized in the stability involving a sum of powers of norms by Aoki [3]. In 1978, Rassias [4] provided a generalization of Hyers Theorem which allows the Cauchy difference to be unbounded. During the last decades, stability problems of various functional equations have been extensively studied and generalized by a number of authors [5–10]. In particular, Rassias [11] introduced the Euler-Lagrange type quadratic functional equation
(1)f(rx+sy)+f(sx-ry)=(r2+s2)[f(x)+f(y)],
for fixed reals r, s with r≠0, s≠0. Also, Jun and Kim [12] proved the Hyers-Ulam-Rassias stability of a Euler-Lagrange type cubic mapping as follows:
(2)f(ax+by)+f(bx+ay)=(a+b)(a-b)2[f(x)+f(y)]+ab(a+b)f(x+y),
where a≠0, b≠0, a±b≠0, for all x,y∈X. Several Euler-Lagrange type functional equations have been investigated by numerous mathematicians; c.f. for example, [13–15].And Rassias [16] investigated stability properties of the following quartic functional equation:
(3)f(x+2y)+f(x-2y)+6f(x)=4f(x+y)+4f(x-y)+24f(y).It is easy to see thatf(x)=x4 is a solution of (3) by virtue of the identity
(4)(x+2y)4+(x-2y)4+x4=4(x+y)4+4(x-y)4+24y4.For this reason, (3) is called a quartic functional equation. Also, Chung and Sahoo [17] determined the general solution of (3) without assuming any regularity conditions on the unknown function. In fact, they proved that the function f:ℝ→ℝ is a solution of (3) if and only if f(x)=A(x,x,x,x), where the function A:ℝ4→ℝ is symmetric and additive in each variable. Lee and Chung [18] introduced a quartic functional equation as follows:
(5)f(ax+y)+f(ax-y)=a2f(x+y)+a2f(x-y)+2a2(a2-1)f(x)-2(a2-1)f(y),
for fixed integer a with a≠0, ±1.In this paper, we consider the following a generalized quartic functional equation:(6)f(ax+by)+f(bx+ay)+12ab(a-b)2f(x-y)=(a2-b2)2[f(x)+f(y)]+12ab(a+b)2f(x+y),
for fixed integers a and b such that a≠0, b≠0, a±b≠0, for all x,y∈X. In fact, the generalized quartic functional equation (6) is following from the spirit of the pioneering Euler-Lagrange quartic functional equation (3) as well as Euler-Lagrange quadratic functional equation (1) introduced by Rassias: see  [16] and  [11], respectively. For the same reason as (1), (2), and (3), we call (6) a Euler-Lagrange-Rassias quartic functional equation. First of all, we obtain the general solution of Euler-Lagrange-Rassias quartic functional equation. To prove the stability problem for the Euler-Lagrange-Rassias quartic functional equation on various quasi-normed spaces, we may consider the following:
(7)f(ax+y)+f(x+ay)+12a(a-1)2f(x-y)=(a2-1)2[f(x)+f(y)]+12a(a+1)2f(x+y),
for fixed integer a with a≠0, a≠±1, for all x,y∈X.We will use the following definitions to prove Hyers-Ulam-Rassias stability for the Euler-Lagrange-Rassias quartic functional equation in the quasi-β-normed and quasi fuzzy β-normed spaces. Let β be a real number with 0<β≤1 and let 𝕂 be either ℝ or ℂ.Definition 1.
LetX be a linear space over a field 𝕂. A  quasiβ-norm  ∥·∥ is a real-valued function on X satisfying the following statements:(1)
∥x∥≥0 for all x∈X and ∥x∥=0 if and only if x=0,(2)
∥λx∥=|λ|β·∥x∥ for all λ∈𝕂 and all x∈X,(3)
there is a constantK≥1 such that ∥x+y∥≤K(∥x∥+∥y∥) for all x,y∈X.The pair(X,∥·∥) is called a quasi-β-normed space if ∥·∥ is a quasi-β-norm on X. The smallest possible K is called the modulus of concavity of ∥·∥. A quasi-β-Banach space is a complete quasi-β-normed space.A quasiβ-norm ∥·∥ is called a (β,p)-norm (0<p≤1) if (3) takes the form ∥x+y∥p≤∥x∥p+∥y∥p for all x,y∈X. In this case, a quasi β-Banach space is called a (β,p)-Banach space; see  [19, 20].In 1984, Katsaras [21] and Wu and Fang [22] independently introduced a notion of a fuzzy norm and they gave the generalization of the Kolmogoroff normalized theorem for a fuzzy topological linear space. Since then, some mathematicians have defined fuzzy metrics and norms on a linear space from various points of view; see [23–27]. In 2003, Bag and Samanta [23] modified the definition of Cheng and Mordeson [28]. Bag and Samanta  [23] introduced the following definition of fuzzy normed spaces. The notion of fuzzy stability of functional equations was given in the paper [29].Definition 2.
LetX be a real vector space. A function N:X×ℝ→[0,1] is called a  fuzzy norm on X if for all x,y∈X and all s,t∈ℝ(N1)
N(x,t)=0 for t≤0;(N2)
x=0 if and only if N(x,t)=1 for all t>0;(N3)
N(cx,t)=N(x,t/|c|) if c≠0;(N4)
N(x+y,s+t)≥min{N(x,s),N(y,t)};(N5)
N(x,·) is a nondecreasing function of ℝ and limt→∞N(x,t)=1;(N6)
forx≠0,  N(x,·) is continuous on ℝ.
The pair (X,N) is called a fuzzy normed vector space.Mirmostafaee [30] introduced a notion for a quasi fuzzy p-normed space as follows.Definition 3.
Bya quasi fuzzy norm, one means a real vector space X, with a fuzzy subset N of X×ℝ and some K≥1 such that all axioms of fuzzy normed space in Definition 2 except (N4) and(N4′)
(8)N(x+y,K(s+t))≥min{N(x,s),N(y,t)}iiiiiiiiiii(x,y∈X,s,t>0)hold.
A quasi fuzzy normed space (X,N) which satisfies(N4′′)
(9)N(x+y,s+tp)≥min{N(x,sp),N(y,tp)}iiiiiiiiiiiiiiiii(x,y∈X,s,t>0),for some 0<p≤1, is called a quasi fuzzyp-norm.Definition 4.
LetX be a real vector space. A quasi fuzzy p-norm N:X×ℝ→[0,1] is called a  quasi fuzzy  (β,p)-norm on X if (N3) in Definition 2 takes the form.(N3′)
(10)N(cx,t)=N(x,t|c|β)(c≠0,0<β≤1).Example 5.
Let(X,∥·∥) be a real normed space. Define
(11)N(x,t)={tt+∥x∥whent>0,t∈ℝ0whent≤0,
where x∈X. Then (X,N) is a quasi fuzzy (β,p)-normed space.
Note that whenp=1, we call the quasi fuzzy (β,p)-norm a quasi fuzzy β-norm.Definition 6.
Let (X,N) be a quasi fuzzy β-normed vector space. A sequence {xn} in X is said to be  convergent or converge if there exists an x∈X such that limn→∞N(xn-x,t)=1 for all t>0. In this case, x is called the  limit of the sequence {xn} and one denotes it by N-limn→∞xn=x.Definition 7.
Let (X,N) be a quasi fuzzy β-normed vector space. A sequence {xn} in X is called  Cauchy if for each ε>0 and each t>0 there exists an n0∈ℕ such that, for all n≥n0 and all integer d>0, one has N(xn+d-xn,t)>1-ε.It is well known that every convergent sequence in a quasi fuzzyβ-normed vector space is Cauchy. If each Cauchy sequence is convergent, then the quasi fuzzy β-normed space is said to be quasi fuzzy complete and the quasi fuzzy β-normed vector space is called a quasi fuzzy Banach space.


## 2. Euler-Lagrange-Rassias Quartic Functional Equations

LetX, Y be real vector spaces. In this section, we will investigate that the functional equation (3) is equivalent to the presented functional equation (6).Lemma 8.
A mappingf:X→Y satisfies the functional equation (3) if and only if f satisfies
(12)f(2x+y)+f(x+2y)=9f(x)+9f(y)+9f(x+y)-f(x-y),
for all x,y∈X.Proof.
It follows from [31, 32].Theorem 9.
A mappingf:X→Y satisfies the functional equation (3) if and only if f satisfies the functional equation (7).Proof.
It is easy to verify thatf(0)=0 by letting x=y=0 in (3). We will show this induction on a. Lemma 8 implies that it is true when a=2, and we may assume it holds for all a. Now, letting x=(a-1)x+y and y=x in (3), we have
(13)f((a+1)x+y)+f((a-3)x+y)=4f(ax+y)+4f((a-2)x+y)+24f(x)-6f((a-1)x+y),
for all x,y∈X. After the switching x and y in the previous equation (13),
(14)f(x+(a+1)y)+f(x+(a-3)y)=4f(x+ay)+4f(x+(a-2)y)+24f(y)-6f(x+(a-1)y),
for all x,y∈X. Adding two equations (13) and (14), we have
(15)f(x+(a+1)y)+f(x+(a+1)y)=-[f((a-3)x+y)+f(x+(a-3)y)]+4[f(ax+y)+f(x+ay)]+4[f((a-2)x+y)+f(x+(a-2)y)]-6[f((a-1)x+y)+f(x+(a-1)y)]+24[f(x)+f(y)],
for all x,y∈X. The induction steps imply that
(16)f(x+(a+1)y)+f(x+(a+1)y)=-12a3f(x-y)-12a2f(x-y)+12a3f(x+y)+52a2f(x+y)+4af(x+y)+2f(x+y)+(a4+4a3+4a2)[f(x)+f(y)]=-12(a+1)((a+1)-1)2f(x-y)+12(a+1)((a+1)+1)2f(x+y)+((a+1)2-1)2[f(x)+f(y)],
for all x,y∈X. Hence we have
(17)f((a+1)x+y)+f(x+(a+1)y)+12(a+1)((a+1)-1)2f(x-y)=((a+1)2-1)2[f(x)+f(y)]+12(a+1)((a+1)+1)2f(x+y),
for all x,y∈X, as desired.Note thatf(ax)=a4f(x) by letting y=0 in (7).Lemma 10.
A mappingf:X→Y satisfies the functional equation (7) if and only if f satisfies the functional equation (6).Proof.
It is easy to show thatf(0)=0 and f(ax)=a4f(x) by putting x=y=0 and y=0 in (7), respectively. By letting y=by in (7), we have
(18)f(ax+by)+f(x+aby)+12a(a-1)2f(x-by)=(a2-1)2[f(x)+f(by)]+12a(a+1)2f(x+by).
Also, switching x and y in the above equation and then adding two equations, we get
(19)f(ax+by)+f(bx+ay)=-[f(abx+y)+f(x+aby)]+(a2-1)2×[f(x)+f(y)+b4f(x)+b4f(y)]+12a(a+1)2[f(bx+y)+f(x+by)]-12a(a-1)2[f(bx-y)+f(x-by)].
Then (7) implies that
(20)f(ax+by)+f(bx+ay)+12ab(a-b)2f(x-y)=(a2-b2)2[f(x)+f(y)]+12ab(a+b)2f(x+y).Corollary 11.
A mappingf:X→Y satisfies the functional equation (3) if and only if f satisfies the functional equation (6).


## 3. Stability in Quasi-β-Normed Spaces

Throughout this section, letX be a quasi-β-normed space and let Y be a quasi β-Banach space with a quasi β-norm ∥·∥Y. Let K be the modulus of concavity of ∥·∥Y. We will investigate the Hyers-Ulam-Rassias stability problem for the functional equation (7). For a given mapping f:X→Y and all fixed integers a with a≠0, a≠±1, let
(21)Daf(x,y)≔f(ax+y)+f(x+ay)+12a(a-1)2f(x-y)-(a2-1)2[f(x)+f(y)]-12a(a+1)2f(x+y),
for x and y in X.Theorem 12.
Suppose that there exists a mappingϕ:X2→ℝ+∶=[0,∞) for which a mapping f:X→Y satisfies f(0)=0 and
(22)∥Daf(x,y)∥Y≤ϕ(x,y),
and the series ∑j=0∞‍(K/|a|4β)jϕ(ajx,ajy) converges for all x,y∈X. Then there exists a unique Euler-Lagrange-Rassias quartic mapping Q:X→Y which satisfies (7) and the inequality
(23)∥f(x)-Q(x)∥Y≤K|a|4β∑j=0∞(K|a|4β)jϕ(ajx,0),
for all x∈X.Proof.
By lettingy=0 in (22) and f(0)=0, we have
(24)∥f(ax)+f(x)+12a(a-1)2f(x)-12a(a+1)2-(a2-1)2f(x)12∥Y=∥f(ax)-a4f(x)∥Y=|a|4β∥f(x)-1a4f(ax)∥Y≤ϕ(x,0),
that is,
(25)∥f(x)-1a4f(ax)∥Y≤1|a|4βϕ(x,0),
for all x∈X. For any positive integer m, we have
(26)∥(1a4)mf(amx)-(1a4)m+1f(am+1x)∥Y≤1|a|4β(1|a|4β)mϕ(amx,0),
for all x∈X. For any positive integers n and m with m<n,
(27)∥(1a4)mf(amx)-(1a4)nf(anx)∥Y≤1Km-11|a|4β∑j=mn-1‍(K|a|4β)jϕ(ajx,0),
for all x∈X. By letting m=0, we have
(28)∥f(x)-(1a4)nf(anx)∥Y≤K|a|4β∑j=0n-1(K|a|4β)jϕ(ajx,0),
for all x∈X and n∈ℕ. Since the right-hand side of the previous inequality tends to 0 as n→∞, {(1/a4)nf(anx)} is a Cauchy sequence in the quasi β-Banach space Y. Thus we may define
(29)Q(x)=limn→∞(1a4)nf(anx),
for all x∈X. Hence we have the inequality (23). Since K≥1, replacing x and y by anx and any, respectively, and dividing by |a|4βn in (22), we have
(30)(1|a|4β)n∥Daf(anx,any)∥Y≤(K|a|4β)nϕ(anx,any),
for all x,y∈X. By taking n→∞, the definition of Q implies that Q satisfies (7) for all x,y∈X; that is, Q is the Euler-Lagrange-Rassias quartic mapping. It is left to show that the quadratic mapping Q is unique. Assume that there exists T:X→Y satisfying (7) and (23). Then
(31)∥T(x)-Q(x)∥Y=(1|a|4β)n∥T(anx)-Q(anx)∥Y≤(1|a|4β)nK(12∥T(anx)-f(anx)∥Y+∥f(anx)-Q(anx)∥Y12)≤2|a|4βn∑j=n∞‍(K|a|4β)jϕ(ajx,0),
for all x∈X. By letting n→∞, we immediately have the uniqueness of Q.Theorem 13.
Suppose that there exists a mappingϕ:X2→ℝ+∶=[0,∞) for which a mapping f:X→Y satisfies f(0)=0 and
(32)∥Daf(x,y)∥Y≤ϕ(x,y),
and the series ∑j=1∞‍(|a|4βK)jϕ(a-jx,a-jy) converges for all x,y∈X. Then there exists a unique Euler-Lagrange-Rassias quartic mapping Q:X→Y which satisfies (7) and the inequality
(33)∥f(x)-Q(x)∥Y≤∑j=1∞(|a|4βK)jϕ(a-jx,0),
for all x∈X.Proof.
Ifx is replaced by (1/a)x in inequality (25), we have
(34)∥f(x)-a4f(1ax)∥Y≤ϕ(x,0),
for all x∈X. The remains of the proof follow from the proof of Theorem 12.


## 4. Stability in Quasi Fuzzyβ-Normed Spaces

Let us fix some notations which will be used throughout this section. We assumeX is a vector space and (Y,N) is a quasi fuzzy β-Banach space. We will prove the Hyers-Ulam-Rassias stability of the functional equation satisfying equation (7) in quasi fuzzy β-Banach space.Theorem 14.
Letϕ:X2→[0,∞) be a function such that for some 0<|α|<|a|4(35)N′(ϕ(ax,0),t)≥N′(αϕ(x,0),t),limn→∞N′(ϕ(anx,any),|a|4nβt)=1,
for all x,y∈X and all t>0. Let f:X→Y be a mapping satisfying f(0)=0 and
(36)N(Daf(x,y),t)≥N′(ϕ(x,y),t),
for all x,y∈X and all t>0.
ThenQ(x)≔N-limn→∞(1/a4n)f(anx) exists for each x∈X and defines a unique Euler-Lagrange-Rassias quartic mapping Q:X→Y such that
(37)N(f(x)-Q(x),t)≥N′(ϕ(x,0),|a|4β-|α|β2t),
for all x∈X and all t>0.Proof.
Lety=0 in inequality (36). Since f(0)=0, we have
(38)N(Daf(x,0),t)=N(1a4f(ax)-f(x),t|a|4β)≥N′(ϕ(x,0),t),
for all x∈X and all t>0. Replacing x by anx in inequality (38),
(39)N(1a4f(an+1x)-f(anx),t|a|4β)≥N′(ϕ(anx,0),t),
that is,
(40)N((1a4)n+1f(an+1x)-(1a4)nf(anx),t|a|4β1|a|4nβ)≥N′(ϕ(anx,0),t),
for all x∈X, t>0 and n≥0. Since N′(ϕ(anx,0),t)≥N′(ϕ(x,0),(t/|α|nβ)),
(41)N((1a4)n+1f(an+1x)-(1a4)nf(anx),t|a|4β1|a|4nβ)≥N′(ϕ(x,0),t|α|nβ).
By letting t=|α|nβt in the previous inequality,
(42)N((1a4)n+1f(an+1x)-(1a4)nf(anx),t|a|4β·(|α||a|4)nβ)≥N′(ϕ(x,0),t),
for all x∈X and all t>0. Hence we get
(43)N((1a4)nf(anx)-f(x),∑j=0n-1t|a|4β·(|α||a|4)jβ)≥min⋃j=0n-1[⋃j=0n-1N(⋃j=0n-1(1a4)j+1f(aj+1x)iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii-(1a4)jf(ajx),t|a|4β·(|α||a|4)jβ⋃j=0n-1)]≥N′(ϕ(x,0),t),
for all x∈X and all t>0. Letting x=amx in the previous inequality, we have
(44)N((1a4)nf(an+mx)-f(amx),∑j=0n-1t|a|4β·(|α||a|4)jβ)≥N′(ϕ(amx,0),t)≥N′(ϕ(x,0),t|α|mβ),
that is,
(45)N(t|a|4mβ∑j=0n-11|a|4β·(|α||a|4)jβ(1a4)n+mf(an+mx)-(1a4)mf(amx),t|a|4mβ∑j=0n-11|a|4β·(|α||a|4)jβ)≥N′(ϕ(x,0),t|α|mβ),
for all x∈X and all t>0. Letting t=|α|mβt, we have
(46)N(t|a|4β∑j=mn+m-1(|α||a|4)jβ(1a4)n+mf(an+mx)-(1a4)mf(amx),t|a|4β∑j=mn+m-1(|α||a|4)jβ)≥N′(ϕ(x,0),t),
for all x∈X,  t>0 and n,m≥0. Hence {(1/a4n)f(anx)} is a Cauchy sequence in the quasi fuzzy β-Banach space (Y,N). Thus, we may define
(47)Q(x)=N-limn→∞1a4nf(anx),
for all x∈X. Hence inequality (43) implies that
(48)N(Q(x)-f(x),t)≥min{N(Q(x)-(1a4)nf(anx),t2),N((1a4)nf(anx)-f(x),t2)}≥N′(ϕ(x,0),t(2/|a|4β)∑j=0n-1(|α|/|a|4)jβ),
for n large enough and all x∈X. Taking the limit as n→∞ and using (N6), we have
(49)N(Q(x)-f(x),t)≥N′(ϕ(x,0),|a|4β-|α|β2t),
for all x∈X. Hence it satisfies inequality (37). Now letting x=anx and y=any in (36),
(50)N(Daf(anx,any),t)≥N′(ϕ(ann,any),t),
for all x∈X and all t>0. This implies that
(51)N((1a4)nDaf(anx,any),t)≥N′(ϕ(ann,any),|a|4βt),
for all x∈X and all t>0. Since N′(ϕ(ann,any),|a|4βt)=1, we may conclude that the mapping Q satisfies (7); that is, Q is the Euler-Lagrange-Rassias quartic mapping. It is left to show that the quartic mapping Q is unique. Assume there is another T:X→Y satisfying (7) and inequality (37). For each x∈X, clearly Q(anx)=a4nQ(x) and T(anx)=a4nT(x) for all n∈ℕ. 							(52)N(T(x)-Q(x),t)=N(1a4nT(anx)-1a4nQ(anx),t)=N(T(anx)-Q(anx),|a|4nβt)≥min{N(T(anx)-f(anx),|a|4nβt2),N(|a|4nβt2f(anx)-Q(anx),|a|4nβt2)}≥N′(ϕ(anx,0),|a|4β-|α|β2·|a|4nβt2)≥N′((|a|4|α|)nβ·|a|4β-|α|β4tϕ(x,0),(|a|4|α|)nβ·|a|4β-|α|β4t),
for all x∈X and t>0. Since 0<|α|<|a|4, we have limn→∞(|a|4/|α|)nβ=∞. Hence N(T(x)-Q(x),t)=1; that is, the mapping Q is unique, as desired.Theorem 15.
Letϕ:X2→[0,∞) be a function such that for some |α|>|a|4(53)N′(ϕ(1ax,0),t)≥N′(αϕ(x,0),t),limn→∞N′(ϕ(a-nx,a-ny),1|a|4nβt)=1,
for all x,y∈X and all t>0. Let f:X→Y be a mapping satisfying f(0)=0 and
(54)N(Daf(x,y),t)≥N′(ϕ(x,y),t),
for all x,y∈X and all t>0.
ThenQ(x):=N-limn→∞a4nf((1/an)x) exists for each x∈X and defines a unique Euler-Lagrange-Rassias quartic mapping Q:X→Y such that
(55)N(f(x)-Q(x),t)≥N′(ϕ(x,0),|α|β-|a|4β2|α|2βt),
for all x∈X and all t>0.Proof.
The techniques are completely similar to the proof of Theorem14. Hence we present some key idea of this proof. Let y=0 in inequality (54). Since f(0)=0, we have
(56)N(Daf(x,0),t)=N(f(ax)-a4f(x),t)≥N′(ϕ(x,0),t),
for all x∈X and all t>0. Replacing x by (1/a)x in inequality (56), we have
(57)N(f(x)-a4f(1ax),t)≥N′(ϕ(1ax,0),t)≥N′(ϕ(x,0),1|α|βt)
or
(58)N(f(x)-a4f(1ax),|α|βt)≥N′(ϕ(x,0),t),
for all x∈X and all t>0. For positive integers n and m,
(59)N(|α|β∑j=mn+m-1(|a|4|α|)jβta4(n+m)f(1an+mx)-a4mf(1amx),|α|β∑j=mn+m-1(|a|4|α|)jβt)≥N′(ϕ(x,0),t),
for all x∈X and t>0. Hence we may conclude that {a4nf((1/an)x)} is a Cauchy sequence in the quasi fuzzy β-Banach space (Y,N). Thus we may define
(60)Q(x)=N-limn→∞a4nf(1anx),
for all x∈X. Also, for any positive integer n, we get
(61)N(a4nf(1anx)-f(x),t)≥N′(ϕ(x,0),t|α|β∑j=0n-1(|a|4/|α|)jβ),
for all x∈X and all t>0. This implies inequality (55).



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*Source: 908168-2013-10-31.xml*