In this article, we present generalized Hyers–Ulam stability results of a cubic functional equation associated with an approximate cubic Lie derivations on convex modular algebras χρ with Δ2-condition on the convex modular functional ρ.

National Research Foundation of KoreaNRF-2016R1D1A3B039309711. Introduction

In 1940, S. M. Ulam [1] raised the question concerning the stability of group homomorphisms. Let G be a group and let G′ be a metric group with the metric d(·,·). Given ε>0, does there exist δ>0 such that if a mapping f:G→G′ satisfies the inequality (1)dfxy,fxfy<δfor all x,y∈G, then there exists a homomorphism F:G→G′ with d(f(x),F(x))<ε for all x∈G? D. H. Hyers [2] has solved the problem of Ulam for the case of additive mappings in 1941. The result was generalized by T. Aoki [3] in 1950, by Th.M. Rassias [4] in 1978, by J. M. Rassias [5] in 1992, and by P. Gǎvruta [6] in 1994. Over the past few decades, many mathematicians have investigated the generalized Hyers–Ulam stability theorems of various functional equations [7–12].

Now, we recall some basic definitions and remarks of modular spaces with modular functions, which are primitive notions corresponding to norms or metrics, as in the following [13–15].

Definition 1.

Let χ be a linear space.

A function ρ:χ→[0,∞] is called a modular if, for arbitrary x,y∈χ,

ρ(x)=0 if and only if x=0,

ρ(αx)=ρ(x) for every scalar α with |α|=1,

ρ(αx+βy)≤ρ(x)+ρ(y) for any scalars α, β, where α+β=1 and α,β≥0;

alternatively, if (3) is replaced by

ρ(αx+βy)≤αρ(x)+βρ(y) for every scalars α, β, where α+β=1 and α,β≥0,

then we say that ρ is a convex modular.

It is well known that a modular ρ defines a corresponding modular space, i.e., the linear space χρ given by (2)χρ=x∈χ:ρλx→0 as λ→0.

Let ρ be a convex modular. Then, we remark the modular space χρ can be a Banach space equipped with a norm called the Luxemburg norm, defined by(3)xρ=infλ>0:ρxλ≤1.

If ρ is a modular on χ, we note that ρ(tx) is an increasing function in t≥0 for each fixed x∈χ; that is, ρ(ax)≤ρ(bx), whenever 0≤a<b. In addition, if ρ is a convex modular on χ, then ρ(αx)≤αρ(x) for all x∈χ and 0≤α≤1. Moreover, we see that ρ(αx)≤αρ(x) for all x∈χ and α≤1.

Remark. (a) In general, we note that ρ(∑i=1nαixi)≤∑i=1nαiρ(xi) for all xi∈χ and αi≥0(i=1,…,n) whenever 0<∑i=1nαi≔α≤1 [14].

(b) Consequently, we lead to ρ(∑i=1nαixi)≤∑i=1nαiρxi for all xi∈χ and 0<∑i=1nαi≔α≤1.

Definition 2.

Let χρ be a modular space and let xn be a sequence in χρ. Then,

{xn} is ρ-convergent to x∈χρ and we write xn→ρx if ρ(xn-x)→0 as n→∞;

{xn} is called ρ-Cauchy in χρ if ρ(xn-xm)→0 as n,m→∞;

a subset K of χρ is called ρ-complete if and only if any ρ-Cauchy sequence in K is ρ-convergent to an element in K.

They say that the modular ρ has the Fatou property if and only if ρ(x)≤liminfn→∞ρ(xn) whenever the sequence xn is ρ-convergent to x. A modular function ρ is said to satisfy the Δ2-condition if there exists κ>0 such that ρ(2x)≤κρ(x) for all x∈χρ.

In 2014, G. Sadeghi [16] has demonstrated generalized Hyers–Ulam stability via the fixed point method of a generalized Jensen functional equation f(rx+sy)=rg(x)+sh(y) in convex modular spaces with the Fatou property satisfying the Δ2-condition with 0<κ≤2. In [15], the authors have proved the generalized Hyers–Ulam stability of quadratic functional equations via the extensive studies of fixed point theory in the framework of modular spaces whose modulars are convex and lower semicontinuous but do not satisfy any relatives of Δ2-conditions (see also [17, 18]). Recently, the authors [14, 19, 20] have investigated stability theorems of functional equations in modular spaces without using the Fatou property and Δ2-condition. In 2001, J. M. Rassias [21] has introduced to study Hyers–Ulam stability of the following cubic functional equation: (4)f2x+y+fx-y+3fy=3fx+y+6fx,which is equivalent to (5)f2x+y+f2x-y=2fx+y+2fx-y+12fx,whose general solution is characterized as f(x)=B(x,x,x) where B is symmetric and additive for each fixed one variable [22]. For this reason, every solution of the cubic functional equation is said to be a cubic mapping.

Now, we say that χρ is called a (convex) modular algebra if the fundamental space χ is an algebra over K=R or C with (convex) modular ρ subject to ρ(ab)≤ρ(a)ρ(b) for all a,b∈χ. A subset K of a convex modular algebra χρ is called ρ-complete if and only if any ρ-Cauchy sequence in K is ρ-convergent to an element in K. Throughout the paper, χρ will be a ρ-complete convex modular algebra and the symbol [a,b] will denote the commutator ab-ba. We say that a mapping f is cubic homogeneous if f(λx)=λ3f(x) for all vectors x and all scalars λ, and a cubic homogeneous mapping f is called a cubic Lie derivation if f([x,y])=[f(x),y3]+[x3,f(y)] for all vectors x,y [23, 24].

In this article, we first investigate generalized Hyers–Ulam stability of the equation(6)f3x-y+fx+y=2f2x-y+12fx+2fy,in ρ-complete convex modular algebras without using the Fatou property and Δ2-condition and then present alternatively generalized Hyers–Ulam stability of (6) using necessarily Δ2-condition without the Fatou property in ρ-complete convex modular algebras.

2. Generalized Hyers–Ulam Stability of (<xref ref-type="disp-formula" rid="EEq1.1">6</xref>)

First of all, we remark that (6) is equivalent to the original cubic functional equation, and so every solution of (6) is a cubic mapping.

For notational convenience, we let the difference operators CEf of cubic equation (6) and CDf of cubic derivation be as follows: (7)CEfλx,λy≔f3λx-λy+fλx+λy-2λ3f2x-y-12λ3fx-2λ3fy,CDfx,y≔fx,y-fx,y3-x3,fyfor all x,y in a linear space X and λ∈Λ≔{λ∈C:λ=1}. In the following, we present a generalized Hyers–Ulam stability via direct method of the system CEf=0 and CDf=0 in ρ-complete convex modular algebras without using both the Fatou property and Δ2-condition.

Theorem 3.

Suppose that a mapping f:χρ→χρ satisfies(8)ρCEfλx,λy≤ϕ1x,y,z,ρCDfx,y≤ϕ2x,yand ϕ1:χρ3→[0,∞), ϕ2:χρ2→[0,∞) are mappings such that(9)Φx,y,z≔∑j=0∞ϕ12jx,2jy,2jz23j<∞,limn→∞ϕ22nx,2ny82n=0for all x,y,z∈χρ and λ∈Λ. If for each x∈χρ the mapping r→f(rx) from R to χρ is continuous, then there exists a unique cubic Lie derivation F1:χρ→χρ which satisfies equation (6) and(10)ρfx-F1x≤116Φx,x,0for all x∈χρ.

Proof.

Putting y=x and λ=1 in (8), we obtain(11)ρ2f2x-16fx≤ϕ1x,x,0,which yields(12)ρf2x-8fx=12ρ2f2x-16fx≤12ϕ1x,x,0,ρfx-f2x8≤18ρf2x-8fx≤116ϕ1x,x,0for all x∈χρ. Since ∑j=0n-11/8j+1≤1, we prove the following functional inequality:(13)ρfx-f2nx23n=ρ∑j=0n-1f2jx23j-f2j+1x23j+1=ρ∑j=0n-1123j+18f2jx-f2j+1x≤∑j=0n-1123j+1ρ8f2jx-f2j+1x≤116∑j=0n-1ϕ12jx,2jx,023jfor all x∈χρ by using the property of convex modular ρ.

Now, replacing x by 2mx in (13), we have(14)ρf2mx23m-f2m+nx23m+n≤116∑j=mm+n-1ϕ12jx,2jx,023jwhich converges to zero as m→∞ by assumption (9). Thus the above inequality implies that the sequence f2nx/23n is ρ-Cauchy for all x∈χρ and so it is convergent in χρ since the space χρ is ρ-complete. Thus, we may define a mapping F1:χρ→χρ as (15)F1x≔ρ-limn→∞f2nx23n⇔limn→∞ρf2nx23n-F1x=0,for all x∈χρ.

Claim 1. F1 is a cubic mapping satisfying approximation (10). In fact, if we put (x,y,z)≔(2nx,2ny,0) in (8) and then divide the resulting inequality by 23n, one obtains(16)ρCEf2nλx,2nλyR·23n≤ρCEf2nλx,2nλyR·23n≤ϕ12nx,2ny,0R·23n→0for all x,y∈χρ, where R≥16λ+3 is a fixed positive real. Thus we figure out by use of the first remark(17)ρ1RCEF1λx,λy=ρ1RCEF1λx,λy-CEf2nλx,2nλyR·23n+CEf2nλx,2nλyR·23n≤1RρF13λx-λy-f2n3λx-λy23n+1RρF1λx+λy-f2nλx+λy23n+2λ3RρF12x-y-f2n2x-y23n+12λ3RρF1x-f2nx23n+2λ3RρF1y-f2ny23n+1RρCEf2nλx,2nλy23nfor all x,y∈χρ, λ∈Λ and all positive integers n. Taking the limit as n→∞, one obtains ρ(1/RCEF1(λx,λy))=0, and so CEF1(λx,λy)=0 for all x,y∈χρ. Hence, taking λ=1 in CEF1(x,y)=0, it follows that F1 satisfies (6) and so it is cubic. On the other hand, since ∑i=0n1/23(i+1)+1/23≤1 for all n∈N, it follows from (12) and the first remark that (18)ρfx-F1x=ρ∑i=0n123i+123f2ix-f2i+1x+f2n+1x23n+1-F12x23≤12∑i=0n123i+1ρCEf2ix,2ix+123ρf2n+1x23n-F12x≤116∑i=0n123iϕ12ix,2ix,0+123ρf2n·2x23n-F12x,without applying the Fatou property of the modular ρ for all x∈χρ and all n∈N, from which we obtain the approximation of f by the cubic mapping F1 as follows: (19)ρfx-F1x≤116∑i=0∞123iϕ12ix,2ix,0=116Φx,x,0for all x∈χρ by taking n→∞ in the last inequality.

Claim 2. F1 is cubic homogeneous. By (17), we have CEF1(λx,λx)=0, which yields F1(2λx)=8λ3F1(x) for all x∈χρ and λ∈Λ. From the assumption that for each x∈χρ the mapping r→f(rx) from R to χρ is continuous, it follows that F1(λx)=λ3F1(x) for all x∈χρ and λ∈R by the same argument as in the paper [4, 25]. Thus, for any nonzero λ∈C(20)F1λx=F12λλλ2x=8λλ3F1λ2x=8λλ3λ23F1x=λ3F1xfor all x∈χρ and λ∈C, which concludes that F1 is cubic homogeneous.

Claim 3. F1 is a cubic Lie derivation. From the second inequality in (9) and the second condition in (8), we arrive at (21)ρ14CDF1x,y=ρ14CDF1x,y-CDf2nx,2ny4·82n+CDf2nx,2ny4·82n≤14ρF1x,y-f22nx,y82n+14ρx3,f2ny8n-x3,F1y+14ρf2nx,y38n-F1x,y3+14·82nρCDf2nx,2nyfor all x,y∈χρ, which tends to zero as n tends to ∞. Therefore, one obtains ρ(1/4CDF1(x,y))=0, and so F1 is a cubic Lie derivation.

Claim 4. F1 is a unique cubic Lie derivation. To show the uniqueness of F1, let us assume that there exists a cubic Lie derivation G1:χρ→χρ which satisfies the inequality (22)ρfx-G1x≤116∑j=0∞ϕ12jx,2jx23j=116Φx,xfor all x∈χρ, but suppose F1(x0)≠G1(x0) for some x0∈X. Then there exists a positive constant ε>0 such that ε<ρ(F1(x0)-G1(x0)). For such given ε>0, it follows from (9) that there is a positive integer n0∈N such that ∑j=n0∞ϕ1(2jx0,2jx0)/23(j+1)<ε. Since F1 and G1 are cubic mappings, we see from the equality F1(2n0x0)=23n0F1(x0) and G1(2n0x0)=23n0G1(x0) that (23)ε<ρF1x0-G1x0=ρF12n0x0-f2n0x023n0+f2n0x0-G12n0x023n0≤123n0ρF12n0x0-f2n0x0+123n0ρf2n0x0-G12n0x0≤123n018∑j=0∞ϕ12j+n0x0,2j+n0x023j=18∑j=n0∞ϕ12jx0,2jx023j<ε,which leads a contradiction. Hence the mapping F1 is a unique cubic Lie derivation near f satisfying approximation (10) on the modular algebra χρ.

As a corollary of Theorem 3, we obtain the following stability result of cubic equation (6) associated with cubic Lie derivation on the Banach algebra χρ, which may be considered as endowed with modular ρ=∥·∥.

Corollary 4.

Suppose χρ is a Banach algebra with norm ·. For given real numbers θ,θi,ϑi≥0, ri<3(i=1,2), and a1+b1<3, a2+b2<6, suppose that a mapping f:χρ→χρ satisfies (24)CEfλx,λy≤θ1xr1+θ2yr2+ϑ1xa1yb1,CDfx,y≤ϑ2xa2yb2for all x,y∈χρ and λ∈Λ, where x,y≠0 whenever ri,ai,bi<0 and that for each x∈χρ the mapping r→f(rx) from R to χρ is continuous. Then there exists a unique cubic Lie derivation F1:χρ→χρ such that (25)ρfx-F1x≤θ1xr1223-2r1+θ2xr2223-2r2+θxa1+b1223-2a1+b1for all x∈χρ, where x≠0 whenever ri,a1+b1<0.

We observe that if the modular ρ satisfies the Δ2-condition, then κ≥1 for nontrivial modular ρ, and κ≥2 for nontrivial convex modular ρ. See [13–16]. In the following theorem, we prove generalized Hyers–Ulam stability of the system CDf=0 and CEf=0 using necessarily Δ2-condition, which permits the existence of ρ-Cauchy sequence in χρ.

Theorem 5.

Let χρ be a ρ-complete convex modular space with Δ2-condition. Suppose there exist two functions φ1,φ2:χρ2→[0,∞) for which a mapping f:χρ→χρ satisfies(26)ρCEfλx,λy≤φ1x,y,Ψx,y≔∑j=1∞k4j2jφ1x2j,y2j<∞,(27)ρCDfx,y≤φ2x,y,limn→∞k6nφ2x2n,y2n=0for all x,y∈χρ and λ∈Λ. If for each x∈χρ the mapping r→f(rx) from R to χρ is continuous, then there exists a unique cubic Lie derivation F2:χρ→χρ satisfying (6) and(28)ρfx-F2x≤14κ2Ψx,xfor all x∈χρ.

Proof.

First, we remark that since ∑j=1∞κ4j/2jφ1(0,0)=Ψ(0,0)<∞ and ρ(CEf(0,0))≤φ1(0,0), we lead to φ1(0,0)=0, CEf(0,0)=0 and so f(0)=0. Thus, it follows from (12) that (29)ρfx-8fx2≤12φ1x2,x2for all x∈χρ. Thus, one obtains the following inequality by the convexity of the modular ρ and Δ2-condition: (30)ρfx-82fx22≤12ρ2fx-2·8fx2+122ρ22·8fx2-22·82fx22≤κ22φ1x2,x2+κ523φ1x22,x22for all x∈χρ. Then using the repeated process for any n≥2, we prove the following functional inequality:(31)ρfx-8nfx2n≤12κ3∑j=1nκ4j2jφ1x2j,x2jfor all x∈χρ. In fact, it is true for n=2. Assume that inequality (31) holds true for n. Thus, using the convexity of the modular ρ, we deduce (32)ρfx-8n+1fx2n+1=ρ122fx-2·8fx2+122·8fx2-2·8n+1fx2n+1≤κ2ρfx-8fx2+κ42ρfx2-8nfx2n+1≤κ212φ1x2,x2+κ42·12κ3∑j=1nκ4j2jφ1x2j+1,x2j+1=12κ3∑j=1n+1κ4j2jφ1x2j,x2j,which proves (31) for n+1. Now, replacing x by 2-mx in (31), we have (33)ρ23mfx2m-23m+nfx2m+n≤κ3mρfx2m-23nfx2m+n≤κ3m2κ3∑j=1nκ4j2jφ1x2j+m,x2j+m≤κ3m2κ3∑j=1nκ4j2jφ1x2j+m,x2j+m·κm2m=12κ3∑j=1nκ4j+m2j+mφ1x2j+m,x2j+m=12κ3∑j=m+1m+nκ4j2jφ1x2j,x2j,which converges to zero as m→∞ by assumption (27). Thus, the sequence 8nfx/2n is ρ-Cauchy for all x∈χρ and so it is ρ-convergent in χρ since the space χρ is ρ-complete. Thus, we may define a mapping F2:χρ→χρ as (34)F2x≔ρ-limn→∞8nfx2n⇔limn→∞ρ8nfx2n-F2x=0,for all x∈χρ.

Claim 1. F2 is a cubic mapping with estimation (28) near f. By Δ2-condition without using the Fatou property, we can see the following inequality: (35)ρfx-F2x≤12ρ2fx-2·8nfx2n+2·8nfx2n-2F2x≤κ2ρfx-8nfx2n+κ2ρ8nfx2n-F2x≤κ2·12κ3∑j=1nκ4j2jφ1x2j,x2j+κ2ρ8nfx2n-F2x≤14κ2∑j=1∞κ4j2jφ1x2j,x2j=14κ2Ψx,xby taking n→∞, which yields approximation (28).

Now, setting (x,y)≔(2-nx,2-ny) in (26) and multiplying the resulting inequality by 8n, we get (36)ρ23nCEf2-nλx,2-nλy≤κ3nφ12-nx,2-ny≤κ3nφ12-nx,2-ny·κn2n=κ4n2nφ12-nx,2-ny,which tends to zero as n→∞ for all x,y∈χρ. Thus, it follows from the first remark that (37)ρ1RCEF2λx,λy=ρ1RCEF2λx,λy-23nRCEfλx2n,λy2n+23nRCEfλx2n,λy2n≤1RρF23λx-λy-23nf3λx-λy2n+2λ3RρF2y-23nfy2n+1RρF2λx+λy-23nfλx+λy2n+2λ3RρF22x-y-23nf2x-y2n+12λ3RρF2x-23nfx2n+1Rρ23nCEfλx2n,λy2nfor all x,y∈χρ, λ∈Λ, and all positive integers n, where R≥16λ3+3 is a fixed real number. Taking the limit as n→∞, one obtains ρ(1/RCEF2(x,y))=0, and thus CEF2(x,y)=0 for all x,y∈χρ. Hence F2:χρ→χρ satisfies (6), and so it is cubic.

Claim 2. F2 is a cubic Lie derivation. By the same proof of Theorem 3, the mapping F2 is a cubic homogeneous mapping. From the last inequality in (27) and the last condition in (26), one obtains that (38)ρ14CDF2x,y=ρ14CDF2x,y-82nCDf2-nx,2-ny4+82nCDf2-nx,2-ny4≤14ρF2x,y-82nf2-2nx,y+14ρ8nx3,f2-ny-x3,F2y+14ρ8nf2-nx,y3-F2x,y3+14ρ82nCDf2-nx,2-ny≤14ρF2x,y-82nf2-2nx,y+14ρ8nx3,f2-ny-x3,F2y+14ρ8nf2-nx,y3-F2x,y3+κ6n4φ22-nx,2-nyfor all x,y∈χρ, from which CDF2(x,y)=0 by taking n→∞ and so F2 is a cubic Lie derivation.

Claim 3. F2 is unique. To show the uniqueness of F2, let us assume that there exists a cubic Lie derivation G2:χρ→χρ which satisfies the approximation (28). Since F2 and G2 are cubic mappings, we see from the equalities 23nF2(2-nx)=F2(x) and 23nG2(2-nx)=G2(x) that (39)ρG2x-F2x=ρ23n+123G2x2n-fx2n+23n+123fx2n-F2x2n≤κ3n+123ρG2x2n-fx2n+κ3n+123ρfx2n-F2x2n≤κ3n+123·12κ2∑j=1∞κ4j2jφ1x2j+n,x2j+n·κn2n≤κ24∑j=1∞κ4j+n2j+nφ1x2j+n,x2j+n=κ24∑j=n+1∞κ4j2jφ1x2j,x2jwhich tends to zero as n→∞ for all x∈χρ. Hence the mapping F2 is a unique cubic Lie derivation satisfying (28).

Remark. In Theorem 5 if χρ is a Banach algebra with norm ρ, and so ρ(2x)=2ρ(x), κ≔2, then we see from (26) and (27) that there exists a unique cubic Lie derivation F2:χρ→χρ, defined as F2(x)=limn→∞23nf(x/2n), x∈χρ, which satisfies (6) and (40)ρfx-F2x≤116∑j=1∞23jφ1x2j,x2jfor all x∈χρ.

As a corollary of Theorem 5, we obtain the following stability result of (6), which generalizes stability result on Banach algebras.

Corollary 6.

Suppose χρ is a Banach algebra with norm · and κ=2. For given real numbers θi,ϑi≥0, ri>3(i=1,2), a1+b1>3, and 6<a2+b2, if a mapping f:χρ→χρ satisfies (41)CEfx,y≤θ1xr1+θ2yr2+ϑ1xa1yb1,CDfx,y≤ϑ2xa2yb2for all x,y∈χρ and λ∈Λ, then there exists a unique cubic Lie derivation F2:χρ→χρ such that (42)fx-F2x≤θ1xr122r1-23+θ2xr222r2-8+ϑ1xa1+b122a1+b1-8for all x∈χρ.

3. Conclusion

We introduce modular algebras with modular ρ over K and obtain stability results of a cubic equation associated with cubic derivations on ρ-complete modular algebras, which generalizes stability results on Banach algebras.

Data Availability

Previously reported data were used to support this study and are available at https://journalofinequalitiesandapplications.springeropen.com/articles/10.1186/s13660-017-1422-z and https://www.hindawi.com/journals/jfs/2015/461719/. These prior studies (and datasets) are cited at relevant places within the text as [13–17].

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Acknowledgments

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2016R1D1A3B03930971).

UlamS. M.HyersD. H.On the stability of the linear functional equationAokiT.On the stability of the linear transformation in Banach spacesRassiasTh. M.On the stability of the linear mapping in Banach spacesRassiasJ. M.On the stability of the Euler-Lagrange functional equationGăvruţaP.A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappingsBorelliC.FortiG. L.On a general HyersUlam stability resultBrzdekJ.PopaD.RasaI.XuB.CzerwikS.On the stability of the quadratic mapping in normed spacesJungS.-M.MinS.Stability of the Wave Equation with a SourceKhanA.ShahK.LiY.KhanT. S.Ulam Type Stability for a Coupled System of Boundary Value Problems of Nonlinear Fractional Differential EquationsSkofF.Local properties and approximation of operatorsKimH.-M.HongY. S.Approximate Cauchy-Jensen type mappings in modular spacesKimH.HongY.Approximate quadratic mappings in modular spacesWongkumK.ChaipunyaP.KumamP.On the generalized UlamHyersRassias stability of quadratic mappings in modular spaces without Δ_{2}-conditionsSadeghiG.A fixed point approach to stability of functional equations in modular spacesKhamsiM. A.Quasicontraction mappings in modular spaces without Δ_{2}-conditionKimC. I.ParkS. W.A fixed point approach to stability of additive functional inequalities in fuzzy normed spacesKimH.ShinH.Approximation of almost cubic mappings by cubic mappings via modular functionalKimH.-M.ShinH.-Y.Refined stability of additive and quadratic functional equations in modular spacesRassiasJ. M.Solution of the Ulam stability problem for cubic mappingsJunK.KimH.The generalized Hyers-Ulam-Rassias stability of a cubic functional equationFošnerAjdaFošnerMajaApproximate Cubic Lie DerivationsKangD.On the stability of cubic Lie *-derivationsJunK.-W.KimH.-M.On the stability of Euler-Lagrange type cubic mappings in quasi-Banach spaces