Gonyostomum semen is a bloom-forming freshwater raphidophyte that is currently on the increase, which concerns water managers and ecologists alike. Much indicates that the recent success of G. semen is linked to its diel vertical migration (DVM), which helps to overcome the spatial separation of optimal light conditions for photosynthesis at the surface of a lake and the high concentration of phosphate in the hypolimnion. I here present data from a field study conducted in Lake Lundebyvannet (Norway) in 2017–2019 that are consistent with the idea that the DVM of G. semen also allows for a hypolimnetic uptake of ammonium. As expected, microbial mineralization of organic matter in a low-oxygen environment led to an accumulation of ammonium in the hypolimnion as long as G. semen was absent. In contrast, a decreasing or constantly lower concentration of hypolimnetic ammonium was found in presence of a migrating G. semen population. In summer of 2019, a short break in the DVM of G. semen coincided with a rapid accumulation of hypolimnetic ammonium, which was equally rapidly decimated when G. semen resumed its DVM. Taken together, these data support the idea that G. semen can exploit the hypolimnetic pool of ammonium, which may be one reason for the recent success of the species and its significant impact on the structure of the aquatic food web.
Citation: Thomas Rohrlack. Hypolimnetic assimilation of ammonium by the nuisance alga Gonyostomum semen[J]. AIMS Microbiology, 2020, 6(2): 92-105. doi: 10.3934/microbiol.2020006
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Gonyostomum semen is a bloom-forming freshwater raphidophyte that is currently on the increase, which concerns water managers and ecologists alike. Much indicates that the recent success of G. semen is linked to its diel vertical migration (DVM), which helps to overcome the spatial separation of optimal light conditions for photosynthesis at the surface of a lake and the high concentration of phosphate in the hypolimnion. I here present data from a field study conducted in Lake Lundebyvannet (Norway) in 2017–2019 that are consistent with the idea that the DVM of G. semen also allows for a hypolimnetic uptake of ammonium. As expected, microbial mineralization of organic matter in a low-oxygen environment led to an accumulation of ammonium in the hypolimnion as long as G. semen was absent. In contrast, a decreasing or constantly lower concentration of hypolimnetic ammonium was found in presence of a migrating G. semen population. In summer of 2019, a short break in the DVM of G. semen coincided with a rapid accumulation of hypolimnetic ammonium, which was equally rapidly decimated when G. semen resumed its DVM. Taken together, these data support the idea that G. semen can exploit the hypolimnetic pool of ammonium, which may be one reason for the recent success of the species and its significant impact on the structure of the aquatic food web.
Hyers [1] made a response to the question of Ulam in the context of Banach spaces in relation to additive mappings and was a considerable step towards further solutions in this area. Note the concept of stability is a major property in the qualitative theory of differential equations. Over the last few years, results have been presented on numerous types of differential equations. The approach proposed by Hyers [1] which provides the additive function is named a direct technique. This technique is a significant and helpful tool used to investigate the stability of different functional equations. In recent years, a number of research monographs and articles have been studied on diverse applications and generalizations of the HUS, like k-additive mappings, differential equations, Navier–Stokes equations, ODEs, and PDEs (see [2,3,4]). Also in recent years, the stability of different (integral and differential, others functional) equations and other subjects (such as C∗-ternary algebras, groups, flows and Banach algebras) have been investigated. Fixed–point methods are useful when examining stability and fixed point theory proposes vital tools for solving problems arising in different fields of functional analysis, like equilibrium problems, differential equations, and dynamical systems.
Assume Banach algebras Q and Q′′. Suppose (Q′,Δ) is a probability measure space and suppose (Q,BQ) and (Q′′,BQ′′) are Borel measurable spaces. Then a map f:Q′×Q→Q′′ is a operator if {℘:f(℘,α)∈ν}∈Δ for each α in Q and ν∈BQ′′. Assume ℧=(℧1,…,℧m) and Ω=(Ω1,…,Ωm),m∈N. Then we have
℧⪯Ω⟺℧ı≤Ωı,ı=1,⋯,m; |
and also
℧→0⟺℧ı→0,ı=1,⋯,m. |
Definition 1.1 ([5]). Let ∇≠∅ is a set and d:∇2→[0,+∞]m,m∈N, is a given mapping. If the following conditions are satisfied, then we say d is a generalized metric on ∇:
(1) For each (g,g′)∈∇×∇, we get
d(g,g′)=(0,⋯,0)⏟m⟺g=g′; |
(2) For each (g,g′)∈∇×∇, we get
d(g′,g)=d(g,g′)⟺g=g′; |
(3) For each g,g′,g′′∈∇, we get
d(g,g′′)+d(g′′,g′)⪰d(g′,g). |
Theorem 1.2 ([5]). Assume the following assumptions:
(1) d:∇2→[0,+∞]m,m∈N, and (∇,d) is a complete generalized metric space.
(2) L:∇→∇ is a strictly contractive mappingwith Lipschitz constant Z<1.
Then for each g∈∇, either
d(Lng,Ln+1g)=m⏞(+∞,⋯,+∞) |
for each n∈N∪{0} or there is a n0∈N such that
(1) d(Lng,Ln+1g)⪯m⏞(+∞,⋯,+∞),∀n≥n0;
(2) The sequence {Lng} converges to a fixed point (g′)∗ of L;
(3) (g′)∗ is the unique fixed point of L in the set ∁={g′∈∇∣d(Ln0g,g′)⪯m⏞(+∞,⋯,+∞)};
(4) d(g′,(g′)∗)⪯11−Zd(g′,Lg′) for each g′∈∁.
We use fixed-point way to study the multi-stability of antiderivations associated with the following inequality:
diag[‖f1(Λ,ε+ζ+ς)−f1(Λ,ε+ς)−f1(Λ,ε+ζ−ς)+f1(Λ,ε−ς))‖,…,‖fn(Λ,ε+ζ+ς)−fn(Λ,ε+ς)−fn(Λ,ε+ζ−ς)+fn(Λ,ε−ς))‖]n×n⪯diag[‖θ1(f1(Λ,ε−ζ+ς)−f1(Λ,ε+ς)−f1(Λ,ε−ζ−ς)+f1(Λ,ε−ς))‖,…,‖θn(fn(Λ,ε−ζ+ς)−fn(Λ,ε+ς)−fn(Λ,ε−ζ−ς)+fn(Λ,ε−ς))‖]n×n | (1.1) |
for each ε,ζ,ς∈Q, Λ∈Q′ with |θ1|,…,|θn|<1.
For this section we refer the reader [6,7]. Assume ℜ(ν) denotes the real part of ν if ν∈C. Also, let
(1) Z+ be the set of the positive integers;
(2) Z− be the negative integer numbers;
(3) R− be the negative real numbers;
(4) R+ be the positive real numbers.
We begin by defining various functions which will be needed later. The gamma function is given by
Γ(X)=∫∞0e−YYX−1dY,ℜ(X)>0,X∈C. |
Euler's functional equation is given by
Γ(X+1)=XΓ(X),ℜ(X)>0,X∈C. |
Theorem 2.1 ([6]).If X∈N∪{0}, then
Γ(X+1)=X!. |
Theorem 2.2 ([6]). Γ(0.5)=π0.5.
The Pochhammer symbol is
(∁)ȷ=ȷ∏ı=1(∁+ı−1)=Γ(∁+ȷ)Γ(∁)={1ȷ=0∁(∁+1)⋯(∁+ȷ−1)ȷ∈N∪{0} |
where ∁∈C and ȷ,ı∈N.
Note that
Γ(∁+ȷ)=∁(∁+1)⋯(∁+ȷ−1)Γ(∁) |
where ȷ∈N∪{0}.
The Gauss hypergeometric series [7] is given by
φⓈ1(X):=2F1(α,B;T;X)=1+αBTX+α(α+1)B(B+1)T(T+1)X22+⋯=∞∑n=0(α)n(B)n(T)nXnn!, | (2.1) |
where α,B,T,X∈C, n∈N∪{0}, and |X|<1.
Consider the Gauss differential equation
(X−X2)d2ωdX2+(T−(α+B+1)X)dωdX−αBω=0, | (2.2) |
where α,B,X∈C, T∈C∖(Z−∪{0}), and |X|<1. The hypergeometric series is a solution of (2.2).
Theorem 2.3 ([6]).Let α,B,T,X∈C and |X|<1. Then
2F1(α,B;T;X)=Γ(T)Γ(B)Γ(T−B)∫10YB−1(1−Y)T−B−1(1−XY)−αdY, |
where ℜ(T)>ℜ(B)>0.
Theorem 2.4. If ℜ(T)>0,|X|<1, and |arg(−X)|<π, then
2F1(α,B;T;X)=Γ(T)Γ(α)Γ(B)12πi∫+i∞−i∞Γ(α+Y)Γ(B+Y)Γ(−Y)Γ(T+Y)(−X)YdY, |
We now present the Clausen hypergeometric series [7] and its properties:
φⓈ2(X):=pFq((α);(T);X)=pFq(α,⋯,αp;T1,⋯,Tq;X)=pFq(α1,⋯,αpT1,⋯,Tq;X)=∞∑k=0(α1)k⋯(αp)k(T1)k⋯(Tq)kXnn!, | (2.3) |
where p,n,q∈N∪{0} and αn,X,Tn∈C.
Now, (2.3) is a solution of the following differential equation
(M(q,η,Tn)ω)(X)−(N(p,η,αn)ω)(X)=0, |
where
(M(q,η,Tn)ω)(X)=(XddX)q∏n=1((XddX)ω(X)+(Tn−1)ω(X))=XddX(q∏n=1((XddX+(Tn−1))ω)(X)), |
and
(N(p,η,αn)ω)(X)=Xp∏n=1(Xdω(X)dX+αnω(X))=Xp∏n=1((XddX+αn)ω)(X) |
and αn,X,Tn∈C, p,n,q∈N∪{0}, and |X|<1,
Theorem 2.5 ([6]).Suppose αn∈C∖(Z−∪{0}):
(1) The series converges only for X=0, if p>q+1.
(2) The series converges absolutely for X∈C, if p<q.
(3) The series converges absolutely for |X|<1 and diverges for |X|=1 and for |X|>1 it converges absolutely for ℜ(∑qk=1Tk−∑pk=1αk)>0, if p=q+1.
Assume the following notation [7]:
Ξ:=−q∑k=1bk+p∑j=1aj, | (2.4) |
σ:=−q∏k=1|bk|−bk+p∏j=1|aj|−aj, | (2.5) |
and
χ:=−p∑j=1κj+q∑k=1ϑk+p−q2, | (2.6) |
where κj,ϑk∈C,k,j∈N,p,q∈N∪{0}, and bk,aj∈R+.
The Wright generalized hypergeometric series is given by
φⓈ3(X):=pWq(X)=pWq((κ1,a1),⋯,(κp,ap)(ϑ1,b1),⋯,(ϑq,bq);X)=pWq((κp,ap)1,p(ϑq,bq)1,q;X)=∞∑s=0{∏pj=1Γ(κj+ajs)}{∏qk=1Γ(ϑk+bks)}Xss!, | (2.7) |
where j,s,k∈N,X∈C,Ξ>−1,κj,ϑk∈C,p,q∈N∪{0}, and bk,aj∈R+.
Theorem 2.6 ([6]).Suppose X∈C,ϑk,κj∈C,j,s,k∈N, bk,aj∈R+, then
(1) (2.7) is absolutely convergent for each valueof |X|=σ and of |X|<σ, and ℜ(χ)>0.5, if Ξ+1=0.
(2) (2.7) is absolutely convergent for X∈C, if Ξ+1>0.
Now, the Wright function is given by
φⓈ4(X):=K(ϑ,b,X)=0W1(−(b,ϑ);X)=∞∑k=01Γ(ϑ+bk)Xkk!, | (2.8) |
where X,ϑ∈C, and b∈R.
Theorem 2.7 ([6]).Now (2.8) for b∈C (b∈Z−∪{0} if ϑ=0) and ϑ>−1 is an entire function of type δ=(1+ϑ)|ϑ|−ϑ1+ϑ, andfinite order p=11+ϑ.
Theorem 2.8 ([6]).Now (2.8) is an entire functionof X for each b∈C and ϑ>−1.
The Wright generalized Bessel function (Bessel-Maitland function) is given by
φⓈ5(X):=J(κ,a,X)=∞∑k=01Γ(κ+1+ak)(−X)kk!=0W1(−(κ+1,b);−X), |
where κ,X∈C, and a∈R.
Theorem 2.9 ([6]).Suppose X∈C,j,s,k∈N, aj,bk∈R+, and κj,ϑk∈C. Then (2.7) is an entire function of X.
Theorem 2.10 ([6]).Suppose b∈R and ϑ∈C.
(1) (2.8) is absolutely convergent for all |X|<1 and of |X|=1, and ℜ(χ)>0.5, if b+1=0.
(2) (2.8) is absolutely convergent for X∈C, if b+1>0.
Theorem 2.11 ([6]).Suppose b>−1,ϑ∈C. Then(2.8) is anentire function of X.
Theorem 2.12 ([6]).Suppose X∈C,j,k,s∈N,κj,ϑk∈C, and aj,bk∈R+. Then
pWq((κ1,1),⋯,(κp,1)(ϑ1,1),⋯,(ϑq,1);X)=∏pj=1Γ(κj)∏pk=1Γ(ϑk)pFq(κ1,⋯,κpϑ1,⋯,ϑq;X), |
where Ξ+1≥0.
The shifted Wright generalized hypergeometric series [6] is given by
φⓈ6(X):=pBq(X)=pBq((κ1,a1;ϑ1,b1),⋯,(κp,ap;ϑp,bp)(ˆκ1,c1;ˆϑ1,d1),⋯,(ˆκp,cp;ˆϑp,dp);X)=pBq((κp,ap;ϑp,bp)1,p(ˆκp,cp;ˆϑp,dp)1,q;X)=∞∑k=0{∏pm=1b(κm+amk;ϑm+bmk)}{∏qn=1b(ˆκn+cnk;ˆϑn+dnk)}Xkk!=∞∑k=0∏pm=1(Γ(κm+amk)Γ(ϑm+bmk))∏qn=1Γ((ˆκn+ˆϑn)+(cn+dn)k)∏pm=1Γ((ϑm+κm)+(bm+am)k)∏qn=1(Γ(ˆκn+cnk)Γ(ˆϑn+dnk))Xkk!, |
where m,n∈N,k∈N∪{0},κm,ϑm,ˆκn,ˆϑn,X∈C,p,q∈N∪{0}, and am,bm,cn,dn∈R+.
We have the following special cases:
0B0=eX,1B0(X)=1B0((κ,a;ϑ,b)−;X)=∞∑k=0b(κ+ak;ϑ+bk)Xkk!=∞∑k=0Γ(κ+ak)Γ(ϑ+bk)Γ[(b+b)k+(ϑ+κ)]Xkk!=2W1((κ,b),(ϑ,b)(ϑ+κ,b+b);X),0B1(X)=0B1(−(κ,a;ϑ,b);X)=∞∑k=01b(κ+ak;ϑ+bk)Xkk!=∞∑k=0Γ[(b+b)k+(ϑ+κ)]Γ(ϑ+bk)Γ(κ+ak)Xkk!=1W2((ϑ+κ,b+b)(κ,b),(ϑ,b);X),1B1(X)=1B1((ˆκ,c;ˆϑ,d)(κ,a;ϑ,b);X)=∞∑k=0b(κ+ak;ϑ+bk)b(ˆκ+ck;ˆϑ+dk)Xkk!=∞∑k=0Γ(κ+ak)Γ(ϑ+bk)Γ[(ˆκ+ˆϑ)+(c+d)k]Γ[(b+b)k+(κ+ϑ)]Γ(ˆϑ+dk)Γ(ˆκ+ck)Xkk!=3W3((κ,b),(ϑ,b),(ˆκ+ˆϑ,c+d)(ˆκ,c),(ˆϑ,d),(ϑ+κ,b+b);X), |
where k∈N∪{0},κm,ϑm,ˆκn,ˆϑn,X∈C, and am,bm,cn,dn∈R+.
Now, we define the Wright generalized hypergeometric series (see [6]) as follows
φⓈ7(X):=[pWq]n(X)=n∑s=0{∏pj=1Γ(κj+ajs)}{∏qk=1Γ(ϑk+bks)}Xss!, |
where X,κj,ϑk∈C,s,j,k,q,p∈N, and aj,bk∈R+.
Let
diag[ρ1,⋯,ρn]n×n=[ρ10…00ρ2⋱⋮⋮⋱⋱00…0ρn]n×n. |
Note that ρ:=diag[ρ1,⋯,ρn]⪯ϱ:=diag[ϱ1,⋯,ϱn] if ρi≤ϱi for each 1≤i≤n.
We denote W[X] as
diag[φⓈ1(X),⋯,φⓈn(X)]n×n. |
A HUR-stability with control functions W[X], is called multi-stability.
We now propose the notion of antiderivations in Banach algebras and introduce the super-multi-stability of antiderivations in algebras Banach, associated to (1.1).
Throughout this section, let Q be a complex Banach algebra and that θ1,⋯,θn∈C∖{0} with |θ1|,…,|θn|<1.
In this subsection, we study the multi stability of the additive (θ1,⋯,θn)-functional inequality (1.1).
Lemma 3.1. Suppose fi:Q′×Q→Q(i=1,…,n∈N) are mappings satisfying fi(Λ,0)=0 and (1.1) for each ε,ζ,ς∈Q, and Λ∈Q′. Then the mappings fi:Q′×Q→Q,(i=1,…,n∈N) are additive (the usual definition is at the end of the proof).
Proof. Assume that fi:Q′×Q→Q(i=1,…,n∈N) satisfies (1.1).
Replacing ζ by −ζ in (1.1), we get
diag[‖f1(Λ,ε−ζ+ς)−f1(Λ,ε+ς)−f1(Λ,ε−ζ−ς)+f1(Λ,ε−ς)‖,…,‖fn(Λ,ε−ζ+ς)−fn(Λ,ε+ς)−fn(Λ,ε−ζ−ς)+fn(Λ,ε−ς)‖]⪯diag[‖θ1(f1(Λ,ε+ζ+ς)−f1(Λ,ε+ς)−f1(Λ,ε+ζ−ς)+f1(Λ,ε−ς))‖,…,‖θn(fn(Λ,ε+ζ+ς)−fn(Λ,ε+ς)−fn(Λ,ε+ζ−ς)+fn(Λ,ε−ς))‖] | (3.1) |
for each ε,ζ,ς∈Q, and Λ∈Q′. According to (1.1) and (3.1) we have
diag[‖f1(Λ,ε+ζ+ς)−f1(Λ,ε+ς)−f1(Λ,ε+ζ−ς)+f1(Λ,ε−ς)‖,…,‖fn(Λ,ε+ζ+ς)−fn(Λ,ε+ς)−fn(Λ,ε+ζ−ς)+fn(Λ,ε−ς)‖]⪯diag[‖θ21(f1(Λ,ε+ζ+ς)−f1(Λ,ε+ς)−f1(Λ,ε+ζ−ς)+f1(Λ,ε−ς))‖,…,‖θ2n(fn(Λ,ε+ζ+ς)−fn(Λ,ε+ς)−fn(Λ,ε+ζ−ς)+fn(Λ,ε−ς))‖] |
and so
fi(Λ,ε+ζ+ς)−fi(Λ,ε+ς)−fi(Λ,ε+ζ−ς)+fi(Λ,ε−ς)=0,i=1,…,n | (3.2) |
for each ε,ζ,ς∈Q,Λ∈Q′, since |θi|<1(i=1,…,n).
Letting ς=ε in (3.2),
fi(Λ,2ε+ζ)−fi(Λ,2ε)−fi(Λ,ζ)=0,i=1,…,n |
for each ε,ζ∈Q,Λ∈Q′. Thus fi(i=1,…,n) are additive.
Throughout the paper, let φji:(Q)3i→[0,∞)i, 1≤i≤n,1≤ji≤n, and n∈N. Notice that M:=diag[φj1,…,φjn] is a matrix valued control function such that φj1(φjn) represents the element at the 1th(nth) row and 1th(nth) column of the matrix M and φji demonstrates the jith given control function.
Theorem 3.2. Let (φj1,…,φjn):(Q×Q×Q)n→[0,∞)n(1≤j1,…,jn≤n), be functions such that there exists an (T1,…,Tn)<(1,…,1)⏟n with
diag[φj1(ε2,ζ2,ς2),…,φjn(ε2,ζ2,ς2)]⪯diag[T12φj1(ε,ζ,ς),…,Tn2φjn(ε,ζ,ς)], | (3.3) |
for all ε,ζ,ς∈Q. Suppose fi:Q′×Q→Q(i=1,…,n) are mappings satisfying fi(Λ,0)=0 and
diag[‖f1(Λ,ε+ζ+ς)−f1(Λ,ε+ς)−f1(Λ,ε+ζ−ς)+f1(Λ,ε−ς)‖+|θ1|φj1⏟1≤j1≤n(ε,−ζ,ς),…,‖fn(Λ,ε+ζ+ς)−fn(Λ,ε+ς)−fn(Λ,ε+ζ−ς)+fn(Λ,ε−ς)‖+|θn|φjn⏟1≤jn≤n(ε,−ζ,ς)]⪯diag[‖θ1(f1(Λ,ε−ζ+ς)−f1(Λ,ε+ς)−f1(Λ,ε−ζ−ς)+f1(Λ,ε−ς))‖+φj1⏟1≤j1≤n(ε,ζ,ς),…,‖θn(fn(Λ,ε−ζ+ς)−fn(Λ,ε+ς)−fn(Λ,ε−ζ−ς)+fn(Λ,ε−ς))‖+φjn⏟1≤jn≤n(ε,ζ,ς)], | (3.4) |
for each ε,ζ,ς∈Q and Λ∈Q′. Then there exist unique additive mappings f′i:Q′×Q→Q such that
diag[‖f1(Λ,ε)−f′1(Λ,ε)‖,…,‖fn(Λ,ε)−f′n(Λ,ε)‖]n×n⪯diag[T12(1−T1)φj1⏟1≤j1≤n(ε2,ε,ε2),…,Tn2(1−Tn)φjn⏟1≤jn≤n(ε2,ε,ε2)]n×n, | (3.5) |
for each ε∈Q, and Λ∈Q′.
Proof. Replacing ζ by −ζ in (3.4), we get
diag[‖f1(Λ,ε−ζ+ς)−f1(Λ,ε+ς)−f1(Λ,ε−ζ−ς)+f1(Λ,ε−ς)‖+|θ1|φj1⏟1≤j1≤n(ε,ζ,ς),…,‖fn(Λ,ε−ζ+ς)−fn(Λ,ε+ς)−fn(Λ,ε−ζ−ς)+fn(Λ,ε−ς)‖+|θn|φjn⏟1≤jn≤n(ε,ζ,ς)]⪯diag[‖θ1(f1(Λ,ε+ζ+ς)−f1(Λ,ε+ς)−f1(Λ,ε+ζ−ς)+f1(Λ,ε−ς))‖+φj1⏟1≤j1≤n(ε,−ζ,ς),…,‖θn(fn(Λ,ε+ζ+ς)−fn(Λ,ε+ς)−fn(Λ,ε+ζ−ς)+fn(Λ,ε−ς))‖+φjn⏟1≤jn≤n(ε,−ζ,ς)], | (3.6) |
for each ε,ζ,ς∈Q, and Λ∈Q′. According to (3.4) and (3.6) we have
diag[‖f1(Λ,ε+ζ+ς)−f1(Λ,ε+ς)−f1(Λ,ε+ζ−ς)+f1(Λ,ε−ς)‖,…,‖fn(Λ,ε+ζ+ς)−fn(Λ,ε+ς)−fn(Λ,ε+ζ−ς)+fn(Λ,ε−ς)‖]n×n⪯diag[φj1⏟1≤j1≤n(ε,ζ,ς),…,φjn⏟1≤jn≤n(ε,ζ,ς)]n×n, | (3.7) |
for each ε,ζ,ς∈Q, and Λ∈Q′.
Letting ε=ς=σ2 and ζ=σ in (3.7), we get
diag[‖f1(Λ,2σ)−2f1(Λ,σ)‖,…,‖fn(Λ,2σ)−2fn(Λ,σ)‖]n×n⪯diag[φj1⏟1≤j1≤n(σ2,σ,σ2),…,φjn⏟1≤jn≤n(σ2,σ,σ2)]n×n, | (3.8) |
for each σ∈Q, and Λ∈Q′.
Let ℏ=(ℏ1,…,ℏn) and ℏ′=(ℏ′1,…,ℏ′n).
Now, consider the set
∇:={ℏ:(Q′×Q)n→Qn:ℏ(Λ,0)=n⏞(0,…,0)} |
and define the generalized metric on ∇ by
d(ℏ,ℏ′)=inf{(μ1,…,μn)∈Rn+:diag[‖ℏ1(Λ,ε)−ℏ′1(Λ,ε)‖,…‖ℏn(Λ,ε)−ℏ′n(Λ,ε)‖]⪯diag[μ1φj1⏟1≤j1≤n(ε2,ε,ε2),…,μnφjn⏟1≤jn≤n(ε2,ε,ε2)],∀ε∈Q,Λ∈Q′}, |
where inf∅=(+∞,…,+∞)⏟n.
Now (∇,d) is complete (also, see [8]).
Let L:=(L1,…,Ln). Now, we consider the linear mapping L:∇→∇ s.t.
Liℏi(Λ,ε):=2ℏi(Λ,ε2),i=1,…,n |
for each ε∈Q, and Λ∈Q′.
Let ℏ,ℏ′∈∇ be given s.t. d(ℏ,ℏ′)=(ε1,…,εn). Then
diag[‖ℏ1(Λ,ε)−ℏ′1(Λ,ε)‖,…,‖ℏn(Λ,ε)−ℏ′n(Λ,ε)‖]⪯diag[ε1φj1⏟1≤j1≤n(ε2,ε,ε2),…,εnφjn⏟1≤jn≤n(ε2,ε,ε2)], |
for each ε∈Q, and Λ∈Q′. Hence
diag[‖L1ℏ1(Λ,ε)−L1ℏ′1(Λ,ε)‖,…,‖Lnℏn(Λ,ε)−Lnℏ′n(Λ,ε)‖]=diag[‖2ℏ1(Λ,ε2)−2ℏ′1(Λ,ε2)‖,…,‖2ℏn(Λ,ε2)−2ℏ′n(Λ,ε2)‖]⪯diag[2ε1φj1⏟1≤j1≤n(ε4,ε2,ε4),…,2εnφjn⏟1≤jn≤n(ε4,ε2,ε4)]⪯diag[T1ε1φj1⏟1≤j1≤n(ε2,ε,ε2),…,Tnεnφjn⏟1≤jn≤n(ε2,ε,ε2)], |
for each ε∈Q, and Λ∈Q′. Thus d(ℏ,ℏ′)=(ε1,…,εn)⏟n implies that
d(Lℏ(Λ,ε),Lℏ′(Λ,ε))⪯(T1ε1,…,Tnεn). |
Hence
d(Lℏ(ε),Lℏ′(ε))⪯(T1,…,Tn)d(ℏ,ℏ′), |
for each ℏ,ℏ′∈∇. According to (3.8), we get
diag[‖f1(Λ,ε)−2f1(Λ,ε2)‖,…,‖fn(Λ,ε)−2fn(Λ,ε2)‖]n×n⪯diag[φj1⏟1≤j1≤n(ε4,ε2,ε4),…,φjn⏟1≤jn≤n(ε4,ε2,ε4)]n×n⪯diag[T12φj1⏟1≤j1≤n(ε2,ε,ε2),…,Tn2φjn⏟1≤jn≤n(ε2,ε,ε2)]n×n, |
for each ε∈Q,Λ∈Q′, so d(f,Lf)⪯(T12,…,Tn2).
According to Theorem 1.2 there exist mappings f′i:Q→Q(i=1,…,n) satisfying the following:
(1) f′ is a fixed point of L, i.e.
f′(Λ,ε)=2f′(Λ,ε2), | (3.9) |
for each ε∈Q, and Λ∈Q′. The mapping f′ is a unique fixed point of L in the set
k={ℏ∈∇:d(f,ℏ)<∞}. |
This implies that f′ is a unique mapping satisfying (3.8) s.t. there exist μ1,…,μn∈(0,∞) satisfying
diag[‖f1(Λ,ε)−f′1(Λ,ε)‖,…,‖fn(Λ,ε)−f′n(Λ,ε)‖]⪯diag[μ1φj1⏟0≤j1≤n(ε2,ε,ε2),…,μnφjn⏟0≤jn≤n(ε2,ε,ε2)], |
for each ε∈Q, and Λ∈Q′.
(2) Since limn→∞d(Lnf,f′)=0,
limn→∞2nfi(Λ,ε2n)=f′i(Λ,ε),∀i=1,…,n | (3.10) |
for each ε∈Q, and Λ∈Q′.
(3) d(f,f′)⪯(11−T1,…,11−Tn)d(f,Lf), which implies
diag[‖f1(Λ,ε)−f′1(Λ,ε)‖,…,‖fn(Λ,ε)−f′n(Λ,ε)‖]n×n⪯diag[T12(1−T1)φj1⏟1≤j1≤n(ε2,ε,ε2),…,Tn2(1−Tn)φjn⏟1≤jn≤n(ε2,ε,ε2)]n×n, |
for each ε∈Q, and Λ∈Q′. According to (3.3) and (3.4) we have
diag[‖f′1(Λ,ε+ζ+ς)−f′1(Λ,ε+ς)−f′1(Λ,ε+ζ−ς)+f′1(Λ,ε−ς)‖,…,‖f′n(Λ,ε+ζ+ς)−f′n(Λ,ε+ς)−f′n(Λ,ε+ζ−ς)+f′n(Λ,ε−ς)‖]=diag[limn→∞2n‖f1(Λ,ε+ζ+ς2n)−f1(Λ,ε+ς2n)−f1(Λ,ε+ζ−ς2n)+f1(Λ,ε−ς2n)‖,…,limn→∞2n‖fn(Λ,ε+ζ+ς2n)−fn(Λ,ε+ς2n)−fn(Λ,ε+ζ−ς2n)+fn(Λ,ε−ς2n)‖]⪯diag[limn→∞2n|θ1|‖f1(Λ,ε−ζ+ς2n)−f1(Λ,ε+ς2n)−f1(Λ,ε−ζ−ς2n)+f1(Λ,ε−ς2n)‖+limn→∞2n(φj1⏟1≤j1≤n(ε2n,ζ2n,ς2n)−θ1φj1⏟1≤j1≤n(ε2n,−ζ2n,ς2n)),…,limn→∞2n|θn|‖fn(Λ,ε−ζ+ς2n)−fn(Λ,ε+ς2n)−fn(Λ,ε−ζ−ς2n)+fn(Λ,ε−ς2n)‖+limn→∞2n(φjn⏟1≤jn≤n(ε2n,ζ2n,ς2n)−θnφjn⏟1≤jn≤n(ε2n,−ζ2n,ς2n))]⪯diag[‖θn(f′1(Λ,ε−ζ+ς)−f′1(Λ,ε+ς)−f′1(Λ,ε−ζ−ς)+f′1(Λ,ε−ς))‖,…,‖θn(f′n(Λ,ε−ζ+ς)−f′n(Λ,ε+ς)−f′n(Λ,ε−ζ−ς)+f′n(Λ,ε−ς))‖] |
for each ε,ζ,ς∈Q, and Λ∈Q′. According to Lemma 3.1, the mapping f′i(i=1,…,n) is additive.
Definition 3.3. Assume Q is a complex Banach algebra. A C-linear mapping G:Q′×Q→Q is called an antiderivation if it satisfies
G(Λ,ε)G(Λ,ζ)=G(Λ,G(Λ,ε)ζ)+G(Λ,εG(Λ,ζ)) |
for each ε,ζ∈Q and Λ∈Q′.
Example 3.4. Suppose Qm is the collection of all polynomials of degree m with complex coefficients and
Q={qm∈Qm|q(Λ,0)=0,m∈N}. |
Define G:Q′×Q→Q by
G(Λ,n∑k=1bkχk)=n∑k=1bkkχk |
and G(Λ,0)=0. Then G is an antiderivation.
Example 3.5. Consider the collection of all continuous functions on R, represented by C(R).
Define G:Q′×C(R)→C(R) by
G(Λ,g(ε))=∫ε0g(τ)dτ |
for each τ∈R. Then G is an antiderivation.
Lemma 3.6. [9]Suppose Q is complex Banach algebra and suppose f:Q′×Q→Q is an additive mapping s.t. f(Λ,Jε)=Jf(Λ,ε) for each J∈T1:={η∈C:|η|=1} and each ε∈Q. Then f is C-linear.
Theorem 3.7. Suppose φj1,…,φjn:Q3→[0,∞), (1≤j1,…,jn≤n), are functions.
(i) If there exist (T1,…,Tn)<(1,…,1) satisfying
diag[φj1⏟1≤j1≤n(εJ,ζJ,ςJ),…,φjn⏟1≤jn≤n(εJ,ζJ,ςJ)]⪯diag[T12φj1⏟1≤j1≤n(2ε,2ζ,2ς),…,Tn2φjn⏟1≤jn≤n(2ε,2ζ,2ς)], | (3.11) |
and if fi:Q′×Q→Q, (i=1,…,n), are mappings satisfying fi(Λ,0)=0 and
diag[‖Jf1(Λ,ε+ζ+ς)−f1(Λ,J(ε+ς))−f1(Λ,J(ε+ζ−ς))+Jf1(Λ,ε−ς)‖+|θ1|φj1⏟1≤j1≤n(ε,−ζ,ς),…,‖Jfn(Λ,ε+ζ+ς)−fn(Λ,J(ε+ς))−fn(Λ,J(ε+ζ−ς))+Jfn(Λ,ε−ς)‖+|θn|φjn⏟1≤jn≤n(ε,−ζ,ς)]⪯diag[‖θ1(Jf1(Λ,ε−ζ+ς)−f1(Λ,J(ε+ς))−f1(Λ,J(ε−ζ−ς))+Jf1(Λ,ε−ς))‖+φj1⏟1≤j1≤n(ε,ζ,ς),…,‖θn(Jfn(Λ,ε−ζ+ς)−fn(Λ,J(ε+ς))−fn(Λ,J(ε−ζ−ς))+Jfn(Λ,ε−ς))‖+φjn⏟1≤jn≤n(ε,ζ,ς)], | (3.12) |
for each J∈T1 and all ε,ζ,ς∈Q,Λ∈Q′, then there exist unique C-linear mappings Gi:Q′×Q→Q,(i=1,…,n), s.t.
diag[‖f1(Λ,ε)−G1(Λ,ε)‖,…,‖fn(Λ,ε)−Gn(Λ,ε)‖]⪯diag[T12(1−T1)φj1⏟1≤j1≤n(ε2,ε,−ε2),…,Tn2(1−Tn)φjn⏟1≤jn≤n(ε2,ε,−ε2)], | (3.13) |
for each ε∈Q,Λ∈Q′.
(ii) In addition, if (T1,…,Tn)<(12,…,12) and fi,(i=1,…,n), are continuous and satisfy fi(Λ,2ε)=2fi(Λ,ε) and
diag[‖f1(Λ,ε)f1(Λ,ζ)−f1(Λ,f1(Λ,ε)ζ)−f1(Λ,εf1(Λ,ζ))‖,…,‖fn(Λ,ε)fn(Λ,ζ)−fn(Λ,fn(Λ,ε)ζ)−fn(Λ,εfn(Λ,ζ))‖]⪯diag[φj1⏟1≤j1≤n(ε,ζ,ε),…,φjn⏟1≤jn≤n(ε,ζ,ε)], | (3.14) |
for each ε,ζ∈Q, then fi:Q′×Q→Q are antiderivations.
Proof. By a similar method used in Theorem 3.2 the proof of (i) is straightforward. Now, we prove (ii).
(ii) Since Gi=fi,(i=1,…,n), are continuous and C-linear, we conclude from (3.11) and (3.14) that
diag[‖G1(Λ,ε)G1(Λ,ζ)−G1(Λ,G1(Λ,ε)ζ)−G1(Λ,εG1(Λ,ζ))‖,…,‖Gn(Λ,ε)Gn(Λ,ζ)−Gn(Λ,Gn(Λ,ε)ζ)−Gn(Λ,εGn(Λ,ζ))‖]=diag[limm→∞4m‖Jm(f1(Λ,ε2mJm)f1(Λ,ζ2mJm)−G1(Λ,f1(Λ,ε2mJm)ζ2mJm)−G1(Λ,ε2mJmf1(Λ,ζ2mJm)))‖,…,limm→∞4m‖Jm(fn(Λ,ε2mJm)fn(Λ,ζ2mJm)−Gn(Λ,fn(Λ,ε2mJm)ζ2mJm)−Gn(Λ,ε2mλmfn(Λ,ζ2mJm)))‖]=diag[limm→∞4m‖Jm(f1(Λ,ε2mJm)f1(Λ,ζ2mJm)−f1(Λ,f1(Λ,ε2mJm)ζ2mJm)−f1(Λ,ε2mJmf1(Λ,ζ2mJm)))‖,…,limm→∞4m‖Jm(fn(Λ,ε2mJm)fn(Λ,ζ2mJm)−fn(Λ,f1(Λ,ε2mJm)ζ2mJm)−fn(Λ,ε2mJmfn(Λ,ζ2mJm)))‖]⪯diag[limm→∞22mφj1⏟1≤j1≤n(ε2mJm,ζ2mJm,ε2mJm),…,limm→∞22mφjn⏟1≤jn≤n(ε2mJm,ζ2mJm,ε2mJm)]⪯diag[limm→∞(2T1)mφj1⏟1≤j1≤n(ε,ζ,ε),…,limm→∞(2Tn)mφjn⏟1≤jn≤n(ε,ζ,ε)], |
for each J∈T1 and each ε,ζ∈Q,Λ∈Q′. Since (2T1,⋯,2Tn)⏟n<(1,⋯,1)⏟n, the C-linear mappings Gi,(i=1,⋯,n), are antiderivations. Thus the mappings fi:Q′×Q→Q,(i=1,⋯,n), are antiderivations.
In this subsection, we investigate the super-multi-stability of continuous antiderivations in Banach algebras.
Theorem 3.8. Consider φj1⏟1≤j1≤n,⋯,φjn⏟1≤jn≤n:Q3→[0,∞).
(i) If there exist (T1,⋯,Tn)≺n⏞(1,⋯,1) satisfying
diag[φj1⏟1≤j1≤n(εJ,ζJ,ςJ),⋯,φjn⏟1≤jn≤n(εJ,ζJ,ςJ)]⪯diag[T12φj1⏟1≤j1≤n(2ε,2ζ,2ς),⋯,Tn2φjn⏟1≤jn≤n(2ε,2ζ,2ς)] | (3.15) |
and if fi:Q′×Q→Q,(i=1,⋯,n), are mappings satisfying fi(Λ,0)=0 and
diag[‖Jf1(Λ,ε+ζ+ς)−f1(Λ,J(ε+ς))−f1(Λ,J(ε+ζ−ς))+Jf1(Λ,ε−ς)‖+|θ1|φj1⏟1≤j1≤n(ε,−ζ,ς),⋯,‖Jfn(Λ,ε+ζ+ς)−fn(Λ,J(ε+ς))−fn(Λ,J(ε+ζ−ς))+Jfn(Λ,ε−ς)‖+|θn|φjn⏟1≤jn≤n(ε,−ζ,ς)]⪯diag[‖θ1(Jf1(Λ,ε−ζ+ς)−f1(Λ,J(ε+ς))−f1(Λ,J(ε−ζ−ς))+Jf1(Λ,ε−ς))‖+φj1⏟1≤j1≤n(ε,ζ,ς),⋯,‖θn(Jfn(Λ,ε−ζ+ς)−fn(Λ,J(ε+ς))−fn(Λ,J(ε−ζ−ς))+Jfn(Λ,ε−ς))‖+φjn⏟1≤jn≤n(ε,ζ,ς)], | (3.16) |
for each J∈C−¯T1 and each ε,ζ,ς∈Q,Λ∈Q′, then there are unique C-linear mappings Gi:Q′×Q→Q,(i=1,⋯,n), s.t.
diag[‖f1(Λ,ε)−G1(Λ,ε)‖,⋯,‖fn(Λ,ε)−Gn(Λ,ε)‖]⪯diag[T12(1−T1)φj1⏟1≤j1≤n(ε2,ε,−ε2),⋯,Tn2(1−Tn)φjn⏟1≤jn≤n(ε2,ε,−ε2)], | (3.17) |
for each ε∈Q,Λ∈Q′.
(ii) Furthermore, if (T1,⋯,Tn)≺n⏞(12,⋯,12), φj1⏟1≤j1≤n,⋯,φjn⏟1≤jn≤n are continuous functions and also fi,(i=1,⋯,n), are continuous and satisfy fi(Λ,2ε)=2fi(Λ,ε) and
diag[‖f1(Λ,ε)f1(Λ,ζ)−f1(Λ,f1(Λ,ε)ζ)−f1(Λ,εf1(Λ,ζ))‖,⋯,‖fn(Λ,ε)fn(Λ,ζ)−fn(Λ,fn(Λ,ε)ζ)−fn(Λ,εfn(Λ,ζ))‖]⪯diag[φj1⏟1≤j1≤n(ε,ζ,ε),⋯,φjn⏟1≤jn≤n(ε,ζ,ε)], |
for each ε,ζ∈Q,Λ∈Q′, then fi:Q′×Q→Q are continuous antiderivations.
Proof. Using the same reasoning as in the proof of Theorem 3.7, we obtain the desired result.
Here, let n=7.
Corollary 3.9. Suppose fi:Q′×Q→Q(i=1,…,n) are mappings satisfying fi(Λ,0)=0 and
diag[‖f1(Λ,ε+ζ+ς)−f1(Λ,ε+ς)−f1(Λ,ε+ζ−ς)+f1(Λ,ε−ς)‖,…,‖fn(Λ,ε+ζ+ς)−fn(Λ,ε+ς)−fn(Λ,ε+ζ−ς)+fn(Λ,ε−ς)‖]⪯diag[‖θ1(f1(Λ,ε−ζ+ς)−f1(Λ,ε+ς)−f1(Λ,ε−ζ−ς)+f1(Λ,ε−ς))‖+(1−|θ1|)φⓈj1⏟1≤j1≤n(‖ε2‖+‖ζ2‖+‖ς2‖),…,‖θn(fn(Λ,ε−ζ+ς)−fn(Λ,ε+ς)−fn(Λ,ε−ζ−ς)+fn(Λ,ε−ς))‖+(1−|θn|)φⓈjn⏟1≤jn≤n(‖ε2‖+‖ζ2‖+‖ς2‖)] |
for each ε,ζ,ς∈Q, and Λ∈Q′. Then there are unique additive mappings f′i:Q′×Q→Q s.t.
diag[‖f1(Λ,ε)−f′1(Λ,ε)‖,…,‖fn(Λ,ε)−f′n(Λ,ε)‖]n×n⪯diag[φⓈj1⏟1≤j1≤n(‖ε‖2),…,φⓈjn⏟1≤jn≤n(‖ε‖2)], |
for each ε∈Q and Λ∈Q′.
Proof. The proof follows from Theorem 3.2 by letting
diag[φj1⏟1≤j1≤n(ε,ζ,ς),…,φjn⏟1≤jn≤n(ε,ζ,ς)]:=diag[φⓈj1⏟1≤j1≤n(‖ε2‖+‖ζ2‖+‖ς2‖),…,φⓈjn⏟1≤jn≤n(‖ε2‖+‖ζ2‖+‖ς2‖)], |
for each ε,ζ,ς∈Q. Choosing (T1,…,Tn)=(47,…,47), we obtain the desired result.
Corollary 3.10. Suppose fi:Q′×Q→Q,(i=1,…,n) are mappings satisfying fi(Λ,0)=0 and
diag[‖f1(Λ,ε+ζ+ς)−f1(Λ,ε+ς)−f1(Λ,ε+ζ−ς)+f1(Λ,ε−ς)‖,…,‖fn(Λ,ε+ζ+ς)−fn(Λ,ε+ς)−fn(Λ,ε+ζ−ς)+fn(Λ,ε−ς)‖]⪯diag[‖θ1(f1(Λ,ε−ζ+ς)−f1(Λ,ε+ς)−f1(Λ,ε−ζ−ς)+f1(Λ,ε−ς))‖+(1−|θ1|)φⓈj1⏟1≤j1≤n(‖εζς‖),…,‖θn(fn(Λ,ε−ζ+ς)−fn(Λ,ε+ς)−fn(Λ,ε−ζ−ς)+fn(Λ,ε−ς))‖+(1−|θn|)φⓈjn⏟1≤jn≤n(‖εζς‖)] |
for each ε,ζ,ς∈Q and Λ∈Q′. Then there are unique additive mappings f′i:Q′×Q→Q,(i=1,…,n), s.t.
diag[‖f1(Λ,ε)−f′1(Λ,ε)‖,…,‖fn(Λ,ε)−f′n(Λ,ε)‖]⪯diag[φⓈj1⏟1≤j1≤n(‖ε‖3),…,φⓈjn⏟1≤jn≤n(‖ε‖3)], |
for each ε∈Q and Λ∈Q′.
Proof. The proof follows from Theorem 3.2 by letting
diag[φj1⏟1≤j1≤n(ε,ζ,ς),…,φjn⏟1≤jn≤n(ε,ζ,ς)]:=diag[φⓈj1⏟1≤j1≤n(‖εζς‖),…,φⓈjn⏟1≤jn≤n(‖εζς‖)], |
for each ε,ζ,ς∈Q and Λ∈Q′. Choosing (T1,…,Tn)=(89,…,89), we obtain the desired result.
Corollary 3.11. Let fi:Q′×Q→Q,(i=1,…,n) be odd mappings satisfying
diag[‖f1(Λ,ε+ζ+ς)−f1(Λ,ε+ς)−f1(Λ,ε+ζ−ς)+f1(Λ,ε−ς)‖,…,‖fn(Λ,ε+ζ+ς)−fn(Λ,ε+ς)−fn(Λ,ε+ζ−ς)+fn(Λ,ε−ς)‖]⪯diag[‖θ1[f1(Λ,ε−ζ+ς)−f1(Λ,ε+ς)−f1(Λ,ε−ζ−ς)+f1(Λ,ε−ς)]‖+(1−|θ1|)φⓈj1⏟1≤j1≤n(‖εζς‖),…,‖θn[fn(Λ,ε−ζ+ς)−fn(Λ,ε+ς)−fn(Λ,ε−ζ−ς)+fn(Λ,ε−ς)]‖+(1−|θn|)φⓈjn⏟1≤jn≤n(‖εζς‖)], | (3.18) |
for each ε,ζ,ς∈Q and Λ∈Q′. Then fi(i=1,…,n) are additive.
Proof. Putting ε=0 in (3.18), we get
diag[‖f1(Λ,ζ+ς)−f1(Λ,ς)−f1(Λ,ζ−ς)+f1(Λ,−ς)‖,…,‖fn(Λ,ζ+ς)−fn(Λ,ς)−fn(Λ,ζ−ς)+fn(Λ,−ς)‖]⪯diag[‖θ1(f1(Λ,−ζ+ς)−f1(Λ,ς)−f1(Λ,−ζ−ς)+f1(Λ,−ς))‖,…,‖θn(fn(Λ,−ζ+ς)−fn(Λ,ς)−fn(Λ,−ζ−ς)+fn(Λ,−ς))‖], | (3.19) |
for each ζ,ς∈Q and Λ∈Q′. Replacing ζ by −ζ in (3.19), we have
diag[‖f1(Λ,−ζ+ς)−f1(Λ,ς)−f1(Λ,−ζ−ς)+f1(Λ,−ς)‖,…,‖fn(Λ,−ζ+ς)−fn(Λ,ς)−fn(Λ,−ζ−ς)+fn(Λ,−ς)‖]⪯diag[‖θ1[f1(Λ,ζ+ς)−f1(Λ,ς)−f1(Λ,ζ−ς)+f1(Λ,−ς)]‖,…,‖θn[fn(Λ,ζ+ς)−fn(Λ,ς)−fn(Λ,ζ−ς)+fn(Λ,−ς)]‖], | (3.20) |
for each ζ,ς∈Q and Λ∈Q′. From (3.19) and (3.20), it follows that
fi(Λ,ζ+ς)−fi(Λ,ς)−fi(Λ,ζ−ς)+fi(Λ,−ς)=0,i=1,…,n |
for each ζ,ς∈Q and Λ∈Q′. Since fi,(i=1,…,n), are odd mappings,
fi(ς+ζ)+fi(ς−ζ)−2fi(ς)=0,i=1,…,n |
for each ζ,ς∈Q and Λ∈Q′. Thus the mappings fi,(i=1,…,n), are additive.
Corollary 3.12. Suppose fi:Q′×Q→Q,(i=1,⋯,n), are mappings satisfying fi(Λ,0)=0 and
diag[‖f1(Λ,ε+ζ+ς)−f1(Λ,ε+ς)−f1(Λ,ε+ζ−ς)+f1(Λ,ε−ς)‖,⋯,‖fn(Λ,ε+ζ+ς)−fn(Λ,ε+ς)−fn(Λ,ε+ζ−ς)+fn(Λ,ε−ς)‖]⪯diag[‖θ1(f1(Λ,ε−ζ+ς)−f1(Λ,ε+ς)−f1(Λ,ε−ζ−ς)+f1(Λ,ε−ς))‖+(1−|θ1|)φⓈj1⏟1≤j1≤n(‖ε4+ζ4+ς4‖),⋯,‖θn(fn(Λ,ε−ζ+ς)−fn(Λ,ε+ς)−fn(Λ,ε−ζ−ς)+fn(Λ,ε−ς))‖+(1−|θn|)φⓈjn⏟1≤jn≤n(‖ε4+ζ4+ς4‖)] |
and
diag[‖f1(Λ,ε)f1(Λ,ζ)−f1(Λ,f1(Λ,ε)ζ)−f1(Λ,εf1(Λ,ζ))‖,⋯,‖fn(Λ,ε)fn(Λ,ζ)−fn(Λ,fn(Λ,ε)ζ)−fn(Λ,εfn(Λ,ζ))‖]⪯diag[φⓈj1⏟1≤j1≤n(‖2ε4+ζ4‖),⋯,φⓈjn⏟1≤jn≤n(‖2ε4+ζ4‖)]n×n, |
for each ε,ζ,ς∈Q,Λ∈Q′. If fi(Λ,2ε)=2fi(Λ,ε) foreach ε,ζ,ς∈Q,Λ∈Q′, and fi,(i=1,⋯,n), are continuous, then the mappings fi:Q′×Q→Q,(i=1,⋯,n), are antiderivations.
Proof. The proof follows from Theorem 3.7 by letting
diag[φj1⏟1≤j1≤n(ε,ζ,ς),⋯,φjn⏟1≤jn≤n(ε,ζ,ς)]:=diag[φⓈj1⏟1≤j1≤n(‖ε4+ζ4+ς4‖),⋯,φⓈjn⏟1≤jn≤n(‖ε4+ζ4+ς4‖)]n×n |
for each ε,ζ,ς∈Q. Choosing (T1,⋯,Tn)=n⏞(817,⋯,817), we obtain the desired result.
In this study, we investigated the concept of antiderivations in Banach algebras and study multi-super-stability of antiderivations in Banach algebras, associated with functional inequalities.
The authors extend their appreciation to the Deanship of Scientific Research at Imam Mohammad Ibn Saud Islamic University (IMSIU) for funding and supporting this work through Research Partnership Program no RP-21-09-08.
The authors declare that they have no competing interests.
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