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Research article

Hypolimnetic assimilation of ammonium by the nuisance alga Gonyostomum semen

  • 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×QQ is a operator if {:f(,α)ν}Δ for each α in Q and νBQ. Assume =(1,,m) and Ω=(Ω1,,Ωm),mN. Then we have

    ΩıΩı,ı=1,,m;

    and also

    0ı0,ı=1,,m.

    Definition 1.1 ([5]). Let is a set and d:2[0,+]m,mN, 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)mg=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,mN, 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 nN{0} or there is a n0N such that

    (1) d(Lng,Ln+1g)m(+,,+),nn0;

    (2) The sequence {Lng} converges to a fixed point (g) of L;

    (3) (g) is the unique fixed point of L in the set ={gd(Ln0g,g)m(+,,+)};

    (4) d(g,(g))11Zd(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+ζς)+f1(Λ,ες)),,fn(Λ,ε+ζ+ς)fn(Λ,ε+ς)fn(Λ,ε+ζς)+fn(Λ,ες))]n×ndiag[θ1(f1(Λ,εζ+ς)f1(Λ,ε+ς)f1(Λ,εζς)+f1(Λ,ες)),,θn(fn(Λ,εζ+ς)fn(Λ,ε+ς)fn(Λ,εζς)+fn(Λ,ες))]n×n (1.1)

    for each ε,ζ,ςQ, ΛQ with |θ1|,,|θn|<1.

    For this section we refer the reader [6,7]. Assume (ν) denotes the real part of ν if νC. 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)=0eYYX1dY,(X)>0,XC.

    Euler's functional equation is given by

    Γ(X+1)=XΓ(X),(X)>0,XC.

    Theorem 2.1 ([6]).If XN{0}, then

    Γ(X+1)=X!.

    Theorem 2.2 ([6]). Γ(0.5)=π0.5.

    The Pochhammer symbol is

    ()ȷ=ȷı=1(+ı1)=Γ(+ȷ)Γ()={1ȷ=0(+1)(+ȷ1)ȷN{0}

    where C and ȷ,ıN.

    Note that

    Γ(+ȷ)=(+1)(+ȷ1)Γ()

    where ȷN{0}.

    The Gauss hypergeometric series [7] is given by

    φ1(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,XC, nN{0}, and |X|<1.

    Consider the Gauss differential equation

    (XX2)d2ωdX2+(T(α+B+1)X)dωdXαBω=0, (2.2)

    where α,B,XC, TC(Z{0}), and |X|<1. The hypergeometric series is a solution of (2.2).

    Theorem 2.3 ([6]).Let α,B,T,XC and |X|<1. Then

    2F1(α,B;T;X)=Γ(T)Γ(B)Γ(TB)10YB1(1Y)TB1(1XY)αdY,

    where (T)>(B)>0.

    Theorem 2.4. If (T)>0,|X|<1, and |arg(X)|<π, then

    2F1(α,B;T;X)=Γ(T)Γ(α)Γ(B)12πi+iiΓ(α+Y)Γ(B+Y)Γ(Y)Γ(T+Y)(X)YdY,

    We now present the Clausen hypergeometric series [7] and its properties:

    φ2(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,qN{0} and αn,X,TnC.

    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)qn=1((XddX)ω(X)+(Tn1)ω(X))=XddX(qn=1((XddX+(Tn1))ω)(X)),

    and

    (N(p,η,αn)ω)(X)=Xpn=1(Xdω(X)dX+αnω(X))=Xpn=1((XddX+αn)ω)(X)

    and αn,X,TnC, p,n,qN{0}, and |X|<1,

    Theorem 2.5 ([6]).Suppose αnC(Z{0}):

    (1) The series converges only for X=0, if p>q+1.

    (2) The series converges absolutely for XC, if p<q.

    (3) The series converges absolutely for |X|<1 and diverges for |X|=1 and for |X|>1 it converges absolutely for (qk=1Tkpk=1αk)>0, if p=q+1.

    Assume the following notation [7]:

    Ξ:=qk=1bk+pj=1aj, (2.4)
    σ:=qk=1|bk|bk+pj=1|aj|aj, (2.5)

    and

    χ:=pj=1κj+qk=1ϑk+pq2, (2.6)

    where κj,ϑkC,k,jN,p,qN{0}, and bk,ajR+.

    The Wright generalized hypergeometric series is given by

    φ3(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,kN,XC,Ξ>1,κj,ϑkC,p,qN{0}, and bk,ajR+.

    Theorem 2.6 ([6]).Suppose XC,ϑk,κjC,j,s,kN, bk,ajR+, then

    (1) (2.7) is absolutely convergent for each valueof |X|=σ and of |X|<σ, and (χ)>0.5, if Ξ+1=0.

    (2) (2.7) is absolutely convergent for XC, if Ξ+1>0.

    Now, the Wright function is given by

    φ4(X):=K(ϑ,b,X)=0W1((b,ϑ);X)=k=01Γ(ϑ+bk)Xkk!, (2.8)

    where X,ϑC, and bR.

    Theorem 2.7 ([6]).Now (2.8) for bC (bZ{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 bC and ϑ>1.

    The Wright generalized Bessel function (Bessel-Maitland function) is given by

    φ5(X):=J(κ,a,X)=k=01Γ(κ+1+ak)(X)kk!=0W1((κ+1,b);X),

    where κ,XC, and aR.

    Theorem 2.9 ([6]).Suppose XC,j,s,kN, aj,bkR+, and κj,ϑkC. Then (2.7) is an entire function of X.

    Theorem 2.10 ([6]).Suppose bR and ϑC.

    (1) (2.8) is absolutely convergent for all |X|<1 and of |X|=1, and (χ)>0.5, if b+1=0.

    (2) (2.8) is absolutely convergent for XC, if b+1>0.

    Theorem 2.11 ([6]).Suppose b>1,ϑC. Then(2.8) is anentire function of X.

    Theorem 2.12 ([6]).Suppose XC,j,k,sN,κj,ϑkC, and aj,bkR+. Then

    pWq((κ1,1),,(κp,1)(ϑ1,1),,(ϑq,1);X)=pj=1Γ(κj)pk=1Γ(ϑk)pFq(κ1,,κpϑ1,,ϑq;X),

    where Ξ+10.

    The shifted Wright generalized hypergeometric series [6] is given by

    φ6(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=0pm=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,nN,kN{0},κm,ϑm,ˆκn,ˆϑn,XC,p,qN{0}, and am,bm,cn,dnR+.

    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 kN{0},κm,ϑm,ˆκn,ˆϑn,XC, and am,bm,cn,dnR+.

    Now, we define the Wright generalized hypergeometric series (see [6]) as follows

    φ7(X):=[pWq]n(X)=ns=0{pj=1Γ(κj+ajs)}{qk=1Γ(ϑk+bks)}Xss!,

    where X,κj,ϑkC,s,j,k,q,pN, and aj,bkR+.

    Let

    diag[ρ1,,ρn]n×n=[ρ1000ρ2000ρn]n×n.

    Note that ρ:=diag[ρ1,,ρn]ϱ:=diag[ϱ1,,ϱn] if ρiϱi for each 1in.

    We denote W[X] as

    diag[φ1(X),,φn(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,,θnC{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×QQ(i=1,,nN) are mappings satisfying fi(Λ,0)=0 and (1.1) for each ε,ζ,ςQ, and ΛQ. Then the mappings fi:Q×QQ,(i=1,,nN) are additive (the usual definition is at the end of the proof).

    Proof. Assume that fi:Q×QQ(i=1,,nN) satisfies (1.1).

    Replacing ζ by ζ in (1.1), we get

    diag[f1(Λ,εζ+ς)f1(Λ,ε+ς)f1(Λ,εζς)+f1(Λ,ες),,fn(Λ,εζ+ς)fn(Λ,ε+ς)fn(Λ,εζς)+fn(Λ,ες)]diag[θ1(f1(Λ,ε+ζ+ς)f1(Λ,ε+ς)f1(Λ,ε+ζς)+f1(Λ,ες)),,θn(fn(Λ,ε+ζ+ς)fn(Λ,ε+ς)fn(Λ,ε+ζς)+fn(Λ,ες))] (3.1)

    for each ε,ζ,ςQ, and ΛQ. According to (1.1) and (3.1) we have

    diag[f1(Λ,ε+ζ+ς)f1(Λ,ε+ς)f1(Λ,ε+ζς)+f1(Λ,ες),,fn(Λ,ε+ζ+ς)fn(Λ,ε+ς)fn(Λ,ε+ζς)+fn(Λ,ες)]diag[θ21(f1(Λ,ε+ζ+ς)f1(Λ,ε+ς)f1(Λ,ε+ζς)+f1(Λ,ες)),,θ2n(fn(Λ,ε+ζ+ς)fn(Λ,ε+ς)fn(Λ,ε+ζς)+fn(Λ,ες))]

    and so

    fi(Λ,ε+ζ+ς)fi(Λ,ε+ς)fi(Λ,ε+ζς)+fi(Λ,ες)=0,i=1,,n (3.2)

    for each ε,ζ,ςQ,ΛQ, 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,ζQ,ΛQ. Thus fi(i=1,,n) are additive.

    Throughout the paper, let φji:(Q)3i[0,)i, 1in,1jin, and nN. 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(1j1,,jnn), 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,ςQ. Suppose fi:Q×QQ(i=1,,n) are mappings satisfying fi(Λ,0)=0 and

    diag[f1(Λ,ε+ζ+ς)f1(Λ,ε+ς)f1(Λ,ε+ζς)+f1(Λ,ες)+|θ1|φj11j1n(ε,ζ,ς),,fn(Λ,ε+ζ+ς)fn(Λ,ε+ς)fn(Λ,ε+ζς)+fn(Λ,ες)+|θn|φjn1jnn(ε,ζ,ς)]diag[θ1(f1(Λ,εζ+ς)f1(Λ,ε+ς)f1(Λ,εζς)+f1(Λ,ες))+φj11j1n(ε,ζ,ς),,θn(fn(Λ,εζ+ς)fn(Λ,ε+ς)fn(Λ,εζς)+fn(Λ,ες))+φjn1jnn(ε,ζ,ς)], (3.4)

    for each ε,ζ,ςQ and ΛQ. Then there exist unique additive mappings fi:Q×QQ such that

    diag[f1(Λ,ε)f1(Λ,ε),,fn(Λ,ε)fn(Λ,ε)]n×ndiag[T12(1T1)φj11j1n(ε2,ε,ε2),,Tn2(1Tn)φjn1jnn(ε2,ε,ε2)]n×n, (3.5)

    for each εQ, and ΛQ.

    Proof. Replacing ζ by ζ in (3.4), we get

    diag[f1(Λ,εζ+ς)f1(Λ,ε+ς)f1(Λ,εζς)+f1(Λ,ες)+|θ1|φj11j1n(ε,ζ,ς),,fn(Λ,εζ+ς)fn(Λ,ε+ς)fn(Λ,εζς)+fn(Λ,ες)+|θn|φjn1jnn(ε,ζ,ς)]diag[θ1(f1(Λ,ε+ζ+ς)f1(Λ,ε+ς)f1(Λ,ε+ζς)+f1(Λ,ες))+φj11j1n(ε,ζ,ς),,θn(fn(Λ,ε+ζ+ς)fn(Λ,ε+ς)fn(Λ,ε+ζς)+fn(Λ,ες))+φjn1jnn(ε,ζ,ς)], (3.6)

    for each ε,ζ,ςQ, and ΛQ. According to (3.4) and (3.6) we have

    diag[f1(Λ,ε+ζ+ς)f1(Λ,ε+ς)f1(Λ,ε+ζς)+f1(Λ,ες),,fn(Λ,ε+ζ+ς)fn(Λ,ε+ς)fn(Λ,ε+ζς)+fn(Λ,ες)]n×ndiag[φj11j1n(ε,ζ,ς),,φjn1jnn(ε,ζ,ς)]n×n, (3.7)

    for each ε,ζ,ςQ, and ΛQ.

    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×ndiag[φj11j1n(σ2,σ,σ2),,φjn1jnn(σ2,σ,σ2)]n×n, (3.8)

    for each σQ, and ΛQ.

    Let =(1,,n) and =(1,,n).

    Now, consider the set

    :={:(Q×Q)nQn:(Λ,0)=n(0,,0)}

    and define the generalized metric on by

    d(,)=inf{(μ1,,μn)Rn+:diag[1(Λ,ε)1(Λ,ε),n(Λ,ε)n(Λ,ε)]diag[μ1φj11j1n(ε2,ε,ε2),,μnφjn1jnn(ε2,ε,ε2)],εQ,ΛQ},

    where inf=(+,,+)n.

    Now (,d) is complete (also, see [8]).

    Let L:=(L1,,Ln). Now, we consider the linear mapping L: s.t.

    Lii(Λ,ε):=2i(Λ,ε2),i=1,,n

    for each εQ, and ΛQ.

    Let , be given s.t. d(,)=(ε1,,εn). Then

    diag[1(Λ,ε)1(Λ,ε),,n(Λ,ε)n(Λ,ε)]diag[ε1φj11j1n(ε2,ε,ε2),,εnφjn1jnn(ε2,ε,ε2)],

    for each εQ, and ΛQ. Hence

    diag[L11(Λ,ε)L11(Λ,ε),,Lnn(Λ,ε)Lnn(Λ,ε)]=diag[21(Λ,ε2)21(Λ,ε2),,2n(Λ,ε2)2n(Λ,ε2)]diag[2ε1φj11j1n(ε4,ε2,ε4),,2εnφjn1jnn(ε4,ε2,ε4)]diag[T1ε1φj11j1n(ε2,ε,ε2),,Tnεnφjn1jnn(ε2,ε,ε2)],

    for each εQ, and ΛQ. Thus d(,)=(ε1,,εn)n implies that

    d(L(Λ,ε),L(Λ,ε))(T1ε1,,Tnεn).

    Hence

    d(L(ε),L(ε))(T1,,Tn)d(,),

    for each ,. According to (3.8), we get

    diag[f1(Λ,ε)2f1(Λ,ε2),,fn(Λ,ε)2fn(Λ,ε2)]n×ndiag[φj11j1n(ε4,ε2,ε4),,φjn1jnn(ε4,ε2,ε4)]n×ndiag[T12φj11j1n(ε2,ε,ε2),,Tn2φjn1jnn(ε2,ε,ε2)]n×n,

    for each εQ,ΛQ, so d(f,Lf)(T12,,Tn2).

    According to Theorem 1.2 there exist mappings fi:QQ(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 εQ, and ΛQ. The mapping f is a unique fixed point of L in the set

    k={:d(f,)<}.

    This implies that f is a unique mapping satisfying (3.8) s.t. there exist μ1,,μn(0,) satisfying

    diag[f1(Λ,ε)f1(Λ,ε),,fn(Λ,ε)fn(Λ,ε)]diag[μ1φj10j1n(ε2,ε,ε2),,μnφjn0jnn(ε2,ε,ε2)],

    for each εQ, and ΛQ.

    (2) Since limnd(Lnf,f)=0,

    limn2nfi(Λ,ε2n)=fi(Λ,ε),i=1,,n (3.10)

    for each εQ, and ΛQ.

    (3) d(f,f)(11T1,,11Tn)d(f,Lf), which implies

    diag[f1(Λ,ε)f1(Λ,ε),,fn(Λ,ε)fn(Λ,ε)]n×ndiag[T12(1T1)φj11j1n(ε2,ε,ε2),,Tn2(1Tn)φjn1jnn(ε2,ε,ε2)]n×n,

    for each εQ, and ΛQ. According to (3.3) and (3.4) we have

    diag[f1(Λ,ε+ζ+ς)f1(Λ,ε+ς)f1(Λ,ε+ζς)+f1(Λ,ες),,fn(Λ,ε+ζ+ς)fn(Λ,ε+ς)fn(Λ,ε+ζς)+fn(Λ,ες)]=diag[limn2nf1(Λ,ε+ζ+ς2n)f1(Λ,ε+ς2n)f1(Λ,ε+ζς2n)+f1(Λ,ες2n),,limn2nfn(Λ,ε+ζ+ς2n)fn(Λ,ε+ς2n)fn(Λ,ε+ζς2n)+fn(Λ,ες2n)]diag[limn2n|θ1|f1(Λ,εζ+ς2n)f1(Λ,ε+ς2n)f1(Λ,εζς2n)+f1(Λ,ες2n)+limn2n(φj11j1n(ε2n,ζ2n,ς2n)θ1φj11j1n(ε2n,ζ2n,ς2n)),,limn2n|θn|fn(Λ,εζ+ς2n)fn(Λ,ε+ς2n)fn(Λ,εζς2n)+fn(Λ,ες2n)+limn2n(φjn1jnn(ε2n,ζ2n,ς2n)θnφjn1jnn(ε2n,ζ2n,ς2n))]diag[θn(f1(Λ,εζ+ς)f1(Λ,ε+ς)f1(Λ,εζς)+f1(Λ,ες)),,θn(fn(Λ,εζ+ς)fn(Λ,ε+ς)fn(Λ,εζς)+fn(Λ,ες))]

    for each ε,ζ,ςQ, and ΛQ. According to Lemma 3.1, the mapping fi(i=1,,n) is additive.

    Definition 3.3. Assume Q is a complex Banach algebra. A C-linear mapping G:Q×QQ 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,ζQ and ΛQ.

    Example 3.4. Suppose Qm is the collection of all polynomials of degree m with complex coefficients and

    Q={qmQm|q(Λ,0)=0,mN}.

    Define G:Q×QQ by

    G(Λ,nk=1bkχk)=nk=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 τR. Then G is an antiderivation.

    Lemma 3.6. [9]Suppose Q is complex Banach algebra and suppose f:Q×QQ is an additive mapping s.t. f(Λ,Jε)=Jf(Λ,ε) for each JT1:={ηC:|η|=1} and each εQ. Then f is C-linear.

    Theorem 3.7. Suppose φj1,,φjn:Q3[0,), (1j1,,jnn), are functions.

    (i) If there exist (T1,,Tn)<(1,,1) satisfying

    diag[φj11j1n(εJ,ζJ,ςJ),,φjn1jnn(εJ,ζJ,ςJ)]diag[T12φj11j1n(2ε,2ζ,2ς),,Tn2φjn1jnn(2ε,2ζ,2ς)], (3.11)

    and if fi:Q×QQ, (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+ζς))+Jf1(Λ,ες)+|θ1|φj11j1n(ε,ζ,ς),,Jfn(Λ,ε+ζ+ς)fn(Λ,J(ε+ς))fn(Λ,J(ε+ζς))+Jfn(Λ,ες)+|θn|φjn1jnn(ε,ζ,ς)]diag[θ1(Jf1(Λ,εζ+ς)f1(Λ,J(ε+ς))f1(Λ,J(εζς))+Jf1(Λ,ες))+φj11j1n(ε,ζ,ς),,θn(Jfn(Λ,εζ+ς)fn(Λ,J(ε+ς))fn(Λ,J(εζς))+Jfn(Λ,ες))+φjn1jnn(ε,ζ,ς)], (3.12)

    for each JT1 and all ε,ζ,ςQ,ΛQ, then there exist unique C-linear mappings Gi:Q×QQ,(i=1,,n), s.t.

    diag[f1(Λ,ε)G1(Λ,ε),,fn(Λ,ε)Gn(Λ,ε)]diag[T12(1T1)φj11j1n(ε2,ε,ε2),,Tn2(1Tn)φjn1jnn(ε2,ε,ε2)], (3.13)

    for each εQ,ΛQ.

    (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[φj11j1n(ε,ζ,ε),,φjn1jnn(ε,ζ,ε)], (3.14)

    for each ε,ζQ, then fi:Q×QQ 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[limm4mJm(f1(Λ,ε2mJm)f1(Λ,ζ2mJm)G1(Λ,f1(Λ,ε2mJm)ζ2mJm)G1(Λ,ε2mJmf1(Λ,ζ2mJm))),,limm4mJm(fn(Λ,ε2mJm)fn(Λ,ζ2mJm)Gn(Λ,fn(Λ,ε2mJm)ζ2mJm)Gn(Λ,ε2mλmfn(Λ,ζ2mJm)))]=diag[limm4mJm(f1(Λ,ε2mJm)f1(Λ,ζ2mJm)f1(Λ,f1(Λ,ε2mJm)ζ2mJm)f1(Λ,ε2mJmf1(Λ,ζ2mJm))),,limm4mJm(fn(Λ,ε2mJm)fn(Λ,ζ2mJm)fn(Λ,f1(Λ,ε2mJm)ζ2mJm)fn(Λ,ε2mJmfn(Λ,ζ2mJm)))]diag[limm22mφj11j1n(ε2mJm,ζ2mJm,ε2mJm),,limm22mφjn1jnn(ε2mJm,ζ2mJm,ε2mJm)]diag[limm(2T1)mφj11j1n(ε,ζ,ε),,limm(2Tn)mφjn1jnn(ε,ζ,ε)],

    for each JT1 and each ε,ζQ,ΛQ. Since (2T1,,2Tn)n<(1,,1)n, the C-linear mappings Gi,(i=1,,n), are antiderivations. Thus the mappings fi:Q×QQ,(i=1,,n), are antiderivations.

    In this subsection, we investigate the super-multi-stability of continuous antiderivations in Banach algebras.

    Theorem 3.8. Consider φj11j1n,,φjn1jnn:Q3[0,).

    (i) If there exist (T1,,Tn)n(1,,1) satisfying

    diag[φj11j1n(εJ,ζJ,ςJ),,φjn1jnn(εJ,ζJ,ςJ)]diag[T12φj11j1n(2ε,2ζ,2ς),,Tn2φjn1jnn(2ε,2ζ,2ς)] (3.15)

    and if fi:Q×QQ,(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+ζς))+Jf1(Λ,ες)+|θ1|φj11j1n(ε,ζ,ς),,Jfn(Λ,ε+ζ+ς)fn(Λ,J(ε+ς))fn(Λ,J(ε+ζς))+Jfn(Λ,ες)+|θn|φjn1jnn(ε,ζ,ς)]diag[θ1(Jf1(Λ,εζ+ς)f1(Λ,J(ε+ς))f1(Λ,J(εζς))+Jf1(Λ,ες))+φj11j1n(ε,ζ,ς),,θn(Jfn(Λ,εζ+ς)fn(Λ,J(ε+ς))fn(Λ,J(εζς))+Jfn(Λ,ες))+φjn1jnn(ε,ζ,ς)], (3.16)

    for each JC¯T1 and each ε,ζ,ςQ,ΛQ, then there are unique C-linear mappings Gi:Q×QQ,(i=1,,n), s.t.

    diag[f1(Λ,ε)G1(Λ,ε),,fn(Λ,ε)Gn(Λ,ε)]diag[T12(1T1)φj11j1n(ε2,ε,ε2),,Tn2(1Tn)φjn1jnn(ε2,ε,ε2)], (3.17)

    for each εQ,ΛQ.

    (ii) Furthermore, if (T1,,Tn)n(12,,12), φj11j1n,,φjn1jnn 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[φj11j1n(ε,ζ,ε),,φjn1jnn(ε,ζ,ε)],

    for each ε,ζQ,ΛQ, then fi:Q×QQ 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×QQ(i=1,,n) are mappings satisfying fi(Λ,0)=0 and

    diag[f1(Λ,ε+ζ+ς)f1(Λ,ε+ς)f1(Λ,ε+ζς)+f1(Λ,ες),,fn(Λ,ε+ζ+ς)fn(Λ,ε+ς)fn(Λ,ε+ζς)+fn(Λ,ες)]diag[θ1(f1(Λ,εζ+ς)f1(Λ,ε+ς)f1(Λ,εζς)+f1(Λ,ες))+(1|θ1|)φj11j1n(ε2+ζ2+ς2),,θn(fn(Λ,εζ+ς)fn(Λ,ε+ς)fn(Λ,εζς)+fn(Λ,ες))+(1|θn|)φjn1jnn(ε2+ζ2+ς2)]

    for each ε,ζ,ςQ, and ΛQ. Then there are unique additive mappings fi:Q×QQ s.t.

    diag[f1(Λ,ε)f1(Λ,ε),,fn(Λ,ε)fn(Λ,ε)]n×ndiag[φj11j1n(ε2),,φjn1jnn(ε2)],

    for each εQ and ΛQ.

    Proof. The proof follows from Theorem 3.2 by letting

    diag[φj11j1n(ε,ζ,ς),,φjn1jnn(ε,ζ,ς)]:=diag[φj11j1n(ε2+ζ2+ς2),,φjn1jnn(ε2+ζ2+ς2)],

    for each ε,ζ,ςQ. Choosing (T1,,Tn)=(47,,47), we obtain the desired result.

    Corollary 3.10. Suppose fi:Q×QQ,(i=1,,n) are mappings satisfying fi(Λ,0)=0 and

    diag[f1(Λ,ε+ζ+ς)f1(Λ,ε+ς)f1(Λ,ε+ζς)+f1(Λ,ες),,fn(Λ,ε+ζ+ς)fn(Λ,ε+ς)fn(Λ,ε+ζς)+fn(Λ,ες)]diag[θ1(f1(Λ,εζ+ς)f1(Λ,ε+ς)f1(Λ,εζς)+f1(Λ,ες))+(1|θ1|)φj11j1n(εζς),,θn(fn(Λ,εζ+ς)fn(Λ,ε+ς)fn(Λ,εζς)+fn(Λ,ες))+(1|θn|)φjn1jnn(εζς)]

    for each ε,ζ,ςQ and ΛQ. Then there are unique additive mappings fi:Q×QQ,(i=1,,n), s.t.

    diag[f1(Λ,ε)f1(Λ,ε),,fn(Λ,ε)fn(Λ,ε)]diag[φj11j1n(ε3),,φjn1jnn(ε3)],

    for each εQ and ΛQ.

    Proof. The proof follows from Theorem 3.2 by letting

    diag[φj11j1n(ε,ζ,ς),,φjn1jnn(ε,ζ,ς)]:=diag[φj11j1n(εζς),,φjn1jnn(εζς)],

    for each ε,ζ,ςQ and ΛQ. Choosing (T1,,Tn)=(89,,89), we obtain the desired result.

    Corollary 3.11. Let fi:Q×QQ,(i=1,,n) be odd mappings satisfying

    diag[f1(Λ,ε+ζ+ς)f1(Λ,ε+ς)f1(Λ,ε+ζς)+f1(Λ,ες),,fn(Λ,ε+ζ+ς)fn(Λ,ε+ς)fn(Λ,ε+ζς)+fn(Λ,ες)]diag[θ1[f1(Λ,εζ+ς)f1(Λ,ε+ς)f1(Λ,εζς)+f1(Λ,ες)]+(1|θ1|)φj11j1n(εζς),,θn[fn(Λ,εζ+ς)fn(Λ,ε+ς)fn(Λ,εζς)+fn(Λ,ες)]+(1|θn|)φjn1jnn(εζς)], (3.18)

    for each ε,ζ,ςQ and ΛQ. 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,ζς)+f1(Λ,ς),,fn(Λ,ζ+ς)fn(Λ,ς)fn(Λ,ζς)+fn(Λ,ς)]diag[θ1(f1(Λ,ζ+ς)f1(Λ,ς)f1(Λ,ζς)+f1(Λ,ς)),,θn(fn(Λ,ζ+ς)fn(Λ,ς)fn(Λ,ζς)+fn(Λ,ς))], (3.19)

    for each ζ,ςQ and ΛQ. Replacing ζ by ζ in (3.19), we have

    diag[f1(Λ,ζ+ς)f1(Λ,ς)f1(Λ,ζς)+f1(Λ,ς),,fn(Λ,ζ+ς)fn(Λ,ς)fn(Λ,ζς)+fn(Λ,ς)]diag[θ1[f1(Λ,ζ+ς)f1(Λ,ς)f1(Λ,ζς)+f1(Λ,ς)],,θn[fn(Λ,ζ+ς)fn(Λ,ς)fn(Λ,ζς)+fn(Λ,ς)]], (3.20)

    for each ζ,ςQ and ΛQ. From (3.19) and (3.20), it follows that

    fi(Λ,ζ+ς)fi(Λ,ς)fi(Λ,ζς)+fi(Λ,ς)=0,i=1,,n

    for each ζ,ςQ and ΛQ. Since fi,(i=1,,n), are odd mappings,

    fi(ς+ζ)+fi(ςζ)2fi(ς)=0,i=1,,n

    for each ζ,ςQ and ΛQ. Thus the mappings fi,(i=1,,n), are additive.

    Corollary 3.12. Suppose fi:Q×QQ,(i=1,,n), are mappings satisfying fi(Λ,0)=0 and

    diag[f1(Λ,ε+ζ+ς)f1(Λ,ε+ς)f1(Λ,ε+ζς)+f1(Λ,ες),,fn(Λ,ε+ζ+ς)fn(Λ,ε+ς)fn(Λ,ε+ζς)+fn(Λ,ες)]diag[θ1(f1(Λ,εζ+ς)f1(Λ,ε+ς)f1(Λ,εζς)+f1(Λ,ες))+(1|θ1|)φj11j1n(ε4+ζ4+ς4),,θn(fn(Λ,εζ+ς)fn(Λ,ε+ς)fn(Λ,εζς)+fn(Λ,ες))+(1|θn|)φjn1jnn(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[φj11j1n(2ε4+ζ4),,φjn1jnn(2ε4+ζ4)]n×n,

    for each ε,ζ,ςQ,ΛQ. If fi(Λ,2ε)=2fi(Λ,ε) foreach ε,ζ,ςQ,ΛQ, and fi,(i=1,,n), are continuous, then the mappings fi:Q×QQ,(i=1,,n), are antiderivations.

    Proof. The proof follows from Theorem 3.7 by letting

    diag[φj11j1n(ε,ζ,ς),,φjn1jnn(ε,ζ,ς)]:=diag[φj11j1n(ε4+ζ4+ς4),,φjn1jnn(ε4+ζ4+ς4)]n×n

    for each ε,ζ,ςQ. 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.


    Acknowledgments



    The author thanks Johnny Kristiansen and Pia Frostad for their assistance in the field and laboratory. The study was supported by NMBU grant 1750051444.

    Conflict of interest



    The author declares no conflict of interests.

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