R E S E A R C H
Open Access
Almost periodic solution of an impulsive
multispecies logarithmic population model
Li Wang
*and Hui Zhang
*Correspondence:
[email protected] Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an, Shaanxi 710072, P.R. China
Abstract
In this paper, some easily verifiable conditions are derived for the existence of almost periodic solution of an impulsive multispecies logarithmic population model in terms of the continuous theorem of coincidence degree theory, which is rarely applied to studying the existence of almost periodic solution of an impulsive differential equation. Our results generalize previous results by Alzabut, Stamov and Sermutlu. Besides, our technique used in this paper can be applied to study the existence of almost periodic solution of an impulsive differential equation with linear impulsive perturbations.
Keywords: coincidence degree theory; almost periodicity; impulsive multispecies logarithmic population model
1 Introduction
Many evolutionary processes in nature are characterized by the fact that their states are subject to sudden changes at certain moments and therefore can be described by an im-pulsive system. Basic theory of imim-pulsive differential equations can be found in the mono-graphs [–]. Gopalsamy [] proposed the following single species logarithmic population model:
N(t) =N(t)a(t) –b(t)lnN(t) –c(t)lnNt–τ(t).
By utilizing the continuous theorem of coincidence degree theory, the existence of pe-riodic solution and almost pepe-riodic solution of this model is investigated in papers [] and [], respectively. In paper [], Alzabut and Abdeljawad consider the existence of pe-riodic solution of the impulsive differential equation
⎧ ⎨ ⎩
N(t) =N(t)(a(t) –b(t)lnN(t) –c(t)lnN(t–τ(t))), t=tk, N(t+
k) =N(tk)+
γkeδk, k= , , . . . .
By employing the contraction mapping principle and Gronwall-Bellman inequality, the existence and exponential stability of positive almost periodic solution of this model are obtained in paper []. Using the same method as [], Yang and Li [] deal with the ex-istence of almost periodic solution of an impulsive two-species logarithmic population model with time-varying delay. By means of the Cauchy matrix, Wang [] gives sufficient
conditions for the existence and exponential stability of periodic and almost periodic so-lution for the delay impulsive logarithmic population:
⎧ ⎨ ⎩
N(t) =N(t)(a(t) –b(t)lnN(t) –c(t)lnN(t–τ(t))), t=tk,
eN(t+k)=e(+pk)N(tk)+ck, k= , , . . . .
Some scholars are interested in impulsive differential equations or the existence of (al-most) periodic solution. Related results can be found in the literature [–]. By using fixed point theory and constructing a suitable Lyapunov function, Chen [] studies the existence, uniqueness and global attractivity of positive periodic solution and almost pe-riodic solution of the delay multispecies logarithmic population model
Ni(t) =Ni(t)
ri(t) –
n
j=
aij(t)lnNj(t) – n
j=
bij(t)lnNj
t–τij(t)
–
n
j= cij(t)
t
–∞kij(t–s)
lnNj(s)ds
, i= , , . . . ,n.
In this paper, we consider the impulsive multispecies logarithmic population model
Ni(t) =Ni(t)
ri(t) –
n
j=
aij(t)lnNj(t) – n
j=
bij(t)lnNj
t–τij(t)
–
n
j= cij(t)
t
–∞kij(t–s)
lnNj(s)ds
, t=tk, (.)
Ni
tk+= ( +dik)Ni(tk), i= , , . . . ,n,k= , , . . . ,
whereri(·),aij(·),bij(·),cij(·),τij(·) are nonnegative almost periodic functions in sense of
Bohr,kij(t–s)≥ and
t
–∞kij(t–s)ds< +∞,{dik}is an almost periodic sequence,{tk}is
an equipotentially almost periodic sequence,r=supt∈R,i,j=,...,n|τi,j(t)|,i,j= , . . . ,n.
By utilizing the continuous theorem of coincidence degree theory, we obtain some suf-ficient conditions for the existence of almost periodic solution of Eq. (.). Our results generalize previous results obtained in [] and are easier to verify than the conditions obtained in paper [].
The continuous theorem of coincidence degree theory has been extensively used to study the existence of periodic solution of a differential equation, regardless of the equa-tion being with impulse or without. Using this theorem, papers [] and [] investigate the existence of almost periodic solution of a population model without impulse. To our best knowledge, the continuous theorem has not been used to prove the existence of almost periodic solution of an impulsive multispecies logarithmic population model. Besides, our technique used in this paper can be applied to study the existence of almost periodic so-lution of an impulsive differential equation with linear impulsive perturbations.
2 Preliminaries
In this section, some lemmas and definitions, which are of importance in proving our main result in Section , will be presented.
Definition .([]) ϕ(·)∈C(R,Rn) is said to be almost periodic in sense of Bohr if∀ε> , there exists a relatively dense setT(ϕ,ε) such that ifτ ∈T(ϕ,ε), then|ϕ(t) –ϕ(t+τ)|<ε
for allt∈R. Denote byap(R,Rn) all such functions.
Definition .([]) Letϕ(·) = (ϕ(·),ϕ(·), . . . ,ϕn(·)) be a piecewise continuous function
with first kind discontinuities at the points of a fixed sequence{tk}. We callϕalmost
pe-riodic if:
() {tk}is equipotentially almost periodic, that is,∀ε> , there exists a relatively dense
set ofε-almost periodic common for any sequences{tjk},tjk=tk+j–tk;
() ∀ε> ,∃δ> such that if the pointst,tbelong to the same interval of continuity and|t–t|<δ, then|ϕ(t) –ϕ(t)|<ε;
() ∀ε> , there exists a relatively dense setT(ϕ,ε)such that ifτ∈T(ϕ,ε), then
|ϕ(t) –ϕ(t+τ)|<εfor allt∈Rwhich satisfy the condition|t–ti|>ε,
i= ,±,±, . . ..
Denote byAP(R,Rn) all such functions.
Now, we introduce some basic notations. Supposef ∈ap(R,Rn) orf ∈AP(R,Rn), we
usefto denote the set of Fourier exponents off,mod(f) to denote the module off,m(f)
to denote the limit mean off. We suppose thatdik andτij(·) in system (.) satisfy the
following conditions: (A) <tk<t( +djk),
<tk<t–τij(t)( +djk)are positive almost periodic functions,
inft∈R<tk<t( +djk) > ,j= , . . . ,n;
(A) –∞t kij(t–s)ln
<tk<s( +djk)dsare almost periodic functions,i,j= , . . . ,n. Thereinto, the definition of<tk<t( +djk) (j= , , . . . ,n) is as follows:
<tk<t
( +djk) =
⎧ ⎨ ⎩
, t∈(–∞,t],
( +dj)· · ·( +djk), t∈(tk,tk+],k= , , , . . . .
Remark There exist a great deal of functions satisfying assumptions (A) and (A). For instance, let{tk}be an arbitrary equipotentially almost periodic sequence,τij(t) =τ > ,
dj,k=dj,k+T, ( +dj,k) > ,k= , , . . . and ( +dj)( +dj) . . . ( +djT) = , then
<tk<t( +djk)
is a positive almost periodic function with discontinuous pointstk, andinft∈R
<tk<t( +
djk) > .
<tk<t–τij(t)( +djk) is also a positive almost periodic function with discontinuous pointstk+τ,i,j= , . . . ,n.
Besides, we suppose thatkij(·) satisfy
t
–∞kij(t–s)ds=
+∞
kij(u)du< +∞and
kij(t) =
⎧ ⎨ ⎩
, t∈[,supi∈Zti+ ],
≥, otherwise,
then–∞t kij(t–s)ln
Before studying the existence of strictly positiveAPsolutions of system (.), we firstly consider the following equation:
yi(t) =yi(t)
Lemma . Suppose that(A)is satisfied,the following results hold: () IfNi(t)∈AP(R,R)is a positive solution of Eq. (.),then
can be obtained easily, we omit it here.
In order to investigate Eq. (.), we takeyi(t) =exi(t), Eq. (.) can be translated to
Obviously, if Eq. (.) has apsolution, then Eq. (.) has strictly positive apsolution. It follows from Lemma . that Eq. (.) has strictly positiveAPsolution. Consequently, we mainly study the existence ofapsolution of Eq. (.). To do so, we firstly summarize a few concepts.
codim ImL<∞andImLis close inZ. IfLis a Fredholm mapping of index zero, there are continuous projectsP:X→X,Q:Z→Zsuch thatImP=KerL,ImL=KerQ=Im(I–Q). It follows thatL|domL∩KerP: (I–P)X→ImLis invertible. We denote the inverse of that map byKp. If is an open subset ofX, the mappingNwill be calledL-compact on if
QN(¯) is bounded andKp(I–Q)N: ¯ →Xis compact. SinceImQis isomorphic toKerL,
there exists an isomorphismJ:ImQ→KerL. The following result is proved in [].
Lemma .(Continuous theorem) Let ⊂X be an open bounded set,let L be a Fredholm mapping of index zero and N be L-compact on ¯.Assume
() For eachλ∈(, ),every solutionxofLx=λNxis such thatx∈/∂ ; () For eachx∈KerL∩∂ ,QNx= ;
() deg(JQN,KerL∩ , )= .
Then Lx=Nx has at least one solution indomL∩ ¯. 3 Main results
In this section, by means of the continuous theorem of coincidence degree theory, we investigate the existence of strictly positiveAPsolution of Eq. (.). To do so, we take
X=x= (x, . . . ,xn)∈ap
R,Rn:mod(xi)⊂mod(F),∀λ∈xi,α>|λ|>α
∪ {},
Z=
z= (z, . . . ,zn)∈AP
R,Rn,
zi(t) areAPfunctions with discontinuous pointstk,
mod(zi)⊂mod(F),∀λ∈zi,α>|λ|>α, ∞
j=
a(λj,zi)< +∞
∪ {},
Z=X=h= (h,h, . . . ,hn)∈Rn
,
whereαandαare given positive constants, andFis a given almost periodic function in sense of Bohr. DefineX=X⊕X,Z=Z⊕Zwith the normφ=max≤i≤nsupt∈R|φi(t)|, φ= (φ,φ, . . . ,φn)∈XorZ.
Lemma . X and Z are Banach spaces equipped with the norm · .
Proof Firstly, we can easily obtain thatXis a Banach space, hence we only need to prove thatZis a Banach space. If{zk= (zk, . . . ,znk)}k⊂Zconverges toz= (z, . . . ,zn) uniformly,
it follows from the properties ofAP functions thatz∈AP(R,Rn). Similar as [], the factmod(zi)⊂mod(F),∀λ∈zi,α>|λ|>αcan be obtained easily. Now, we assert that ∞
j=|a(λj,zi)|< +∞,i= , . . . ,n. If not,∀M> ,∃m> and¯isuch that
m
j=
a(λj,z¯i)>M. (.)
Since{zk}is a Cauchy sequence, for /m,∃Ksuch that for anym,¯ n¯>K,
a(λ,z¯i,m¯) –a(λ,z¯i,n¯)= lim
T→+∞
T
T
–T
z¯i,m¯(t) –z¯i,¯n(t)e–iλtdt<
Sincezkconverges tozuniformly,
forλl∈z¯iand
l,∃nl>K, s.t.,a(λl,z¯i)<a(λl,z¯i,nl)+
l,l= , , . . . ,m. (.)
Sincen,n, . . . ,nm>K, from (.)
a(λj,z¯i,nj)<a(λj,z¯i,n)+
m, j= , , . . . ,m. (.)
Hence, combining (.)-(.)
M<
m
j=
a(λj,z¯i)< m
j=
a(λj,z¯i,nj)+ j<
m
j=
a(λj,z¯i,nj)+
<
m
j=
a(λj,z¯i,n)+ +
m m <
∞
j=
a(λj,z¯i,n)+ < +∞.
The arbitrariness ofMleads to a contradiction, so the assertion holds. Therefore,Zis a Banach space. Similarly, we can obtain thatZis a Banach space. The proof is complete.
Lemma . Let
L:X→Z, L(x,x, . . . ,xn) =
dx
dt, dx
dt , . . . , dxn
dt
,
then L is a Fredholm mapping of index zero.
Proof Obviously,KerL=X. We proveImL=Z. Firstly, for anyϕ+ϕ=ϕ∈ImL⊂Z,
ϕ∈Z, ϕ∈Z, since t
ϕ(s)ds∈ap(R,Rn), that is, t
ϕ(s)ds+ t
ϕ(s)ds∈ap(R,Rn). From [] we know tϕ(s)ds∈ap(R,Rn), then ϕ= . Henceϕ∈Z,ImL⊂Z. Sec-ondly, for anyϕ= (ϕ,ϕ, . . . ,ϕn)∈Z, without loss of generality, we supposeϕ= . Since
t
ϕ(s)ds∈ap(R,Rn), furthermore
t
ϕi(s)ds–m(
t
ϕi(s)ds)=ϕi, i= , , . . . ,n,
thentϕi(s)ds–m(
t
ϕi(s)ds) is the primitive ofϕiinX,ϕ∈ImL. Therefore,Z⊂ImL. To sum up,Z=ImL. Besides, one can easily show thatImLis closed inZ
dim KerL=n=codim ImL.
Therefore,Lis a Fredholm mapping of index zero.
Remark Iff ∈ap(R,Rn) and∀λ∈f,|λ|>α> , thenf has anapprimitive function. It
does not hold for anAP function. From [] we know that iff ∈AP(R,Rn),∀λ∈ f, α>|λ|>α> ,
∞
i=|a(λi,f)|< +∞, thenf has anAP primitive function. That is the
Set
then we have the following.
Lemma . N is L-compact on ¯ ( is an open,bounded subset of X).
Proof Firstly, it is easy to show thatPandQare continuous projectors such that
ImP=KerL, ImL=Im(I–Q) =KerQ,
Obviously, QN and (I–Q)N are continuous. In fact, Kp is also continuous. For any
andmod(zi)⊂mod(F), thenmod(
t
zi(s)ds)⊂mod(F). Therefore, there exists <δ<ε such thatT(F,δ)⊂T(tzi(s)ds,ε). Letlbe the inclusion interval ofT(F,δ), thenl≥l(ε).
For anyt∈/[,l], there existsξ∈T(F,δ)⊂T(tzi(s)ds,ε) such thatt+ξ ∈[,l]. Hence,
by the definition of almost periodic function, we have
sup
Hence, we can conclude that Kp is continuous, and consequently, Kp(I–Q)N is also
continuous. In addition, we also have Kp(I–Q)N is uniformly bounded in ¯, QN(¯)
is bounded and Kp(I –Q)N is equicontinuous in ¯. Since (I–Q)Nx∈Z =ImL and Kp(I–Q)Nx=(I–Q)Nx, ∀x∈ , thenmod(Kp(I–Q)Nx) =mod((I–Q)Nx)⊂mod(F). For
anyε> ,∃δ> such thatT(F,δ)⊂T(Kp(I–Q)Nx,ε),∀x∈ , henceKp(I–Q)Nis
equi-almost periodic in , According to [], we can conclude thatKp(I–Q)N¯ is compact,
thusNisL-compact on ¯.
Noticing Lemmas .-., for Eq. (.), we have the following result.
Theorem . If(A)and(A)are satisfied,m(jn=(aij(t) +bij(t) +cij(t)
t
–∞kij(t–s)ds))=
,i= , , . . . ,n,then Eq. (.)has at least one strictly positive almost periodic solution.
Proof From the analysis above, we know that in order to prove the existence of strictly positiveAPsolution of Eq. (.), we only need to investigate the existence ofapsolution of Eq. (.).
Define the isomorphismJ:ImQ→KerLas an identity mapping. We search for an ap-propriate bounded open subset for the application of Lemma .. Corresponding to the operator equationLx=λNx,λ∈(, ), we have
Ifx∈Xis a solution of system (.), taking the limit mean for system (.), we obtain
then
Combining (.)-(.), we obtain
where
M= |m(r¯i(t))|
|m(nj=(aij(t) +bij(t) +cij(t)
t
–∞kij(t–s)ds))|
+ +
t∗+τ¯
t∗
¯ri(s)ds.
Take ={x∈X,x ≤M + }, then it is clear that verifies all the requirements in Lemma ., hence Eq. (.) has at least one almost periodic solution in and the proof is
complete.
Remark In this paper, we investigate the existence of almost periodic solution of Eq. (.) in terms of the continuous theorem of coincidence degree theory. The novelty of this paper is not only the method but also the results. In fact, ifdik= ,cij= ,i=j= , Eq. (.) can
be rewritten as
N(t) =N(t)r(t) –a(t)lnN(t) –b(t)lnNt–τ(t).
Paper [] obtains the existence of almost periodic solution of the above equation based on the assumptionsm(a(t) +b(t))= . Obviously, our results generalize previous results obtained in paper [].
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.
Acknowledgements
The authors would like to thank the Editor and the referees for their constructive suggestions on improving the presentation of the paper. This work was supported by the Fundamental Research Funds for the Central Universities (3102014JC-Q01087), the National Natural Science Foundation of China (No. 31300310).
Received: 9 October 2014 Accepted: 29 January 2015
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