Some results on Ricatti Equations, Floquet Theory and Applications
Anderson L. A. de Araujo, Abílio Lemos, Alexandre M. Alves, Kennedy M. Pedroso
aa r X i v : . [ m a t h . C A ] O c t Some results on Ricatti Equations, Floquet Theory andApplications
Anderson L. A. de Araujo, Ab´ılio Lemos, Alexandre Miranda AlvesUniversidade Federal de Vi¸cosa, CCE, Departamento de Matem´aticaAvenida PH Rolfs, s/nCEP 36570-900, Vi¸cosa, MG, BrasilE-mail: [email protected], [email protected], [email protected] and
Kennedy Martins PedrosoUniversidade Federal de Juiz de Fora, Departamento de Matem´aticaRua Jos´e Louren¸co Kelmer, s/n, Bairro S˜ao PedroCEP: 36036-900, Juiz de Fora, MG, BrasilE-mail: [email protected]
Abstract . In this paper, we present two new results to the classical Floquet theory, whichprovides the Floquet multipliers for two classes of the planar periodic system. One these resultsprovides the Floquet multipliers independently of the solution of system. To demonstrate theapplication of these analytical results, we consider a cholera epidemic model with phage dynamicsand seasonality incorporated.
AMS Subject Classification 2010 . Floquet theory; stability; epidemic models.
Keywords . 34C25; 34D20; 92D30. 1
Introduction
The Floquet theory is concerned with the study of the linear stability of differential equationswith periodic coefficients, see [2]. This theory focuses on the concept of Floquet multipliers andoffers a powerful means to analyze nonautonomous, periodic differential equations. However, itis very difficult to determine the Floquet multipliers of general linear periodic systems. Exceptfor a few special cases, which include second-order scalar equations and systems of Hamiltoniantype or canonical forms, very little is known about the analysis of Floquet multipliers.For a nonlinear periodic system, if it has a nonconstant periodic solution, its stability canbe analyzed by linearization about the periodic solution. The variational system then becomesa linear periodic system, and its Floquet multipliers provide useful information on the stabilityof the periodic solution.Consider the planar system(1) ˙ u = p ( t ) u + p ( t ) v ˙ v = p ( t ) u + p ( t ) v, where the p ij are continuous real valued T -periodic functions with T >
0. By Floquet theory,see [2, 5], there are solutions to the system (1), say, ϕ = ( u , v ) and ϕ = ( u , v ), and realnumbers λ and λ (not necessarily distinct) that satisfy ϕ ( t + T ) = λ ϕ ( t ) ϕ ( t + T ) = λ ϕ ( t ) . The solutions ϕ and ϕ are called normal solutions and the numbers λ and λ are called Floquet multipliers . Set φ ( t ) = (cid:20) u ( t ) u ( t ) v ( t ) v ( t ) (cid:21) and P ( t ) = (cid:20) p ( t ) p ( t ) p ( t ) p ( t ) (cid:21) , it is a well-known fact that λ and λ are the eigenvalues of the matrix φ − (0) φ ( T ) and that(2) λ λ = exp Z T trace P ( t ) dt. We observe that, if λ = λ , then ϕ and ϕ are linearly independent. In this case, knowledgeof the Floquet multipliers and the values of ϕ and ϕ for 0 < t < T gives information for everysolution of (1) for all t . The calculation of the Floquet multipliers is not routine, since one doesnot generally know even one nontrivial solution of (1). A procedure for obtaining the Floquetmultipliers and the corresponding normal solutions for (1) is possible when an associate Riccatiequation to the system (1) presents a periodic solution. Following Proctor, see [7], making thechange of coordinate u = z + σz and v = z in (1), where σ is a solution of the Riccati equation(3) ˙ x = c ( t ) + b ( t ) x + a ( t ) x with a ( t ) = − p ( t ), b ( t ) = p ( t ) − p ( t ) and c ( t ) = p ( t ), that is,24) ˙ x = p ( t ) + [ p ( t ) − p ( t )] x − p ( t ) x , the differential equation in z can be integrated, thus providing the following result: Proposition 1.1 [7, Theorem 3.2](1) If ϕ = ( ϕ , ϕ ) is a solution of (1), then σ = ϕ ϕ is a solution of (3) on any interval onwhich ϕ does not vanish.(2) If σ is a solution of (3) on an interval I containing the number k , then (5) u ( t ) = σ ( t ) exp R tk p ( s ) σ ( s ) + p ( s ) dsv ( t ) = exp R tk p ( s ) σ ( s ) + p ( s ) ds is a solution of (1)on I .(3) If σ is a solution of (3) with period nT and f is the mean value of p σ + p over the period nT , where n is a positive integer, then exp ( nT f ) is a Floquet multiplier for (1) for the period nT and (5) is a normal solution of (1) corresponding to this multiplier. Thus, for n = 1, λ = exp Z T p ( s ) σ ( s ) + p ( s ) ds is a Floquet multiplier associated to solution ϕ . We observe that the formula given in (2)provides us another multiplier.Now the question is: how to guarantee the existence of a periodic solution to the equation(3)? The next result, proved by Mokhtarzadeh, Pounark and Razani, provides an answer to thequestion above. Before enunciating it, we define the following function. G : [0 , T ] × [0 , T ] −→ R , such that G ( t, s ) = − exp R T b ( r ) dr exp R ts b ( r ) dr, ≤ s ≤ t ≤ T exp R T b ( r ) dr − exp R T b ( r ) dr exp R ts b ( r ) dr, ≤ t ≤ s ≤ T. Proposition 1.2 [Theorem . , [6]] Let a ( t ) , b ( t ) and c ( t ) be continuous T -periodic functionswith R T b ( t ) dt = 0 . Set (6) M = sup ≤ t,s ≤ T | G ( t, s ) | and N = sup ≤ t ≤ T (cid:12)(cid:12)(cid:12)(cid:12)Z T G ( t, s ) c ( s ) ds (cid:12)(cid:12)(cid:12)(cid:12) and suppose (7) Z T | a ( ξ ) | dξ ≤ M N .
Then, x ′ = c ( t ) + b ( t ) x + a ( t ) x has at least a T -periodic solution.
3n [6, Theorem 3.2], the authors defined the following Banach space, X = { φ ; φ is a T -periodic continuous real functions on R } . For φ ∈ X , defined | φ | ∞ = sup ≤ t ≤ T | φ ( t ) | and(8) Ω = { φ ∈ X ; | φ − ψ | ∞ ≤ N } , where ψ : [0 , T ] −→ R is defined by(9) ψ ( t ) = Z T G ( t, s ) c ( s ) ds. It is easy to see that Ω is closed, bounded and convex subset of X . They also defined theoperator S : Ω −→ X by(10) S ( φ )( t ) = Z T G ( t, s ) (cid:2) a ( s ) φ ( s ) + c ( s ) (cid:3) ds. By assumptions (6), (7) and using Schauder Fixed Point Theorem, the authors proved theexistence of x ∈ Ω, such that, S ( x ) = x , that is, for all t ∈ [0 , T ] x ( t ) = Z T G ( t, s ) (cid:2) a ( s ) x ( s ) + c ( s ) (cid:3) ds. As a consequence of the Propositions 1.1 and 1.2 we state the following result.
Theorem 1.3
Suppose(i) R T p ( t ) − p ( t ) dt = 0 ;(ii) R T | p ( t ) | dt ≤ / M N .Then, the Floquet multipliers of the system (1) are λ = exp Z T p ( t ) + p ( t ) σ ( t ) dt and λ = exp Z T p ( t ) − p ( t ) σ ( t ) dt, where σ ( t ) is a T -periodic solution of equation (4). Now, we state one of the main results of this work, which is related to the existence of aperiodic solution to the Ricatti equation, which satisfies R T a ( t ) x ( t ) dt = 0, as a consequence ofthe coincidence degree theory, proposed by R. E. Gaines and J. L. Mawhin [3]. Theorem 1.4
Consider < T and let a ( t ) , b ( t ) and c ( t ) be continuous T -periodic functionswith (11) Z T a ( t ) dt = 0 , | a ( t ) | ≤ A, < b ≤ b ( t ) , Z T c ( t ) dt = 0 where b, A > are constants. Then, the Ricatti equation (13) x ′ = c ( t ) + b ( t ) x + a ( t ) x has at least one nontrivial x T - periodic solution that satisfies (14) Z T a ( t ) x ( t ) dt = 0 and | x | ∞ ≤ b A .
As a consequence of the Theorem 1.4, we obtain the Floquet multipliers explicitly. In thiscase, they do not depend on the solution of the Riccati equation.
Theorem 1.5
Suppose(i) R T p ( t ) dt = R T p ( t ) dt = 0 and | p ( t ) | ≤ A , where A > is a constant;(ii) p ( t ) − p ( t ) ≥ b > , where b is a constant.Then the Floquet multipliers of the system (1) are λ = exp Z T p ( t ) dt and λ = exp Z T p ( t ) dt. We observe that the Proposition 1.2 guarantees the existence of a T -periodic solution, butdoes not explicitly provide such a solution. Therefore, the Floquet multipliers provided byTheorem 1.3 cannot be obtained explicitly. On the other hand, the Theorem 1.5 providesFloquet multipliers that can be obtained explicitly.In what follows, starting from section 2, we present some concepts and results of coincidencedegree theory. In section 3, we prove the theorem 1.3, 1.4 and 1.5. In section 4, we presentsome stability results of linear and nonlinear systems in terms of Floquet multipliers. Finally,in section 5, we apply Theorem 1.3 in a mathematical model for cholera. The method to be used in this paper involves the applications of the continuation theorem ofcoincidence degree. In order to make this presentation as self-contained as possible, we introducea few concepts and results about the coincidence degree. For further details, see R. E. Gainesand J. L. Mawhin [3].
Definition 2.1
Let X , Y be real Banach spaces, L : DomL ⊂ X → Y be a linear mapping.The mapping L is said to be a Fredholm mapping of index zero, if dim KerL = codimImL < + ∞ and ImL is closed in Y . L is a Fredholm mapping of index zero, then there are continuous projectors P : X → X and Q : Y → Y , such that ImP = KerL and
KerQ = ImL = Im ( I − Q ) . It follows that the restriction L P of L to DomL ∩ KerP : ( I − P ) X → ImL is invertible. Denotethe inverse of L P by K P . Definition 2.2
A continuous mapping N : X → Y is said to be L -compact on Ω , if Ω is anopen bounded subset of X , QN (Ω) is bounded and K P ( I − Q ) N : Ω → X is compact. Since
ImQ is isomorphic to
KerL , there is an isomorphism J : ImQ → KerL . We shall beinterested in proving the existence of solutions for the operator equation(15) Lx = N x, where a solution is an element of
DomL ∩ Ω which verifies (15).The following results is due to R. E. Gaines and J. L. Mawhin [3].
Proposition 2.3 (Mawhin’s Continuation Theorem)
Let L be a Fredholm mapping of in-dex and let N be L -compact on ¯Ω . Suppose that(i) For each λ ∈ (0 , , x ∈ ∂ Ω Lx = λN x. (ii) QN x = 0 for each x ∈ KerL ∩ ∂ Ω and deg ( J QN, Ω ∩ KerL, = 0 , where J : ImQ → KerL is an isomorphism.Then, the equation Lx = N x has at least one solution in
DomL ∩ ¯Ω . Proof.
Due to ( i ) and ( ii ), the Proposition 1.2 guarantees that the equation (4) has a T -periodicsolution σ ( t ). Thus, by Proposition 1.1, a Floquet multiplier of the system (1) is λ = exp Z T p ( t ) + p ( t ) σ ( t ) dt. Now, we use the equation (2) to determine the other multiplier, which is λ = exp Z T p ( t ) − p ( t ) σ ( t ) dt. .2 Proof of Theorem 1.4 Proof.
Consider the following Banach spaces X = { x | x ∈ C ( R , R ) , x ( t + T ) = x ( t ) , for all t ∈ R } ∩ (cid:26) x | Z T a ( t ) x ( t ) dt = Z T c ( t ) x ( t ) dt = 0 (cid:27) , and Y = { x | x ∈ C ( R , R ) , x ( t + T ) = x ( t ) , for all t ∈ R } , with the norm k x k X = k x k Y = | x | ∞ , where | x | ∞ = max t ∈ [0 ,T ] | x ( t ) | .Define a linear operator L : DomL ⊂ X → Y by setting DomL = { x | x ∈ X, x ′ ∈ C ( R , R ) } and for x ∈ DomL , Lx = x ′ . We also define a nonlinear operator N : X → Y by setting N x = c ( t ) + b ( t ) x + a ( t ) x . It is not difficult to see that, by (11),
KerL = R , and ImL = (cid:26) y | y ∈ Y, Z T y ( s ) ds = 0 (cid:27) . Thus the operator L is a Fredholm operator with index zero.Define the continuous projector P : X → KerL and the averaging projector Q : Y → Y bysetting P x ( t ) = x (0)and Qy ( t ) = 1 T Z T y ( s ) ds. Hence,
ImP = KerL and
KerQ = ImL . Denoting by K P : ImL → DomL ∩ KerP theinverse of L | DomL ∩ KerP , we have K P y ( t ) = Z t y ( s ) ds. Then QN : X → Y and K P ( I − Q ) N : X → X read QN x = 1 T Z T b ( s ) x ( s ) ds + 1 T Z T a ( s ) x ( s ) ds, P ( I − Q ) N x ( t ) = Z t c ( s ) ds + Z t b ( s ) x ( s ) ds + Z t a ( s ) x ( s ) ds − tQN x. Clearly, QN and K P ( I − Q ) N are continuous. By using Arzela-Ascoli theorem, it is notdifficult to prove that K P ( I − Q ) N (Ω) is compact for any open bounded set Ω ⊂ X . Indeed,let L > | x | ∞ ≤ L , for each x ∈ Ω. Notice that QN (Ω) is bounded and | QN x | ∞ ≤| b | ∞ L + | a | ∞ L , for all x ∈ Ω. Hence, K P ( I − Q ) N (Ω) is uniformly bounded and | K P ( I − Q ) N x | ∞ ≤ T ( | c | ∞ + | b | ∞ L + | a | ∞ L ) , ∀ x ∈ Ω . Now, let t , t ∈ [0 , T ], if we suppose t < t , we obtain | K P ( I − Q ) N x ( t ) − K P ( I − Q ) N x ( t ) | = (cid:12)(cid:12)(cid:12)R t t c ( s ) ds + R t t b ( s ) x ( s ) ds + R t t a ( s ) x ( s ) ds − ( t − t ) QN x (cid:12)(cid:12)(cid:12) ≤ | c | ∞ + | b | ∞ L + | a | ∞ L ) | t − t | , ∀ x ∈ Ω . Therefore, K P ( I − Q ) N (Ω) is a equicontinuous set of C ([0 , T ])(hence of X ). By using Arzela-Ascoli theorem, K P ( I − Q ) N (Ω) is compact. Therefore, N is L -compact on Ω with any openbounded set Ω ⊂ X .As b A >
0, we consider(16) Ω d := { x ∈ X || x | ∞ < b A } , that is an open set in X .Notice that(17) b ( t ) − a ( t ) b A > . Indeed, b ( t ) − a ( t ) b A ≥ b − A. b A = b − b b > . Notice that(18) b ( t ) + a ( t ) b A > . Indeed, b ( t ) + a ( t ) b A ≥ b − A b A = b − b b > . Let 0 < λ < x such that x ′ = λc ( t ) + λb ( t ) x + λa ( t ) x . By multiplying by x and integrand of 0 to T , we have0 = Z T x ′ xdt = λ Z T c ( t ) xdt + λ Z T b ( t ) x + a ( t ) x dt. Z T c ( t ) xdt + Z T x ( b ( t ) + a ( t ) x ) dt = Z T x ( b ( t ) + a ( t ) x ) dt. By (16), if x ∈ ∂ Ω d , we have | x | ∞ = b A , and we obtain0 ≥ Z T x ( b ( t ) − | a ( t ) || x | ∞ ) dt. By (12), we have 0 ≥ Z T x ( b − A b A ) dt ≥ Z T x b dt > . But this is a contradiction. Therefore, the condition (1) of Proposition 2.3 holds for Ω c . Take x ∈ ∂ Ω d ∩ KerL . Thus, we have x = − b A or x = b A .If x = − b A , by (17), we have b ( t ) − a ( t ) b A > . Hence,(19)
QN x = 1 T Z T − b A (cid:18) b ( t ) − a ( t ) b A (cid:19) dt < . If x = b A , by (18), we have b ( t ) + a ( t ) b A > . Hence,(20)
QN x = 1 T Z T b A (cid:18) b ( t ) + a ( t ) b A (cid:19) dt > . Then, for each x ∈ ∂ Ω d ∩ KerL , we have(21)
QN x = 1 T Z T x ( b ( t ) + a ( t ) x ) dt = 0 . Therefore, the condition (2) of Proposition 2.3 holds for Ω d .Define a continuous function H ( x, µ ) by setting H ( x, µ ) = (1 − µ ) x + µ T Z T x ( b ( t ) + a ( t ) x ) dt, µ ∈ [0 , . It follows from (19), (20) and (21) that H ( x, µ ) = 0 , for all x ∈ ∂ Ω d ∩ KerL. deg ( QN, Ω d ∩ KerL,
0) = deg (cid:18) T Z T x ( b ( t ) + a ( t ) x ) dt, Ω d ∩ KerL, (cid:19) = deg ( x, Ω d ∩ KerL,
0) = − = 0 . In view of all the discussions above, we conclude from Proposition 2.3 that the equation (13)has a solution in
DomL ∩ ¯Ω d . Proof.
Due to ( i ) and ( ii ), the Theorem 1.4 guarantees that the equation (4) has a T -periodicsolution σ ( t ) and R T p ( t ) σ ( t ) dt = 0. Thus, by Proposition 1.1, a Floquet multiplier of thesystem (1) is λ = exp Z T p ( t ) dt. Now, we use the equation (2) to determine the other multiplier, which is λ = exp Z T p ( t ) dt. The following theorem provides details about the stability of the system (1) in terms ofFloquet exponents. We refer to Theorem 7.2 on page 120 of Hale’s book [4].
Theorem 4.1 (i) A necessary and sufficient condition that the system (1) is uniformly stableis that the Floquet multipliers of the system (1) have modulii ≤ and the ones with modulii = 1 have multiplicity .(ii) A necessary and sufficient condition that the system (1) is uniformly asymptotically stableis that all Floquet multipliers of the system (1) have modulii < . Now, consider the system(22) ˙ X = A ( t ) X + F ( t, X ) , where A ( t ) is an n × n continuous matrix function, and F ( t, X ) is continuous in t and X andLipschitz-continuous in X for all t ∈ R and X in a neighbourhood of X = 0. Moreover, weassume that(23) lim | X |→ | F ( t, X ) || X | = 0 uniformly in t. Notice that the condition in (23) implies that X = 0 is a solution to system (22). Then, we havethe following theorem on the behaviour of the trivial solution X = 0. This result is an extendedversion of [4, Theorem 2.4] or [9, Theorem 7.2].10 heorem 4.2 If the trivial solution X = 0 of the system ˙ X = A ( t ) X is uniformly asymptoti-cally stable for t ≥ , then the trivial solution of (22) is also uniformly asymptotically stable. Ifthe trivial solution X = 0 of the system ˙ X = A ( t ) X is unstable, then the trivial solution of (22)is also unstable. In the following example, we study the stability of a planar system, whose Floquet multipliersare obtained by Theorem 1.5.
Example 1
Consider the planar system (24) ˙ u = ( m ( t ) − A ) u + sin( t ) v ˙ v = a ( t ) u + ( m ( t ) − B ) v, where m ( t ) is a continuous function and π -periodic, a ( t ) = ( α + 1) sin( t ) β + cos( t ) ,β > and A − B > . The Riccati equation associated with this system is y ′ = − a ( t ) y + ( A − B ) y + sin( t ) . Since R π a ( t ) dt = R π sin( t ) dt = 0 , b ( t ) = A − B > and a ( t ) is limited, according to Theorem1.4, that there is a π -periodic solution σ ( t ) with R π a ( t ) σ ( t ) dt = 0 . Now, by Theorem 1.5, theFloquet multipliers are λ = exp Z π p ( t ) dtλ = exp Z π p ( t ) dt, that is, λ = exp Z π ( m ( t ) − A ) dtλ = exp Z π ( m ( t ) − B ) dt. Therefore, λ = e − πA e R π m ( t ) dt and λ = e − πB e R π m ( t ) dt . Observe that, if R π m ( t ) dt < πA and R π m ( t ) dt < πB , then the system (24), according toTheorem 4.1, is uniformly asymptotically stable. In this section, we propose the analysis of a mathematical model for cholera dynamics withseasonal oscillation, studied by [8]. Cholera is a severe intestinal infection caused by the bac-terium
Vibrio cholerae . Many epidemic models have been published, but Code¸co [1] was thefirst to explicitly incorporate bacterial dynamics into a SIR epidemiological model.11he following new model is a significant extension of Code¸co’s model [1] that incorporatesthe phage dynamics and the seasonal oscillation of cholera transmission, as proposed in [8]:(25) dSdt = n ( H − S ) − d BK + B S, dIdt = d BK + B S − rI, dBdt = eI − mB − δ B ˜ K + B P, dPdt = ξ I + κ B ˜ K + B P − ν P, where • S is the susceptible human population, • I is the infectious human population, • B and P are the concentrations of the pathogen (i.e. vibrio) and the phage, respectivelyin the contaminated water, • the total human population, H , is assumed to be a constant, • n denotes the natural human birth/death rate, • d denotes the human contact rate to the vibrio, • δ is the death rate of the bacteria due to phage predation, • κ is the growth rate of the phage due to feeding on the vibrio, • e and ξ are the rates of human contribution (e.g. by shedding) to the pathogen and thephage, respectively, • m and ν are the natural death rates of the vibrio and the phage, respectively.In addition, r = n + γ with γ as the recovery rate, and K and ˜ K as the half saturation ratesof the vibrio in the interaction with human and phage, respectively.In [8], the authors investigated the impact of seasonality on cholera dynamics, particularitywhen they examined the periodic variation of three parameters, m , e and d , and applied theresults from the Floquet theory in the analysis, as follows. Firstly, the authors consider thatthe parameter m is a positive periodic function of time, m ( t ), which represents a seasonalvariation of the extinction rate of the vibrio. On the second scenario, the parameter e is setas a positive periodic function, e ( t ), which represents a seasonal oscillation of the per capitacontamination rate, i.e., the unit rate of human contribution (e.g. shedding) to the pathogen inthe environment. On the third scenario, the parameter d is set as a positive periodic function, d ( t ), which represents a seasonal variation of the contact rate. For an accurate analysis of thedynamics in each previous case, see [8].In this paper, we consider the following scenario, by setting the parameters m , e and d aspositive periodic functions m ( t ), e ( t ) and d ( t ) simultaneously.12t is clear that E = ( H, , ,
0) is the unique disease free equilibrium (DFE) of the system.For ease of discussion, we translated the DFE to the origin via a change of variable by S = HS .Then, with linearization at (0 , , , dSdt = − nS + d ( t ) HK B + (cid:16) d ( t ) B ( H − S ) K + B − d ( t ) HK B (cid:17) , dIdt = − rI + d ( t ) HK B + (cid:16) d ( t ) B ( H − S ) K + B − d ( t ) HK B (cid:17) , dBdt = e ( t ) I − m ( t ) B − δ B ˜ K + B P, dPdt = ξ I + κ B ˜ K + B P − ν P. Thus, system (26) can be written in a compact form˙ X = A ( t ) X + F ( t, X ) , with X = ( ¯ S, I, B, P ) T and the matrix(27) A ( t ) = − n d ( t ) HK − r d ( t ) HK e ( t ) − m ( t ) 00 ξ − ν . By [8], it is straightforward to check that lim | X |→ | F ( t,X ) || X | = 0 uniformly in t . Based on Theorem4.2, we only need to consider the periodic linear system(28) ˙ X = A ( t ) X, where the matrix A ( t ) is given by (27).Notice that the matrix A ( t ) has a block tridiagonal structure. From [8, Theorem 2.5 andCorollary 2.5], two Floquet exponents of the system (28) are given by n and ν ; the other twoFloquet exponents are determined by the matrix block (cid:18) − r d ( t ) HK e ( t ) − m ( t ) (cid:19) . Hence, its stabilitydepends on the 2 × Y = (cid:18) − r d ( t ) HK e ( t ) − m ( t ) (cid:19) Y. We consider m ( t ), e ( t ) and d ( t ), such that(30) A = max ≤ t ≤ T d ( t ) , (31) E = max ≤ t ≤ T e ( t )and(32) m = min ≤ t ≤ T m ( t ) , EM AT HK < min { m , r } and(34) Z T e ( t ) dt ≤ M N , where M and N are defined in (6).We have the following lemma to describe the stability of the sub-system (29). Lemma 5.1
Suppose (33), (34) and r = T R T m ( t ) dt . Then, the trivial solution of (29) isasymptotically stable. Proof.
We observe that m ( t ), e ( t ) and a ( t ) satisfies the assumptions of Theorem 1.3, with R T ( p − p ) dt = − T r + R T m ( t ) dt = 0 and R T p ( t ) dt = R T e ( t ) dt ≤ MN . Then, it followsfrom Theorem 1.3 that the Floquet multipliers of the system (29) are λ = exp (cid:18)Z T ( − r − e ( t ) σ ( t )) dt (cid:19) and λ = exp (cid:18)Z T ( − m ( t ) + e ( t ) σ ( t )) dt (cid:19) , where σ ( t ) is a T -periodic solution of equation (4). Since σ ∈ Ω (Ω defined in (8)), we obtain (cid:12)(cid:12)(cid:12)(cid:12) σ ( t ) − Z T G ( t, s ) a ( s ) HK ds (cid:12)(cid:12)(cid:12)(cid:12) < N.
Therefore, | σ ( t ) | ≤ Z T | G ( t, s ) || a ( s ) HK | ds + N ≤ M AT HK .
Hence, by (33), we obtain − r − e ( t ) σ ( t ) ≤ − r + 2 EM AT HK < − m ( t ) + e ( t ) σ ( t ) ≤ − m + 2 EM AT HK < . Notice that all Floquet multipliers of the system (29) have modulii <
1. By Theorem 4.1-( ii ),the trivial solution X = 0 of the system (29) is uniformly asymptotically stable for t ≥ Theorem 5.2
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