Stability of differential equations with piecewise constant arguments of generalized type

Nonlinear Analysis 68 (2008) 794–803 www.elsevier.com/locate/na

Stability of differential equations with piecewise constant arguments of generalized type M.U. Akhmet ∗ Department of Mathematics and Institute of Applied Mathematics, Middle East Technical University, 06531 Ankara, Turkey Received 20 July 2006; accepted 17 November 2006

Abstract In this paper we continue to consider differential equations with piecewise constant argument of generalized type (EPCAG) [M.U. Akhmet, Integral manifolds of differential equations with piecewise constant argument of generalized type, Nonlinear Anal. TMA 66 (2007) 367–383]. A deviating function of a new form is introduced. The linear and quasilinear systems are under discussion. The structure of the sets of solutions is specified. Necessary and sufficient conditions for stability of the zero solution are obtained. Our approach can be fruitfully applied to the investigation of stability, oscillations, controllability and many other problems of EPCAG. Some of the results were announced at The International Conference on Hybrid Systems and Applications, University of Louisiana, Lafayette, 2006. c 2006 Elsevier Ltd. All rights reserved.

MSC: 34A36; 34D20; 34A30 Keywords: Stability; Quasilinear systems; Piecewise constant argument of generalized type

1. Introduction and preliminaries Let Z, N and R be the sets of all integers, natural and real numbers, respectively. Denote by k · k the Euclidean norm in Rn , n ∈ N. Fix two real valued sequences θi , ζi , i ∈ Z, such that θi < θi+1 , θi ≤ ζi < θi+1 for all i ∈ Z, |θi | → ∞ as |i| → ∞. In the present paper we shall consider the following two equations: z 0 (t) = A0 (t)z(t) + A1 (t)z(γ (t)),

(1)

z 0 (t) = A0 (t)z(t) + A1 (t)z(γ (t)) + f (t, z(t), z(γ (t))),

(2)

and

where z ∈ Rn , t ∈ R, γ (t) = ζi , if t ∈ [θi , θi+1 ), i ∈ Z. The following assumptions will be needed throughout the paper:

∗ Corresponding address: Department of Mathematics, Middle East Technical University, 06531 Ankara, Turkey. Fax: +90 312 210 12 82.

E-mail address: [email protected]. c 2006 Elsevier Ltd. All rights reserved. 0362-546X/$ - see front matter doi:10.1016/j.na.2006.11.037

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(C1) A0 , A1 ∈ C(R) are n × n real valued matrices; (C2) f (t, x, y) ∈ C(R × Rn × Rn ) is an n × 1 real valued function; (C3) f (t, x, y) satisfies the condition k f (t, x1 , y1 ) − f (t, x2 , y2 )k ≤ L(kx1 − x2 k + ky1 − y2 k),

(3)

where L > 0 is a constant, and the condition f (t, 0, 0) = 0, (C4) (C5) (C6) (C7) (C8)

t ∈ R;

(4)

matrices A0 , A1 are uniformly bounded on R; infR kA1 (t)k > 0; ¯ i ∈ Z; there exists a number θ¯ > 0 such that θi+1 − θi ≤ θ, there exists a number θ > 0 such that θi+1 − θi ≥ θ, i ∈ Z; there exists a positive real number p such that i(t0 , t) lim =p t→∞ t − t0 uniformly with respect to t0 ∈ R, where i(t0 , t) denotes the number of points θi in the interval (t0 , t).

The theory of differential equations with piecewise constant argument (EPCA) of the type dx(t) = f (t, x(t), x([t])), dt

(5)

or dx(t) = f (t, x(t), x(2[(t + 1)/2])), (6) dt where [·] signifies the greatest integer function, was initiated in [5] and has been developed by many authors [1,4,6,10–17]. Applications of EPCA are discussed in [3,9,16]. They are hybrid equations, in that they combine the properties of both continuous systems and discrete equations. The novelty of our paper is that we find a class of the systems which in their properties are very close to ordinary differential equations. We believe that our proposals may stimulate new ideas advancing the theory and adding to the previous significant achievements in that direction. The novel idea of our paper is that systems (1) and (2) are EPCA of general type (EPCAG). Indeed if we take ζi = θi = i, i ∈ Z, then (1) takes the form of (5), and (1) and (2) take the form of (6) if θi = 2i − 1, ζi = 2i, i ∈ Z. The particular case of EPCAG, when ζi = θi , i ∈ Z, is considered in [2]. The existing method of investigation of EPCA, as proposed by its founders [5,16], is based on the reduction of EPCA to discrete equations. A new approach proposed in [2] is based on the construction of an equivalent integral equation. We consider the initial value problem in the general form, that is when t0 is an arbitrary real number, not necessarily one of the moments θi . One can easily see that Eqs. (1) and (2) have the form of functional differential equations: z 0 (t) = A0 (t)z(t) + A1 (t)z(ζi ),

(7)

z (t) = A0 (t)z(t) + A1 (t)z(ζi ) + f (t, z(t), z(ζi )),

(8)

0

respectively, if t ∈ [θi , θi+1 ), i ∈ Z. That is, these systems have the structure of a continuous dynamical system within the intervals [θi , θi+1 ), i ∈ Z. In our paper we assume that the solutions of the equation are continuous functions. But the deviating function γ (t) is discontinuous. Hence, in general, the right-hand sides of (1) and (2) have discontinuities at moments θi , i ∈ Z. Summarizing, we consider the solutions of the equations as functions, which are continuous and continuously differentiable within intervals [θi , θi+1 ), i ∈ Z. We use the following definition, which is a version of a definition from [13], modified for our general case. Definition 1.1. A continuous function z(t) is a solution of (1) and (2) on R if: (i) the derivative z 0 (t) exists at each point t ∈ R with the possible exception of the points θi , i ∈ Z, where the one-sided derivatives exist; (ii) the equation is satisfied for z(t) on each interval (θi , θi+1 ), i ∈ Z, and it holds for the right derivative of z(t) at the points θi , i ∈ Z.

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2. The fundamental matrix Let I be an n × n identity matrix. Denote by X (t, s), X (s, s) = I, s ∈ R, the fundamental matrix of solutions of the system x 0 (t) = A0 (t)x(t)

(9)

which is associated with systems (1) and (2). We introduce a matrix-function Mi (t), i ∈ Z, Z t Mi (t) = X (t, ζi ) + X (t, s)A1 (s)ds, ζi

useful in what follows. From now on we make the assumption: (C9) For every fixed i ∈ Z, det[Mi (t)] 6= 0, ∀t ∈ [θi , θi+1 ]. Theorem 2.1. Assume that condition (C1) is fulfilled. For every (t0 , z 0 ) ∈ R × Rn there exists a unique solution z(t) = z(t, t0 , z 0 ) of (1) in the sense of Definition 1.1 such that z(t0 ) = z 0 if and only if condition (C9) is valid. Proof. Let us first prove the sufficiency of (C9). Fix a (t0 , z 0 ) ∈ R × Rn . Without loss of generality assume that θi ≤ ζi < t0 ≤ θi+1 for a fixed i ∈ Z. We consider only the construction of the solution for decreasing t, since forward continuation can be investigated in a similar manner. We shall define a function ψ(t) : (−∞, t0 ] → Rn in a specific way, which guarantees that z(t) = ψ(t), t ∈ (−∞, t0 ]. Condition (C9) implies that the equation Z ζi vi = X (ζi , t0 )z(t0 ) + X (ζi , s)A1 (s)vi ds t0

can be uniquely solved with respect to vi . Indeed, since we have that   Z t0 I+ X (ζi , s)A1 (s)ds vi = X (ζi , t0 )z(t0 ), ζi

then multiplying both parts of the last expression by X (t0 , ζi ), we obtain vi = Mi−1 (t0 )z(t0 ). Denote by ψ(t) : [θi , t0 ] → Rn the unique solution of the equation z 0 (t) = A0 (t)z(t) + A1 (t)vi .

(10)

One can easily see that ψ(ζi ) = vi . Consider now the interval [θi−1 , θi ]. Again, by condition (C9), the equation Z ζi−1 vi−1 = X (ζi−1 , θi )ψ(θi ) + X (ζi−1 , s)A1 (s)vi−1 ds θi

is uniquely solvable with respect to vi−1 . Let ψ(t) be equal to the solution of the equation z 0 (t) = A0 (t)z(t) + A1 (t)vi−1 ,

(11)

on [θi−1 , θi ] with the initial data (θi , ψ(θi )). Obviously, the solution exists and is unique, and ψ(ζi−1 ) = vi−1 . Assume that we have defined the function ψ(t) on the interval [θ j , t0 ], j < i − 1. Then, the equation Z ζ j−1 v j−1 = X (ζ j−1 , θ j )ψ(θ j ) + X (ζ j−1 , s)A1 (s)v j−1 ds θj

is uniquely solvable with respect to v j−1 . We assume that ψ(t) is a solution of the equation z 0 (t) = A0 (t)z(t) + A1 (t)v j−1 ,

(12)

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on [θ j−1 , θ j ], with the initial data (θ j , ψ(θ j )). Consequently, the function ψ(t) could be continued to −∞ by induction. One can easily see that ψ(t) is the unique solution of (2) on (−∞, t0 ] by construction. Assume that condition (C9) is not true for some fixed i ∈ Z and ξ ∈ [θi , θi+1 ]. That is, det Mi (ξ ) = 0. Definitely, ξ 6= ζi . If z(t) = z(t, ξ, z(ξ )) is a solution, then z(ξ ) = Mi (ξ )z(ζi ), and z(ζi ) could not be defined uniquely. This fact proves the necessity of (C9). The theorem is proved.  The last theorem is of major importance for our paper. It arranges the correspondence between points (t0 , z 0 ) ∈ R × Rn and the solutions of (1) in the sense of Definition 1.1, and there exists no solution of the equation out of the correspondence. Using this assertion we can say that the definition of the IVP for EPCAG is similar to the problem for ordinary differential equations. In particular, the dimension of the space of all solutions is n. Hence, the investigation of problems considered in our paper does not need to be supported by results from the theory of functional differential equations [9,18–20], despite the fact that EPCAG are equations with deviated arguments. The following assertion useful in some particular cases is implied by the proof of the last theorem. Theorem 2.2. Assume that condition (C1) is fulfilled, and a number t0 ∈ R, θi ≤ t0 < θi+1 , is fixed. For every z 0 ∈ Rn there exists a unique solution z(t) = z(t, t0 , z 0 ) of (1) in the sense of Definition 1.1 such that z(t0 ) = z 0 if and only if det[Mi (t0 )] 6= 0 and det[M j (t)] 6= 0 for t = θ j , θ j+1 , j ∈ Z. System (1) is a differential equation with a delay argument. That is why it is reasonable to suppose that the initial “interval” must consist of more than one point. The following arguments show that in our case we need only one initial moment. Indeed, assume that (t0 , z 0 ) is fixed, and θi ≤ t0 < θi+1 for a fixed i ∈ Z. We suppose that t0 6= ζi . The solution satisfies, on the interval [θi , θi+1 ], the following functional differential equation: z 0 (t) = A0 (t)z + A1 (t)z(ζi ).

(13)

Formally we need the pair of initial points (t0 , z 0 ) and (ζi , z(ζi )) to proceed with the solution, but since z 0 = Mi (t0 )z(ζi ), where matrix Mi (t0 ) is nonsingular, we can say that the initial condition z(t0 ) = z 0 is sufficient to define the solution. Theorem 2.1 implies that the set of the solutions of (1) is an n-dimensional linear space. Hence, for a fixed t0 ∈ R there exists a fundamental matrix of solutions of (1), Z (t) = Z (t, t0 ), Z (t0 , t0 ) = I , such that dZ = A0 (t)Z (t) + A1 (t)Z (γ (t)). dt Let us construct Z (t). Without loss of generality assume that θi < t0 < ζi for a fixed i ∈ Z, and define the matrix only for increasing t, as the construction is similar for decreasing t. We have Z (ζi ) = Mi−1 (t0 )I = Mi−1 (t0 ). Hence, on the interval [t0 , θi+1 ], Z (t, t0 ) = Mi (t)Mi−1 (t0 ). −1 −1 Then Z (ζi+1 ) = Mi+1 (θi+1 )Z (θi+1 ) = Mi+1 (θi+1 )Mi (θi+1 )Mi−1 (t0 ), and then Z (t, t0 ) = Mi+1 (t)Z (ζi+1 ) = −1 Mi+1 (t)Mi+1 (θi+1 )Mi (θi+1 )Mi−1 (t0 ) if t ∈ [θi+1 , θi+2 ]. One can continue by induction to obtain " # i+1 Y −1 Z (t) = Ml (t) Mk (θk )Mk−1 (θk ) Mi−1 (t0 ), (14) k=l

if t ∈ [θl , θl+1 ], for arbitrary l > i. One can easily see that Z (t, s) = Z (t)Z −1 (s),

t ≥ s,

(15)

and a solution z(t), z(t0 ) = z 0 , (t0 , z 0 ) ∈ R × Rn , of (1) is equal to z(t) = Z (t, t0 )z 0 ,

t ∈ R.

(16)

3. Solutions of the quasilinear system Let us consider system (2). One can easily see that (C4)–(C7) imply the existence of positive numbers M, m and M¯ such that m ≤ kZ (t, s)k ≤ M, kX (t, s)k ≤ M¯ if t, s ∈ [θi , θi+1 ], i ∈ Z. From now on we make the assumption

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M.U. Akhmet / Nonlinear Analysis 68 (2008) 794–803

(C10) 2 M¯ L(1 + M)θ¯ < 1, ¯ + M)L θe ¯ M¯ L(1+M)θ¯ < 1, and the expression κ(L) = Then we can see that M(1 introduced. The following assumption is also needed

Me M L(1+M)θ ¯ ¯ M¯ L(1+M)θ¯ 1− M(1+M)L θe ¯

¯

can be

(C11) 2 M¯ L θ¯ κ(L)(1 + M) < m. The following lemma is the most important auxiliary result of the paper. Lemma 3.1. Assume that conditions (C1)–(C7), (C9)–(C11) are fulfilled, and fix i ∈ Z. Then for every (ξ, z 0 ) ∈ [θi , θi+1 ] × Rn there exists a unique solution z(t) = z(t, ξ, z 0 ) of (8) on [θi , θi+1 ]. Proof. Existence. Fix i ∈ Z. We assume without loss of generality that θi ≤ ζi < ξ ≤ θi+1 . Define a norm kz(t)k0 = max[θi ,θi+1 ] kz(t)k, and take z 0 (t) = Z (t, ξ )z 0 and a sequence   Z ζi z k+1 (t) = Z (t, ξ ) z 0 + X (ζi , s) f (s, z k (s), z k (ζi ))ds ξ

Z

t

+ ζi

X (t, s) f (s, z k (s), z k (ζi ))ds,

k ≥ 0.

The last expression implies that ¯ k+1 Mkz 0 k. kz k+1 (t) − z k (t)k0 ≤ [2 M¯ L(1 + M)θ] Thus, there exists a unique solution z(t) = z(t, ξ, z 0 ) of the equation   Z ζi z(t) = Z (t, ξ ) z 0 + X (ζi , s) f (s, z(s), z(ζi ))ds ξ

Z

t

+ ζi

X (t, s) f (s, z(s), z(ζi ))ds,

(17)

which is, also, solution of (8) on [θi , θi+1 ]. Existence is proved. j j Uniqueness. Denote by z j (t) = z(t, ξ, z 0 ), z j (ξ ) = z 0 , j = 1, 2, the solutions of (8), where θi ≤ ξ ≤ θi+1 . Without loss of generality, we assume that ξ ≤ ζi . It is sufficient to check that for every t ∈ [θi , θi+1 ], z 01 6= z 02 implies z 1 (t) 6= z 2 (t). We have that   Z ζi 2 1 z 1 (t) − z 2 (t) = Z (t, ξ ) (z 0 − z 0 ) + X (ζi , s)[ f (s, z 1 (s), z 1 (ζi )) − f (s, z 2 (s), z 2 (ζi ))]ds ξ

t

Z +

ζi

X (t, s)[ f (s, z 1 (s), z 1 (ζi )) − f (s, z 2 (s), z 2 (ζi ))]ds.

Hence, kz 1 (t) − z 2 (t)k ≤ Mkz 02 − z 01 k + M¯ L θ¯ (1 + M)kz 1 (ζi ) − z 2 (ζi )k Z t + M¯ L(1 + M) kz 1 (s) − z 2 (s)kds . ξ

The Gronwall–Bellman Lemma yields that kz 1 (t) − z 2 (t)k ≤ [Mkz 02 − z 01 k + M¯ L θ¯ (1 + M)kz 1 (ζi ) − z 2 (ζi )k]e M L(1+M)θ . ¯

¯

Particularly, ¯ ¯ kz 1 (ζi ) − z 2 (ζi )k ≤ [Mkz 02 − z 01 k + M¯ L θ¯ (1 + M)kz 1 (ζi ) − z 2 (ζi )k]e M L(1+M)θ .

Then, kz 1 (ζi ) − z 2 (ζi )k ≤ κ(L)kz 02 − z 01 k.

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M.U. Akhmet / Nonlinear Analysis 68 (2008) 794–803

Hence, kz 1 (t) − z 2 (t)k ≤ κ(L)kz 02 − z 01 k.

(18)

Assume to the contrary that there exists t ∈ [θi , θi+1 ] such that z 1 (t) = z 2 (t). Then Z ζi Z (t, ξ )(z 01 − z 02 ) = Z (t, ξ ) X (ζi , s)[ f (s, z 1 (s), z 1 (ζi )) − f (s, z 2 (s), z 2 (ζi ))]ds ξ

Z

t

+ ζi

X (t, s)[ f (s, z 1 (s), z 1 (ζi )) − f (s, z 2 (s), z 2 (ζi ))]ds.

(19)

We have that kZ (t, ξ )(z 02 − z 01 )k ≥ mkz 02 − z 01 k.

(20)

Moreover, (18) implies that

Z ζi

Z (t, ξ ) X (ζi , s)[ f (s, z 1 (s), z 1 (ζi )) − f (s, z 2 (s), z 2 (ζi ))]ds

ξ

Z t

+ X (t, s)[ f (s, z 1 (s), z 1 (ζi )) − f (s, z 2 (s), z 2 (ζi ))]ds

ζi ≤ 2 M¯ L θ¯ κ(L)(1 + M)kz 02 − z 01 k.

(21)

Finally, one can see that (C11), (20) and (21) contradict (19). The lemma is proved.



Remark 3.1. Inequality (18) implies the continuous dependence of solutions of (2) on the initial value. Theorem 3.1. Assume that conditions (C1)–(C7), (C9)–(C11) are fulfilled. Then for every (t0 , z 0 ) ∈ R × Rn there exists a unique solution z(t) = z(t, t0 , z 0 ) of (2) in the sense of Definition 1.1 such that z(t0 ) = z 0 . Proof. We prove the theorem only for decreasing t, but one can easily see that the proof is similar for increasing t. Let us assume without loss of generality that θi ≤ ζi < t0 ≤ θi+1 for some i ∈ Z. Using Lemma 3.1 for ξ = t0 one can check that solution z(t) = z(t, t0 , z 0 ) of (1) exists on [ζi , t0 ] as a solution of Eq. (8) and is unique. Then conditions (C1)–(C3) imply that z(t) can be continued to t = θi , as it is a solution of the system of ordinary differential equations z 0 = A0 (t)z(t) + A1 (t)z(ζi ) + f (t, z(t), z(ζi )) on [θi , θi+1 ). Next, using the lemma again we can continue z(t) from t = θi to t = ζi−1 , and then to t = θi−1 . Since θi → −∞ as i → −∞, the induction completes the proof.  Lemma 3.2. Assume that conditions (C1)–(C7), (C9)–(C11) are fulfilled. Then the solution z(t) = z(t, t0 , z 0 ), (t0 , z 0 ) ∈ R × Rn , of (2) is a solution on R of the following integral equation   Z ζi z(t) = Z (t, t0 ) z 0 + X (t0 , s) f (s, z(s), z(γ (s)))ds t0

+

k= j−1 X

Z (t, θk+1 )

k=i

Z

t

+ ζj

Z

ζk+1 ζk

X (θk+1 , s) f (s, z(s), z(γ (s)))ds

X (t, s) f (s, z(s), z(γ (s)))ds,

(22)

where θi ≤ t0 ≤ θi+1 and θ j ≤ t ≤ θ j+1 , i < j. Proof. We shall prove the lemma only for θi < t0 < θi+1 < t ≤ θi+2 . All other cases can be proved analogously. Consider at first t ∈ [θi , θi+1 ]. The solution uniquely satisfies the following equation Z t Z t z(t) = X (t, ζi )z(ζi ) + X (t, s)A1 (s)z(ζi )ds + X (t, s) f (s, z(s), z(γ (s)))ds. ζi

ζi

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M.U. Akhmet / Nonlinear Analysis 68 (2008) 794–803

Using the last expression one can easily see that   Z t0 −1 z(ζi ) = Mi (t0 ) z 0 − X (t0 , s) f (s, z(s), z(γ (s)))ds . ζi

Hence,  Z z(t) = Mi (t)Mi−1 (t0 ) z 0 −

t0

 Z X (t0 , s) f (s, z(s), z(γ (s)))ds +

ζi

t ζi

X (t, s) f (s, z(s), z(γ (s)))ds,

and z(θi+1 ) =

Mi (θi+1 )Mi−1 (t0 ) θi+1

Z +

ζi



t0

Z z0 −

X (t0 , s) f (s, z(s), z(γ (s)))ds

ζi



X (θi+1 , s) f (s, z(s), z(γ (s)))ds.

Then for t ∈ [θi+1 , θi+2 ], z(t) = X (t, ζi+1 )z(ζi+1 ) +

t

Z

X (t, s)A1 (s)z(γ (s))ds +

ζi+1

t

Z

ζi+1

X (t, s) f (s, z(s), z(γ (s)))ds,

and z(ζi+1 ) =

−1 Mi+1 (θi+1 )



z(θi+1 ) −

θi+1

Z

ζi+1



X (θi+1 , s) f (s, z(s), z(γ (s)))ds .

Hence, z(t) =

−1 Mi+1 (t)Mi+1 (θi+1 )

Z

t

+ ζi+1

 × Z

z(θi+1 ) −

θi+1 ζi t

+ ζi+1

θi+1

Z

ζi+1

X (θi+1 , s) f (s, z(s), z(γ (s)))ds



Z z0 −

t0 ζi

X (t0 , s) f (s, z(s), z(γ (s)))ds

X (θi+1 , s) f (s, z(s), z(γ (s)))ds −

θi+1

Z

ζi+1



X (θi+1 , s) f (s, z(s), z(γ (s)))ds



−1 X (t, s) f (s, z(s), z(γ (s)))ds = Mi+1 (t)Mi+1 (θi+1 )Mi−1 (θi+1 )Mi−1 (t0 )z 0

−1 + Mi+1 (t)Mi+1 (θi+1 )Mi−1 (θi+1 )Mi−1 (t0 )

+



−1 X (t, s) f (s, z(s), z(γ (s)))ds = Mi+1 (t)Mi+1 (θi+1 )

Mi−1 (θi+1 )Mi−1 (t0 )

+ Z



−1 Mi+1 (t)Mi+1 (θi+1 )

= Z (t, t0 )z 0 + Z (t, t0 )

Z

Z

ζi+1

ζi ζi

Z

ζi

X (t, s) f (s, z(s), z(γ (s)))ds

t0

X (θi+1 , s) f (s, z(s), z(γ (s)))ds +

Z

t ζi+1

X (t, s) f (s, z(s), z(γ (s)))ds

X (t0 , s) f (s, z(s), z(γ (s)))ds

t0

+ Z (t, θi+1 ) The lemma is proved.

Z

ζi+1 ζi

X (θi+1 , s) f (s, z(s), z(γ (s)))ds +

Z

t ζi+1

X (t, s) f (s, z(s), z(γ (s)))ds.



4. Stability In this section we assume that conditions (C1)–(C10) are fulfilled, and, hence, all solutions of the considered systems are defined on the whole real axis, and their integral curves do not intersect each other. For the stability

M.U. Akhmet / Nonlinear Analysis 68 (2008) 794–803

801

investigation we consider the systems on R+ = [0, ∞). Definitions of Lyapunov stability for the solutions of both discussed systems can be given in the same way as for ordinary differential equations. Let us formulate only one of them. Definition 4.1. The zero solution of (1) ((2)) is stable if for an arbitrary positive  there exists a positive number δ = δ(t0 , ) such that kx(t, t0 , x0 )k < , t ≥ t0 ≥ 0, if kx0 k < δ. Let Z (t) be a fundamental matrix of (1). We can prove the following assertions, using representations (15) and (16) in exactly the same way as theorems for ordinary differential equations [7,8]. Theorem 4.1. The zero solution of (1) is stable if and only if Z (t) is bounded on t ≥ 0. Theorem 4.2. The zero solution of (1) is asymptotically stable if and only if Z (t) → 0, as t → ∞. Theorem 4.3. The zero solution of (1) is uniformly stable if and only if there exists a number M > 0 such that kZ (t)Z −1 (s)k ≤ M, t ≥ s ≥ 0. Theorem 4.4. The zero solution of (1) is uniformly asymptotically stable if and only if there exist two positive numbers N and ω such that kZ (t)Z −1 (s)k ≤ N e−ω(t−s) , t ≥ s ≥ 0. On the basis of the last theorems we can formulate the following theorems which provide sufficient conditions for the stability of linear systems. Theorem 4.5. Assume that conditions (C1)–(C6) are fulfilled and kMk−1 (θk )Mk−1 (θk )k ≤ 1, k ∈ N. Then the zero solution of (1) is stable. Theorem 4.6. Assume that conditions (C1)–(C6) are fulfilled and there exists a nonnegative number κ < 1 such that kMk−1 (θk )Mk−1 (θk )k ≤ κ, k ∈ N; then the zero solution of (1) is asymptotically stable. Theorem 4.7. Assume that conditions (C1)–(C7) are fulfilled. The zero solution of (1) is uniformly stable if kMk−1 (θk )Mk−1 (θk )k ≤ 1, k ∈ N. Theorem 4.8. Assume that conditions (C1)–(C5), (C8) are fulfilled. The zero solution of (1) is uniformly stable if kMk−1 (θk )Mk−1 (θk )k ≤ 1, k ∈ N. Theorem 4.9. Assume that conditions (C1)–(C7) are fulfilled. The zero solution of (1) is uniformly asymptotically stable if there exists a nonnegative number κ < 1 such that kMk−1 (θk )Mk−1 (θk )k ≤ κ, k ∈ N. Theorem 4.10. Assume that conditions (C1)–(C5), (C8) are fulfilled. The zero solution of (1) is uniformly asymptotically stable if there exists a nonnegative number κ < 1 such that kMk−1 (θk )Mk−1 (θk )k ≤ κ, k ∈ N. Interesting results comparable to the last theorems can be found in [16]. Example 4.1. Consider the following EPCAG: x 0 (t) = αx(t) + βx(γ (t)),

(23)

where α, β are fixed real constants, and the identification function γ (t) is defined by sequences θi = κi, ζi = θi + κ1 , i ∈ Z, where κ > 0, κ > κ1 > 0, are fixed numbers. We will find conditions on the coefficients and the sequences for providing uniformly asymptotic stability for the zero solution. One can evaluate that Z t β Mi (t) = eα(t−ζi ) + eα(t−s) βds = eα(t−ζi ) + [eα(t−ζi ) − 1]. α ζi Then Mi (θi ) = e−ακ1 +

β −ακ1 [e − 1]. α

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M.U. Akhmet / Nonlinear Analysis 68 (2008) 794–803

Moreover, if we define κ2 = κ − κ1 , then Mi−1 (θi ) = eακ2 +

β ακ2 [e − 1], α

and Mi−1 (θi )Mi−1 (θi ) =

eακ2 + βα [eακ2 − 1]

e−ακ1 + βα [e−ακ1 − 1]

.

On the basis of Theorem 4.10 either of the inequalities   β β ακ2 −ακ1 >2 , (a) −β > α > 0, [e + e ] 1+ α α and   β β [eακ2 + e−ακ1 ] 1 + >2 α α

(b) α < 0, −β > α,

is sufficient for the zero solution to be uniformly asymptotically stable. It is of particular interest to consider the case when κ1 = 0, and, hence, κ2 = κ in the equation. Then conditions (a) and (b) are transformed to (a0 ) −β > α > 0,

eακ <

β −α , β +α

and eακ <

(b0 ) α < 0, −β > α,

β −α . β +α

Let us obtain an evaluation of the fundamental solution z(t, t0 ) of the equation by using (14). Define eακ2 + β [eακ2 − 1] α ξ = . e−ακ1 + β [e−ακ1 − 1] α

Assume that condition (a) is valid; then ξ < 1 and min

Mi (t) = e−ακ1 +

max

Mi (t) = eακ2 +

θi ≤t≤θi+1 θi ≤t≤θi+1

Hence, if we define ( N = max e

ακ2

β −ακ1 [e − 1] α

β ακ2 [e − 1]. α

 −1 ) β ακ2 β −ακ1 −ακ1 + [e − 1], e + [e − 1] e− ln ξ , α α

then |z(t, t0 )| ≤ N e

ln ξ κ

(t−t0 )

,

t ≥ t0 .

M Denote γ (L) = 1−2 M¯ L(1+M) . θ¯ We may make the following assumption ¯ (C12) 2K e2ωθ M¯ L max(1, γ (L)) < ω.

Theorem 4.11. Assume that conditions (C1)–(C7), (C9)–(C12) are fulfilled and the zero solution of (1) is uniformly asymptotically stable. Then the zero solution of (2) is uniformly asymptotically stable.

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M.U. Akhmet / Nonlinear Analysis 68 (2008) 794–803

Proof. If z(t) = z(t, t0 , z 0 ) is a solution of (2), then by (17), (22) and assuming, without loss of generality, that t0 < ζi ≤ · · · ζ j < t, we have that Z ζi 2 M¯ L max(1, γ (L))kz(s)kds kz(t)k ≤ K e−ω(t−t0 ) kz 0 k + K e−ω(t−t0 ) t0

+

k= j X

ζk+1

K e−ω(t−θk+1 )

Z

Z

t

ζk

k=i

≤ K e−ω(t−t0 ) kz 0 k +

2 M¯ L max(1, γ (L))kz(s)kds +

Z

t ζj

2 M¯ L max(1, γ (L))kz(s)kds

2K e−ω(t−s−2θ) M¯ L max(1, γ (L))kz(s)kds. ¯

t0

If we denote u(t) ≡ kz(t)kK eωt , then the last inequality implies that Z t ¯ 2K e2ωθ M¯ L max(1, γ (L))u(s)ds. u(t) ≤ K u(t0 ) + t0

Now, by virtue of Gronwell–Bellmann Lemma, we obtain 2ωθ¯ M ¯L

kz(t)k ≤ K e(−ω+2K e

max(1,γ (L)))(t−t0 )

kz 0 k.

The last inequality, in conjunction with (C11), proves that the zero solution is uniformly asymptotically stable. The theorem is proved.  References [1] A.R. Aftabizadeh, J. Wiener, J.- M. Xu, Oscillatory and periodic solutions of delay differential equations with piecewise constant argument, Proc. Amer. Math. Soc. 99 (1987) 673–679. [2] M.U. Akhmet, Integral manifolds of differential equations with piecewise constant argument of generalized type, Nonlinear Anal. TMA 66 (2007) 367–383. ¨ [3] M.U. Akhmet, H. Oktem, S.W. Pickl, G.W. Weber, An anticipatory extension of malthusian model, in: D.M. Dubois (Ed.), CASYS 2005 — Seventh International Conference, in: AIP Conference Proceedings, vol. 839, The American Institute of Physics, 2006, pp. 260–264. [4] A. Alonso, J. Hong, R. Obaya, Almost periodic type solutions of differential equations with piecewise constant argument via almost periodic type sequences, Appl. Math. Lett. 13 (2000) 131–137. [5] K.L. Cooke, J. Wiener, Retarded differential equations with piecewise constant delays, J. Math. Anal. Appl. 99 (1984) 265–297. [6] K.L. Cooke, J. Wiener, An equation alternately of retarded and advanced type, Proc. Amer. Math. Soc. 99 (1987) 726–732. [7] C. Corduneanu, Principles of Differential and Integral Equations, Chelsea, New York, 1977. [8] P. Hartman, Ordinary Differential Equations, Wiley, New York, 1964. [9] K. Gopalsamy, Stability and Oscillations in Delay Differential Equations, Kluwer Academic Publishers Group, Dordrecht, 1992. [10] K. Gopalsamy, M.R.S. Kulenovi´c, G. Ladas, On a logistic equation with piecewise constant arguments, Differential Integral Equations 4 (1991) 215–223. [11] I. Gy¨ori, G. Ladas, Oscillation Theory of Delay Differential Equations. With Applications, Oxford University Press, New York, 1991. [12] Y. Muroya, Persistence, contractivity and global stability in logistic equations with piecewise constant delays, J. Math. Anal. Appl. 270 (2002) 602–635. [13] G. Papaschinopoulos, Linearization near the integral manifold for a system of differential equations with piecewise constant argument, J. Math. Anal. Appl. 215 (1997) 317–333. [14] G. Seifert, Almost periodic solutions of certain differential equations with piecewise constant delays and almost periodic time dependence, J. Differential Equations 164 (2000) 451–458. [15] J. Wiener, Differential equations with piecewise constant delays, in: V. Lakshmikantham (Ed.), Trends in the Theory and Practice of Nonlinear Differential Equations, Marcel Dekker, New York, 1983, pp. 547–580. [16] J. Wiener, Generalized Solutions of Functional Differential Equations, World Scientific, Singapore, 1993. [17] J. Wiener, V. Lakshmikantham, A damped oscillator with piecewise constant time delay, Nonlinear Stud. 7 (2000) 78–84. [18] T.A. Burton, Stability and Periodic Solutions of Ordinary and Functional Differential Equations, Academic Press, Inc., New York, 1985. [19] R.D. Driver, Ordinary and Delay Differential Equations, Springer-Verlag, New York, 1977. [20] J. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York, 1977.