Almost periodic solutions for an impulsive delay model of price fluctuations in commodity markets

Nonlinear Analysis: Real World Applications 12 (2011) 3170–3176

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Nonlinear Analysis: Real World Applications journal homepage: www.elsevier.com/locate/nonrwa

Almost periodic solutions for an impulsive delay model of price fluctuations in commodity markets G.Tr. Stamov a , J.O. Alzabut b,∗ , P. Atanasov c , A.G. Stamov d a

Department of Mathematics, Technical University—Sofia, 8800 Sliven, Bulgaria

b

Department of Mathematics and Physical Sciences, Prince Sultan University, P.O. Box 66388 Riyadh 11586, Saudi Arabia

c

Research Center on the Company, the Organizations and the Inheritance, Universite de Limoges, 87031 Limoges, France

d

Faculty of Economics and Business, University of Amsterdam, 1018 WB Amsterdam, Netherlands

article

info

Article history: Received 19 February 2011 Accepted 12 May 2011

abstract In the present paper, we shall consider the following impulsive delay system for modeling the price fluctuations in single-commodity markets: p˙ (t ) = F (p(t ), p(t − h))p(t ), t ̸= τk , p(t ) = ϕ0 (t ), t ∈ [t0 − h, t0 ], ∆p(t ) = Ik (p(t )), t = τk , k ∈ Z.

 Keywords: Almost periodic solution Lyapunov’s functions Razumikhin techniques Price fluctuations in single-commodity markets

Sufficient conditions are established for the existence of almost periodic solutions for this system. Piecewise continuous functions of the Lyapunov type as well as the Razumikhin technique have been utilized to prove our main results. © 2011 Elsevier Ltd. All rights reserved.

1. Introduction Impulsive delay differential equations have conspicuously occupied a great part of researchers’ interests for well over the last three decades. Indeed, it has been recently recognized that these equations do only generalize the corresponding theory of impulsive differential equations but also provide better mathematical descriptions for many real life applications. The publications [1–15] are devoted to the theory and applications of impulsive differential equations with or without delay. Specifically, the dynamics of the economy is one of most actively developing research areas that can be represented by using impulsive delay differential equations of a certain type [16–18]. In particular, it has been noticed that these equations can provide adequate visualization for modeling the process of price fluctuations in single-commodity markets. Early authors often attributed these fluctuations to random factors such as weather change for agricultural commodities [19–21]. Other authors, however, speculated that fluctuations might be caused by dynamical characteristics of unstable economic systems [22–25]. Apart from some diversities in the authors’ beliefs regarding this discussion, their work and that of others has played a fundamental role in the development of theory of nonlinear dynamics [26–31]. Searching the literature, one can realize that there has been intensive work regarding the study of periodic impulsive dynamical systems with or without delay; see for instance the Refs. [32–37] in which the existence of periodic solutions has been the main concern of the authors. Although it is known to be a natural generalization to the periodicity, the notion of almost periodicity has rarely been considered. The reader can easily figure out that a few results exist in this direction [38–42].

∗

Corresponding author. E-mail address: [email protected] (J.O. Alzabut).

1468-1218/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.nonrwa.2011.05.016

G.Tr. Stamov et al. / Nonlinear Analysis: Real World Applications 12 (2011) 3170–3176

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The purpose of this paper is to study the almost periodic behavior of solutions for the impulsive delay model for price fluctuations in commodity markets. Piecewise continuous functions of the Lyapunov type as well as the Razumikhin technique have been utilized to prove the main results. 2. Problem statement; essential notation, definitions and lemmas For a single market commodity, there are three effective variables: the quantity demanded D, and the quantity supplied S and its price p. It has been realized that there exist definite relationships among these three variables. These relationships, which are called the demand curve and the supply curve, are occasionally modeled by a demand function D = D(p) and a supply function S = S (p); both are dependent on the price variable p. If the rate of price changes with respect to time is assumed to be proportional to D and S, then the formulation of the following system proves helpful: 1 dp p dt

= f (D, S ).

(1)

In particular, one can easily deduce that for linear demand and supply functions, system (1) has a dynamically stable solution of exponential type. Indeed, let D = a − bp and S = −c + dp be given where a, b, c , d ∈ R+ = [0, ∞). If f has the form f = α(D − S ), α > 0 where D − S is the excess demand, then system (1) becomes dp dt

= α(a + c − p(b + d)) = pα



a+c p



− (b + d) .

(2)

The complementary and particular solutions of (2) can be easily attained. In [43], Belair and Mackey considered Eq. (1) in order to study the dynamics of price, production and consumption for a particular commodity with price dependent delays. In [44], Mackey developed a price adjustment model for a singlecommodity market with state dependent production and storage delays. Conditions for the equilibrium price to be stable are derived in terms of a variety of economic parameters. A particular case of the general model was considered by Farahani and Grove in [45]. Indeed, they proposed a system of the form p′ (t )

a

cpm (t − h)

, t ≥ 0, (3) (t ) d + pm (t − h) where a, b, c , d, m, h ∈ R+ and n ≥ 1. Under the initial condition p(t ) = ϕ(t ), t ∈ [−h, 0], the authors established p(t )

=

b+

pn

−

necessary and sufficient conditions for the existence of positive solutions of system (3). They proved, moreover, that these solutions oscillate to the unique positive equilibrium solution of (3). For the case where t − h = g (t ), system (3) was studied in [46]. Indeed, the author proved the existence of a unique positive bounded solution of (3). A more general system of the form



p˙ (t ) = F p(t ), p(t − h) p(t ), p(t ) = ϕ(t ), t ∈ [−h, 0],





t ∈ R, h ∈ R+

(4)

was investigated by Rus and Iancu in [47]. The authors proved the existence and uniqueness of the equilibrium solution of the system and established some relations between this solution and the coincidence points. If at certain moments in time the price is subject to short-term perturbations, then it is natural to expect the existence of an ‘‘irregular’’ solution for system (4). Indeed, the solution must have some jumps and these jumps will follow a specific pattern. Recently, it has been realized that impulsive delay differential equations provide the most adequate description for these phenomena. These equations have been widely used in many fields such as physics, chemistry, biology, population dynamics and industrial robotics as a control. The abrupt changes in the prices which may be caused by the environment, the competitive market or the government, can affect the transient behavior of the market. The result of this paper suggests that such a policy would be highly destabilizing and would either destabilize a previously stable market situation or exacerbate the instability of a market through an increase in the amplitude and period of oscillations in commodity prices. Let R be the one-dimensional Euclidean space with the   norm |·|, Bν = {x ∈ R : |x| ≤ ν}, ν > 0 and suppose that Λ ⊂ Bν where Λ ̸= ø. Let B = {τk } : τk ∈ R, τk < τk+1 , k ∈ Z be the set of all sequences unbounded and strictly increasing with (1)

(2)

distance ρ({τk }, {τk }) where Z = {. . . , −3, −2, −1, 0, +1, +2, +3, . . .}. Define the spaces PC [R, R] = ϕ : R → R, ϕ is a piecewise continuous function with points of   discontinuity at the first kind τk , {τk } ∈ B at which ϕ(τk − 0) and ϕ(τk + 0) exist and ϕ(τk − 0) = ϕ(τk ) and PC 1 [R, R] = ϕ : R → R, ϕ is a function that is continuously differentiable everywhere



except at the points τk , {τk } ∈ B at which ϕ(τ ˙ k − 0) and ϕ(τ ˙ k + 0) exist, and ϕ(τ ˙ k − 0) = ϕ(τ ˙ k) . Suppose that ϕ0 ∈ PC [R, Λ] and ‖ϕ0 ‖ = supt ∈R |ϕ0 (t )|. We shall consider the following impulsive delay model for price fluctuations in commodity markets:



   p˙ (t ) = F p(t ), p(t − h) p(t ), t ̸= τk , p(t ) = ϕ (t ), t ∈ [t − h, t ], 1p(t ) = 0I (p(t )), t 0= τ , k0 ∈ Z, k k where:

(5)

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(i) t0 ∈ R, F : Λ × Λ → R, F (0, 0) = 0. (ii) 1p(t ) = p(t + 0) − p(t − 0), Ik ∈ C [Λ, R], Ik (0) = 0 and p + Ik (p) are invertible in Bν . The functions Ik characterize the magnitude of the impulse effect at the moments τk , and p(τk − 0) and p(τk + 0) are respectively the price levels before and after the impulse moments {τk } where {τk } ∈ B. System (5) was first considered by Emmenegger and Stamova in [48]. They obtained sufficient conditions for the stability, uniform stability and asymptotic stability of the solutions. One of the most important behaviors of solutions which has been a main object of investigation among authors is the periodic behavior of the solution. To consider periodic environmental and commodity markets factors, it is natural to study system (5) subject to periodic coefficients. The assumption of periodicity of the parameters in the system is a way of incorporating the environment and market periodicity (e.g. seasonal effects of commodity condition, market supplies, etc.). On the other hand, upon considering long-term dynamical behavior, it has been found that the periodic parameters often turn out to experience certain perturbations that may lead to a change in character. Thus, the investigation of almost periodic behavior is considered to be more in accordance with reality. A very basic and important problem associated with the study of price fluctuation in an almost periodic environment is the existence and asymptotic stability of almost periodic solutions. Such a question also arises in many other situations. In this paper, we shall study the almost periodic behavior for solutions of system (5). Piecewise continuous functions of the Lyapunov type as well as the Razumikhin technique have been utilized to prove our main results. Let p(t ) = p(t ; t0 , ϕ0 ) be the solution of system (5). These solutions are piecewise continuous functions with points of discontinuity at the first kind at t = τk , k ∈ Z, at which they are continuous from the left, that is, at the moments of impulse effect τk the following relations are valid: p(τk − 0) = p(τk ) and p(τk + 0) = p(τk ) + Ik (p(τk )),

k ∈ Z.

If for some positive integer j we have τk < τj + h < τk+1 , k ∈ Z, then in the interval [τj + h, τk+1 ], the solution of system (5) coincides with the solution of the system



y˙ (t ) = F y(t ), p(t − h + 0) y(t ), y(τj + h) = p(τj + h).





If τj + h = τk for j ∈ Z, k ∈ Z, then in the interval [τj + h, τk+1 ] the solution of system (5) coincides with the solution of the system



y˙ (t ) = F y(t ), p(t − h + 0) y(t ), y(τj + h) = p(τj + h) + Ik (p(τj + h)).





If the points p(τk ) + Ik (p(τk )) ̸∈ Λ, then the solution of system (5) is not defined. Since the solutions of (5) are piecewise functions we adopt the following definitions for almost periodicity. Before stating these definitions, here is some essential notation. For T , P ∈ B, let s(T ∪ P ) : B → B be a map such that the set s(T ∪ P ) forms a strictly increasing sequence. For D ⊂ R, suppose that θε (D) = {t + ε, t ∈ D} and Fε (D) = ∩{θε (D)} for ε > 0. By φ = (ϕ(t ), T ), we shall denote an element from the space PC × B. For every sequence of real numbers {αn }, n = 1, 2, . . . , θαn φ mean the sets {ϕ(t + αn ), T − αn } ⊂ PC × B where T − αn = {τk − αn , k ∈ Z, n = 1, 2, . . .}. Definition 2.1 ([4]). The set of sequences {τkl }, τkl = τk+l − τk , k ∈ Z, l ∈ Z, is said to be uniformly almost periodic if for any ε > 0 there exists a relatively dense set in R of ε -almost periods common to all of the sequences {τkl }. Lemma 2.2 ([4]). The set of sequences {τkl } is uniformly almost periodic if and only if from each infinite sequence of shifts {τk − αn }, k ∈ Z, n = 1, 2, . . . , αn ∈ R, wecanchooseasubsequenceconvergentin B. Definition 2.3. The sequence {φn }, φn = (ϕn (t ), Tn ) ∈ PC × B, is convergent to φ where φ = (ϕ(t ), T ) and (ϕ(t ), T ) ∈ PC × B if and only if for any ε > 0 there exists n0 > 0 such that for n ≥ n0 it follows that

ρ(T , Tn ) < ε,

‖ϕn (t ) − ϕ(t )‖ < ε

hold uniformly for t ∈ R \ Fε (s(Tn ∪ T )). Definition 2.4. The function ϕ ∈ PC [R, R] is said to be an almost periodic piecewise continuous function with points  ′ of discontinuity of the first kind τk , {τk } ∈ B, if for every sequence of real numbers αm it follows that there exists a ′ subsequence {αn } , αn = αm , such that θαn φ is a compact in PC × B. n Introduce the following assumptions: H1. H2. H3. H4.

The function F : Λ × Λ → R is uniformly continuous. The sequence of functions {Ik (p)}, k ∈ Z, is almost periodic uniformly with respect to p ∈ Λ, Λ ⊂ R. The function ϕ0 ∈ PC[R,R] is almost periodic. The set of sequences τkl , τkl = τk+l − τk , k ∈ Z, l ∈ Z, {τk } ∈ B, is uniformly almost periodic.

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′ Let the assumptions H1–H4 hold and let αm be an arbitrary sequence of real numbers. Then there exists a subsequence ′ {αn } , αn = αmn , such that the sequence ϕ0 (t + αn ) is convergent uniformly to the function ϕ0α (t ) and the set of sequences





  {τk − αn }, k ∈ Z, is convergent to the sequence τkα uniformly with respect to k ∈ Z as n → ∞. By kni , we shall denote the sequence of integer numbers such that the subsequence {τk+ni } is convergent to τkα uniformly with respect to k as i → ∞. In view of H2, it follows that there exists a subsequence of the sequence {kni }, such that the sequence {Ik+kni (p)} is convergent uniformly to the limit denoted by Ikα (p). ′ By the foregoing arguments and for an arbitrary sequence {αm }, system (5) ‘‘moves’’ to the system    α α p˙ (t ) = F p(t ), p(t − h) p(t ), t ̸= τk , (6) p(t ) = ϕ α (t ), t ∈ [t − h, t ], 1p(t ) = 0I α (p(τ α )), 0t = τ α ,0 k ∈ Z. k k k Throughout the remainder of the paper, we shall denote the set of all systems of form (6) by mod(F , ϕ0 , Ik , τk ). The following definition is provided as stated in [5]. Definition 2.5 ([5]). The zero solution of system (6) is said to be: (a) Uniformly stable if (∀ε > 0) (∃δ > 0) (∀ t0 ∈ R) (∀ϕ0 ∈ PC [R, R] ∩ Bδ ) (∀ t > t0 ) : |p(t ; t0 , ϕ0 )| < ε . (b) Uniformly attractive if (∃ λ > 0) (∀ε > 0) (∃T > 0) (∀t0 ∈ R) (∀ϕ0 ∈ PC [R, R] ∩ Bλ ) (∀t > t0 + T ) : |x(t ; t0 , ϕ0 )| < ε . (c) Uniformly asymptotically stable if it is uniformly stable and uniformly attractive. We shall consider, together with system (5), the following system: u˙ = g (t , u), t ≥ t0 , t ̸= τk , 1u(τk ) = γk (u(τk )), τk ≥ t0 , k ∈ Z, u(t0 + 0) = u0 , t0 ∈ R,



(7)

where g ∈ C [R × R+ , R+ ], γk ∈ C [R+ , R+ ] and u + γk (u) are invertible in Bν , {τk} ∈ B.  Consider the sets Gk = (t , x, y) ∈ R × R × R : τk−1 < t < τk , k ∈ Z , G = ∪k∈Z Gk , Q = a ∈ C [R+ , R+ ]: a is strictly increasing in R+ and a(0) = 0 , V0 =





V ∈ C [G, R+ ], and there exist the limits V (τk − 0, p0 , q0 ) and

V (τk + 0, p0 , q0 ), (p0 , q0 ) ∈ Bν × Bν where V is locally Lipshitz continuous with a constant L ∈ Bν × Bν and W0 =  W ∈ C [R × R, R+ ], W (t , 0) = 0, t ̸= τkα , and there exist the limits W (τkα − 0, p0 ) and V (τkα + 0, p0 ), p0 ∈ Bν where W





is locally Lipshitz continuous along p . Suppose that V ∈ V0 , t > t0 , t ̸= τk , p ∈ PC [R, R], q ∈ PC [R, R], and W ∈ W0 , t > t0 , t ̸= τkα . Introduce the following functions:      1  D+ V (t , p(t ), q(t )) = lim sup V t + δ, p(t ) + δ F p(t ), p(t − h) p(t ), q(t ) + δ F q(t ), q(t − h) q(t ) δ→0

δ

 − V t , p(t ), q(t ) 

and D+ W (t , p(t )) = lim sup δ→0

1

δ







W t + δ, p(t ) + δ F α t , p(t ), p(t − h) p(t ) − W t , p(t )







.

We shall also need the following class of functions:





Ł = (p, q) : p, q ∈ PC [R, Λ], V s, p(s), q(s) ≤ V t , p(t ), q(t )







for t − h ≤ s ≤ t , t ≥ t0 and V ∈ V0 . In the proof of the main results we shall use the following lemmas. Lemma 2.6. Let the following conditions hold: 1. 2. 3. 4. 5.

The function g : (t0 , ∞)× R+ → R+ is continuous in each of the sets (τk−1 , τk ]× R+ , k ∈ Z and g (t , 0) = 0 for t ∈ (t0 , ∞). γk ∈ C [R+ , R+ ], γk (0) = 0 and ψk (u) = u + γk (u), k ∈ Z are nondecreasing with respect to u. The maximal solution r (t ; t0 , u0 ) of system (7) is defined in the interval (t0 , ∞). The solutions p(t ) = p(t ; t0 , ϕ0 ) and q(t ) = q(t ; t0 , ϕ0 ) of system (5) are such that p(t ), q(t ) ∈ PC [R, Λ] ∩ PC 1 [R, Λ]. The function V ∈ V0 is such that V (t0 + 0, ϕ0 , ϕ0 ) ≤ u0 and D+ V (t , p(t ), q(t )) ≤ g (t , V (t , p(t ), q(t ))),

t ̸= τk

and V (t + 0, p(t ) + Ik (p(t )), q(t ) + Ik (q(t ))) ≤ ψk (V (t , p(t ), q(t ))), where t = τk , k ∈ Z for each t > t0 and p, q ∈ Ł. Then V t , p(t ; t0 , ϕ0 ), q(t ; t0 , ϕ0 ) ≤ r (t ; t0 , u0 ) as t ≥ t0 .





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The proof of Lemma 2.6 is analogous to that of Lemma 2 in [11] and hence is omitted. Lemma 2.7. Let the following conditions hold: 1. Assumptions H1–H4 are met. 2. For any  system  (4)there exist  functions   W ∈ W0 , a, b ∈ Q such that: i. a |p(t )| ≤ W t , p(t ) ≤ b |p(t )| , t ∈ R, p(t ) ∈ PC [R, R]; ii. for any t > t0 , p ∈ PC [(t0 , ∞), R] for which W s, p(s) ≤ W t , p(t ) , s ∈ [t0 , t ] the following inequalities hold:

       D+ W t , p(t ) ≤ −cW t , p(t ) , t ̸= τkα , c = const > 0     W t + 0, p(t ) + Ikα (p(t )) ≤ W t , p(t ) , t = τkα , k ∈ Z. 

and

Then the zero solution of (6) is uniformly asymptotically stable. The proof of Lemma 2.7 is analogous to that of Corollary 1 in [11] and hence is omitted. 3. The main results Theorem 3.1. Let the following conditions hold: 1. Assumptions H1–H4 are met. 2. The functions V ∈ V0 and a, b ∈ Q are such that a |p(t ) − q(t )| ≤ V t , p(t ), q(t ) ≤ b |p(t ) − q(t )| ,













(8)

where p(t ) ∈ PC [R, Λ] and q(t ) ∈ PC [R, Λ]. 3. The inequalities D+ V t , p(t ), q(t ) ≤ −cV t , p(t ), q(t ) ,









t ̸= τk , c = const > 0

(9)

and V t + 0, p(t ) + Ik (p(t )), q(t ) + Ik (q(t )) ≤ V t , p(t ), q(t ) ,









(10)

where t = τk , p, q ∈ Ł, k ∈ Z, hold. 4. There exists a solution p(t ; t0 , ϕ0 ) of (5) such that

|p(t ; t0 , ϕ0 )| < ν1 ,

where t ≥ t0 , ν1 < ν.

Then there exists a unique almost periodic solution ω(t ) for system (5) such that: I. |ω(t )| ≤ ν1 . II. mod(ω(t ), τk ) ⊂ mod(F , ϕ0 , Ik , τk ). III. ω(t ) is uniformly asymptotically stable. Proof. Let {αi } be an arbitrary sequence of real numbers such that αi → ∞ as i → ∞ and {αi } ‘‘moves’’ systems (5)–(6). For any real number β , let i0 = i0 (β) be the smallest value of i such that αi0 + β ≥ t0 . Since |p(t ; t0 , ϕ0 )| < ν1 , ν1 < ν , for all t ≥ t0 , then p(t + αi ; t0 , ϕ0 ) ∈ Bν1 for t ≥ β, i ≥ i0 . Let I ⊂ (β, ∞) be compact. Then for any ε > 0, choose an integer n0 (ε, β) ≥ i0 (β) so large that for l ≥ i ≥ n0 (ε, β) and t ∈ (β, ∞) it follows that b(2ν1 )e−c (β+αi −t0 ) < a(ε),

(11)

where c = const > 0.   Consider the function V σ , p(σ ), p(σ + αl − αi ) . In view of (9), (11) and Lemma 2.6, it follows that

    V t + αi , p(t + αi ), p(t + αl ) ≤ e−c (t +αi −t0 ) V t0 , p(t0 ), p(t0 + αl − αi ) < a(ε) for σ > t0 and (x(σ ), x(σ + αl − αi )) ∈ Ł. From (6), therefore, we have

|p(t + αi ) − p(t + αl )| < ε for l ≥ i ≥ n0 (ε, β) and t ∈ I. Consequently, there exists a function ω(t ) such that p(t + αi ) − ω(t ) → 0 for i → ∞. Since β is arbitrary it follows that ω(t ) is defined uniformly on t ∈ I. We show that ω(t ) is a solution of (6). As p(t + αj ) ∈ Bν1 , it follows that there exists n1 (ε) > 0 such that if l ≥ i ≥ n1 (ε), then

        F p(t + αi ), p(t + αi − h) p(t + αi ) − F p(t + αl ), p(t + αl − h) p(t + αl ) < ε

G.Tr. Stamov et al. / Nonlinear Analysis: Real World Applications 12 (2011) 3170–3176

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and

|˙p(t + αi ) − p˙ (t + αl )| ≤ ε,

t + αi ̸= τkα ,

which shows that limi→∞ p˙ (t + αi ) exists uniformly on all compact subsets of R. Then limi→∞ p˙ (t + αi ) = ω( ˙ t ), and

 ω( ˙ t ) = lim F (p(t + αi ), p(t + αi − h))p(t + αi ) − F (ω(t ), ω(t + αi − h))ω(t ) i→∞  + F (ω(t ), ω(t + αi − h))ω(t ) = F α (t , ω(t ), ω(t − h)), t ̸= τkα .

(12)

On the other hand, it follows that

ω(τkα + 0) − ω(τkα − 0) = lim (p(τkα + αi + 0) − p(τkα + αi − 0)) i→∞

= lim Ikα (p(τkα + αi )) = Ikα (ω(τkα )), i→∞

for t + αi = τkα .

(13)

By virtue of H3, we get that for the sequence {αi } there exists a subsequence {αn }, αn = αin , such that the sequence {ϕ0 (t + αn )} is convergent uniformly to the function ϕ0α . From (12) and (13), it follows that ω(t ) is a solution of (6). We show that ω(t ) is an almost periodic function. Let the sequence {αi } move system (5) to mod(F , ϕ0 , Ik , τk ). For any ε > 0 there exists m0 (ε) > 0 such that if l ≥ i ≥ m0 (ϕ), then e−c αi b(2ν1 ) <

a(ε) 2

and

 a(ε)        , F p(κ + αi ), p(κ + αi − h) p(κ + αi ) − F p(κ + αl ), p(κ + αl − h) p(κ + αl ) < 2L

a(ε)

where p ∈ PC [(t0 , ∞), R] and c = const > 0. For each fixed t ∈ R, let τε = 2L be the translation number of F such that t + τε ≥ 0. Consider the function V (τε + σ , ω(σ ), ω(σ + αl − αi )), where t ≤ σ ≤ t + αi . It follows that







D+ V τε + σ , ω(σ ), ω(σ + αl − αi ) − cV τε + σ , ω(σ ), ω(σ + αl − αi )



       + L F α ω(σ ), ω(σ − h) ω(σ ) − F α ω(σ ), ω(τε + σ − h)    a(ε) . ≤ −cV τε + σ , ω(σ ), ω(σ + αl − αi ) +

(14)

2

On the other hand,



V τε + τkα , ω(τkα ) + Ikα (ω(τkα )), ω(τkα + αl − αi ) + Ikα (ω(τkα + αl − αi ))



≤ V (τε + τkα , ω(τkα ), ω(τkα + αl − αi )).

(15)

In view of (14), (15) and Lemma 2.6, we obtain a(ε) V (τε + t + αi , ω(t + αi ), ω(t + αl )) ≤ e−c αi V (τε + t , ω(t ), ω(t + αi − αl )) + < a(ε). 2 By the last inequality, we get

|ω(t + αi ) − ω(t + αl )| < ε,

l ≥ i ≥ m0 (ε).

(16)

From the definition of the sequence {αi }, we have

ρ(τk + αi , τk + αl ) < ε for l ≥ i ≥ m0 (ε). In light of (16) and the last inequality, we obtain that the sequence ω(t + αi ) is convergent uniformly to the function ω(t ). Assertions I and II of Theorem 3.1 are clearly obtained. We shall prove assertion III. Let ω(t ) be an arbitrary solution of (6). Set u(t ) = ω(t ) − ω(t ),

     g α (t , u(t )) = F α u(t ) + ω(t ), u(t ) + ω(t − h) u(t ) + ω(t ) − F α ω(t ), ω(t − h) ω(t ) and

γkα (u) = Ikα (u + ω) − Ikα (u).

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G.Tr. Stamov et al. / Nonlinear Analysis: Real World Applications 12 (2011) 3170–3176

Consider the system

   u˙ = g α t , u(t ) , t ̸= τkα , (17) 1u(τkα ) = γkα (u(τkα )), k ∈ Z, u(t + 0 ) = u , t ∈ R . 0 0 0   Suppose that W (t , u(t )) = V t , ω(t ), ω(t ) + u(t ) . Then by Lemma 2.7, it follows that the zero solution u(t ) = 0 of system (17) is uniformly asymptotically stable for t0 ≥ 0 and hence ω(t ) is uniformly asymptotically stable. The proof of Theorem 3.1 is complete.



Remark 3.2. The results of this paper show that by means of appropriate impulsive perturbations one can control the almost periodic dynamic behavior of system (5). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48]

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