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Existence of almost periodic solutions for strongly stable nonlinear impulsive differential–difference equations
Nonlinear Analysis: Hybrid Systems 6 (2012) 818–823
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Existence of almost periodic solutions for strongly stable nonlinear impulsive differential–difference equations G.T. Stamov a , I.M. Stamova b , J.O. Alzabut c,∗ a
Department of Mathematics, Technical University of Sofia, 8800 Sliven, Bulgaria
b
Department of Mathematics, Bourgas Free University, Bourgas, Bulgaria
c
Department of Mathematics and Physical Sciences, Prince Sultan University, P.O. Box 66833, Riyadh 11586, Saudi Arabia
article
info
Article history: Received 14 February 2011 Accepted 16 August 2011 Keywords: Almost periodic solutions Lyapunov’s function Markoff’s sets
abstract Sufficient conditions are established for the existence of almost periodic solutions for strongly stable nonlinear impulsive differential–difference equations. The investigations are carried out by means of piecewise continuous functions of Lyapunov type and by using Markoff’s sets. We provide an example to demonstrate the effectiveness of our results. © 2011 Elsevier Ltd. All rights reserved.
1. Introduction Impulsive differential equations (IDE’s for short) have conspicuously occupied a great part among researchers’ interests in the last two decades [1–4]. However, it has been recently recognized that impulsive differential–difference equations (IDDE’s for short) do not only generalize the corresponding theory of IDE’s but also provide more mathematical description for many real world phenomena. Indeed, these equations adequately model processes which are characterized by the fact that their motions depend on the previous history as well as at certain moments of time they undergo sudden changes whose duration is almost negligible. Such generalization for the notion of IDE’s motivates us to study and analyze different types of classical problems for IDDE’s. In particular, the widespread application of IDDE’s in depicting lots of dynamical models requires the formulation of effective criteria for the concepts of stability and almost periodic oscillatory behavior for their solutions; see for instance [5–13] for stability and [14–18] for almost periodicity results. Noticeably, a few results exist in the direction of investigating almost periodic solutions for IDDE’s. In the present paper, sufficient conditions are established for the existence of almost periodic solutions for strongly stable nonlinear impulsive differential–difference equations. The investigations are carried out by means of piecewise continuous functions which are analogous to Lyapunov’s functions and by using Markoff’s sets [11,19]. We provide an example to demonstrate the effectiveness of our results. 2. Preliminary notes Let Rn be the n-dimensional Euclidean space with the norm |x| = the origin and R+ = [0, ∞). Let t0 ∈ R and h > 0.
∗
Corresponding author. Tel.: +966 543431972. E-mail address: [email protected] (J.O. Alzabut).
1751-570X/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.nahs.2011.08.002
∑n
i=1
x2i
12
, Ω be a bounded domain in Rn containing
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We consider the nonlinear impulsive differential–difference equation of the form
x˙ (t ) = f (t , x(t ), x(t − h)), t ̸= τk , 1x(t ) = Ik (x(t )), t = τk , k ∈ Z ,
(1)
where x ∈ Rn , f : R × Ω × Ω → Rn , 1x(t ) = x(t + 0) − x(t ), Ik : Ω → Rn , k ∈ Z and the impulse moments {τk } form a strictly increasing sequence such that limk→±∞ τk = ±∞. For ease of exposition throughout the rest of the paper, we adopt the following notations: Gk = {(t , x) ∈ R × Ω : τk−1 < t < τk }, k ∈ Z ; ρ(x, y) = |x − y| is the distance between x ∈ Rn and y ∈ Rn ; Bα (a) = {x ∈ Ω , |x − a| < α}, a ∈ Rn , α = const > 0 and Bα = Bα (0); Γ = {(t , x) ∈ R × Bα }; PC (t0 ) is the space of all functions ϕ : [t0 − h, t0 ] → Ω having points of discontinuity at τ1 , τ2 , . . . , τs ∈ (t0 − h, t0 ) of the first kind and are left continuous at these points and ‖ϕ‖ = supt ∈[t0 −h,t0 ] |ϕ(t )| is the norm of the function ϕ ∈ PC (t0 ). Let ϕ0 ∈ PC (t0 ). Denote by x(t ) = x(t ; t0 , ϕ0 ), x ∈ Rn the solution of Eq. (1) satisfying the initial conditions
x(t ; t0 , ϕ0 ) = ϕ0 (t ), t0 − h ≤ t ≤ t0 , x(t0 + 0; t0 , ϕ0 ) = ϕ0 (t0 ),
(2)
and by J + (t0 , ϕ0 ) and J − (t0 , ϕ0 ) the maximal intervals of the type [t0 , β) and (α, t0 ), respectively, (α ≥ −∞, β ≤ +∞) in which the solution x(t ; t0 , ϕ0 ) is defined. The solution x(t ) = x(t ; t0 , ϕ0 ) of the initial value problem (1), (2) is characterized as follows. For t0 − h ≤ t ≤ t0 , the solution x(t ) satisfies the initial conditions (2). For t > t0 , the solution x(t ; t0 , ϕ0 ) of problem (1), (2) is a piecewise continuous function with points of discontinuity at the first kind at t = τk , k ∈ Z at which it is continuous from the left, that is, the following relations hold. x(τk − 0) = x(τk ),
x(τk + 0) = x(τk ) + 1x(τk ) = x(τk ) + Ik (x(τk )).
Introduce the following conditions: H1. H2. H3. H4. H5.
τk+1 > τk , k ∈ Z and limk→±∞ τk = ±∞; f ∈ C [R × Ω × Ω , Rn ] and f (t , 0, 0) = 0, t ∈ R;
The function f is Lipschitz continuous with respect to its second and third arguments in R × Ω × Ω ; Ik ∈ C [Ω , Rn ] and Ik (0) = 0, k ∈ Z ; E + Ik : Ω → Ω , k ∈ Z , where E is the identity in Ω .
If conditions H1–H5 are fulfilled, then the standard existence and uniqueness theorems guarantee that, for each point (t0 , ϕ0 ) ∈ R × PC (t0 ), there exists a unique solution x(t ; t0 , ϕ0 ), t > t0 , of Eq. (1) which satisfies the initial conditions (2). In this case, J + (t0 , ϕ0 ) = [t0 , ∞) and J − (t0 , ϕ0 ) = (α, t0 ); see [5] for more details. Definition 1 ([20]). The continuous function f (t , x, y) is said to be almost periodic in t uniformly with respect to x ∈ X and y ∈ Y , X ⊆ Rn , Y ⊆ Rn if for any ε > 0, it is possible to find a number l(ε) such that, for any t ∈ R there exists a number τ , |τ | < l(ε) and for any x ∈ X , y ∈ Y |f (t + τ , x, y) − f (t , x, y)| < ε. The number τ is called an ε -translation number of the function f (t , x, y). Definition 2 ([20]). The sequence {Ik (x)}, Ik : X → Rn , k ∈ Z , is said to be almost periodic in k uniformly with respect to x ∈ X , X ⊆ Rn if for any ε > 0, it is possible to find l(ε) > 0 such that, for any k ∈ Z and x ∈ X there exists a number q, |q| < l(ε) and |Ik+q (x) − Ik (x)| < ε . The number q is called an ε -translation number for the sequence {Ik (x)}.
j
j
Definition 3 ([1]). The set of sequences τk , τk = τk+j − τk , k ∈ Z , j ∈ Z , is said to be uniformly almost periodic if for arbitrary ε > 0, there exists a relatively dense set of ε -almost periods common for any sequences. Definition 4 ([1]). A piecewise continuous function ϕ : R → Rn with discontinuity of the first kind at the points t = τk is said to be almost periodic if
j
j
(i) the set of sequences τk , τk = τk+j − τk , k ∈ Z , j ∈ Z is uniformly almost periodic; (ii) for any ε > 0 there exists a real number δ > 0 such that, if the points t ′ and t ′′ belong to one and the same interval of continuity of ϕ(t ) and satisfy the inequality |t ′ − t ′′ | < δ , then |ϕ(t ′ ) − ϕ(t ′′ )| < ε ; (iii) for any ε > 0 there exists a relatively dense set T such that, if τ ∈ T , then |ϕ(t + τ ) − ϕ(t )| < ε for all t ∈ R satisfying the condition |t − τk | > ε, k ∈ Z . The elements of T are called ε -almost periods. Introduce the following conditions: H6. The function f (t , x, y) is almost periodic in t ∈ R uniformly with respect to x and y, x ∈ Ω , y ∈ Ω ; H7. The sequence {Ik (x)}, Ik : Ω → Rn , k ∈ Z , is almost periodic uniformly with respect to x, x ∈ Ω ; j j H8. The set of sequences {τk }, τk = τk+j − τk , k ∈ Z , j ∈ Z , is uniformly almost periodic and infk τk1 = θ > 0.
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We provide the definitions of stability as stated in the paper [9]. Definition 5. The zero solution x(t ) ≡ 0 of Eq. (1) is said to be strongly stable if (∀ε > 0) (∃δ > 0) (∀ t0 ∈ R) (∀ϕ0 ∈ PC (t0 ) : ‖ϕ0 ‖ < δ) (∀t ∈ R) : |x(t ; t0 , ϕ0 )| < ε . Definition 6. An arbitrary solution x(t ) = x(t ; t0 , x0 ) of Eq. (1) is said to be strongly stable if (∀ε > 0) (∀η > 0) (∃δ > 0) (∀t1 ∈ R, ∀t2 ∈ R, ρ(x(t1 ), x(t2 )) < δ) (∀t ∈ R, |t + ti − τk | > η, i = 1, 2, k ∈ Z ) : ρ(x(t + t1 ), x(t + t2 )) < ε . We introduce the classes V0 and V1 for piecewise continuous functions. Definition 7. The function V : Γ → R+ belongs to the class V0 if the following conditions hold: (i) V is continuous in each of the sets Gk and V (t , 0) = 0 for t ∈ R; (ii) For each k ∈ Z and each point x0 ∈ Bα there exist the finite limits V (τk − 0, x0 ) =
lim
(t ,x)→(τk ,x0 ) (t ,x)∈Gk
V (t , x),
V (τk + 0, x0 ) =
lim
(t ,x)→(τk ,x0 ) (t ,x)∈Gk+1
V (t , x)
and the equality V (τk − 0, x0 ) = V (τk , x0 ) holds. Definition 8. The function V : Γ → R belongs to the class V1 if V ∈ V0 and V is continuously differentiable in each of the sets Gk . For convenience of use in the forthcoming analysis, we shall use the following classes of functions: PC [R, Bα ] = {σ : R → Rn : σ (t ) is continuous everywhere except possibly at some points t = τk at which σ (τk − 0) and σ (τk + 0) exist and σ (τk − 0) = σ (τk ), k ∈ Z } and PC 1 [R, Bα ] = {σ ∈ PC [R, Rn ] : σ (t ) is continuously differentiable everywhere, except possibly at some points t = τk at which σ˙ (τk − 0) and σ˙ (τk + 0) exist and σ˙ (τk − 0) = σ˙ (τk ), k ∈ Z }. For each function V ∈ V1 , we define V˙ (t , x) =
n ∂ V (t , x) − ∂ V ( t , x) + fi (t , x, y) ∂t ∂ xi i=1
for (t , x) ∈ Gk , k ∈ Z , x ∈ Ω . If x(t ) is a solution of Eq. (1), then d V (t , x(t )) = V˙ (t , x(t )), t ∈ R, t ̸= τk . dt We shall also use the class K of all continuous and strictly increasing functions µ : R+ → R+ such that, µ(0) = 0. Definition 9 ([19]). The set S , S ⊂ R, is said to be (i) ∆ − m set if from every m + 1 real numbers t0 , t1 , . . . , tm , one can find i ̸= j such that ti − tj ∈ S. (ii) symmetric ∆ − m set if S is ∆ − m set symmetric with respect to 0. Lemma 1 ([19]). Every symmetric ∆ − m set is relatively dense. 3. The main results We are in a position to state and prove our main results. Theorem 1. Let conditions H1–H8 be fulfilled. Then any strongly stable bounded solution of (1) is almost periodic. Proof. Let x = x(t ; t0 , x0 ) be a unique bounded solution of Eq. (1) satisfying the initial conditions
x(t ; t0 , ϕ0 ) = ϕ0 (t ), t0 − h ≤ t ≤ t0 , x(t0 + 0; t0 , ϕ0 ) = ϕ0 (t0 ).
Let ε > 0 be given, δ(ε) > 0 and the points a1 , a2 , . . . , aN , where al ∈ Rn and l ∈ {1, . . . , N } are such that, for t ∈ R, t > t0 it follows that x(t ) ∈ B δ (al ). If t0 , . . . , tN are given real numbers, then for some i ̸= j and some l ∈ {1, . . . , N }, we get 2
ρ(x(ti ), al ) <
δ(ε) 2
,
ρ(x(tj ), al ) <
δ(ε) 2
.
It follows that ρ(x(ti ), x(tj )) < δ(ε). On the other hand, the solution x(t ) is strongly stable, that is, for any η > 0 it follows that ρ(x(t + ti ), x(t + tj )) < ε , where t ∈ R and |t + ti −τk | > η, |t + tj −τk | > η, k ∈ Z . Thus, for t ∈ R and |t + ti − tj −τk | > η, we have ρ(x(t + ti − tj ), x(t )) < ε . Consequently, ti − tj is ε -almost period of the solution x(t ). Let T be the set of every ε-almost periods of x(t ). Then for any sequence of numbers t0 , . . . , tN from above it follows that there exists i ̸= j such that, ti − tj ∈ T . In view of Definition 9, we deduce that T is symmetric ∆ − N set and in virtue of Lemma 1, it follows that T is a relatively dense set. Therefore, x(t ) is an almost periodic function. This completes the proof of Theorem 1.
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Let x(t ) be a solution of (1). Set z = x − x and consider the equation
z˙ = g (t , z (t ), z (t − h)), t ̸= τk , 1z = Jk (z ), t = τk , k ∈ Z ,
(3)
where g (t , z (t ), z (t − h)) = f (t , z (t ) + x(t ), z (t − h) + x(t − h)) − f (t , x(t ), x(t − h)) and Jk (z ) = Ik (z + x) − Ik (x). Lemma 2 ([5]). Let conditions H1–H5 be fulfilled and let the function V ∈ V1 be such that (i) V˙ (t , x) ≡ 0 for t ∈ R, t ̸= τk , x ∈ Ω0 , where
Ω0 = {x ∈ PC 1 [R, Bα ] : V (s, x(s)) ≤ V (t , x(t )), t − h ≤ s ≤ t }. (ii) V (τk + 0, x + Ik (z )) = V (τk , x), k ∈ Z , z ∈ Bα . Then V (t , x(t ; t0 , ϕ0 )) = V (t0 , ϕ0 (t0 )), t ∈ R. Theorem 2. Let the following assumptions be fulfilled: 1. Conditions H1–H8 hold. 2. There exist functions V ∈ V1 and µ, ν ∈ K such that (i) µ(|z |) ≤ V (t , z ) ≤ ν(|z |), (t , z ) ∈ Γ ; (ii) V˙ (t , z ) ≡ 0, for (t , z ) ∈ Γ , t ̸= τk ; (iii) V (τk + 0, z + Ik (z )) = V (τk , z ), k ∈ Z , z ∈ Bα . Then solution x(t ) of (1) is almost periodic. Proof. Let 0 < ε < α be given and δ = δ(ε) < min ε, ν −1 (µ(ε)) . If z (t ) = z (t ; t0 , ϕ0 ) is a solution of Eq. (3) such that, t0 ∈ R, (t0 , ϕ0 ) ∈ R × PC (t0 ) and ‖ϕ0 ‖ < δ . Then, by Lemma 2, we have
V (t , z (t ; t0 , ϕ0 )) = V (t0 , ϕ0 (t0 )),
t ∈ R.
Therefore, from (i) it follows that
µ(|z |) ≤ V (t , z (t )) = V (t0 + 0, ϕ0 (t0 )) ≤ ν(|ϕ0 (t0 )|) ≤ ν(‖ϕ0 ‖) < ν(δ(ϵ)) < µ(ε). Consequently |z (t ; t0 , ϕ0 )| < ε for t ∈ R, that is, the zero solution of (3) is strongly stable. Therefore, x(t ) is strongly stable and in virtue of Theorem 1 it follows that x(t ) is almost periodic. The proof of Theorem 2 is complete. Definition 10 ([5]). The zero solution of Eq. (3) is said to be uniformly stable to the right (to the left), if for any ε > 0 there exists δ(ε) > 0 such that, if t0 ∈ R and ϕ0 ∈ PC (t0 ) : ‖ϕ0 ‖ < δ , then |z (t ; t0 , ϕ0 )| < ε for all t ≥ t0 (for all t < t0 ), where z (t ; t0 , ϕ0 ) is a solution of (3) such that z (t0 + 0) = ϕ0 (t0 ). Lemma 3. The zero solution of Eq. (3) is uniformly stable to the left if and only if for any ε > 0 the following inequality holds.
γ (ε) = inf {|z (t ; t0 , ϕ0 )| : t0 ∈ R, |ϕ0 (t0 )| ≥ ε} > 0. The proof of Lemma 3 is carried out by following the same arguments used to prove Lemma 2.3 in [14]. Hence, it is omitted. Lemma 4 ([5]). The zero solution of Eq. (3) is strongly stable if and only if it is stable to the left and to the right at the same time. Example 1. Consider the linear impulsive differential–difference system
x˙ = A(t )x(t ) + B(t )x(t − h), 1x = Bk x, t = τk , k ∈ Z ,
t ̸= τk ,
(4)
where A(t ), B(t ) are n × n matrices, the elements of which are almost periodic continuous functions for t ∈ R, {Bk } is an almost periodic sequence of constant matrices such that det(E + Bk ) ̸= 0 (E is the identity matrix) and for the points τk condition H8 be fulfilled. Let Uk (t , s) (t , s ∈ (τk−1 , τk ]) be the Cauchy matrix for the linear system x˙ (t ) = A(t )x(t ),
τk−1 < t ≤ τk , k ∈ Z .
Therefore, the solution of system (4) can be represented by the following integral equation x(t ; t0 , x0 , ϕ0 ) = x(t ) = W (t , t0 )ϕ0 (t0 ) +
∫
t
W (t , s)B(s)x(s − h)ds, t0
t > t0 ,
(5)
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where
W (t , s) =
Uk (t , s) as t , s ∈ (τk−1 , τk ], Uk+1 (t , τk + 0)(E + Bk )Uk (τk , s) as τk−1 < s ≤ τk < t ≤ τk+1 , −1 U k (t , τk )(E + Bk ) Uk+1 (τk + 0, s) as τk−1 < t ≤ τk < s ≤ τk+1 , i +1 ∏ Uk+1 (t , τk + 0) (E + Bj )Uj (τj , τj−1 + 0)(E + Bi )Ui (τi , s)
(6)
j =k
as τi−1 < s ≤ τi < τk < t ≤ τk+1 , k−1 ∏ Ui (t , τi ) (E + Bj )−1 Uj+1 (τj + 0, τj+1 )(E + Bk )−1 Uk+1 (τk + 0, s) j =i as τi−1 < t ≤ τi < τk < s ≤ τk+1 ,
is the solving operator of the system
x˙ (t ) = A(t )x(t ), t ̸= τk , 1x(τk ) = Bk x(τk ).
In view of (5), we get
[
ϕ0 (t0 ) = W (t , t0 + 0) x(t ) −
∫
t
]
W (t , s)B(s)x(s − h)ds . t0
Hence, for any ε > 0 and |ϕ0 (t0 )| ≥ ε , we have
∫ t ε ≤ |ϕ0 (t0 )| ≤ |W (t , t0 )| x(t ) − W (t , s)B(s)x(s − h)ds t0 [ ] ∫ t −1 ≤ |W (t , t0 )| |x(t )| + ‖ϕ0 ‖ |W (t , s)| |B(s)|ds −1
t0
and
−1 − ‖ϕ0 ‖ |x(t ; t0 , x0 )| ≥ ε |W −1 (t , t0 )|
∫
t
|W (t , s)| |B(s)|ds. t0
−1
On the other hand, for t = t0 and |ϕ0 (t0 )| = ε , we have |x(t ; t0 , x0 )| = ε |W −1 (t , t0 )|
−1 γ (ε) = inf ε |W −1 (t , t0 )| − ‖ϕ0 ‖
∫
t
. Hence
|W (t , s)| |B(s)|ds : t ≥ t0 .
t0
Therefore, by applying Lemma 3, we conclude that the zero solution of system (4) is uniformly stable to the left if and only if the functions |W −1 (t , s)| and |B(s)| are bounded on the set s ≤ t < ∞. Moreover, it is clear that the zero solution of (4) is uniformly stable to the right if and only if the functions ‖W (t , s)‖ and |B(s)| are bounded on the set s ≤ t < ∞. Thus, in virtue of Lemma 4, the zero solution of system (4) is strongly stable if and only if the functions Uk (t , s) are bounded for t ∈ R. Consequently, an arbitrary solution x(t ) of system (4) is bounded and strongly stable. Theorem 1 guarantees that the solution x(t ) of (4) is almost periodic. Consider the following scalar impulsive differential equations
u˙ = ω1 (t , u), t ̸= τk , 1u = Pk (u), t = τk , k ∈ Z ,
(7)
v˙ = ω2 (t , v), t ̸= τk , 1v = Pk (v), t = τk , k ∈ Z ,
(8)
and
where ω1 : [t0 − T , t0 ] × R+ → R+ , Pk : R+ → R+ , T is constant such that, t0 > T and ω2 : [t0 , ∞] × R+ → R+ . We denote by u+ : [t0 − T , t0 ] → R+ , u+ (t ) = u+ (t ; t0 , u0 ), the maximal solution of Eq. (7), for which u+ (t0 − 0) = u0 ∈ R+ , and by v + : [t0 , ∞) → R+ , v + (t ) = v + (t ; t0 , u0 ) the maximal solution of Eq. (8) for which v + (t0 + 0) = v0 . Introduce the following conditions:
H9. ω1 ∈ C [t0 − T , t0 ] × R+ , R+
and ω1 (t , 0) = 0, t ∈ [t0 − T , t0 ];
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H10. ω2 ∈ C [t0 , ∞) × R+ , R+
and ω2 (t , 0) = 0, t ∈ [t0 , ∞);
H11. Pk ∈ C [R+ , R+ ] and Pk (0) = 0, k ∈ Z . Definition 11 ([9]). The zero solution u(t ) = 0 of Eq. (7) is said to be uniformly stable to the left if for any ε > 0 there exists η(ε) > 0 such that, if t0 ∈ R and u0 ∈ R+ : 0 ≤ u0 < η, then u+ (t ; t0 , u0 ) < ε for t ∈ [t0 − T , t0 ]. Definition 12 ([9]). The zero solution v(t ) = 0 of Eq. (8) is said to be uniformly stable to the right if for any ε > 0 there exists η(ε) > 0 such that, if t0 ∈ R and u0 ∈ R+ : 0 ≤ u0 < η, then v + (t ; t0 , u0 ) < ε for t ∈ [t0 , ∞]. Let Ω1 and Ω2 be defined as follows:
Ω1 =
x ∈ PC
1
[t0 − T , t0 ], Bα : V (s, x(s)) ≤ V (t , x(t )), t − h ≤ s ≤ t , t ∈ [t0 − T , t0 ]
and
Ω2 =
x ∈ PC
1
[t0 , ∞), Bα : V (s, x(s)) ≤ V (t , x(t )), t − h ≤ s ≤ t , t ∈ [t0 , ∞) .
Therefore, we can formulate the following result. Theorem 3. Let the following assumptions be fulfilled: 1. Conditions H1–H11 hold. 2. There exist functions V ∈ V1 and µ, ν ∈ K such that (i) µ(|z |) ≤ V (t , z ) ≤ ν(|z |), (t , z ) ∈ Γ ; (ii) ω1 (t , V (t , z )) ≤ V˙ (t , z ), t ∈ [t0 − T , t0 ], t ̸= τk , z ∈ Ω1 ; (iii) V˙ (t , z ) ≤ ω2 (t , V (t , z )), t ∈ [t0 , ∞), t ̸= τk , z ∈ Ω2 . 3. The zero solution u(t ) ≡ 0 (v(t ) ≡ 0) of Eq. (7) (Eq. (8)) is uniformly stable to the left (to the right); 4. The functions u + Pk (u), k ∈ Z are monotone increasing in R+ ; 5. V (τk + 0, z + Jk (z )) = V (τk , z ) + Pk (V (τk , z )); 6. The solution x(t ) of Eq. (1) is bounded. Then the solution x(t ) of Eq. (1) is almost periodic. Proof. By employing conditions H1–H11 and the result in [5], it follows that the zero solution of Eq. (1) is strongly stable, that is, the solution x(t ) is strongly stable. Therefore, in virtue of Theorem 1, it follows that x(t ) of (1) is almost periodic. The proof of Theorem 3 is finished. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
A.M. Samoilenko, N.A. Perestyuk, Impulsive Differential Equations, World Scientific, Singapore, 1995. D.D. Bainov, P.S. Simeonov, Theory of Impulsive Differential Equations: Periodic Solutions and Applications, Longman, Harlow, 1993. V. Lakshmikantham, D.D. Bainov, P.S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989. D.D. Bainov, P.S. Simeonov, Impulsive Differential Equations, World Scientific, Singapore, 1995. D.D. Bainov, I.M. Stamova, Strong stability of impulsive differential–difference equations, Panamer. Math. J. 9 (1999) 87–95. J. Diblik, N. Koksch, Existence of global solutions of delayed differential equations via retract approach, Nonlinear Anal. 64 (2006) 1153–1170. L. Hatvani, On the strong stability by Liapunov’s direct method, Acta Sci. Math. (Szeged) 37 (1975) 237–256. Z. Luo, J. Shen, Stability results for impulsive functional differential equations with infinite delays, J. Comput. Appl. Math. 131 (2001) 55–64. G.K. Kulev, D.D. Bainov, Strong stability of impulsive systems, Internat. J. Theoret. Phys. 27 (1988) 745–755. J. Yan, J. Shen, Impulsive stabilization of impulsive functional differential equations by Lyapunov–Razumikhin functions, Nonlinear Anal. 37 (1999) 245–255. I.M. Stamova, Vector Lyapunov functions for practical stability of nonlinear impulsive functional differential equations, J. Math. Anal. Appl. 325 (2007) 612–623. I.M. Stamova, G.T. Stamov, Lyapunov–Razumikhin method for impulsive functional differential equations and applications to the population dynamics, J. Comput. Appl. Math. 130 (2001) 163–171. M.U. Akhmet, J.O. Alzabut, A. Zafer, Perron’s theorem for linear impulsive differential equations with distributed delay, J. Comput. Appl. Math. 193 (1) (2006) 204–218. S. Ahmad, G.T. Stamov, Almost periodic solutions of n-dimensional impulsive competitive systems, Nonlinear Anal. RWA 10 (3) (2009) 1846–1853. G.Tr. Stamov, Existence of almost periodic solutions for strong stable impulsive differential equations, IMA J. Math. Control 18 (2001) 153–160. G.Tr. Stamov, Impulsive cellular networks and almost periodicity, Proc. Japan Acad. Ser. A Math. Sci. 80 (10) (2004) 198–203. G.T. Stamov, J.O. Alzabut, Almost periodic solutions for abstract impulsive differential equations, Nonlinear Anal. 72 (5) (2010) 2457–2464. J.O. Alzabut, J.J. Nieto, G.T. Stamov, On existence of almost periodic solutions of impulsive delay hematopoiesis model, Bound. Value Probl. 2009 (2009) Article ID 127510. A. Markoff, Stabilitat in Liapounoffschen sinne und Fastperiodizitat, Math. Z. 36 (1933) 708–738. T. Yoshizawa, Stability theory by Lyapunov’s second method, J. Math. Soc. Japan (1966).
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Nonlinear Analysis: Hybrid Systems journal homepage: www.elsevier.com/locate/nahs
Existence of almost periodic solutions for strongly stable nonlinear impulsive differential–difference equations G.T. Stamov a , I.M. Stamova b , J.O. Alzabut c,∗ a
Department of Mathematics, Technical University of Sofia, 8800 Sliven, Bulgaria
b
Department of Mathematics, Bourgas Free University, Bourgas, Bulgaria
c
Department of Mathematics and Physical Sciences, Prince Sultan University, P.O. Box 66833, Riyadh 11586, Saudi Arabia
article
info
Article history: Received 14 February 2011 Accepted 16 August 2011 Keywords: Almost periodic solutions Lyapunov’s function Markoff’s sets
abstract Sufficient conditions are established for the existence of almost periodic solutions for strongly stable nonlinear impulsive differential–difference equations. The investigations are carried out by means of piecewise continuous functions of Lyapunov type and by using Markoff’s sets. We provide an example to demonstrate the effectiveness of our results. © 2011 Elsevier Ltd. All rights reserved.
1. Introduction Impulsive differential equations (IDE’s for short) have conspicuously occupied a great part among researchers’ interests in the last two decades [1–4]. However, it has been recently recognized that impulsive differential–difference equations (IDDE’s for short) do not only generalize the corresponding theory of IDE’s but also provide more mathematical description for many real world phenomena. Indeed, these equations adequately model processes which are characterized by the fact that their motions depend on the previous history as well as at certain moments of time they undergo sudden changes whose duration is almost negligible. Such generalization for the notion of IDE’s motivates us to study and analyze different types of classical problems for IDDE’s. In particular, the widespread application of IDDE’s in depicting lots of dynamical models requires the formulation of effective criteria for the concepts of stability and almost periodic oscillatory behavior for their solutions; see for instance [5–13] for stability and [14–18] for almost periodicity results. Noticeably, a few results exist in the direction of investigating almost periodic solutions for IDDE’s. In the present paper, sufficient conditions are established for the existence of almost periodic solutions for strongly stable nonlinear impulsive differential–difference equations. The investigations are carried out by means of piecewise continuous functions which are analogous to Lyapunov’s functions and by using Markoff’s sets [11,19]. We provide an example to demonstrate the effectiveness of our results. 2. Preliminary notes Let Rn be the n-dimensional Euclidean space with the norm |x| = the origin and R+ = [0, ∞). Let t0 ∈ R and h > 0.
∗
Corresponding author. Tel.: +966 543431972. E-mail address: [email protected] (J.O. Alzabut).
1751-570X/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.nahs.2011.08.002
∑n
i=1
x2i
12
, Ω be a bounded domain in Rn containing
G.T. Stamov et al. / Nonlinear Analysis: Hybrid Systems 6 (2012) 818–823
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We consider the nonlinear impulsive differential–difference equation of the form
x˙ (t ) = f (t , x(t ), x(t − h)), t ̸= τk , 1x(t ) = Ik (x(t )), t = τk , k ∈ Z ,
(1)
where x ∈ Rn , f : R × Ω × Ω → Rn , 1x(t ) = x(t + 0) − x(t ), Ik : Ω → Rn , k ∈ Z and the impulse moments {τk } form a strictly increasing sequence such that limk→±∞ τk = ±∞. For ease of exposition throughout the rest of the paper, we adopt the following notations: Gk = {(t , x) ∈ R × Ω : τk−1 < t < τk }, k ∈ Z ; ρ(x, y) = |x − y| is the distance between x ∈ Rn and y ∈ Rn ; Bα (a) = {x ∈ Ω , |x − a| < α}, a ∈ Rn , α = const > 0 and Bα = Bα (0); Γ = {(t , x) ∈ R × Bα }; PC (t0 ) is the space of all functions ϕ : [t0 − h, t0 ] → Ω having points of discontinuity at τ1 , τ2 , . . . , τs ∈ (t0 − h, t0 ) of the first kind and are left continuous at these points and ‖ϕ‖ = supt ∈[t0 −h,t0 ] |ϕ(t )| is the norm of the function ϕ ∈ PC (t0 ). Let ϕ0 ∈ PC (t0 ). Denote by x(t ) = x(t ; t0 , ϕ0 ), x ∈ Rn the solution of Eq. (1) satisfying the initial conditions
x(t ; t0 , ϕ0 ) = ϕ0 (t ), t0 − h ≤ t ≤ t0 , x(t0 + 0; t0 , ϕ0 ) = ϕ0 (t0 ),
(2)
and by J + (t0 , ϕ0 ) and J − (t0 , ϕ0 ) the maximal intervals of the type [t0 , β) and (α, t0 ), respectively, (α ≥ −∞, β ≤ +∞) in which the solution x(t ; t0 , ϕ0 ) is defined. The solution x(t ) = x(t ; t0 , ϕ0 ) of the initial value problem (1), (2) is characterized as follows. For t0 − h ≤ t ≤ t0 , the solution x(t ) satisfies the initial conditions (2). For t > t0 , the solution x(t ; t0 , ϕ0 ) of problem (1), (2) is a piecewise continuous function with points of discontinuity at the first kind at t = τk , k ∈ Z at which it is continuous from the left, that is, the following relations hold. x(τk − 0) = x(τk ),
x(τk + 0) = x(τk ) + 1x(τk ) = x(τk ) + Ik (x(τk )).
Introduce the following conditions: H1. H2. H3. H4. H5.
τk+1 > τk , k ∈ Z and limk→±∞ τk = ±∞; f ∈ C [R × Ω × Ω , Rn ] and f (t , 0, 0) = 0, t ∈ R;
The function f is Lipschitz continuous with respect to its second and third arguments in R × Ω × Ω ; Ik ∈ C [Ω , Rn ] and Ik (0) = 0, k ∈ Z ; E + Ik : Ω → Ω , k ∈ Z , where E is the identity in Ω .
If conditions H1–H5 are fulfilled, then the standard existence and uniqueness theorems guarantee that, for each point (t0 , ϕ0 ) ∈ R × PC (t0 ), there exists a unique solution x(t ; t0 , ϕ0 ), t > t0 , of Eq. (1) which satisfies the initial conditions (2). In this case, J + (t0 , ϕ0 ) = [t0 , ∞) and J − (t0 , ϕ0 ) = (α, t0 ); see [5] for more details. Definition 1 ([20]). The continuous function f (t , x, y) is said to be almost periodic in t uniformly with respect to x ∈ X and y ∈ Y , X ⊆ Rn , Y ⊆ Rn if for any ε > 0, it is possible to find a number l(ε) such that, for any t ∈ R there exists a number τ , |τ | < l(ε) and for any x ∈ X , y ∈ Y |f (t + τ , x, y) − f (t , x, y)| < ε. The number τ is called an ε -translation number of the function f (t , x, y). Definition 2 ([20]). The sequence {Ik (x)}, Ik : X → Rn , k ∈ Z , is said to be almost periodic in k uniformly with respect to x ∈ X , X ⊆ Rn if for any ε > 0, it is possible to find l(ε) > 0 such that, for any k ∈ Z and x ∈ X there exists a number q, |q| < l(ε) and |Ik+q (x) − Ik (x)| < ε . The number q is called an ε -translation number for the sequence {Ik (x)}.
j
j
Definition 3 ([1]). The set of sequences τk , τk = τk+j − τk , k ∈ Z , j ∈ Z , is said to be uniformly almost periodic if for arbitrary ε > 0, there exists a relatively dense set of ε -almost periods common for any sequences. Definition 4 ([1]). A piecewise continuous function ϕ : R → Rn with discontinuity of the first kind at the points t = τk is said to be almost periodic if
j
j
(i) the set of sequences τk , τk = τk+j − τk , k ∈ Z , j ∈ Z is uniformly almost periodic; (ii) for any ε > 0 there exists a real number δ > 0 such that, if the points t ′ and t ′′ belong to one and the same interval of continuity of ϕ(t ) and satisfy the inequality |t ′ − t ′′ | < δ , then |ϕ(t ′ ) − ϕ(t ′′ )| < ε ; (iii) for any ε > 0 there exists a relatively dense set T such that, if τ ∈ T , then |ϕ(t + τ ) − ϕ(t )| < ε for all t ∈ R satisfying the condition |t − τk | > ε, k ∈ Z . The elements of T are called ε -almost periods. Introduce the following conditions: H6. The function f (t , x, y) is almost periodic in t ∈ R uniformly with respect to x and y, x ∈ Ω , y ∈ Ω ; H7. The sequence {Ik (x)}, Ik : Ω → Rn , k ∈ Z , is almost periodic uniformly with respect to x, x ∈ Ω ; j j H8. The set of sequences {τk }, τk = τk+j − τk , k ∈ Z , j ∈ Z , is uniformly almost periodic and infk τk1 = θ > 0.
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We provide the definitions of stability as stated in the paper [9]. Definition 5. The zero solution x(t ) ≡ 0 of Eq. (1) is said to be strongly stable if (∀ε > 0) (∃δ > 0) (∀ t0 ∈ R) (∀ϕ0 ∈ PC (t0 ) : ‖ϕ0 ‖ < δ) (∀t ∈ R) : |x(t ; t0 , ϕ0 )| < ε . Definition 6. An arbitrary solution x(t ) = x(t ; t0 , x0 ) of Eq. (1) is said to be strongly stable if (∀ε > 0) (∀η > 0) (∃δ > 0) (∀t1 ∈ R, ∀t2 ∈ R, ρ(x(t1 ), x(t2 )) < δ) (∀t ∈ R, |t + ti − τk | > η, i = 1, 2, k ∈ Z ) : ρ(x(t + t1 ), x(t + t2 )) < ε . We introduce the classes V0 and V1 for piecewise continuous functions. Definition 7. The function V : Γ → R+ belongs to the class V0 if the following conditions hold: (i) V is continuous in each of the sets Gk and V (t , 0) = 0 for t ∈ R; (ii) For each k ∈ Z and each point x0 ∈ Bα there exist the finite limits V (τk − 0, x0 ) =
lim
(t ,x)→(τk ,x0 ) (t ,x)∈Gk
V (t , x),
V (τk + 0, x0 ) =
lim
(t ,x)→(τk ,x0 ) (t ,x)∈Gk+1
V (t , x)
and the equality V (τk − 0, x0 ) = V (τk , x0 ) holds. Definition 8. The function V : Γ → R belongs to the class V1 if V ∈ V0 and V is continuously differentiable in each of the sets Gk . For convenience of use in the forthcoming analysis, we shall use the following classes of functions: PC [R, Bα ] = {σ : R → Rn : σ (t ) is continuous everywhere except possibly at some points t = τk at which σ (τk − 0) and σ (τk + 0) exist and σ (τk − 0) = σ (τk ), k ∈ Z } and PC 1 [R, Bα ] = {σ ∈ PC [R, Rn ] : σ (t ) is continuously differentiable everywhere, except possibly at some points t = τk at which σ˙ (τk − 0) and σ˙ (τk + 0) exist and σ˙ (τk − 0) = σ˙ (τk ), k ∈ Z }. For each function V ∈ V1 , we define V˙ (t , x) =
n ∂ V (t , x) − ∂ V ( t , x) + fi (t , x, y) ∂t ∂ xi i=1
for (t , x) ∈ Gk , k ∈ Z , x ∈ Ω . If x(t ) is a solution of Eq. (1), then d V (t , x(t )) = V˙ (t , x(t )), t ∈ R, t ̸= τk . dt We shall also use the class K of all continuous and strictly increasing functions µ : R+ → R+ such that, µ(0) = 0. Definition 9 ([19]). The set S , S ⊂ R, is said to be (i) ∆ − m set if from every m + 1 real numbers t0 , t1 , . . . , tm , one can find i ̸= j such that ti − tj ∈ S. (ii) symmetric ∆ − m set if S is ∆ − m set symmetric with respect to 0. Lemma 1 ([19]). Every symmetric ∆ − m set is relatively dense. 3. The main results We are in a position to state and prove our main results. Theorem 1. Let conditions H1–H8 be fulfilled. Then any strongly stable bounded solution of (1) is almost periodic. Proof. Let x = x(t ; t0 , x0 ) be a unique bounded solution of Eq. (1) satisfying the initial conditions
x(t ; t0 , ϕ0 ) = ϕ0 (t ), t0 − h ≤ t ≤ t0 , x(t0 + 0; t0 , ϕ0 ) = ϕ0 (t0 ).
Let ε > 0 be given, δ(ε) > 0 and the points a1 , a2 , . . . , aN , where al ∈ Rn and l ∈ {1, . . . , N } are such that, for t ∈ R, t > t0 it follows that x(t ) ∈ B δ (al ). If t0 , . . . , tN are given real numbers, then for some i ̸= j and some l ∈ {1, . . . , N }, we get 2
ρ(x(ti ), al ) <
δ(ε) 2
,
ρ(x(tj ), al ) <
δ(ε) 2
.
It follows that ρ(x(ti ), x(tj )) < δ(ε). On the other hand, the solution x(t ) is strongly stable, that is, for any η > 0 it follows that ρ(x(t + ti ), x(t + tj )) < ε , where t ∈ R and |t + ti −τk | > η, |t + tj −τk | > η, k ∈ Z . Thus, for t ∈ R and |t + ti − tj −τk | > η, we have ρ(x(t + ti − tj ), x(t )) < ε . Consequently, ti − tj is ε -almost period of the solution x(t ). Let T be the set of every ε-almost periods of x(t ). Then for any sequence of numbers t0 , . . . , tN from above it follows that there exists i ̸= j such that, ti − tj ∈ T . In view of Definition 9, we deduce that T is symmetric ∆ − N set and in virtue of Lemma 1, it follows that T is a relatively dense set. Therefore, x(t ) is an almost periodic function. This completes the proof of Theorem 1.
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Let x(t ) be a solution of (1). Set z = x − x and consider the equation
z˙ = g (t , z (t ), z (t − h)), t ̸= τk , 1z = Jk (z ), t = τk , k ∈ Z ,
(3)
where g (t , z (t ), z (t − h)) = f (t , z (t ) + x(t ), z (t − h) + x(t − h)) − f (t , x(t ), x(t − h)) and Jk (z ) = Ik (z + x) − Ik (x). Lemma 2 ([5]). Let conditions H1–H5 be fulfilled and let the function V ∈ V1 be such that (i) V˙ (t , x) ≡ 0 for t ∈ R, t ̸= τk , x ∈ Ω0 , where
Ω0 = {x ∈ PC 1 [R, Bα ] : V (s, x(s)) ≤ V (t , x(t )), t − h ≤ s ≤ t }. (ii) V (τk + 0, x + Ik (z )) = V (τk , x), k ∈ Z , z ∈ Bα . Then V (t , x(t ; t0 , ϕ0 )) = V (t0 , ϕ0 (t0 )), t ∈ R. Theorem 2. Let the following assumptions be fulfilled: 1. Conditions H1–H8 hold. 2. There exist functions V ∈ V1 and µ, ν ∈ K such that (i) µ(|z |) ≤ V (t , z ) ≤ ν(|z |), (t , z ) ∈ Γ ; (ii) V˙ (t , z ) ≡ 0, for (t , z ) ∈ Γ , t ̸= τk ; (iii) V (τk + 0, z + Ik (z )) = V (τk , z ), k ∈ Z , z ∈ Bα . Then solution x(t ) of (1) is almost periodic. Proof. Let 0 < ε < α be given and δ = δ(ε) < min ε, ν −1 (µ(ε)) . If z (t ) = z (t ; t0 , ϕ0 ) is a solution of Eq. (3) such that, t0 ∈ R, (t0 , ϕ0 ) ∈ R × PC (t0 ) and ‖ϕ0 ‖ < δ . Then, by Lemma 2, we have
V (t , z (t ; t0 , ϕ0 )) = V (t0 , ϕ0 (t0 )),
t ∈ R.
Therefore, from (i) it follows that
µ(|z |) ≤ V (t , z (t )) = V (t0 + 0, ϕ0 (t0 )) ≤ ν(|ϕ0 (t0 )|) ≤ ν(‖ϕ0 ‖) < ν(δ(ϵ)) < µ(ε). Consequently |z (t ; t0 , ϕ0 )| < ε for t ∈ R, that is, the zero solution of (3) is strongly stable. Therefore, x(t ) is strongly stable and in virtue of Theorem 1 it follows that x(t ) is almost periodic. The proof of Theorem 2 is complete. Definition 10 ([5]). The zero solution of Eq. (3) is said to be uniformly stable to the right (to the left), if for any ε > 0 there exists δ(ε) > 0 such that, if t0 ∈ R and ϕ0 ∈ PC (t0 ) : ‖ϕ0 ‖ < δ , then |z (t ; t0 , ϕ0 )| < ε for all t ≥ t0 (for all t < t0 ), where z (t ; t0 , ϕ0 ) is a solution of (3) such that z (t0 + 0) = ϕ0 (t0 ). Lemma 3. The zero solution of Eq. (3) is uniformly stable to the left if and only if for any ε > 0 the following inequality holds.
γ (ε) = inf {|z (t ; t0 , ϕ0 )| : t0 ∈ R, |ϕ0 (t0 )| ≥ ε} > 0. The proof of Lemma 3 is carried out by following the same arguments used to prove Lemma 2.3 in [14]. Hence, it is omitted. Lemma 4 ([5]). The zero solution of Eq. (3) is strongly stable if and only if it is stable to the left and to the right at the same time. Example 1. Consider the linear impulsive differential–difference system
x˙ = A(t )x(t ) + B(t )x(t − h), 1x = Bk x, t = τk , k ∈ Z ,
t ̸= τk ,
(4)
where A(t ), B(t ) are n × n matrices, the elements of which are almost periodic continuous functions for t ∈ R, {Bk } is an almost periodic sequence of constant matrices such that det(E + Bk ) ̸= 0 (E is the identity matrix) and for the points τk condition H8 be fulfilled. Let Uk (t , s) (t , s ∈ (τk−1 , τk ]) be the Cauchy matrix for the linear system x˙ (t ) = A(t )x(t ),
τk−1 < t ≤ τk , k ∈ Z .
Therefore, the solution of system (4) can be represented by the following integral equation x(t ; t0 , x0 , ϕ0 ) = x(t ) = W (t , t0 )ϕ0 (t0 ) +
∫
t
W (t , s)B(s)x(s − h)ds, t0
t > t0 ,
(5)
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where
W (t , s) =
Uk (t , s) as t , s ∈ (τk−1 , τk ], Uk+1 (t , τk + 0)(E + Bk )Uk (τk , s) as τk−1 < s ≤ τk < t ≤ τk+1 , −1 U k (t , τk )(E + Bk ) Uk+1 (τk + 0, s) as τk−1 < t ≤ τk < s ≤ τk+1 , i +1 ∏ Uk+1 (t , τk + 0) (E + Bj )Uj (τj , τj−1 + 0)(E + Bi )Ui (τi , s)
(6)
j =k
as τi−1 < s ≤ τi < τk < t ≤ τk+1 , k−1 ∏ Ui (t , τi ) (E + Bj )−1 Uj+1 (τj + 0, τj+1 )(E + Bk )−1 Uk+1 (τk + 0, s) j =i as τi−1 < t ≤ τi < τk < s ≤ τk+1 ,
is the solving operator of the system
x˙ (t ) = A(t )x(t ), t ̸= τk , 1x(τk ) = Bk x(τk ).
In view of (5), we get
[
ϕ0 (t0 ) = W (t , t0 + 0) x(t ) −
∫
t
]
W (t , s)B(s)x(s − h)ds . t0
Hence, for any ε > 0 and |ϕ0 (t0 )| ≥ ε , we have
∫ t ε ≤ |ϕ0 (t0 )| ≤ |W (t , t0 )| x(t ) − W (t , s)B(s)x(s − h)ds t0 [ ] ∫ t −1 ≤ |W (t , t0 )| |x(t )| + ‖ϕ0 ‖ |W (t , s)| |B(s)|ds −1
t0
and
−1 − ‖ϕ0 ‖ |x(t ; t0 , x0 )| ≥ ε |W −1 (t , t0 )|
∫
t
|W (t , s)| |B(s)|ds. t0
−1
On the other hand, for t = t0 and |ϕ0 (t0 )| = ε , we have |x(t ; t0 , x0 )| = ε |W −1 (t , t0 )|
−1 γ (ε) = inf ε |W −1 (t , t0 )| − ‖ϕ0 ‖
∫
t
. Hence
|W (t , s)| |B(s)|ds : t ≥ t0 .
t0
Therefore, by applying Lemma 3, we conclude that the zero solution of system (4) is uniformly stable to the left if and only if the functions |W −1 (t , s)| and |B(s)| are bounded on the set s ≤ t < ∞. Moreover, it is clear that the zero solution of (4) is uniformly stable to the right if and only if the functions ‖W (t , s)‖ and |B(s)| are bounded on the set s ≤ t < ∞. Thus, in virtue of Lemma 4, the zero solution of system (4) is strongly stable if and only if the functions Uk (t , s) are bounded for t ∈ R. Consequently, an arbitrary solution x(t ) of system (4) is bounded and strongly stable. Theorem 1 guarantees that the solution x(t ) of (4) is almost periodic. Consider the following scalar impulsive differential equations
u˙ = ω1 (t , u), t ̸= τk , 1u = Pk (u), t = τk , k ∈ Z ,
(7)
v˙ = ω2 (t , v), t ̸= τk , 1v = Pk (v), t = τk , k ∈ Z ,
(8)
and
where ω1 : [t0 − T , t0 ] × R+ → R+ , Pk : R+ → R+ , T is constant such that, t0 > T and ω2 : [t0 , ∞] × R+ → R+ . We denote by u+ : [t0 − T , t0 ] → R+ , u+ (t ) = u+ (t ; t0 , u0 ), the maximal solution of Eq. (7), for which u+ (t0 − 0) = u0 ∈ R+ , and by v + : [t0 , ∞) → R+ , v + (t ) = v + (t ; t0 , u0 ) the maximal solution of Eq. (8) for which v + (t0 + 0) = v0 . Introduce the following conditions:
H9. ω1 ∈ C [t0 − T , t0 ] × R+ , R+
and ω1 (t , 0) = 0, t ∈ [t0 − T , t0 ];
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H10. ω2 ∈ C [t0 , ∞) × R+ , R+
and ω2 (t , 0) = 0, t ∈ [t0 , ∞);
H11. Pk ∈ C [R+ , R+ ] and Pk (0) = 0, k ∈ Z . Definition 11 ([9]). The zero solution u(t ) = 0 of Eq. (7) is said to be uniformly stable to the left if for any ε > 0 there exists η(ε) > 0 such that, if t0 ∈ R and u0 ∈ R+ : 0 ≤ u0 < η, then u+ (t ; t0 , u0 ) < ε for t ∈ [t0 − T , t0 ]. Definition 12 ([9]). The zero solution v(t ) = 0 of Eq. (8) is said to be uniformly stable to the right if for any ε > 0 there exists η(ε) > 0 such that, if t0 ∈ R and u0 ∈ R+ : 0 ≤ u0 < η, then v + (t ; t0 , u0 ) < ε for t ∈ [t0 , ∞]. Let Ω1 and Ω2 be defined as follows:
Ω1 =
x ∈ PC
1
[t0 − T , t0 ], Bα : V (s, x(s)) ≤ V (t , x(t )), t − h ≤ s ≤ t , t ∈ [t0 − T , t0 ]
and
Ω2 =
x ∈ PC
1
[t0 , ∞), Bα : V (s, x(s)) ≤ V (t , x(t )), t − h ≤ s ≤ t , t ∈ [t0 , ∞) .
Therefore, we can formulate the following result. Theorem 3. Let the following assumptions be fulfilled: 1. Conditions H1–H11 hold. 2. There exist functions V ∈ V1 and µ, ν ∈ K such that (i) µ(|z |) ≤ V (t , z ) ≤ ν(|z |), (t , z ) ∈ Γ ; (ii) ω1 (t , V (t , z )) ≤ V˙ (t , z ), t ∈ [t0 − T , t0 ], t ̸= τk , z ∈ Ω1 ; (iii) V˙ (t , z ) ≤ ω2 (t , V (t , z )), t ∈ [t0 , ∞), t ̸= τk , z ∈ Ω2 . 3. The zero solution u(t ) ≡ 0 (v(t ) ≡ 0) of Eq. (7) (Eq. (8)) is uniformly stable to the left (to the right); 4. The functions u + Pk (u), k ∈ Z are monotone increasing in R+ ; 5. V (τk + 0, z + Jk (z )) = V (τk , z ) + Pk (V (τk , z )); 6. The solution x(t ) of Eq. (1) is bounded. Then the solution x(t ) of Eq. (1) is almost periodic. Proof. By employing conditions H1–H11 and the result in [5], it follows that the zero solution of Eq. (1) is strongly stable, that is, the solution x(t ) is strongly stable. Therefore, in virtue of Theorem 1, it follows that x(t ) of (1) is almost periodic. The proof of Theorem 3 is finished. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
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