#### Compactness criteria and new impulsive functional dynamic equations on time scales

Agarwal, RP, Bohner, M: Basic calculus on time scales and some of its applications. Results Math.
Compactness criteria and new impulsive functional dynamic equations on time scales
Chao Wang 2
Ravi P Agarwal 1
Donal O'Regan 0
0 School of Mathematics, Statistics and Applied Mathematics, National University of Ireland , Galway , Ireland
1 Department of Mathematics, Texas A&M University-Kingsville, 700 University Blvd. , Kingsville, TX 78363-8202 , USA
2 Department of Mathematics, Yunnan University , Kunming, Yunnan 650091 , People's Republic of China
In this paper, we introduce the concept of -sub-derivative on time scales to define ε-equivalent impulsive functional dynamic equations on almost periodic time scales. To obtain the existence of solutions for this type of dynamic equation, we establish some new theorems to characterize the compact sets in regulated function space on noncompact intervals of time scales. Also, by introducing and studying a square bracket function [x(·), y(·)] : T → R on time scales, we establish some new sufficient conditions for the existence of almost periodic solutions for ε-equivalent impulsive functional dynamic equations on almost periodic time scales. The final section presents our conclusion and further discussion of this topic.
relatively compact; existence; impulsive functional dynamic equations; almost periodic time scales
1 Introduction
The theory of calculus on time scales (see [, ] and references cited therein) was initiated
by Stefan Hilger in (see []) to unify continuous and discrete analysis. In particular
time the theory of scales unifies the study of differential and difference equations, and the
qualitative analysis of dynamic equations on time scales is of particular importance (see
[–]).
Time scales can be used to describe different natural phenomena in our real world and
most changes in nature are inundated with periodic and almost periodic natural
phenomena, so almost periodic problems of functional dynamic equations are important (see [–
]) (typical examples are time intervals around a celestial body motion, the climate change
during a year, the frequency of a tidal flood or an earthquake, etc.). We always study
periodic or almost periodic problems assuming that the time scale T is periodic, that is, we
always suppose the functions that describe the status of the object are periodic or almost
periodic on periodic time scales (i.e., the status of the object is the same or almost the
same after an accurately chosen interval; see [] and its references). However, this is not
always the case. Since the status of the object is frequently disturbed by its immediate
state, the object’s status is not always the same or almost the same after a precisely equal
time interval. In fact frequently the object’s status with periodicity or almost periodicity
will be always the same or almost the same after ‘an almost equivalent time interval’, and
we say the object has ‘double almost periodicity’. To describe such a situation, recently, the
authors introduced a type of time scales called ‘almost periodic time scales’ which can be
adopted to accurately describe the status of the assigned object which is almost the same
after an almost equivalent time interval (see [–]), and the authors studied the almost
periodic dynamical behavior of impulsive delay dynamic models on almost periodic time
scales.
Although the authors have this effective way to describe ‘double almost periodicity’, the
models proposed in [, ]) can be generalized with delays. In this paper, we investigate
a new general type of impulsive functional dynamic equation and obtain some sufficient
conditions for the existence of solutions. We now recall some ideas in [] that will be used
and improved in this paper. Using measure theory on time scales, the authors obtained
some properties of almost periodic time scales, and they established a new class of delay
dynamic equations which includes all almost periodic dynamic equations on periodic time
scales if we assume that almost periodic time scales are equal to periodic time scales (see
[], Section ). Let ε = E{T, ε}, T ε = {T–τ : –τ ∈ E{T, ε}}, where E{T, ε} is ε-translation
number set of T. Consider two types of delay dynamic equations.
Type I. For t ∈ T ∩ (∪–τ T ε ),
x (t) = g t, x t ± τ (t) , τ : T →
ε.
Type II. For t ∈ T ∩ (∪–τ T ε ),
x (t) = g t,
b
a
x(t ± θ )
ε θ , θ ∈ [a, b] ε .
(.)
(.)
Observe that if T is a periodic time scale, ε will turn into the periodicity set of T, and then
the dynamic equations (.) and (.) will include all delay dynamic equations in Section
from []. As a result the above two types of dynamic equations are more general than
in the literature. Moreover, (.) and (.) are ‘shaky slightly’ since ε is arbitrary, which is
motivated by almost periodic time scales, and such a ‘shake’ occurs in the time variable.
As a result this type of delay dynamic equation can describe the ‘double periodicity’ of the
status of the object. Therefore, the existence of solutions for such a new type of dynamic
equations with ‘sight vibration’ is significant not only for the theory of dynamic equations
on time scales but also for practical applications.
Motivated by the above theoretical and practical significance, in this paper, we propose a
general type of ε-equivalent impulsive functional dynamic equations with ‘sight vibration’
on almost periodic time scales as follows:
and obtain the existence of solutions (including almost periodic solutions) on almost
periodic time scales.
This paper is organized as follows. In Section , we recall some necessary definitions
of almost periodic time scales and introduce the concept of a -sub-derivative on time
scales to define ε-equivalent impulsive functional dynamic equations on almost periodic
time scales. In Section , we establish some new theorems to characterize the compact
sets in regulated function space on noncompact intervals of time scales, which play an
important role in establishing the existence of solutions for ε-equivalent impulsive
functional dynamic equations with such a ‘sight vibration’. In Section , we propose a type of
function [x(·), y(·)] (see Definition .) on time scales and obtain some basic properties of
it. Using this function [x(·), y(·)], we establish some sufficient conditions for the existence
of almost periodic solutions for such a class of ε-equivalent impulsive functional dynamic
equations on almost periodic time scales. In Section , we present conclusions and further
discussion of this topic.
2 Preliminaries
In this section, we first recall some basic definitions and lemmas, which will be used.
Let T be a nonempty closed subset (time scale) of R. The forward and backward jump
operators σ , ρ : T → T and the graininess μ : T → R+ are defined, respectively, by
σ (t) = inf{s ∈ T : s > t},
ρ(t) = sup{s ∈ T : s < t},
μ(t) = σ (t) – t.
A point t ∈ T is called left-dense if t > inf T and ρ(t) = t, left-scattered if ρ(t) < t,
rightdense if t < sup T and σ (t) = t, and right-scattered if σ (t) > t. If T has a left-scattered
maximum M, then Tk = T\{M}; otherwise Tk = T. If T has a right-scattered minimum m, then
Tk = T\{m}; otherwise Tk = T.
Definition . A function f : T → R is right-dense continuous provided it is continuous
at right-dense point in T and its left-side limits exist at left-dense points in T. If f is
continuous at each right-dense point and each left-dense point, then f is said to be a continuous
function on T.
Definition . For y : T → R and t ∈ Tk , we define the delta derivative of y(t), y (t), to be
the number (if it exists) with the property that, for a given ε > , there exists a
neighborhood U of t such that
y σ (t) – y(s) – y (t) σ (t) – s < ε σ (t) – s
for all s ∈ U. Let y be right-dense continuous, and if Y (t) = y(t), then we define the delta
integral by
a
t
y(s) s = Y (t) – Y (a).
Definition . A function p : T → R is called regressive provided + μ(t)p(t) = for all
t ∈ Tk . The set of all regressive and rd-continuous functions p : T → R will be denoted by
R = R(T) = R(T, R). We define the set R+ = R+(T, R) = {p ∈ R : + μ(t)p(t) > , ∀t ∈ T}.
An n × n-matrix-valued function A on a time scale T is called regressive provided
T
I + μ(t)A(t) is invertible for all t ∈ ,
and the class of all such regressive and rd-continuous functions is denoted, similar to the
above scalar case, by R = R(T) = R(T, Rn×n).
Definition . If r is a regressive function, then the generalized exponential function er
is defined by
er(t, s) = exp
ξμ(τ) r(τ )
τ
s
ξh(z) =
z,
if h = .
for all s, t ∈ T, with the cylinder transformation
For more details as regards dynamic equations on time scales, we refer the reader to
[–].
In the following, we give some basic definitions and results of almost periodic time
scales. For more details, one may consult [–].
Let τ be a number, and we set the time scales:
where
Let
+∞
i=–∞
T :=
Define the distance between two time scales, T and Tτ by
d T, Tτ = max sup αi – αiτ , sup βi – βiτ ,
i∈Z i∈Z
αiτ := inf α ∈ Tτ : |αi – α|
and
βiτ := inf β ∈ Tτ : |βi – β| .
:= τ ∈ R : T ∩ Tτ = ∅ = {}.
(.)
Next, for an arbitrary time scale T, we can introduce the following new definition.
Definition . Let S = {s˜ ∈ T : s˜ is a right- or left-scattered point in T}. For any s ∈ and
any right-scattered point t in T ∩ Ts, define dt = infs˜∈S |t – s˜|. Let the forward jump operator
σs(t) = inf{sˆ ∈ T ∩ Ts : sˆ > t}, t ∈ T ∩ Ts. We introduce a piecewise graininess function μs :
T ∩ Ts → R+ as follows:
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎧⎪⎪⎪⎪⎪⎪ μμ,((tt)++ddt)σ,(t),
μs(t) = ⎨
t is a right-dense point in T ∩ Ts,
t is a right-scattered point in both T ∩ Ts and T,
and dt = , dσ (t) > ,
t is a right-scattered point in T ∩ Ts and a right-dense
point in T, and dt > , dσ (t) = ,
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ μμ((tt)+, dt) + dσ (t), ttapnoiissdinaadtrrtiii,nggdhhTσtt(--ta)ssncc=aadttttd.eetrr,eedddσ (ppt)oo>iinntt, iinn bTo∩thTTs a∩nTdsaarnigdhTt-,dense
ϕ s (tˆ) = lim ϕ(σs(t)) – ϕ(tˆ)
t→tˆ σs(t) – tˆ
exists for t, tˆ ∈ T ∩ Ts (see Figures -).
In the following, we introduce some basic definitions of almost periodic time scales.
Definition . ([, ]) A subset S of is called relatively dense if there exists a positive
number L ∈ such that [a, a + L] ∩ S = ∅ for all a ∈ . The number L is called the
inclusion length.
Definition . ([, ]) We say T is an almost periodic time scale if for any give ε > ,
there exists a constant l(ε) > such that each interval of length l(ε) contains a τ (ε) ∈
such that
d T, Tτ < ε,
i.e., for any ε > , the set
E{T, ε} = τ ∈
: d Tτ , T < ε
is relatively dense in . Now τ is called the ε-translation number of T and l(ε) is called the
inclusion length of E{T, ε}, E{T, ε} is called the ε-translation set of T, and for simplicity,
we use the notation E{T, ε} := ε. Note that sup T = +∞, inf T = –∞.
According to Definition ., we obtain the following useful lemmas.
Lemma . Let T be an almost periodic time scale. If τ, τ ∈
(τ, τˆ+ τ] such that ξ ∈ ε, where
ε, then there exists ξ ∈
⎧
(τ, τˆ+ τ] = ⎨ (τ, τ + τ], τ < τ + τ,
⎩ [τ + τ, τ), τ > τ + τ.
Proof From the condition of the lemma, we obtain
d T, Tτ+τ < d T, Tτ + d Tτ , Tτ+τ < ε.
Case I. If d(T, Tτ+τ ) < ε, then ξ = τ + τ, and we get the desired result.
Case II. Let ε > d(T, Tτ+τ ) > ε. Note (.), let
f (x) = d T, Tx , x ∈ (τ, τˆ+ τ],
and note f (x) is continuous on R. Now, let F(x) = f (x) – ε, and we obtain
F(τ) = f (τ) – ε < ,
F(τ + τ) = f (τ + τ) – ε > ,
so, there exists some ξ ∈ (τ, τˆ+ τ) such that F(ξ ) = . From the continuity of F , we see
that there exists some ξ ∈ (τˆ, ξ ) such that F(ξ) < , i.e., f (ξ) < ε. This completes the
proof.
Remark . From Lemma ., we can see that ε is an infinite number set.
Lemma . For any ε > and τ ∈ ε, if t ∈ T ∩ Tτ , then there exists τ ∈ ε with τ > τ
such that t ∈ T ∩ Tτ .
Proof From Lemma ., there exists ξ > τ such that d(T, Tξ ) < ε.
Case I. If for t ∈ T ∩ Tτ , we have t ∈ T ∩ Tξ , and then τ = ξ ∈ ε.
Case II. If for t ∈ T ∩ Tτ , we have t ∈/ T ∩ Tξ , then t ∈/ Tξ and
d Tτ , Tξ < d T, Tτ + d T, Tξ < ε.
Then we denote γ = inft∈Tξ |t – t| + ε, and we get
ti∈nTfξ |t – t| ≤ d Tξ , T < ε,
so we have γ < ε. Therefore, by (.), we obtain
t ∈inTfτ αiξ + γ – t < ε and
t ∈inTfτ βiξ + γ – t < ε,
so
αiξ + γ – αiτ < ε and
βiξ + γ – βiτ < ε,
(.)
and so we obtain
Hence,
αiξ + γ > αiτ – ε and
βiξ + γ > βiτ – ε.
d T, Tξ+γ = max sup αi – αiξ + γ , sup βi – βiξ + γ
i∈Z i∈Z
≤ max sup αi – αiτ – ε , sup βi – βiτ – ε
i∈Z i∈Z
= d T, Tτ < ε.
Hence, we have ξ + γ ∈ ε and t ∈ T ∩ Tξ+γ . Hence, we can take τ = ξ + γ > τ such
that t ∈ T ∩ Tτ . This completes the proof.
Remark . From Lemma ., one see that if τ ∈
ε : t ∈ T ∩ Tτ } is an infinite number set.
Let μτ : Tτ → R+ be the graininess function of Tτ , and we obtain
ε and t ∈ T ∩ Tτ , then the set {τ ∈
μτ (t + τ ) = ⎨⎧ μ(t), t + τ ∈/ T,
T
⎩ μ(t + τ ), t + τ ∈ .
(.)
Thus, from (.), we can simplify Definition . as follows.
Definition . Let μ : T → R+ be a graininess function of T. We say T is an almost
periodic time scale if for any ε > , the set
∗ = τ ∈
: μ(t + τ ) – μ(t) < ε, ∀t ∈ T ∩ T–τ
is relatively dense in .
Definition . ([, ]) Let T be an almost periodic time scale, i.e., T satisfies
Definition .. A function f ∈ C(T × D, En) is called an almost periodic function in t ∈ T
uniformly for x ∈ D if the ε-translation set of f
E{ε, f , S} = τ ∈ ε : f (t + τ , x) – f (t, x) < ε, for all (t, x) ∈ T ∩ T–τ
× S
is a relatively dense set in ε for all ε > ε > and for each compact subset S of D; that is,
for any given ε > ε > and each compact subset S of D, there exists a constant l(ε, S) >
such that each interval of length l(ε, S) contains a τ (ε, S) ∈ E{ε, f , S} such that
f (t + τ , x) – f (t, x) < ε, for all (t, x) ∈ T ∩ T–τ
× S.
This τ is called the ε-translation number of f and l(ε, S) is called the inclusion length of
E{ε, f , S}.
Remark . Note that Definition . from [] is equivalent to Definition .. For
Definition . from [], we note the following:
(.)
(.)
(.)
(i) for all t ∈ T ∩ (∪T ε ), there exists –τ ∈ ε such that t ∈ T ∩ T–τ . From
Remark ., we know that the set {–τ ∈ ε : t ∈ T ∩ T–τ } is an infinite number set.
(ii) d(T ∩ (∪T ε ), T) < ε.
(iii) According to (i) from Remark ., for all t ∈ T ∩ (∪T ε ), there exists an infinite
number set ¯ ⊂ ε such that t + τ ∈ T.
Definition . ([, ]) Let T be an almost periodic time scale and assume that {τi} ⊂ T
satisfying the derived sequence {τij}, i, j ∈ Z, is equipotentially almost periodic. We call a
function ϕ ∈ PCrd(T, Rn) almost periodic if:
(i) for any ε > , there is a positive number δ = δ(ε) such that if the points t and t
belong to the same interval of continuity and |t – t | < δ, then ϕ(t ) – ϕ(t ) < ε;
(ii) for any ε > ε > , there is a relative dense set of ε-almost periods such that if
τ ∈ ⊂ ε , then ϕ(t + τ ) – ϕ(t) < ε for all t ∈ T ∩ (∪T ε ) which satisfy the
Z
condition |t – τi| > ε, i ∈ .
Remark . From Definition ., if T is a periodic time scale, then one can take a
periodicity set ˜ ⊂ of T, such that μs(t) = μ(t) for all t ∈ T ∩ Ts = T, s ∈ ˜ , dt = dσ (t) = . If
T is an almost periodic time scale from [], then one can take a ε-translation number set
ε ⊂ of T, such that |μs(t) – μ(t)| < ε for all t ∈ T ∩ Ts, s ∈ ε, i.e., for all right-scattered
and right-dense points t ∈ T ∩ Ts, one can obtain dt, dσ (t) < ε.
According to Definition . and Remark ., we can introduce a concept of ε-equivalent
impulsive functional dynamic equations on almost periodic time scales as follows.
Definition . Let T be an almost periodic time scale. Consider the following impulsive
functional dynamic equations with sub-derivative x –s (t) on T ∩ T–s:
⎩
⎧⎨ x –s (t) = f (t, xt),
t ∈ T ∩ T–s, t = tk, k ∈ Z,
Z
x(tk) = Ik(x(tk)), t = tk, k ∈ ,
⎨⎧ x (t) = f (t, xt), t = tk, k ∈ Z,
⎩ x(tk) = Ik(x(tk)), t = tk, k ∈ Z,
where t ∈ T ∩ (∪T ε ).
where –s ∈ , xt(s) = x(t + s). We say the functional dynamic equations
⎧⎨ x (t) = f (t, xt), t ∈ T ∩ T–s, t = tk, k ∈ Z,
⎩ x(tk) = Ik(x(tk)), t = tk, k ∈ Z,
are ε-equivalent impulsive functional dynamic equations for (.) if –s ∈
that (.) can also be written as
ε ⊂
. Note
Remark . Let T be an almost periodic time scale. According to Theorem . and
Remark . from [], one can observe that we say the dynamic equations
⎩
⎧⎨ x (t) = f (t, xt),
t ∈ T, t = tk, k ∈ ,
Z
Z
x(tk) = Ik(x(tk)), t = tk, k ∈ ,
have an almost periodic solution on the almost periodic time scale T if (.) has an almost
periodic solution on T ∩ T–s for any –s ∈ ε.
Remark . According to Theorems . and . from [], for (.), if we let xt(τ (t)) =
x(t – τ (t)), τ : T → ε, then it becomes
⎧⎨ x (t) = f (t, x(t – τ (t))), t = tk, k ∈ Z,
⎩ x(tk) = Ik(x(tk)), t = tk, k ∈ Z,
if we let xt(θ ) = ab x(t + θ )
ε θ , and then it becomes
⎩
⎨⎧ x (t) = f (t, ab x(t + θ )
x(tk) = Ik(x(tk)),
Z
ε θ ), t = tk, k ∈ ,
Z
t = tk, k ∈ .
Lemma . ([]) Let A be a compact convex subset of a locally convex (linear topological)
space and f be a continuous map of A into itself. Then f has a fixed point.
3 Characterizations of compact sets in a regulated functional space on time
scales
In this section, we introduce some new definitions and establish new characterization
results of compact sets in functional spaces on time scales which will play an important
role in studying abstract discontinuous dynamic equations on time scales.
First, let δL+∞, δR+∞ : [T, +∞)T → R+ ∪ {}. Similar to [], we can extend the -gauge
for [a, b]T to [T, +∞)T.
Definition . We say δ+∞ = (δL+∞, δR+∞) is a -gauge for [T, +∞)T provided δL+∞(t) >
on (T, +∞)T and δL+∞(T) ≥ , δR+∞(t) > on [T, +∞)T and δR+∞(t) ≥ μ(t) for all t ∈
[T, +∞)T.
For a -gauge, δ+∞, we always assume δL+∞(T) ≥ (we will sometimes not even point
this out).
For T ∈ T and a Banach space (X, · ), let
G [T, +∞)T, X := x : [T, +∞)T → X;
s, t < +∞ and
Endow G([T, +∞)T, X) with the norm x ∞ = supt∈[T,+∞)T x(t) .
Lemma . (G([T, +∞)T, X), · ∞) is a Banach space.
Proof Let {xn} be an arbitrary Cauchy sequence in G, i.e., for any ε > , there exists N such
that n, m > N implies
xn(t) – xm(t) < ε for all t ∈ [T, +∞)T.
xn(t) – x(t) < ε for all t ∈ [T, +∞)T.
Since X is a Banach space, for each t ∈ [T, +∞)T, {xn(t)} ⊂ X is a Cauchy sequence so
xn(t) → x(t). Hence, let m → +∞ in (.), and we have
Furthermore, since {xn} ⊂ G, for any ε > and t ∈ [T, +∞)T, there exists δ > , t ∈ (t –
δL+∞(t), t)T with
(.)
(.)
(.)
Similarly, for t ∈ (t, t + δR+∞(t))T, we also obtain
x(t) – x t–
≤ x(t) – xn(t) + xn(t) – xn t–
+ xn t– – x t–
≤ ε.
Therefore, from (.) and (.), we obtain x ∈ G. Hence, G is a Banach space.
In the following, we will introduce the definition of a partition P for [T, +∞)T.
Definition . A partition P for [T, +∞)T is a division of [T, +∞)T denoted by
P = T = tP ≤ η ≤ tP ≤ · · · ≤ tnP– ≤ ηn ≤ tnP ≤ · · · < · · ·
with tiP > tiP– for i = , , . . . and ti, ηi ∈ T. We call the points ηi tag points and the points ti
end points.
Definition . If δ+∞ is a -gauge for [T, +∞)T, then we say a partition P is δ+∞-fine if
ηi – δL+∞(ηi) ≤ tiP– < tiP ≤ ηi + δR+∞(ηi)
for i = , , . . . .
Remark . From Definition ., one can observe that if a partition P is δ+∞-fine for
[T, +∞)T, then for any closed interval [a, b]T ⊂ [T, +∞)T, there must exist a δ-fine
partition P∗ and P∗ ⊂ P .
Definition . A set A ⊂ G([T, +∞)T, X) is called uniformly equi-regulated, if it has the
following property: for every ε > and t ∈ [T, +∞)T, there is a δ+∞ = (δL+∞, δR+∞) such
that
(a) If x ∈ A, t ∈ [T, +∞)T and t – δL+∞(t) < t < t, then x(t–) – x(t ) < ε.
(b) If x ∈ A, t ∈ [T, +∞)T and t < t < t + δR+∞(t), then x(t+) – x(t ) < ε.
From Definition ., we obtain the following theorem.
Theorem . A set A ⊂ G([T, +∞)T, X) is uniformly equi-regulated, if and only if, for
every ε > , there is a δ+∞-fine partition P :
T = tP < tP < tP < · · · < tnP < · · · < · · ·
such that
x t – x(t) ≤ ε,
for every x ∈ A and [t, t ]T ⊂ (tjP–, tjP )T, j = , , . . . .
Proof Let ε > be given and let
D = ξ ; ξ ∈ (T, +∞)T
such that there is a partition P :
T = tP < tP < · · · < tkP = ξ
(.)
for which (.) holds with j = , , . . . , k.
(i) If A ⊂ G([T, +∞)T, X) is uniformly equi-regulated, then there is a δR+∞(T) > such
that
x ξ– – x(t) <
ε
for every x ∈ A and t ∈ (T, T + δR+∞(T))T. Denote ξ = T + δR+∞(T), T = tP < tP = ξ.
Thus, for [t, t ]T ⊂ (T, ξ)T and x ∈ A, the inequalities
≤ x(t) – x T+
+ x t – x T+
≤ ε,
holds and we have ξ ∈ D.
Let ξ > ξ > T. Since x ∈ A, then there is a δL+∞(ξ) such that
for every x ∈ A and t ∈ (ξ – δL+∞(ξ), ξ)T ∩ [T, +∞)T.
Let ξ˜ ∈ (ξ – δL(ξ), ξ)T and a partition T = tP < tP < tP < · · · < tkP = ξ˜ be such that (.)
holds with j = , , . . . , k. Denote tkP+ = ξ. Then for [t, t ]T ⊂ (tkP , tkP+)T and x ∈ A, we have
≤ x(t) – x ξ–
+ x t – x ξ–
≤ ε
which implies ξ ∈ D. Thus we use the same argument as before to find that ξi ∈ (ξi–, +∞)T
such that ξi ∈ D, i = , , . . . . Hence ξ∞ = sup D = +∞ and we are finished.
(ii) Reciprocally, for any given ε > , there is a δ+∞-fine partition P : T = tP < tP < tP <
· · · < tkP < · · · < · · · such that
x(t) – x(T) < , t ∈ [T, +∞)T.
Let C be the set of all τ ∈ (T, T]T such that there exists Kτ > such that
x(t) – x(T) ≤ Kτ ,
for any x ∈ A and t ∈ [T, τ ]T.
for every x ∈ A and [t, t ]T ⊂ (tjP–, tjP )T, j = , , . . . .
Let ηj be a tag of (tj–, tj)T. Since this partition is δ+∞-fine, we have (tjP–, tjP )T ⊂ (ηj –
δ+∞(ηj), ηj + δR+∞(ηj))T. Therefore, the inequality (.) holds, for t, t ∈ (ηj – δL+∞(ηj), ηj +
L
δ+∞(ηj))T. Taking t = ηj– and t ∈ (ηj – δL+∞(ηj), ηj]T, then the inequality (.) remains true.
R
Also, if t = ηj+ and t ∈ [ηj, ηj + δR+∞(ηj))T, the inequality (.) is fulfilled. Then, from
Definition ., it follows that A is uniformly equi-regulated. This completes the proof.
Definition . Let A ⊂ G([T, +∞)T, X). We say A is uniformly Cauchy if for any ε > ,
there exist T ∈ (T, +∞)T and a δ+∞ = (δL+∞, δR+∞)-fine partition P :
tP = T < tP < tP < · · · < tnP < · · · < · · ·
such that:
(a) If x ∈ A, t , t ∈ [T, +∞)T and t – δL+∞(t) < t < t, then x(t–) – x(t ) < ε.
(b) If x ∈ A, t , t ∈ [T, +∞)T and t < t < t + δR+∞(t), then x(t+) – x(t ) < ε.
(c) If x ∈ A, t ∈ [a , b ]T ⊂ (tjP–, tjP )T, t ∈ [a , b ]T ⊂ (tiP–, tiP )T, i, j = , , . . . , then
x(t ) – x(t ) < ε.
Remark . From Definition ., one can observe that if A ⊂ G([T, +∞)T, X) is
uniformly Cauchy, then there exists T ∈ (T, +∞)T such that A is uniformly equi-regulated
on [T, +∞)T.
Theorem . Assume that a set A ⊂ G([T, +∞)T, X) is uniformly equi-regulated and
uniformly Cauchy, and for any t ∈ [T, +∞)T, there is a number βt such that, for x ∈ A,
x t+ – x(t) ≤ βt, t ∈ [T, +∞)T.
Then there is a constant K > such that x(t) – x(T) ≤ K , for every x ∈ A and t ∈
[T, +∞)T.
Proof Since A is uniformly Cauchy, according to Definition ., there exists T ∈ (T, +∞)T
such that
(.)
(.)
(.)
Since A is uniformly equi-regulated, there is a δR+∞(T) such that
for every x ∈ A and t ∈ (T, T + δR+∞(t)]T. This fact together with the hypothesis implies
that
x(t) – x(T) ≤ x(t) – x T+
+ x T+ – x(T) ≤ + βT := KT+δ+∞ ,
x(t) – x(T) ≤ x(t) – x τ–
+ x τ– – x(τ ) + x(τ ) – x(T) ≤ + + Kτ = + Kτ ,
for every x ∈ A and t ∈ (τ , τ)T. Also,
x τ– – x(T) ≤ x τ– – x(τ ) + x(τ ) – x(T) ≤ + Kτ .
These inequalities and this hypothesis imply that
x(τ) – x(T) ≤ x(τ) – x τ–
+ x τ– – x(T) ≤ βτ + + Kτ .
(.)
Thus τ ∈ C, where Kτ = βτ + + Kτ .
If τ < T, then, since A is uniformly equi-regulated, there is a δR+∞(τ) > such that
x(t) – x τ+
≤ , for any x ∈ A and t ∈ τ, τ + δR(τ) T,
which implies
x(t) – x(T) ≤ x(t) – x τ+
+ x τ+ – x(τ) + x(τ) – x(T)
≤ + βτ + Kτ = Kτ+δR+∞(τ),
for t ∈ (τ, τ + δR+∞(τ)]T and x ∈ A. Thus τ + δR+∞(τ) ∈ C, which contradicts the fact that
τ = sup C. Therefore, τ = T. Hence, by (.), we have
x(T) – x(T) ≤ Kτ .
Combining with (.), we have
x(t) – x(T) ≤ x(t) – x(T) + x(T) – x(T) ≤ + Kτ ,
for t ∈ (T, +∞)T. Then we can get the desired result.
Now, we give some sufficient conditions to guarantee that A is relatively compact in
G([T, +∞)T, X).
Theorem . Let A ⊂ G([T, +∞)T, X) be uniformly equi-regulated and uniformly
Cauchy, for every t ∈ [T, +∞)T, and let the set {x(t); x ∈ A} be relatively compact in X.
Then the set A is relatively compact in G([T, +∞)T, X).
Proof Since A is uniformly Cauchy, ∀x ∈ A, by (a), (b) from Definition ., for any ε > ,
there exists T ∈ (T, +∞)T, there is a δ+∞ = (δL+∞, δR+∞)-fine partition P:
tP = T < tP < tP < · · · < tnP < · · · < · · ·
for each j = , , , . . . . Similarly, from (.), (.), and (.), we also have
x T
+
– x tjP+
+
≤ x T
– x t
+ x tjP+ – x t
+ x t
– x t
< ε,
for each j = , , , . . . . Hence, from (.), we obtain
x T + δ+∞(T) – x(t) < ε, t ∈ tjP– + δ+∞ tjP– , tjP – δ+∞ tP
R R L j
T.
From (.) and (.), we get
x T + δR+∞(T) – x(t) = x tjP– – x(t) + x T + δR+∞(T) – x tjP–
< ε, t ∈ tjP – δL+∞ tjP , tjP T.
Similarly, from (.) and (.), we also obtain
x tjP+ – x(t) = x T
+
– x(t) + x T
+
– x tjP+
< ε, t ∈ tjP , tjP + δ+∞ tP
R j
so we obtain
x T+ – x(t) < ε, t ∈ tjP , tjP + δR+∞ tjP T.
(.)
Further, since A is uniformly equi-regulated on [T, +∞)T, given ε > , there is a
δ+∞fine partition P:
tP = T < tP < · · · < tKP = T,
such that
x t – x(t) < ε,
xn tjP , xn T + δR+∞(T) , xn T+ , n ∈ N
xnk tjP , xnk T + δR+∞(T) , xnk T+ , k ∈ N
for every [t, t ]T ⊂ (tjP– , tjP )T, j ∈ {, , . . . , K }. Obviously, P = P ∪ P is a δ+∞-fine
partition for [T, +∞)T.
Let {xn; n ∈ N} be a given sequence. By assumption, the set
is relatively compact in X for every j = , , , . . . , K . Therefore, we can find a subsequence
of indices {nk; k ∈ N} ⊂ {n; n ∈ N} such that the set
is also relatively compact in X for every j = , , . . . , K . Using this fact, we can find the
elements {yj; j = , , , . . . , K , K + , K + } ⊂ X such that yj = limk→∞ xnk (tjP ), yK+ =
limk→∞ xnk (T + δR+∞(T)) and yK+ = limk→∞ xnk (T+). Therefore, there exists N ∈ N such
that, for every k > N , we have
for any j = , , . . . , K . Similarly, we also have
ε
xnq T + δR+∞(T) – yK+ < ,
or t ∈ (tjP– , tjP )T for some j ∈ {, , . . . , K } and, in this case, we have
Moreover, from (.), let t ∈ [T, +∞)T, we also obtain
xnk (t) – xnq (t) ≤ xnk (t) – xnq T + δR+∞(T)
+ xnq (t) – xnk T + δR+∞(T)
+ xnq T + δR+∞(T) – xnk T + δR+∞(T)
ε ε ε
< + + < ε,
for t ∈ [tjP– + δR+∞(tjP–), tjP – δL+∞(tjP )]T, j ∈ {, , . . .}.
Similarly, from (.), we also obtain
xnk (t) – xnq (t) ≤ xnk (t) – xnq T + δR+∞(T)
+ xnq (t) – xnk T + δR+∞(T)
for t ∈ (tjP , tjP + δR+∞(tjP )]T, j ∈ {, , , . . .}.
Thus, for t ∈ [T, +∞)T, the sequence
xnk (t); k ∈ N ⊂ X
is a Cauchy sequence. Since X is complete, limk→∞ xnk (t) exists. Hence, any {xn} ⊂ A has
a convergent subsequence which means that A is a relatively compact set. The proof is
complete.
In the following, let X = Rn; we will give some sufficient conditions to guarantee that
A ⊂ G([T, +∞)T, Rn) is relatively compact.
Theorem . Let a set A ⊂ G([T, +∞)T, Rn). If A is relatively compact in the sup-norm
topology, then it is uniformly equi-regulated. If A is uniformly equi-regulated and uniformly
Cauchy, satisfying (.), then A is relatively compact in G([T, +∞)T, Rn).
Proof A subset A of a Banach space X is relatively compact if and only if it is totally
bounded, i.e., for every ε > , there is a finite ε-net F for A, i.e., such a subset F =
{x, x, . . . , xk} of X that, for every x ∈ A, there is xn ∈ F satisfying x – xn ≤ ε.
and
(i) Assume that A is relatively compact. Then it is bounded by a constant C, and evidently
(.) is satisfied with βt = C for every t ∈ [T, +∞)T.
Let t ∈ [T, +∞)T and ε be given. Let {x, x, . . . , xk} ⊂ G([T, +∞)T, Rn) be a finite
ε/net for A. For every n = , , . . . , k, there is a δn+∞ = (δL+∞,n, δR+∞,n) such that
for t ∈ t, t + δR+∞,n(t) T ∩ [T, +∞)T
ε
xn t– – xn(t) <
for t ∈ t – δL+∞,n(t), t T ∩ [T, +∞)T.
Denote δ+∞ = (min≤n≤k(δL+∞,n(t)), min≤n≤k(δR+∞,n(t))) = (δL+∞(t), δR+∞(t)).
For arbitrary x ∈ A, we can find xn such that x – xn ∞ ≤ ε/ for every t ∈ (t, t +
δR+∞(t))T ∩ [T, +∞)T, and we have the inequality
and similarly, |x(t–) – x(t)| < ε for t ∈ (t – δL+∞(t), t)T.
(ii) Assume that A is uniformly equi-regulated, (.) holds, i.e., there exists α˜ > such
that |x(T)| ≤ α˜ for every x ∈ A.
From Theorem ., there is K such that |x(t) – x(T)| ≤ K for any x ∈ A and t ∈
[T, +∞)T. Hence, |x(t)| ≤ |x(t) + x(T)| + |x(T)| ≤ K + α˜ . If we denote γ˜ = K + α˜ , then
x ≤ γ˜ for x ∈ A. Since A is uniformly bounded in Rn, A is a sequentially compact closed
set in Rn, i.e., A is relatively compact in Rn. From Theorem ., one can obtain A is a
relatively compact set in G([T, +∞)T, Rn). This completes the proof.
Remark . If A˜ is uniformly bounded, equi-continuous and uniformly Cauchy, then A˜ ⊂
A. Hence, one can observe that Lemma from [] is just a particular case of Theorem ..
Remark . For a, b ∈ T, let
G [a, b]T, X := x : [a, b]T → X;
sl→imt+ x(s) = x t+ and sl→imt– x(s) = x t– exist and are finite ,
and according to each x ∈ G, we can construct the set G([a, +∞)T, X), satisfying, for each
x˜ ∈ G:
x˜(t) = ⎨⎧ x(t), t ∈ [a, b]T,
⎩ x(b–), t ∈ (b, +∞)T.
One can immediately see that G and G are topological homeomorphic, i.e., there exists
a homeomorphic mapping f : G → G such that f (x) = x˜ and f –(x˜) = x. Since for any
set A ⊂ G, according to the construction of the set G, one can see that f (A) is uniformly
Cauchy. If the set {f (x); x ∈ A} is relatively compact in X, then, by Theorem ., the relative
compactness of f (A) is decided by the uniformly equi-regulatedness of f (A). However, for
all t ∈ (b, +∞)T, obviously, f (A) is uniformly equi-regulated. Thus the uniformly
equiregulatedness of f (A) on [a, +∞)T is actually decided by the equi-regulatedness of A on
[a, b]T. Therefore, if A is equi-regulated on [a, b]T, then f (A) ⊂ G is relatively compact,
i.e., A is relatively compact in G.
From Theorem . and Remark ., we can obtain the following corollaries.
Corollary . Let X = Rn. A set A ⊂ G is relatively compact if and only if it is
equiregulated and for every t ∈ [a, b]T, the set {x(t); x ∈ A} is bounded in Rn.
Proof If A ⊂ G is relatively compact, then A is totally bounded in Rn, and then for every
t ∈ [a, b]T, {x(t); x ∈ A} is bounded in Rn. Moreover, by Remark ., there exists a
homeomorphic mapping f : G → G such that f (A) ⊂ G is relatively compact, which means
that f (A) ⊂ G is equi-regulated on [a, b]T according to Theorem .. Since f is continuous
and a one-to-one mapping, A is equi-regulated on [a, b]T.
If A ⊂ G is equi-regulated and for every t ∈ [a, b]T, the set {x(t); x ∈ A} is bounded in
Rn, then f (A) ⊂ G is equi-regulated and for every t ∈ [a, +∞)T, the set {f (x); f (x) ∈ f (A)}
is bounded in Rn, according to Theorem ., f (A) is relatively compact in G, i.e., A is
relatively compact in G. This completes the proof.
Corollary . Let A ⊂ G be equi-regulated, and for every t ∈ [a, b]T, let the set {x(t); x ∈
A} be relatively compact in X. Then the set A is relatively compact in G([a, b]T, X).
Proof From the assumption of this corollary and Remark ., by Theorem ., we can
see that the set f (A) is relatively compact in G([a, +∞)T, X), i.e., the set A is relatively
compact in G([a, b]T, X). This completes the proof.
Remark . In fact, if we let T = R, Corollaries . and . can include Corollary . from
[] and Theorem . from [], respectively.
Similarly, for T¯ ∈ T and a Banach space (X, · ), let
G (–∞, T¯ ]T, X := x : (–∞, T¯ ]T → X;
sl→imt+ x(s) = x t+ and sl→imt– x(s) = x t– exist and are finite,
From Definition ., we can also introduce a δ–∞ and extend the -gauge for [a, b]T to
(–∞, T¯]T, and then we can repeat the same above discussion, and the following theorems
can also be obtained (we omit the proofs).
Theorem . Assume that a set A ⊂ G((–∞, T¯ ]T, X) is uniformly equi-regulated and
uniformly Cauchy, and for any t ∈ (–∞, T¯ ]T, there is a number βt such that, for x ∈ A,
x t+ – x(t) ≤ βt, t ∈ (–∞, T¯ ]T.
(.)
Then there is a constant K > such that x(t) – x(T¯ ) ≤ K , for every x ∈ A and t ∈
(–∞, T¯ ]T.
Theorem . Let A ⊂ G((–∞, T¯ ]T, X) be uniformly equi-regulated and uniformly
Cauchy, for every t ∈ (–∞, T¯ ]T, let the set {x(t); x ∈ A} be relatively compact in X. Then
the set A is relatively compact in G((–∞, T¯ ]T, X).
Theorem . Let a set A ⊂ G((–∞, T¯ ]T, Rn). If A is relatively compact in the sup-norm
topology, then it is uniformly equi-regulated. If A is uniformly equi-regulated and uniformly
Cauchy, satisfying (.), then A is relatively compact in G((–∞, T¯ ]T, Rn).
For the more general case, for a Banach space (X, · ), let
G (–∞, +∞)T, X := x : (–∞, +∞)T → X;
sl→imt+ x(s) = x t+ and sl→imt– x(s) = x t– exist and are finite,
– ∞ < s, t < +∞ and
sup
t∈(–∞,+∞)T
From Definition ., we can also introduce a δ±∞ and extend the -gauge for [a, b]T to
(–∞, +∞)T, since for any T > T¯ , (–∞, +∞)T = (–∞, T¯ ]T ∪ [T¯ , T]T ∪ [T, +∞)T, and
then we can repeat the above discussion, and the above similar theorems can also be
obtained (we omit these similar statements here).
Let
G[T, +∞)T := x; x ∈ PCrd [T, +∞)T, Rn and
where PCrd([T, +∞)T, Rn) is the set formed by all rd-piecewise continuous functions (one
can consult Definition . from []). Endow G with the norm x = supt∈[T,+∞)T |x(t)|,
and note (G, · ) is a Banach space.
Next, we will establish some theorems to guarantee that A ⊂ G([T, +∞)T, Rn) is
uniformly Cauchy and uniformly equi-regulated.
Now, we give a new definition called ‘equi-absolutely continuity’ for the function set
A ⊂ G, which will be used in the following lemma’s proof.
Definition . Let A ⊂ G. We say A is equi-absolutely continuous if for any f ∈ A and
ε > , there exists δ > , such that for any finite mutually disjoint open interval (xi, yi)T ⊂
[T, +∞)T (i = , , . . . , n)
implies
n
i=
n
i=
(yi – xi) < δ
f (xi) – f (yi) < ε.
Lemma . Let A ⊂ G([T, +∞)T, Rn) be uniformly bounded and
Proof From the condition of the theorem, since x is -differentiable at [T, +∞)T\Z and
there exists M > such that |x (t)| ≤ M, according to Corollary . from [], we can
obtain for all t ∈ [T, +∞)T\Z , x(t) satisfies the Lipschitz condition
and
N
j=
N
j=
x(t) – x(t) ≤ M|t – t|,
∀t, t ∈ [T, +∞)T\Z .
So for each N ∈ Z+, we can obtain
which implies
tj – tj <
ε
M ,
tj, tj T ⊂ [T, +∞)T,
x tj – x tj
< M
tj – tj < ε,
N
j=
i.e., A is equi-absolutely continuous on [T, +∞)T\Z . Hence, from (.) and (.), for
all t ∈ [T, +∞)T\Z , A is uniformly equi-regulated. For all t ∈ Z , since μ (Z ) = , from
(.) and (.), we can take δ+∞ = (δL+∞, δR+∞) and δL, δR < Mε , so
N
j=
N
j=
x t+ – x(t) <
x tj – x tj
< ε if t ∈ t, t + δR+∞(t) T
x t– – x(t) <
x tj – x tj
< ε if t ∈ t – δL+∞(t), t T,
and thus A is uniformly equi-regulated on [T, +∞)T. According to Theorem ., for any
closed interval [aP , bP ]T, there is a δ+∞-fine partition P:
aP := T = tP < tP < tP < · · · < tNP := bP
such that
x t – x t
≤ ,
(.)
(.)
for every x ∈ A and [t , t ]T ⊂ (tjP–, tjP )T, j = , , . . . , N . Hence, we have
x tjP–
– x tjP–+
≤ x tjP–
– x t
+ x t – x t
+ x t
– x tjP–+
≤ ,
where t ∈ (tjP – δ+∞(tjP ), tjP )T, t ∈ (tjP–, tjP– + δ+∞(tjP–))T.
L R
Similarly, for any closed interval [aP , bP ]T, there is a δ+∞-fine partition P:
aP := bP = tP < tP < tP < · · · < tNP := bP
such that
,
for every x ∈ A and [t , t ]T ⊂ (tjP– , tjP )T, j = , , . . . , N . Hence, we have
x tjP–
– x tj–
P+
≤ x tjP–
– x t
+ x t – x t
+ x t
– x tj–
P+
≤
,
where t ∈ (tjP – δ+∞(tjP ), tjP )T, t ∈ (tjP– , tjP– + δ+∞(tjP– ))T.
L R
We can repeat the above process, then for each i = , , . . . , we can see that, for any closed
for every x ∈ A and [t , t ]T ⊂ (tjP–i, tjPi )T, j = , , . . . , N . Hence, we have
x tjPi– – x tjP–i+
≤ x tjPi–
– x t
+ x t – x t
+ x t
interval [aPi , bPi ]T, there is a δi+∞-fine partition Pi:
aPi := bPi– = tPi < tPi < tPi < · · · < tNPi := bPi
such that
i=i j=
+∞
i=i
+∞
i=
i
< +∞,
which implies that, for any ε > , there exists i > such that
which implies that, for any ε > , there exists T ≥ tPi and a partition
x tjP– – x tjP–+
=
Moreover, since x is -differentiable almost everywhere on [T, +∞)T, we obtain
This completes the proof.
Theorem . Let A ⊂ G([T, +∞)T, Rn) be uniformly bounded and
Z = t ∈ [T, +∞)T : x is not -differentiable at t
and μ (Z ) = for all x ∈ A, i.e., x is -differentiable at [T, +∞)T\Z and there exists
M > such that |x (t)| ≤ M for all x ∈ A. Then there exists T > such that A is uniformly
Cauchy on [T, +∞)T.
Proof According to Lemma ., for any ε > , there exists T > such that, for any x ∈ A,
the following is fulfilled:
[T,+∞)T\Z
Hence, for any t, t ∈/ Z , t, t > T, we obtain
This completes the proof.
t
t
x (s) s <
[T,+∞)T\Z
In the following, we will give the following useful corollaries.
Corollary . Let A ⊂ G([T, +∞)T, Rn) be uniformly bounded and
and μ (Z ) = for all x ∈ A, and there exists M > such that |x (t)| ≤ M for all x ∈ A.
Then A is relatively compact in G.
Proof According to Theorem ., A is uniformly equi-regulated, uniformly Cauchy.
Further, since A is uniformly bounded, so it satisfies (.), by Theorem ., we get the desired
result immediately.
Let
BC[T, +∞)T := x ∈ BC [T, +∞)T, Rn and
where BC([T, +∞)T, Rn) denotes the set of all bounded continuous functions on [T,
+∞)T. Then we can obtain the following corollary.
Corollary . Let A ⊂ BC[T, +∞)T be uniformly bounded and for all x ∈ A, x is
differentiable and there exists M > such that |x (t)| < M. Then A is relatively compact in
BC.
Proof Since x is -differentiable on [T, +∞)T, A is equi-absolutely continuous on
[T, +∞)T, for any t, t ∈ [T, +∞)T, we can obtain
+∞
Thus, for any ε > , there exists T > , and we have t > t > T implies
t
t
x t – x t
=
x (s) s < ε,
i.e., A is uniformly Cauchy. According to Theorem ., A is relatively dense in BC. This
completes the proof.
Remark . Note that if for all x ∈ A ⊂ BC, x has uniformly bounded -derivatives, then
one can see that A is equi-absolutely continuous, which will lead to that A is uniformly
Cauchy. Hence, the uniformly boundedness of A and the uniformly boundedness of
derivatives functions of A can guarantee A is relatively compact.
4 Existence of solutions for impulsive functional dynamic equations
In this section, we introduce some new definitions and give some new methods to obtain
some sufficient conditions for the existence of solutions for a class of ε-equivalent
impulsive functional dynamic equations on almost periodic time scales. We always assume that
sup T = +∞ and inf T = –∞.
Definition . For arbitrary functions x, y : T → Rn, we define the function [x(·), y(·)]μ :
T → R to be the number (provided it exists) with the property that, for any given ε > ,
there exists a neighborhood U of t (i.e., U = (t – δ, t + δ)T for some δ > ) such that
x(t) + σ (t) – s y(t) – x(t) – σ (t) – s x(·), y(·) μ < ε σ (t) – s
for all s ∈ U.
Remark . In Definition ., if t is a right-dense point, μ(t) = , then one can obtain
let s – t = h → +, and (.) can be written as
x(t), y(t) = hl→im+ |x(t) + hy(ht)| – |x(t)| .
If t is a right-scattered point, μ(t) > , then one can get
x(t), y(t) μ = |x(t) + μ(tμ)y(t()t)| – |x(t)| .
In the following, we will consider the following impulsive system:
⎨⎧ x (t) = A(t, x) + g(t), t = tk,
x(tk) = I˜k(x(tk)),
Z
t = tk, k ∈ ,
where A ∈ PCrd(T × Rn, Rn), g ∈ PCrd(T, Rn), I˜k ∈ C(Rn, Rn).
Next, we will give the following theorem to guarantee that (.) has a unique global
solution.
Theorem . Let T be a time scale with the bounded graininess function μ. In the system
(.), if for any ε > , there exists a neighborhood U of t such that
d˜ x(t) + σ (t) – s A(t, x) + g(t) , Br() < ε σ (t) – s , s ∈ U,
for all (t, x) ∈ (tk–, tk)T × Br(), k ∈ Z, where d˜ (z, Br()) denotes the distance from z to Br()
and Br() = {x ∈ PCrd(T, Rn) : x ≤ r, r is a constant}. Then for each tn ∈ T (n ∈ N), the
Cauchy problem with the initial value x(tn) = un for (.) has a unique global solution x(t)
on T such that x ∈ Br() for all t ∈ .
T
Proof Consider the following form of (.):
⎨⎧ x (t) = A˜x(t) + A(t, x) – A˜ x(t) + g(t), t = tk,
x(tk) = I˜k(x(tk)),
Z
t = tk, k ∈ ,
where A˜ = A(t, ). From Theorem . from [], for any t ∈ T, we can find k ∈ Z, tk– <
t ≤ tk , for t ∈ [t, tk)T, there is a unique solution
x(t) = e A˜(t, t)x(t) +
e A˜(t, s) A(s, x) – A˜x(s) + g(s)
s,
and by using x(tk+) – x(tk–) = I˜k(x(tk)), we obtain
x tk+ = e A˜(tk, t)x(t) +
e A˜(tk, s) A(s, x) – A˜ x(s) + g(s)
s + I˜k x(tk) ,
(.)
(.)
(.)
(.)
(.)
x(t) = e (t, t )x t
k
˜
A
˜
e (t, s) A(s, x) – Ax(s) + g(s)
s
k
= e (t, t ) e (t , t )x(t ) +
k k
˜
A
˜
e (t , s) A(s, x) – Ax(s) + g(s)
k
˜
s + I x(t )
k k
˜
e (t, s) A(s, x) – Ax(s) + g(s)
˜
e (t, s) A(s, x) – Ax(s) + g(s)
s
s
= e (t, t )x(t ) +
˜
A
˜
e (t, s) A(s, x) – Ax(s) + g(s)
˜
A
and then we have
Repeating this procedure, we get Hence, for t
t , we know that (.) has a unique solution satisfying
˜
A
σ
˜
e t, (s) A(s, x) – Ax(s) + g(s)
s
˜
A
x(t) = e (t, t )x(t ) +
˜
e (t, s) A(s, x) – Ax(s) + g(s)
s +
˜
A
e (t, t )I˜ x(t ) .
k k k
t <t <t
k
˜
A
˜
A
+
˜
A
+
t
t
k
t
k
˜
A
˜
A
˜
A
˜
A
+
t
k
k
t
t
t
˜
A
˜
A
t <t <t
k
˜
A
e (t, t )I˜ x(t ) .
k k k
that is,
that is,
ε
˜
d x(t), B () < .
r
˜
d x
σ
ε
(t) , B () < .
r
Therefore, (.) has a unique global solution u (t) on
n
. Furthermore, combining with
(.), for all t
, t ) , k
T
, i.e., t = t , (.) can be changed into the following:
k
T
˜
d x(t) +
(t) – s x (t), B () <
r
ε σ
(t) – s .
Case . If t is a right-dense point, from (.), we obtain
˜
d x(t) + h
lim
h
→
x(t + h) – x(t)
h
ε
, B () < h,
r
Case . If t is a right-scattered point, from (.), we get
μ
˜
d x(t) + (t)
σ
x( (t)) – x(t)
μ
, B () <
r
εμ
εμ
,
μ
sup
t
T
μ
(t),
(.)
(.)
(.)
(.)
Thus, if ρ(t) < t, that is, ρ(t) is a right-scattered point, we have σ (ρ(t)) = t, by (.), one
has
d˜ x(t), Br() = d˜ x σ ρ(t) , Br() < ε.
If ρ(t) = t, that is, t is a left-dense point, so there must exist a right-dense point s, for any
ε > , there exists a δL+∞ > such that s ∈ (t – δL+∞(t), t)T implies
d˜ x(s), x(t) < ε.
Hence, by (.), we can easily get
d˜ x(t), Br() < d˜ x(t), x(s) + d˜ x(s), Br() < ε + ε = ε,
Z
t = tk , k ∈ .
For the impulsive points tk , k ∈ Z, since (tk – δL+∞(tk ), tk )T ⊂ (t – δL+∞(t), t + δR+∞(t))T,
where |t – tk | < min{δL+∞(t), δR+∞(t)}, then
d˜ x(tk ), Br() < d˜ x(tk ), x tk–
+ d˜ x tk– , x(s) + d˜ x(s), Br()
where s ∈ (tk – δL+∞(tk ), tk )T ⊂ (t – δL+∞(t), t + δR+∞(t))T, k ∈ Z. This completes the proof.
D+ u(t)
= u(t), u (t) μ,
where D+|u(t)| denotes the right derivative of |u(t)| at t. Further, for any
a ∈ R+ ∪ {}, we have [u(t), u (t)]μ ≤ D+|u(t)| .
Proof If a = , then aμ = , from Remark ., the results (i)-(iii) are obvious. If a ∈ R+, we
can conclude the following:
(i) In fact, if t is a right-dense point, by (.), the result is obvious. Let t be a
right-scattered point, then
x(t), y(t) aμ =
≤ |x(t) + aμa(μt)(yt)(t) – x(t)| = y(t) .
(ii) If t is a right-dense point, by (.), the result is obvious. Let t be a right-scattered
point, then
x(t), y(t) + z(t) aμ =
|x(t) + aμ(t)(y(t) + z(t))| – |x(t)|
aμ(t)
=
| x(t) + aμ(t)y(t) + x(t) + aμ(t)z(t)| – |x(t)|
aμ(t)
= x(t), y(t) aμ + x(t), z(t) aμ.
u(t), u (t) μ =
Further, for a ∈ R+, we can obtain
u(t), u (t) aμ =
This completes the proof.
From Definition ., we obtain the following theorem, which guarantees that (.) has a
unique global solution on T.
Theorem . Let A ∈ PCrd(T × Rn, Rn), g ∈ PCrd(T, Rn) be bounded, A(t, ) = for all
t ∈ T and Ik () = , be relatively dense in R. Suppose that there exist positive numbers
p > M > , r > such that |g(t)| ≤ M, p > Mr and
x – y, A(t, x) – A(t, y) aμ ≤ –p|x – y|
(.)
for t ∈ (tk–, tk), k ∈ Z, –p ∈ R+, a ∈ R+ ∪ {}, |x(t)| ≤ r, |y(t)| ≤ r, and
I˜k(x) – I˜k(y) ≤ I˜|x – y|,
and h˜ = inf(tk – tk–) > ln(+I˜) , I˜ is a Lipschitz constant. Then (.) has a unique solution
M
u(t) with respect to the set of solution of (.) in Cr, where Cr = {ϕ ∈ PCrd(T, Rn) : |ϕ(t)| ≤
r for all t ∈ T}. Moreover, if v(t) is any solution of (.) such that, for some t ∈ , |v(t)| ≤
T
Mp , then |v(t)| ≤ r and
v(t) – u(t) ≤ v(t) – u(t)
( + I˜)e–p(t, t)
for all t ≥ t.
Proof We fix a vector u ∈ Rn with |u| ≤ Mp . For each positive tn ∈ T(n ∈ N), we consider
the Cauchy problem for (.) with the initial value x(tn) = u.
For each x ∈ Rn with |x| = r, for t ∈ (tk–, tk)T, k ∈ Z, (.) and (ii) in Lemma . imply
x, A(t, x) + g(t) μ ≤ x, A(t, x) μ + g(t) ≤ –p|x| + M = –pr + M < .
So for any ε > , there exists a neighborhood of U such that
x + σ (t) – s A(t, x) + g(t) – |x| < x + σ (t) – s A(t, x) + g(t) – |x|
– σ (t) – s x, A(t, x) + g(t) μ
< ε σ (t) – s ,
which implies that
d˜ x + σ (t) – s A(t, x) + g(t) , Br() < ε σ (t) – s ,
by Theorem ., and we see that (.) has a unique global solution un(t) on T such that
un ∈ Br() for all t ∈ T. Moreover, we can show that |un(t)| ≤ r for all t ∈ T. In fact, (.)
and (iii) from Lemma . imply
D+ un(t)
= un(t), A t, un(t) + g(t) μ ≤ –p un(t) + g(t) ≤ –p un(t) + M
(.)
for t ∈ (tk–, tk)T, k ∈ Z. By (.), solving the differential inequality (.), we obtain
un(t) ≤ un tn e–p t, tn + M
t
n e–p t, σ (s)
t
s +
e–p(t, tk)I˜k un(tk)
tn<tk<t
M M
≤ p e–p t, tn + – p
≤ Mp + – I˜e–pθ ≤ Mp + – e I˜p(θ , ) := r
e–p t, tn – e–p(t, t) + I˜
e–p(t–tk)
tn<tk<t
for all t ∈ T, we always denote that supk e p(tk+, tk) := e p(θ , ). According to Corollary .,
there exists a subsequence {unk } such that unk → u on T as k → +∞. Therefore, there
T
exists a solution u(t) of (.) such that |u(t)| ≤ r for all t ∈ .
We next show that this solution u(t) is unique with respect to the set of solutions of (.)
in Cr. Let v(t) be another solution of (.) in Cr and let w(t) = |u(t) – v(t)| for t ∈ T. Then
by (iii) from Lemma . and (.), we have
D+w (t) = u(t) – v(t), A t, u(t) – A t, v(t) μ
for t ∈ (tk–, tk)T. This implies that w(t) is a nonincreasing function on (tk–, tk)T. Hence,
for t ∈ T, t ≥ t, we obtain
t
w(t) ≤ w(t) – p
w(s) s + I˜
w(tk),
and by Gronwall-Bellman’s inequality on time scales, from (.), we obtain
(.)
(.)
w(t) – w(t + h) < εh.
w(t) – w(t + h) < εh
Suppose, for contradiction, that w(t) > for some t ∈ T. Since w(t) is bounded and is
increasing as t decreases, limt→–∞ w(t) exists. For is relatively dense in R, which implies
that, for each ε > and h > , –h ∈ , there exists t ≤ , t ∈ T ∩ T–h such that t + h < t
and
Set h ≤ h and –h ∈
such that t ∈ T ∩ T–h . Then we have either
w(t) – w(t + h) < εh.
w(tn) – w(tn + hn) < εhn, for n ≥ .
or
w(t + h) – w(t + h) < εh.
Hence there exists t ∈ T ∩ T–h (t = t or t + h) such that t, t + h ∈ [t, t + h]T and
Repeating this procedure, we have sequences {tn}, {tn +hn} such that tn, tn +hn ∈ [t, t +h]T
and
On the other hand, by (.), we have
w(tn + hn) ≤ w(tn)
( + I˜)e–p(tn + hn, tn)
tn<tk<tn+hn
≤ w(tn + hn) + εhn
≤ w(tn + hn)
tn<tk<tn+hn
for n ≥ . Therefore, it follows that
w(t) ≤ w(tn + hn) ≤
–
hn
hn
≤ –
≤ –
tn<tk<tn+hn ( + I˜)e–phn
tn<tk<tn+hn ( + I˜)e–p(tk+–tk)
ε
hn
Next, we will show that
( + I˜)e–p(tk+–tk) < .
( + I˜)e–p(tn + hn, tn)
tn<tk<tn+hn
( + I˜)e–p(tn + hn, tn) + εhn
tn<tk<tn+hn ( + I˜)e–p(tn + hn, tn)
ε
ε (n ≥ ).
(.)
(.)
tn<tk<tn+hn
tn<tk<tn+hn
e–p(tk+–tk) = e–p(tk+–tk) < e–phn < .
Case I. If [tn, tn + hn]T does not contain any tk, k ∈ Z, then tk+ – tk > hn, (.) becomes
Case II. If [tn, tn + hn]T contains n impulsive points tk, k ∈ S = {, , . . . , n}, then
tn<tk<tn+hn ep(tk+–tk)
tn<tk<tn+hn eM(tk+–tk)
( + I˜)
<
( + I˜)n
tn<tk<tn+hn e(tk–tk–)M
<
+ I˜ n
eh˜M
+ I˜ n
eh˜M
ln( + I˜)
< since h˜ >
M
.
Hence, from (.), let n → ∞, we have w(t) = . This leads to a contradiction. The last
assertion of this theorem follows easily from the preceding argument. The proof is
complete.
In the following, let T be an almost periodic time scale. For convenience, we give some
notation, BC denotes the set of Rn-valued functions that is continuous and bounded on
(–∞, s] ε , s ∈ ε; for each φ ∈ BC we define φ ε = sups∈(–∞,s] ε |φ(s)|. Then (BC, · ε)
is a real Banach space for any fixed ε > .
Remark . From the definition of φ ε, if we introduce a set
˜ = φ : φ ε ≤ r, ∀ε > , r is a constant ,
then for ∀φ ∈ ˜ , one can immediately obtain that φ ε ≤ φ ε ≤ r if ε < ε.
Let x(t) be an Rn-valued function that is continuous and bounded on T, we define for
each t ∈ T ∩ (∪T ε ), xt(s) = x(t + s) for –s ∈ ε. Clearly, xt ∈ BC for each t ∈ T ∩ (∪T ε ).
We now consider the ε-equivalent impulsive functional dynamic equation
F(t, x, φ) is the Rn-valued function defined on T × Rn × BC which satisfies the following
conditions:
(H) for each r > , there exists M(r) > and N (r) > such that
F(t, , φ) ≤ M(r) and
F(t, x, φ) ≤ N (r)
Ik(x) – Ik(y) ≤ N˜ (r)|x – y|,
Ik() = ,
for all t ∈ T, |x| ≤ r and φ ε ≤ r(φ ∈ BC). Moreover, for any x, y ∈ Rn,
F t, xk(t), ytk → F t, x(t), yt
as k → ∞ for t ∈ T ∩ ∪T ε .
Remark . In (H), piecewise continuity of F(t, y, xt) in (t, y) on T × Rn implies piecewise
continuity of F(t, x(t), xt) in t on T. Moreover, if F is Lipschitz continuous in x uniformly
for t ∈ T, φ ∈ BC( φ ε ≤ r) with Lipschitz constant K , then we can take N (r) = Kr + M(r).
Theorem . Suppose that (H)-(H) are satisfied and
F(t, x, φ) – F(t, y, φ) ≤ K |x – y|,
where K is a Lipschitz constant. Then there exists a solution u(t) of (.) such that
|u(t)| ≤ r for all t ∈ T ∩ (∪T ε ), and this solution is unique in Cr∗|T∩(∪T ε ). Here r is as
in (H), Cr∗ = {ϕ ∈ PCrd(T, Rn) : ϕ(t) is piecewise continuous and |ϕ(t)| ≤ r for all t ∈ T},
and Cr∗|T∩(∪T ε ) = {ϕ|T∩(∪T ε ) : ∀ϕ ∈ Cr∗}.
Proof Let N (r) be as in (H) and set Sr = {f ∈ PCrd(T, Rn) : |f (t)| ≤ r and |f (t)| ≤ Kr +
M(r)}. Obviously, Sr is a compact convex subset of PCrd(T, Rn). We set
A(t, x, φ) = F(t, x, φ) – F(t, , φ) and
B(t, φ) = F(t, , φ).
Define a mapping T : Sr → Sr as follows: x(t) = Tf (t) is the unique solution in Cr of
⎧⎨ x (t) = A(t, x(t), ft) + B(t, ft), t ∈ T, t = tk,
⎩ x(tk) = Ik(x(tk)), t = tk, k ∈ Z,
(.)
where Cr is as in Theorem . and f ∈ Sr. Such a solution x(t) exists by Theorem . and
satisfies |x(t)| ≤ r for all t ∈ T. Since
x (t) = F(t, x, φ) – F(t, , φ) + F(t, , φ)
≤ F(t, x, φ) – F(t, , φ) + F(t, , φ) ≤ Kr + M(r)
it follows that x ∈ Sr.
We show next that T is continuous on Sr. Let f k and f be in Sr such that f k → f as k → ∞.
Put xk = Tf k and x = Tf . We show that each sequence of {xk} contains a subsequence
which converges to x. This will show that T is continuous at f ∈ Sr. Let us denote this
arbitrary subsequence again by {xk} for simplicity. Since xk ∈ Sr, {xk} is -differentiable
almost everywhere on T and uniformly bounded on T. According to Corollary ., there
exists a subsequence {xkj } of {xk} and a w ∈ Sr such that xkj (t) → w(t) as j → ∞ uniformly
on T. Let t ∈ T ∩ (∪T ε ) and let
w∗(t) = w(t) +
F s, w(s), fs
s +
Ik w(tk)
for t ∈ T ∩ ∪T ε .
Then we have
xkj (t) – w∗(t) = xkj (t) +
s – w(t) –
F s, w(s), fs
s
t
t
t
+
+
t<tk<t
t
F s, xkj (s), fskj
t<tk<t
t
t
N˜ (r) xkj (tk) – w(tk)
Ik xkj (tk) –
Ik w(tk)
≤ xkj (t) – w(t) +
F s, xkj (s), fskj – F s, w(s), fs
s
for t ∈ T ∩ (∪T ε ). By (H), (H), (H), and the Lebesgue dominated convergence theorem
we can see that xkj (t) → w∗(t) as j → ∞ for t ∈ T, i.e., w∗(t) = w(t) for t ∈ T ∩ (∪T ε ).
Hence, w is a solution for (.) on Cr∗|T∩(∪T ε ). By the uniqueness of solutions of (.), it
follows that w(t) = x(t) for t ∈ T ∩ (∪T ε ). Thus we conclude that T is continuous at each
f ∈ Sr . From Lemma ., there exists a u ∈ Sr such that Tu = u, i.e.,
⎨⎧ u (t) = A(t, u(t), ut ) + B(t, ut) = F(t, u(t), ut), t ∈ T ∩ (∪T ε ), t = tk ,
⎩ u(tk ) = Ik (u(tk )), t = tk , k ∈ Z.
D+ u(t) – v(t)
= u(t) – v(t), F t, u(t), ut – F t, v(t), vt μ
≤ –p u(t) – v(t) + L ut – vt ε
for all t ∈ (tk–, tk )T ∩ (∪T ε ), k ∈ Z. Let ψ (t) = |u(t) – v(t)|, then we obtain
(.)
⎧
⎨ D+ψ (t)
⎩ ψ (tk+) ≤ ψ (tk–) + N˜ (r)ψ (tk ).
≤ –pψ (t) + L ψt ε,
Then for t ∈ [tk , tk+)T ∩ (∪T ε ), k ∈ Z, we get
ψ (t) ≤ ψ tk+ e–p(t, tk ) +
e–p t, σ (s) L ψs ε s,
thus we obtain
tk
t
tk
ψ (t) ≤ ψ tk– e–p(t, tk ) +
e–p t, σ (s) L ψs ε s + N˜ (r)ψ (tk )e–p(t, tk ),
ψ tk+ ≤ ψ tk+ e–p(tk+, tk ) +
–
e–p tk+, σ (s) L ψs ε s.
Note that u(t) – v(t) = ut() – vt(), from (.) and (.), for t ∈ [t + s, t + s)T ∩ (∪T ε ),
solving the differential inequality (.), we have
ψ (t + s) ≤ ψ (t + s)
t+s<tk <t+s
+
+
t+s<tk <t+s tk <tj<t+s
t+s
e–p(t + s, tk ) N˜ (r)ψ (tk )
e–p t + s, σ (θ ) L ψθ ε θ ,
for all t ∈ [t, +∞)T ∩ (∪T ε ), –s ∈
of θ . It therefore follows that
ε . Note ψθ ε = uθ – vθ ε is nondecreasing function
ut – vt ε ≤
t+s<tk <t+s
N˜ (r)
– e–p(θ , )
+ Lp – e–p(t + s, t + s)
ut – vt ε
+ Lp
for all t ∈ [t, +∞)T ∩ (∪T ε ), and this implies
for all t ∈ [t, +∞)T ∩ (∪T ε ), where
M˜ =
+ Lp .
Consequently, for each t ∈ T ∩ (∪T ε ), we have
– M˜ t+s<tk<t+s
ut – vt ε ≤
ut – vt ε ≤ ut – vt ε ≤
e–p(t + s, t + s) ut – vt ε
– M˜ t+s<tk<t+s
for all t ∈ [t, +∞)T ∩ (∪T ε ). Letting t → +∞ we have ut – vt ε = and this implies in
particular u(t) = v(t). The proof is complete.
We next consider the existence of almost periodic solutions of (.) when F satisfies
the following condition:
(H) F(t, x, φ) is almost periodic in t uniformly for (x, φ) in bounded closed subsets of Rn ×
BC.
Theorem . Suppose (H)-(H) are satisfied and the following are fulfilled:
(i) |F(t, x, φ) – F(t, y, φ)| ≤ K |x – y|, ∀x, y ∈ Rn, where K is a Lipschitz constant;
(ii) the impulsive operator sequence {Ik} is an almost periodic sequence;
(iii) (p–L)(–e–p(θ,)) < , where supk e–p(tk+, tk) := e–p(θ , ).
p(–e–p(θ,))
Then (.) has a unique almost periodic solution u(t) in Cr∗|T∩(∪T ε ) for < r < . Moreover,
if p – L ≥ pLr+rM(r) and v(t) is any solution of (.) with vt ε ≤ Mp(r) for some t ∈ T ∩
(∪T ε ), then vt ε ≤ r and
p
ut – vt ε ≤ p – L e–p(t, t)
e–p(t + s, t + s) ut – vt ε
Proof Let u(t) be the unique solution of (.) in Cr∗|T∩(∪T ε ) obtained in Theorem ..
For each ε > , there exists a positive number l(ε) such that any interval of length l(ε)
contains a τ = τ (ε) ∈ E{ε, F} for which
F t + τ , u(t + τ ), ut+τ – F t, u(t + τ ), ut+τ < ε
for all t ∈ (tk–, tk)T ∩ T–τ , τ ∈ ε , k ∈ Z. By (iii) in Lemma . and (i), (ii) in (H), we have
D+ u(t + τ ) – u(t)
= u(t + τ ) – u(t), F t + τ , u(t + τ ), ut+τ – F t, u(t), ut μ
≤ –p u(t + τ ) – u(t) + L ut+τ – ut
+ F t + τ , u(t + τ ), ut+τ – F t, u(t + τ ), ut+τ
≤ –p u(t + τ ) – u(t) + L ut+τ – ut ε + ε
for all t ∈ (tk–, tk)T ∩ T–τ , –τ ∈ ε , k ∈ Z.
For all t ∈ T ∩ T–s, –s ∈ (–∞, ] ε . From Lemma ., we can find β > and β ∈ ε such
that –s + β ∈ ε . Thus we obtain
d T–τ–s+β , T < d T–τ–s+β , T–s+β + d T–s+β , T < ε.
Let ϕ˜(t) = |u(t + τ ) – u(t)|, (.) can be written in the form
D+ϕ˜ (t) ≤ –pϕ˜(t) + L ϕ˜t ε + ε.
Moreover, since {Ik} is an almost periodic sequence, there exists a q > such that |Ik+q –
Ik| < ε. Hence, we obtain
ϕ˜ tk+ – ϕ˜ tk– ≤ Ik+q u(tk+q) – Ik u(tk) < ε.
Then for t ∈ [tk, tk+)T ∩ (∪T ε ), k ∈ Z, we get
tk
t
t
tk
ϕ˜(t) ≤ ϕ˜ tk+ e–p(t, tk) +
e–p t, σ (s) L ϕ˜s ε + ε
s,
thus we obtain
ϕ˜(t) ≤ ϕ˜ tk– e–p(t, tk) +
e–p t, σ (s) L ϕ˜s ε + ε
s + εe–p(t, tk),
ϕ˜ tk–+ ≤ ϕ˜ tk+ e–p(tk+, tk) +
e–p tk+, σ (s) L ϕ˜s ε + ε
s.
Under (.), for all t ∈ T ∩ (∪T ε ), using (.) and (.), we can solve the differential
inequality (.) and obtain
ϕ˜(t + s) ≤ ϕ˜(t + s – β)
e–p(t + s, t + s – β)
t+s–β<tk<t+s
+
+
t+s–β<tk<t+s tk<tj<t+s
t+s
t+s–β θ<tk<t+s
e–p(t + s, tk) ε
e–p t + s, σ (θ ) L ϕ˜θ ε + ε
θ
for all t ∈ T ∩ (∪T ε ). Note ϕ˜θ ε = uθ+τ – uθ ε is nondecreasing in θ and u ∈
Cr∗|T∩(∪T ε ). It follows from this that
ut+τ – ut ε ≤
e–p(t + s, t + s – β) ut+τ–β – ut–β ε
t+s–β<tk<t+s
+ – e–εp(θ , )
+ ε,
+ ε – e–p(t + s, t + s – β)
ut+τ – ut ε
so we can obtain
t+s–β<tk<t+s
Then
p
ut+τ – ut ε ≤ p – L t+s–β<tk<t+s
re–p(t + s, t + s – β)
+ ((– –e–ep–(pθ(,θ,))()p)ε–pL) .
Hence, for < r < , we can choose a sufficiently large β > such that
re–p(t + s, t + s – β) < ε.
ut+τ – ut ε ≤
p –p L + –– ee––pp((θθ,, ))
ε for all t ∈ T ∩ ∪T ε .
Since ε is arbitrary and ε > ε > , from the condition (iii) from Theorem ., we obtain
p –p L + –– ee––pp((θθ,, ))
ε > ε > ,
by Definition ., u(t) is an almost periodic function on the almost periodic time scale T.
vNtexεt,≤wMep(ra)sfsourmseo mtheattp∈–TL∩ ≥(∪pTLr+εr M).(rT) .heLne,t bvy(t()ii)b,e(iiain)yinsLoelumtimona o.f,(an.d)(iis)uicnh( Htha)t,
for t ∈ (tk, tk+)T ∩ (∪T ε ), we have
≤ v(t), F t, v(t), vt – F(t, , ) μ + F(t, , )
≤ –p v(t) + L vt + M(r).
(.)
D+ v(t)
= v(t), F t, v(t), vt μ
v tk+ – v tk– ≤ Ik v(tk) – Ik() < Lr.
Moreover, note Ik satisfies the Lipschitz condition for each k. Hence, we can obtain
Then for t ∈ [tk , tk+)T ∩ (∪T ε ), k ∈ Z, we get
thus we obtain
v(t) ≤ v tk+ e–p(t, tk ) +
e–p t, σ (s) L vs ε + M(r)
s,
tk
t
t
tk
v(t) ≤ v tk– e–p(t, tk ) +
e–p t, σ (s) L vs ε + M(r)
s + Lre–p(t, tk ),
v tk–+ ≤ v tk+ e–p(tk+, tk ) +
e–p tk+, σ (s) L vs ε + M(r)
s.
Using (.) and (.), we can solve the differential inequality (.) and obtain
v(t + s) ≤ v(t + s)
t+s θ<tk<t+s
e–p(t + s, tk ) Lr
e–p t + s, σ (θ ) L vθ ε + M(r)
θ
for all t ∈ T ∩ (∪T ε ), –s ∈ (–∞, ] ε , which implies that
vt ε ≤
e–p(t + s, t + s) vt ε
t+s<tk<t+s
+ Lp – e–p(t, t) vt ε +
M(r)
p
t+s<tk<t+s
L
e–p(t + s, t + s) vt ε + p vt ε + Lr +
e–p(t + s, t + s) vt ε +
Lrp + M(r)
p – L ≤ r.
and thus
Obviously,
F(t, x, xt) = – + cc((tt))μ(t) x(t) +
sin √t + cos t
+ c(t)μ(t)
xt,
Ik x(k) = k–k x(k),
Z
k ∈ .
Consequently, vt ε ≤ r for all t ∈ T ∩ (∪T ε ). The estimate (.) follows from the same
argument as in the proof of Theorem .. The proof is complete.
Example . Let T be an almost periodic time scale with Z ⊂ T and c ∈ PCrd(T, R+).
Consider the following ε-equivalent impulsive dynamic equation:
⎧⎨ x (t) = – +cc(t(t)μ)(t) x(t) + sin+√c(tt)+μc(ot)s t xt, t = tk = k, k ∈ Z,
x(k) = k–k x(k),
t = tk = k.
One can check that (.) is equivalent to the following impulsive dynamic equation:
⎧⎨ x (t) = –c(t)xσ (t) + (sin √ + cos t)xt, t = tk = k, k ∈ Z,
x(k) = k–k x(k),
D+ x(t) – y(t)
= x(t) – y(t)
= x(t) – y(t), x (t) – y (t) μ
Hence, by (iii) from Lemma . in this paper and using Lemma . from [], for t ∈
T ∩ (∪T ε ), we can calculate that
= x(t) – y(t), F(t, x, xt) – F(t, y, yt ) μ
≤ x(t) – y(t), – + cc((tt))μ(t) x(t) – y(t) +
≤ sign xσ (t) – yσ (t) x (t) – y (t)
= sign xσ (t) – yσ (t) –c(t) xσ (t) – yσ (t) + (sin √t + cos t)(xt – yt )
= –c(t) xσ (t) – yσ (t) + | sin √
= –c(t) μ(t) x(t) – y(t)
+ x(t) – y(t)
t + cos t| xt – yt ε
+ | sin √
t + cos t| xt – yt ε.
(xt – yt )
μ
Also, for |x| ≤ r, we can easily obtain
D+ x(t) – y(t)
≤ – + cc((tt))μ(t) x(t) – y(t) +
| sin √t + cos t| xt – yt ε.
+ c(t)μ(t)
r r
F(t, , φ) ≤ + c(t)μ(t) ≤ inft∈T∩(∪T ε ) + c(t)μ(t)
≤ r + ln := M(r)
and
and
Further, we can obtain
( + c(t))r ( + supt∈T∩(∪T ε ) c(t))r
F(t, x, φ) ≤ + c(t)μ(t) ≤ inft∈T∩(∪T ε ) + c(t)μ(t)
:= N (r).
Ik (x) – Ik (y) ≤ N˜ (r)|x – y|,
Ik () = ,
N˜ (r) = + r
h˜ = inf(tk – tk–) = >
ln( + N˜ (r))
=
M(r)
ln( + r)
r + ln
,
∀r > .
Thus we can take L = inft∈T∩(∪T ε ) +c(t)μ(t) < such that there exists
p > max + ln , , + r > max
r
such that
In addition, we can obtain
– eN˜–p(r(θ) , ) + Lp < –+e–rp + Lp < –+pr + p < p < since p > .
(pp–(L–)(e––pe(–θp,(θ),))) < (––ee––pp((θθ,, ))) < ––ee––pp((θθ,,)) < .
Therefore, all the conditions from Theorem . are satisfied, then (.) has a unique
almost periodic solution u(t) in Cr∗|T∩(∪T ε ) for < r < .
5 Conclusion and further discussion
Since impulsive dynamic equations with ‘slight vibration’ can be established on almost
periodic time scales and describe many natural phenomena precisely, we propose a new
type of ε-equivalent impulsive dynamic equations on almost periodic time scales to reflect
such a ‘slight vibration’, which will contribute to the theory of dynamic equations on time
scales and practical applications in the real world. From this paper, one can observe that
ε-equivalent impulsive dynamic equations are ‘little fuzzy’ in time variables, thus there is
a theoretical and practical significance to obtain the existence of solutions for this type of
dynamic equations.
To study the existence of solutions for ε-equivalent impulsive functional dynamic
equations on almost periodic time scales, we established several theorems in the paper (see
Sections and ) to obtain some new existence results for solutions. These results are
also new when T = R and T = Z. This is the first investigation for this new type of
functional dynamic equations with ‘slight vibration’ and several new methods are provided.
However, many problems remain to be studied. From Definition ., we introduce the
concept of -sub-derivatives on time scales, which means that (.) is the most general
functional dynamic equations with ‘arbitrary vibration’ in time variables. In Theorem .,
we provide sufficient conditions to guarantee the existence of solutions for (.) with
‘arbitrary vibration’. Nevertheless, since this type of dynamic equations has complicated
sub-derivatives.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the manuscript and typed, read and approved the final manuscript.
Acknowledgements
This work is supported by Tian Yuan Fund of NSFC (No. 11526181), Yunnan Province Education Department Scientific
Research Fund Project of China (No. 2014Y008), and Yunnan Province Science and Technology Department Applied Basic
Research Project of China (No. 2014FB102).
x(t) - x t