Electronic Journal of Differential Equations, Vol. 2005(2005), No. 35, pp. 1–7.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu (login: ftp)
BOUNDENESS AND STEPANOV’S ALMOST PERIODICITY OF
SOLUTIONS
ZUOSHENG HU
Abstract. In this paper, we establish a necessary condition of Stepanov’s
almost periodicity of solutions for general linear almost periodic systems, and
then we construct an example of linear system in which all solutions are
bounded, but any non-trivial solution is not Stepanov’s almost periodic.
1. Introduction
It is well-known that for (Bohr) almost periodic differential equations
x0 = A(t)x + f (t),
(1.1)
boundedness of solutions does not imply their almost periodicity. Conley and Miller
[2] gave an example of an equation (1.1) with n = 1 where a bounded solution is
not almost periodic. In [5], Mingarelli, Pu and Zheng constructed an example,
for each n > 1, of an equation (1.1) with almost periodic coefficients in which
there exists a bounded solution which is not almost periodic. In [4], Hu and Mingarelli constructed a class of linear almost periodic systems in which all solutions
are bounded but there still exists no any non-trivial solution which is almost periodic. The question arising naturally is whether boundedness of solutions can imply
their Stepanov’s almost periodicity which is a weaker almost periodicity defined by
Stepanov (see [1] for the details). As far as we know, this is an open problem. In
this paper, in order to solve this open problem, we establish a necessary condition
of Stepanov’s almost periodicity of solutions for general linear almost periodic systems, and then we construct an example of linear system in which all solutions are
bounded, but any non-trivial solution is not Stepanov’s almost periodic. So, the
answer of this problem is negative.
For completeness, we recall the definition of the Stepanov norm Sl (f ) of a function f ∈ Lloc
1 (R, X). The quantity
Z
1 t+l
Sl (f ) = sup
kf (s)kds
t∈R l t
where l > 0 is some constant, is the Stepanov norm (or Sl -norm) of f .
Replacing the supremum norm by the Sl -norm in the definition of continuity
(respectively, uniform continuity, boundedness) of f , we can introduce the concept
2000 Mathematics Subject Classification. 34C27, 34A30.
Key words and phrases. Boundedness; Stepanov’s almost periodicity.
c
2005
Texas State University - San Marcos.
Submitted January 17, 2005. Published March 24, 2005.
1
2
Z. HU
EJDE-2005/35
of Sl -continuity (respectively, Sl -uniform continuity, Sl -boundedness) of f . For
example, we call f ∈ Lloc
1 (R, X) Sl -bounded if there exists a constant M > 0 such
that Sl (f ) ≤ M . It is easy to show that Sl -boundedness (Sl -continuity, Sl -uniform
continuity) is not dependent on the constant l (see [1]). So, we simply call such
functions S-bounded, S-continuous, and S-uniformly continuous whenever these
notions apply.
We define Sl (t, f ) as follows:
Z
1 t+l
Sl (t, f ) =
kf (s)kds for all t ∈ R.
(1.2)
l t
From (1.2) we have that for any t, s ∈ R,
Sl (t, fs ) = Sl (t + s, f ).
where fs is the translate of f . We use Sl C(R, X) to denote the set of all Sl continuous functions. Obviously, C(R, X) ⊂ Sl C(R, X). Because l can be taken
any positive real number, we simply denote Sl C(R, X) by SC(R, X). As in the
case of Bohr almost periodic functions, we introduce the definition of a Stepanov
almost periodic function.
Definition 1.1. Let f ∈ Sl C(R, X). If for any sequence {αn } ⊂ R, there exist a
subsequence {αn0 } of {αn } and a function g ∈ Sl C(R, X) such that
lim Sl (t, fα0n − g) = 0,
n→∞
uniformly on R,
then f is called Sl -almost periodic on R.
We use the notation in [3]. α = {αn } is a sequence of real numbers. α0 ⊂ α
means that α0 = {αn0 } ⊂ {αn } is a subsequence of α. For f, g ∈ R, STα f = g means
that there exist a sequence α and a real number l > 0 such that
Z
1 t+l
kf (s + αn ) − g(s)kds = 0,
(1.3)
lim
n→∞ l t
pointwise for t ∈ R and U STα f = g means that (1.3) holds uniformly on t ∈ R.
Now we give the definition of the uniform Stepanov hull of a Stepanov almost
periodic function.
Definition 1.2. Let f ∈ SC(R, X). The set
{g ∈ SC(R, X) : there exists a sequence {αn } ⊂ R such that U STα f = g}
is called the uniform Stepanov hull, or simply uniform S-hull, and is denoted by
SH(f ).
Obviously, for any f ∈ SC(R, X), SH(f ) is not empty since f ∈ SH(f ).
2. Necessary Condition
Consider the general system of linear differential equations
x0 = A(t)x + f (t),
n
(2.1)
where x ∈ R , A(t) is an n × n matrix function, and f (t) is an n-dimensional vector
function, defined on R. Throughout this paper, we assume that A(t) and f (t) are
all Bohr’s almost periodic on R.
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BOUNDENESS AND STEPANOV’S ALMOST PERIODICITY
3
Lemma 2.1. Suppose that φ(t) is a solution of (2.1) and that there exist a sequence α = {αn }, g ∈ SH(f ), B ∈ SH(A) and ϕ ∈ SH(φ) such that U STα A =
B, U STα f = g and U STα φ = ϕ. Then there exist a subsequence α0 of α and a
solution ϕ̃(t) of
x0 = B(t)x + g(t),
(2.2)
such that U STα0 φ = ϕ̃ on R. If φ is bounded, so is ϕ̃ with the same bound.
Proof. By the assumption, there is a constant l > 0 such that
Z
1 t+l
|A(s + αn ) − B(s)|ds = 0,
lim
n→∞ l t
Z
1 t+l
lim
|f (s + αn ) − g(s)|ds = 0,
n→∞ l t
Z
1 t+l
lim
|φ(s + αn ) − ϕ(s)|ds = 0,
n→∞ l t
Rl
uniformly on R. In particular, for each t ∈ R it follows that limn→∞ 0 |φ(s + t +
αn ) − ϕ(t + s)|ds = 0; in other words, the sequence {φ(t + s + αn )} in L1 [0, l]
converges to ϕ(t + s), and hence the sequence converges to ϕ(t + s) in the sense
of measure on [0, l]. Since t ∈ R is arbitrary, the sequence of measurable functions
{φ(τ +αn )} converges to ϕ(τ ) in the sense of measure on R; hence from a well known
result in the theory of Lebesgue measure we know that there is a subsequence α0
of α such that limn→∞ φ(τ + αn0 ) = ϕ(τ ) a.e. on R. Take a point t0 in R such that
limn→∞ φ(t0 + αn0 ) = ϕ(t0 ). Notice that
Z t
lim
|A(s + αn0 ) − B(s)|ds = 0,
n→∞
t0
Z
t
lim
n→∞
t0
Z t
lim
n→∞
|f (s + αn0 ) − g(s)|ds = 0,
|φ(s + αn0 ) − ϕ(s)|ds = 0,
t0
locally uniformly for t ∈ R (which means the uniformity of convergence on any
finite interval in R). Since φ is a solution of (2.1), we get the following relations:
Z t
0
0
φ(t + αn ) = φ(t0 + αn ) +
{A(s + αn0 )φ(s + αn0 ) + f (s + αn0 )}ds,
(2.3)
t0
for t ∈ R, n = 1, 2, . . . . Note that A(t) and f (t) are bound on R because of Bohr’s
almost periodicity of A(t) and f (t). Then, by Gronwall’s inquality one can see that
{φ(t + αn0 )} is uniformly bounded on each finite interval in R. From these facts we
see that
Z t
Z t
lim
{A(s + αn0 )φ(s + αn0 ) + f (s + αn0 )}ds =
{B(s)ϕ(s) + g(s)}ds,
n→∞
t0
t0
locally uniformly for t ∈ R. Define a (continuous) function ϕ̃(t) by
Z t
ϕ̃(t) = ϕ(t0 ) +
{B(s)ϕ(s) + g(s)}ds, t ∈ R.
t0
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Z. HU
EJDE-2005/35
From (2.3) we see that limn→∞ φ(t + αn0 ) = ϕ̃(t), locally uniformly for t ∈ R.
Consequently, it follows that ϕ(t) ≡ ϕ̃(t) a.e. on R. Hence
Z
Z
1 t+l
1 t+l
lim
|φ(s + αn0 ) − ϕ̃(s)|ds = lim
|φ(s + αn0 ) − ϕ(s)|ds = 0,
n→∞ l t
n→∞ l t
uniformly on R, which shows U STα0 φ = ϕ̃. Furthermore, we get
Z t
ϕ̃(t) = ϕ(t0 ) +
{B(s)ϕ(s) + g(s)}ds
t0
t
Z
{B(s)ϕ̃(s) + g(s)}ds,
= ϕ̃(t0 ) +
t ∈ R,
t0
which shows that ϕ̃ is a solution of (2.2). The last conclusion is obvious. This
completes the proof of Lemma.
Remark. According to Lemma 2.1, if φ is a solution of (2.1) and assumptions are
satisfied, we can simply say that U STα φ is a solution of (2.2).
Theorem 2.2. Let A(t) be Bohr’s almost periodic on R. If x(t) is a non-trivial
Stepanov’s almost periodic solution of the equation
x0 = A(t)x,
(2.4)
then for any l > 0,
1
t∈R l
Z
t+l
kx(s)kds > 0
inf
(2.5)
t
Proof. On the contrary, suppose that there exists a real number l > 0 such that
(2.5) does not hold, i. e.
Z
1 t+l
lim
kx(s)kds = 0.
(2.6)
t∈R l t
Then we will find a contradiction. From (2.6), we can pick up a sequence α = {αn }
such that
Z
1 αn +l
lim
kx(s)kds = 0
n→∞ l α
n
or
Z
1 l
lim
kx(s + αn )kds = 0.
(2.7)
n→∞ l 0
Since A(t) is almost periodic on R and x(t) is Stepanov’s almost periodic on R,
we can extract a subsequence α0 ⊂ α, B(t) ∈ H(A) and y(t) ∈ SH(x) such that
y = U STα0 x and B = U Tα0 A. By Lemma 2.1, there exist a subsequence α00 ⊂ α0
and a solution ỹ of the equation
y 0 = B(t)y
(2.8)
ỹ = U STα00 x
(2.9)
such that
on R. On the other hand, we have
Z
Z
Z
1 l
1 l
1 l
00
|ỹ(s)|ds ≤
|ỹ(s) − x(s + αn )|ds +
|x(s + αn00 )|ds
l o
l 0
l 0
(2.10)
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BOUNDENESS AND STEPANOV’S ALMOST PERIODICITY
5
Let n → ∞, we obtain that
Z
1 l
|ỹ(s)|ds = 0
l 0
from (2.7) and (2.9). And thus, there exists at least one t0 ∈ [0, l] such that
ỹ(t0 ) = 0. Since ỹ(t) is a solution of (2.8), we have ỹ(t) = 0 for all t ∈ R. Then
(2.9) implies
Z
1 τ +l
kx(s + αn00 )kds = 0
lim
n→∞ l τ
uniformly for τ ∈ R; hence
Z
Z
1 l
1 l
kx(t + s)kds = lim
kx(t − α00 + s + αn00 )kds
n→∞ l 0
l 0
00
Z
1 t−α +l
kx(s + αn00 )kds = 0
= lim
n→∞ l t−α00
and consequently, x(t + s) = 0 on [0, l] for any t ∈ R. Since t ∈ R is arbitrary,
we must have x(t) ≡ 0 on R, which is a contradiction to the fact that x(t) is a
non-trivial solution of (2.8). This completes the proof of this theorem.
Corollary 2.3. Let a(t) be a scalar almost periodic function defined on R. If each
bounded solution of the equation
x0 = a(t)x
is Stepanov’s almost periodic on R, then
Z t
sup
a(s)ds < ∞
t∈R
(2.12)
0
implies that for any real number l > 0,
Z
inf
sup
t∈R
(2.11)
s∈[t,t+l]
s
a(τ )dτ > −∞.
(2.13)
0
Rt
Proof. Suppose that (2.12) holds. Then x(t) = exp 0 a(s)ds is a non-trivial
bounded solution of (2.11), so it is Stepanov’s almost periodic. By Theorem 2.2,
for any l > 0,
Z
1 t+l R s a(τ )dτ
inf
e0
ds > 0., .
(2.14)
t∈R l t
Now we show that for any l > 0,
Z s
inf sup
a(τ )dτ > −∞.
t∈R s∈[t,t+l]
0
Otherwise, we can pick up a sequence {tn } ⊂ R such that
Z s
sup
a(τ )dτ ≤ −n, n = 1, 2, . . . .
s∈[tn ,tn +l]
0
So,
Z
s
a(τ )dτ ≤ −n
0
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Z. HU
EJDE-2005/35
for all s ∈ [tn , tn + l], n = 1, 2, . . . . Hence,
1
l
tn +l R
s
Z
e
0
a(τ )dτ
ds ≤
tn
1
l
Z
tn +l
e−n ds = e−n ,
n = 1, 2, . . . .
tn
This implies that
1
lim
n→∞ l
Z
tn +l R
s
e
0
a(τ )dτ
ds = 0
tn
This contradicts (2.14) and the proof of this corollary is completed.
Corollary 2.4. Let a(t) be an almost periodic function defined on R. If there exists
Rt
a positive constant M such that 0 a(s)ds ≤ M and
Z
1 t+l R s a(τ )ds
e0
ds = 0
t∈R l t
Rt
then the function exp 0 a(s)ds is not Stepanov’s almost periodic on R.
inf
Example. According to Corollary 2.4, we can construct many examples of equations whose solutions are bounded, but not Stepanov’s almost periodic on R. To
construct such an example, let n ≥ 3 and define


t ∈ [0, 1] ∪ [2n−1 − 1, 2n−1 ]
0
gn (t) = −n/(2n−1 − 1) t ∈ [2, 2n−1 − 2]


linear
t ∈ [1, 2] ∪ [2n−1 − 2, 2n−1 − 1].
Now, extend gn (t) to be odd and periodic with period 2n . Then, gn (t) satisfies
Z t
gn (s)ds ≤ 0, for all t ∈ R;
(2.15)
0
sup |gn (t)| =
t∈R
Z
t
gn (s)ds = −n
0
2n−1 − 3
,
2n−1 − 1
n
2n−1
−1
;
(2.16)
for all t ∈ [2n−1 − 1, 2n−1 ]
(2.17)
for each n ∈ Z + . Since
∞
X
sup |gn (t)| =
n=3 t∈R
∞
X
n
< ∞,
n−1 − 1
2
n=3
the function
g(t) =
∞
X
gn (t)
n=3
is almost periodic on R and
Z
t
g(s)ds =
0
∞ Z
X
n=3
0
t
gn (s)ds ≤ 0,
t ∈ R.
(2.18)
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BOUNDENESS AND STEPANOV’S ALMOST PERIODICITY
7
Now, let l = 1, tn = 2n−1 − 1, then
Z
Z 2n−1 R
s
1 tn +l R s g(τ )dτ
e0
ds =
e 0 g(τ )dτ ds
l tn
2n−1 −1
Z 2n−1 R
s
≤
e 0 gn (τ )dτ ds
(2.19)
2n−1 −1
Z
2n−1
n−1
−n 2n−1 −3
2
−1
=
e
ds
2n−1 −1
n−1
−n 2n−1 −3
=e
2
−1
for each n ∈ Z + . So,
1
lim
n→∞ l
Z
1
inf
t∈R l
Z
tn +l R
s
e
0
g(τ )dτ
ds = 0.
tn
This implies
t+l R
s
e
g(τ )dτ
ds = 0.
By Corollary 2.4, the function exp 0 g(s)ds is not Stepanov,s almost periodic on
R. Therefore, all non-trivial solutions of equation
0
t
Rt
x0 = g(t)x
are not Stepanov’s almost periodic on R, but they are all bounded on R because
(2.18) holds.
Acknowledgement. The author thanks the referee for his/her critical review of
the original manuscript and for the changes suggested; in particular, for the proof
of Lemma 2.1. The author also wants to thank Professor Mingarelli, at Carleton
University, for his help and suggestions.
References
[1] A. S. Besicovitch; Almost Periodic Functions, Dover, New York, 1954.
[2] C. C. Conley and R. K. Miller; Asymptotic stability without uniform stability: almost periodic
coefficients, J. Deffrentail Equations, 1(1965) 333–336.
[3] A. M. Fink, Almost periodic differential equations, Lecture Notes in Mathematics, Vol. 377,
Springer-Verlay, New York, 1974.
[4] Z. Hu and A. B. Mingarelli; On a question in theory of almost periodic differential equations,
Proc. Amer. Math. Soc. Vol. 127 (1999), pp. 2665-2670.
[5] Angelo B. Mingarelli, F. Q. Pu , L. Zheng; A counter-example in the theory of almost periodic
differential equations, Rocky Mountain J. of Math. 25(1995), 437–440.
School of Mathematics and Statistics, Carleton University, Ottawa, Ontario, K1S
5B6, Canada
E-mail address: zshhu@yahoo.com