Electronic Journal of Differential Equations, Vol. 2004(2004), No. 17, pp. 1–8.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu (login: ftp)
CLASSIFICATION AND EXISTENCE OF POSITIVE SOLUTIONS
TO NONLINEAR DYNAMIC EQUATIONS ON TIME SCALES
CAN-YUN HUANG & WAN-TONG LI
Abstract. A classification scheme is given for the eventually positive solutions to a class of second order nonlinear dynamic equations, in terms of their
asymptotic magnitudes. Also we provide necessary and/or sufficient conditions
for the existence of positive solutions.
1. Introduction
The study of dynamic equations on time scales, which has recently received a lot
of attention, was introduced by Stefan Hilger in his Ph.D. thesis in 1988 in order
to unify continuous and discrete analysis [4]. Since Hilger formed the definition of
derivatives and integrals on time scales, several authors have expended on various
aspects of the new theory, see the paper by Agarwal et al. [2] and the book by
Bohner and Peterson [3].
In recent years there has been much research activity concerning the oscillation
and nonoscillation of some different equations on time scales, we refer the reader
to the papers [13, 14, 15]. However, to the best of our knowledge, there have not
appeared any results concerned with asymptotic behavior and existence of solutions
of dynamic equations on time scales. In this paper, we classify positive solutions of
the second order nonlinear dynamic equation
y ∆∆ (t) + r(t)f (y σ (t)) = 0
for t ∈ T
(1.1)
according to the limiting behavior and then provide sufficient and/or necessary
conditions for their existence, where r ∈ Crd ([t0 , +∞) ∩ T, [0, ∞), r 6 ≡0 for t ∈ T,
t0 > 0 and f (y) > 0 is nondecreasing for any y ∈ R − {0}.
We note that if T = R, then (1.1) becomes the differential equation
y 00 (t) + r(t)f (y(t)) = 0
for t ∈ R.
(1.2)
The asymptotic behavior of solutions of (1.2) has been studied by several authors
under different conditions, see Naito [10, 11]. If T = Z, then (1.1) becomes the
2000 Mathematics Subject Classification. 34B15, 39A10.
Key words and phrases. Time scale, nonlinear dynamic equation, positive solution.
c
2004
Texas State University - San Marcos.
Submitted December 16, 2003. Published February 6, 2004.
Both authors were supported by the Key Research and Development Program for
Outstanding Groups at Lanzhou University of Technology.
W.-T. Li was supported by grant 10171040 from the NNSF of China, and by grant
ZS011-A25-007-Z from the NSF of Gansu Province of China.
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C. Y. HUANG AND W.T. LI
EJDE-2004/17
difference equation
∆2 (yn ) + rn f (yn+1 ) = 0
for n ∈ Z,
(1.3)
which has been discussed in detail by many authors, one can refer to [5, 6, 7, 8, 9].
Since we are interested in asymptotic behavior of positive solutions of (1.1), we
suppose that the time scale T is a time scale interval of the form [t0 , ∞) ∩ T and
shall employ t ≥ t0 to denote [t0 , ∞) ∩ T unless otherwise stated. Here a solution
2
of (1.1) means that y ∈ Crd
and satisfies (1.1) for t ≥ t0 .
2. Some preliminaries
In this section, we give a short introduction to the time scale calculus. For the
explanation and results we refer to the book by Bohner and Peterson [3].
Definition. The forward jump operator σ(t) and ρ(t) are defined by
σ(t) := inf{s ∈ T : s > t} and ρ(t) := sup{s ∈ T : s < t}.
The graininess µ of the time scale is defined by µ(t) = σ(t) − t. Naturally, σ(t) =
t + µ(t) ≥ t.
Definition. A point t ∈ T is said to be left-dense if ρ(t) = t, right-dense if σ(t) = t,
left-scattered if ρ(t) < t and right-scattered if σ(t) > t.
A function f : T → R is said to be rd-continuous if it is continuous at each
right-dense point and if there exists a finite left-hand sided limit at all left-dense
points.
Definition. For a function f : T → R, the (delta) derivative is defined by
(
lims→t f (σ(t))−f (s) , if µ(t) = 0,
f (σ(t)) − f (s)
∆
= f (σ(t))−f (t) t−s
f (t) = lim
s→t
σ(t) − s
,
if µ(t) > 0.
σ(t)−t
The function f is said to be differentiable if its derivative exists.
The derivative of f and the jump operator σ are related by the formula
f (σ(t)) = f σ (t) = f (t) + f ∆ (t)µ(t)
for t ∈ T.
(2.1)
∆
Lemma 2.1 ([3]). The function f (t) is increasing in t if f (t) > 0.
From this lemma, we can say that f is decreasing, nondecreasing and nonincreasing if f ∆ (t) < 0, f ∆ (t) ≥ 0 and f ∆ (t) ≤ 0 for t ∈ T, respectively.
Definition (Antiderivative). A function F : T → R is called an antiderivative of
f : T → R provided
F ∆ (t) = f (t) for all t ∈ T.
If a function f : T → R is rd-continuous, its antiderivative exists, denoted by
Z t
F (t) =
f (s)∆s for t0 and t ∈ T,
t0
where
Z t
(R t
f (s)∆s =
t0
f (s)ds,
if T = R;
s∈[t0 ,t) µ(s)f (s), if [t0 , t] consists of only isolated points.
t0
P
Obviously, if f (t) is differentiable, its antiderivative exists, and
Z t
f ∆ (s)∆s.
f (t) − f (t0 ) =
t0
(2.2)
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POSITIVE SOLUTIONS TO NONLINEAR DYNAMIC EQUATIONS
3
For the sake of convenience, we still call the antiderivative F (t) of f (t) as the
integral of f (t), antiderivative calculus as integral calculus, respectively. Similarly,
infinite integral is defined as
Z ∞
Z t
∆
f (s)∆s = lim
f ∆ (s)∆s.
t→∞
t0
t0
If T = R, then σ(t) = ρ(t) = t, µ(t) ≡ 0, f = f 0 and
Z b
Z b
f (s)∆s =
f (s)ds.
∆
a
a
If T = Z, then σ(t) = t + 1, ρ(t) = t − 1, µ(t) ≡ 1, f ∆ = ∆f , and
Z b
b−1
X
f ∆ (s)∆s =
f (t) for a < b.
a
t=a
3. Main Results
Let y(t) be a positive solution of (1.1). Then from (1.1) we have
y ∆∆ (t) = −r(t)f (y σ (t)) ≤ 0,
which, by Lemma 2.1, implies that y ∆ (t) is nonincreasing. Thus we claim that
y ∆ (t) ≥ 0
for t ≥ t0 .
If not, there exists a sufficiently large t1 ≥ t0 , such that y ∆ (t) < −c, where c > 0 is
a constant. Hence, for t > t1 , we obtain
Z t
Z t
∆
y(t) − y(t1 ) =
y (s)∆s <
(−c)∆s = −c(t − t1 ).
t1
t1
This means that lim y(t) = −∞, which contradicts y(t) > 0.
Since y ∆ (t) is nonincreasing and y ∆ (t) ≥ 0 for t ≥ t0 , then there are positive
constants α and β such that
α ≤ y(t) ≤ βt
for t ≥ t0 .
In view of the above considerations, we may now make the following classifications.
Let Ω be the set of all non-oscillatory solutions of (1.1) and Ω+ be the subset of Ω
containing those which are ultimately positive. Then any non-oscillatory solution
in Ω+ must belong to one of the following three sets:
C[max] = y ∈ Ω+ : lim y ∆ (t) = α > 0 ;
t→+∞
+
C[int] = y ∈ Ω : lim y(t) = ∞ and lim y ∆ (t) = 0 ;
t→+∞
t→+∞
C[min] = y ∈ Ω+ : lim y(t) = β > 0 .
t→+∞
In the following, we will give several necessary and/or sufficient conditions for
the existence of positive solutions of (1.1).
Theorem 3.1. Equation (1.1) has a positive solution of class C[max] if and only
if
Z +∞
r(s)f (bσ(s)) < ∞ for some b > 0.
(3.1)
t0
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C. Y. HUANG AND W.T. LI
EJDE-2004/17
Proof. Let y(t) ∈ C[max] of (1.1), then
lim y ∆ (t) = α > 0
t→+∞
for t ≥ t0 .
Hence there exist a sufficiently large t1 such that
1
3
α < y ∆ (t) < α
2
2
for t ≥ t1 ,
so that
3
1
αt < y(t) < αt for t > t2 > t1 .
2
2
Set b = α/2. Then the nondecreasing property of f implies
f (y(t)) ≥ f (bt) and f (y σ (t)) ≥ f (bσ(t)).
(3.2)
Integrating both sides of (1.1) from t1 to t, we see
Z t
y ∆ (t1 ) − y ∆ (t) =
r(s)f (y σ (s))∆s.
t1
Taking limits on both sides of the above equality, we get
Z t
lim
r(s)f (y σ (s))∆s = y ∆ (t1 ) − α,
t→∞
t2
which implies
Z
∞
r(s)f (y σ (s))∆s < ∞.
(3.3)
t1
From (3.2) and (3.3), it follows that
Z ∞
r(s)f (bσ(s))∆s < ∞.
t2
Conversely, assume that (3.1) holds. Then there exists a large number T such that
Z ∞
b
r(s)(f (bσ(s)))∆s <
for t ≥ T.
(3.4)
2
t
Consider the sequence {xn }∞
0 defined by x0 (t) = b/2 for t ≥ T and
Z Z
b 1 t ∞
xn+1 (t) = +
r(s)f (σ(s)xσn (s))∆s∆τ
2
t T τ
(3.5)
for t ≥ T , n = 0, 1, 2, · · · . In view of (3.4), the sequence {xn (t)}∞
0 is well defined.
In fact,
Z Z
b 1 t ∞
b
r(s)f ( σ(s))∆s∆τ
x1 (t) = +
2
t T τ
2
Z
b t−T ∞
≤ +
r(s)f (bσ(s))∆s∆τ
2
t
Z ∞ T
b
b
b
≤ +
r(s)f (bσ(s))∆s < + = b
2
2
2
T
and x1 (t) ≥ x0 (t) for t ≥ T . By induction and the nondecreasing property of f , we
have
xn+1 (t) ≥ xn (t) for t ≥ T, n = 0, 1, 2, · · · .
(3.6)
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5
∞
Next, we prove {xn (t)}0 is bounded for t ≥ T . Since
b
< b and x1 (t) < b,
2
so if we assume xn (t) < b for t ≥ T , then σ(s)xσn (s) < bσ(s) and
Z Z
b 1 t ∞
r(s)f (σ(s)xσn (s))∆s∆τ
xn+1 (t) = +
2
t T τ
Z Z
b 1 t ∞
≤ +
r(s)f (bσ(s))∆s∆τ
2
t T τ
Z ∞
b
≤ +
r(s)f (bσ(s))∆s < b for t ≥ T,
2
T
x0 (t) =
(3.7)
which, by induction implies {xn (t)}∞
0 is bounded for t ≥ T . In view of (3.6), we
∗
know {xn (t)}∞
0 is pointwise convergent to some function x (t). By means of the
Lebesgue’s dominated convergence theorem, see [1, 12], we obtain
Z Z
b 1 t ∞
∗
r(s)f (σ(s)xσ (s))∆s∆τ for t ≥ T
x (t) = +
2
t T τ
and
b
2
≤ x∗ (t) < b. Setting y(t) = tx∗ (t), i.e.,
Z tZ ∞
b
y(t) = t +
r(s)f (y σ (s))∆s∆τ for t ≥ T,
2
T τ
obviously, y(t) is a solution of (3.1) and belongs to C[max]. The proof is complete.
Theorem 3.2. Equation (1.1) has a positive solution of class C[min] if and only
if
Z +∞ Z +∞
r(s)f (d)∆s∆τ < ∞ for some d > 0.
(3.8)
t0
τ
Proof. Let y(t) ∈ C[min] for (1.1), then
lim y(t) = β > 0
lim y ∆ (t) = 0 for t ≥ t0 .
and
t→+∞
t→+∞
Hence there exists a sufficient large t1 such that
1
3
β < y(t) < β for t ≥ t1 .
2
2
Set d = β/2, then the nondecreasing property implies
and f (y σ (t)) > f (d)
f (y(t)) > f (d)
for t > t1 .
Integrating both sides of (1.1) from t to ∞ for t > t1 , we obtain
Z ∞Z ∞
β − y(t) =
r(s)f (y σ (s))∆s∆τ,
t
(3.9)
τ
which implies
Z
∞
Z
∞
r(s)f (d)∆s∆τ < ∞.
t1
τ
i.e., (3.8) holds. The rest of the proof is similar to that of Theorem 3.1, we omit it
here.
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C. Y. HUANG AND W.T. LI
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Theorem 3.3. If (1.1) has a positive solution in C[int], then
Z ∞
r(s)f (a)∆s < ∞ for some a > 0 ,
t
Z ∞ Z ∞0
r(s)f (bσ(s))∆s∆τ = ∞ for every b > 0.
t0
(3.10)
(3.11)
τ
Proof. Let y ∈ C[int], be a solution of (1.1), then limt→+∞ y(t) = ∞ and limt→+∞ y ∆ (t) =
0. Hence there exist two positive constants a, b and a sufficient large t1 > t0 such
that a < y(t) < bt for t > t1 , which, in view of the nondecreasing property of f ,
implies that
f (y(t)) ≥ f (a) and f (y σ (t)) ≤ f (a),
and f (y σ (t)) ≤ f (bσ(t))
f (y(t)) ≤ f (bt)
for t > t1 .
From equation (1.1), we have
Z t
y ∆ (t) +
r(s)f (y σ (s))∆s = y ∆ (t1 ) for t > t1 .
(3.12)
(3.13)
t1
In view of limt→+∞ y ∆ (t) = 0, (3.13) yields
Z ∞
r(s)f (y σ (s))∆s = y ∆ (t1 ),
t1
R∞
and so t1 r(s)f (a)∆s < ∞, which implies (3.10) holds.
Furthermore, since limt→+∞ y ∆ (t) = 0, we obtain
Z ∞
r(s)f (y σ (s))∆s = y ∆ (s) for s > t1 .
(3.14)
s
Integrating on both sides of (3.14) from t1 to t, we obtain
Z tZ ∞
y(t) − y(t1 ) =
r(s)f (y σ (s))∆s∆τ
t1
τ
Z tZ
≤
∞
r(s)f (bσ(s))∆s∆τ for t > t1 .
t1
τ
Hence, (3.12) and limt→+∞ y(t) = ∞ imply
Z ∞Z ∞
r(s)f (bσ(s))∆s∆τ = ∞.
t1
The proof is complete.
τ
Theorem 3.4. Equation (1.1) has a positive solution in C[int] provided that
Z ∞
r(s)f (aσ(s))∆s < ∞ for some a > 0
(3.15)
t0
Z ∞Z ∞
r(s)f (b)∆s∆τ = ∞ for every b > 0.
(3.16)
t0
τ
Proof. In view of (3.15) and (3.16), there exist two positive constants a, b and a
sufficiently large t1 such that
Z Z
b
b 1 t ∞
< a,
+
r(s)f (aσ(s))∆s∆τ < a for t ≥ t1 .
(3.17)
t
t
t t1 τ
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POSITIVE SOLUTIONS TO NONLINEAR DYNAMIC EQUATIONS
7
∞
Consider the sequence {xn (t)}0 defined by x0 (t) = 0 and
Z Z
b 1 t ∞
r(s)f (σ(s)xσn (s))∆s∆τ for t ≥ t1 , n = 0, 1, · · · .
xn+1 (t) = P xn (t) = +
t t t1 τ
∞
It is easy to see that {xn (t)}0 is well defined. In fact,
x1 (t) =
b
< a, xσ1 (s) < a
t
for t ≥ t1
and
x2 (t) =
≤
b 1
+
t
t
Z tZ
∞
b 1
+
t
t
t1
Z t
Z
∞
r(s)f (σ(s)xσ1 (s))∆s∆τ
τ
r(s)f (aσ(s))∆s∆τ < a for t ≥ t1 .
t1
τ
Furthermore, if we assume that xn (t) < a for t ≥ t1 , then xσn (t) = xn (σ(t)) < a
and
Z Z
b 1 t ∞
xn+1 (t) = +
r(s)f (σ(s)xσn (s))∆s∆τ
t
t t1 τ
Z Z
b 1 t ∞
≤ +
r(s)f (aσ(s))∆s∆τ < a for t ≥ t1 ,
t
t t1 τ
which, by induction, shows that {xn (t)}∞
0 is bounded, i.e.,
0 ≤ xn (t) < a
for t ≥ t1 , n = 0, 1, · · · .
(3.18)
In view of x0 (t) ≤ x1 (t) and the nondecreasing property of f , we have
xn+1 (t) ≥ xn (t)
for t ≥ t1 , n = 0, 1, · · · .
(3.19)
Hence, Lebesgue’s dominated convergence theorem [1, 12] implies that
Z Z
a 1 t ∞
r(s)f (σ(s)x∗ (σ(s)))∆s∆τ for t ≥ t1 .
x∗ (t) = +
t
t t1 τ
Set y(t) = tx∗ (t). Then
Z tZ
y(t) = a +
t1
∞
r(s)f (y σ (s))∆s∆τ
for t ≥ t1 .
τ
It is easily verified that y(t) is a solution of (1.1) and belongs to C[int]. The proof
is complete.
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Can-Yun Huang
Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou,
Gansu 730050, China
Wan-Tong Li
Department of Mathematics, Lanzhou University, Lanzhou, Gansu 730000, China
E-mail address: wtli@lzu.edu.cn