Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2009, Article ID 328479, 17 pages
doi:10.1155/2009/328479
Research Article
Periodic Solution of Second-Order
Hamiltonian Systems with a Change Sign
Potential on Time Scales
You-Hui Su1, 2 and Wan-Tong Li2
1
School of Mathematics and Physical Sciences, Xuzhou Institute of Technology,
Xuzhou, Jiangsu 221008, China
2
School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China
Correspondence should be addressed to You-Hui Su, suyouhui@xzit.edu.cn
Received 26 November 2008; Accepted 4 April 2009
Recommended by Yong Zhou
This paper is concerned with the second-order Hamiltonian system on time scales T of the form
uΔΔ ρtμbt|ut|μ−2 ut∇Ht, ut 0, Δ-a.e. t ∈ 0, T T , u0−uT uΔ ρ0−uΔ ρT 0, where 0, T ∈ T. By using the minimax methods in critical theory, an existence theorem of
periodic solution for the above system is established. As an application, an example is given to
illustrate the result. This is probably the first time the existence of periodic solutions for secondorder Hamiltonian system on time scales has been studied by critical theory.
Copyright q 2009 Y.-H. Su and W.-T. Li. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
1. Introduction
The theory of calculus on time scales was introduced by Stefan Hilger in his Ph.D. thesis in
1988 1. It cannot only unify discrete and continuous calculus but also exhibit much more
complicated dynamics on time scales 2–6. In particular, dynamic equations on time scales
have many important applications, such as, in the study of biological, heat transfer, stock
market, and epidemic models 2, 5, 7–9. Consequently, it has been attracted considerable
amount of interest and is now a hot topic of still fairly theoretical exploration in mathematics.
Recently, for the existence problems of positive solutions for dynamic equations on
time scales, some authors have obtained many results; for details, see 10–21 and the
references therein. To the best of our knowledge, there is no work on the existence of periodic
solutions for second-order Hamiltonian systems on time scales. In particular, there is very
little work 22–24 on the existence of solutions of dynamic equations on time scales by using
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critical theory. Now, it is natural to use critical theory to consider the existence of periodic
solutions for second-order Hamiltonian systems on time scales.
We make the blanket assumption that 0, T are points in T, for an interval 0, T T , we
always mean 0, T ∩ T. Other types of intervals are defined similarly. We say that a property
holds for Δ-a.e. t ∈ A ⊂ T or Δ-a.e. on A ⊂ T, whenever there exists a set E ⊂ A with null
Lebesgue Δ-measure such that this property holds for every t ∈ A \ E. We refer the reader to
3, 24, 25 for a broad introduction on Lebesgue Δ-measure.
In this paper, motivated by references 26, 27, we consider the following second-order
Hamiltonian system on time scales T of the form
uΔΔ ρt μbt|ut|μ−2 ut ∇Ht, ut 0, Δ-a.e. t ∈ 0, T T ,
u0 − uT uΔ ρ0 − uΔ ρT 0,
1.1
where T > 0, μ > 2, b ∈ C0, T T , R, H : 0, T T × Rn → R, t, x → Ht, x is measurable in
t for every x ∈ Rn and continuously differentiable in x for Δ-a.e. t ∈ 0, T T and ∇Ht, u Du Ht, u. By using the minimax methods in critical theory, we establish the existence of at
least one nonzero solution for the problem 1.1. Our results are even new for the special cases
of difference equation and include the results of Tang and Wu 27 for differential equation.
Moreover, we prove some lemmas, which will be very important in proving the existence
of periodic solutions in HT1 T spaces for many other second-order Hamiltonian systems on
time scales. As an application, an example is given to illustrate the result.
There is a solution u of problems 1.1; we mean u : Tκ → Rn which is delta
differential; uΔ and uΔΔ are both continuous Δ-a.e. on Tκ ∩ Tκ , and u satisfies problems 1.1.
Now, we present some basic definitions which can be found in 3–5, 28. Another
excellent source on dynamical systems on measure chains is the book 6.
A time scale T is a nonempty closed subset of R. For t ∈ T, the forward and back jump
operators σ, ρ : T → T are well defined, respectively, by
σt inf{s ∈ T : s > t},
ρt sup{s ∈ T : s < t}.
1.2
In this definition, one puts inf ∅ : sup T and sup ∅ : inf T , where ∅ denotes the empty set.
A point t ∈ T is called left-dense if ρt t, left-scattered if ρt < t, right-dense if σt t,
and right-scattered if σt > t.
If T has a right-scattered minimum m, define Tκ T − {m}; otherwise, set Tκ T.
If T has a left-scattered maximum M, define Tκ T−{M}; otherwise, set Tκ T. The forward
graininess is μt : σt − t. Similarly, the backward graininess is νt : t − ρt.
If f : T → R is a function and t ∈ Tκ , then the delta derivative 4 of f at the point t
is defined to be the number f Δ t provided it exists with the property that, for any > 0,
there is a neighborhood U ⊂ T of t such that
fσt − fs − f Δ tσt − s ≤ |σt − s|,
∀ s ∈ U.
1.3
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If f : T → R and t ∈ Tκ , then the nabla derivative of f at the point t is defined
by the number f ∇ t provided it exists with the property that, for any > 0, there is a
neighborhood U ⊂ T of t such that
f ρt − fs − f ∇ t ρt − s ≤ ρt − s, ∀ s ∈ U.
1.4
We refer the reader to 25 for measure on time scales; absolutely continuous on time
scales can be found in 29. We now provide the definition in 24, 30 and simply summarize
the main points, which also be described in 31.
Let a : inf{s : s ∈ T} and b : sup{s : s ∈ T}, defined a function E : a, b → R by
Et : sup{s ∈ T : s ≤ t},
t ∈ a, b.
1.5
Now, suppose that f : Tκ → R is arbitrary function, if f ◦ E is measurable on the real
interval a, b in the usual Lebesgue senses, then we say f is measurable; if f ◦ E is integrable
on the real interval a, b in the usual Lebesgue senses, then we say f is integrable. Let L1 T
denote the set of such integrable functions on T. Furthermore, for any f ∈ L1 T, we defined
the integral of f by
t
t
f ◦ Edτ
fΔ :
s
for s, t ∈ T,
1.6
s
with the norm defined by
f 1 L T
b
f Δ for f ∈ L1 T.
1.7
a
t
We use the notation s fΔ to denote the Lebesgue integral of a function f between s, t ∈ T
when it is defined. That is, we use the same notation for the Lebesgue-type integral defined
in 24, 30 as is commonly used in the time scale literature for a Riemann-type integral
defined in terms of antiderivatives. A detailed discussion of the Lebesgue-type integral and
it’s relationship with the usual time scale integral is given in 24, 30. With the Lebesgue
integral defined, denote
L2 T : f ∈ L1 T : |f|2 ∈ L1 T ,
f 2 L T
b
1/2
2
|f| Δ
1.8
for f ∈ L T.
2
a
It is shown in 31 that L2 T is completed with respect to the norm fL2 T .
1
T by
Next, define the norm · on Crd
f 2
2
f 2 f Δ 2
L T
L Tκ 1/2
1
for f ∈ Crd
T.
1.9
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1
T with respect to the norm · see 31,
The space H 1 T is the completion of Crd
1
Definition 4.1 and Remark 4.2. The space H T is a time scale analog to the usual Sobolev
space H 1 I on a real interval I.
We refer the reader to 32 for an introduction on basic properties of Sobolev’s spaces
on bounded time scales.
Remark 1.1. If we replace u : T → R with u : T → Rn , then the above discussion still holds.
The rest of the paper is organized as follows. In Section 2, we list some lemmas, which
are important in proving the existence of periodic solutions. By applying these lemmas, we
establish the existence of periodic solutions for problem 1.1. In the final section, an example
is given to illustrate our main result.
2. Some Lemmas
In this section, to interpret Hamiltonian systems on time scales in a functional-analytic
setting, we introduce some lemmas, which will be used in the rest of the paper and be very
important in proving the existence of periodic solutions in HT1 T spaces for second-order
Hamiltonian systems on time scales.
Let HT1 T be the Hilbert space given by
HT1 T u : 0, T T −→ Rn | u is absolutely continuous, u0 uT ,
uΔ t ∈ L2 0, T Tκ , Rn ,
2.1
with the norm defined by
u T
2
|ut| Δ T
0
1/2
Δ
2
|u t| Δ
for u ∈ HT1 T.
0
2.2
Moreover, we define
uL1 T T
0
|ut|Δ, uL2 T T
0
1/2
2
|ut| Δ
,
u∞ sup |u|.
2.3
t∈0,T T
We also define inner product on HT1 T by
u, v T
ut · vt uΔ t · vΔ t Δ.
2.4
u
t ut − ut,
2.5
0
For u ∈ HT1 T, let
1
ut T
T
utΔ,
0
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1 T {u ∈ H 1 T | ut 0}.
1 T be the subspace of H 1 T given by H
and let H
T
T
T
T
In the following, we will prove several lemmas which are very important in proving
the existence of periodic solutions for problem 1.1.
Lemma 2.1. Let p ∈ R be such that p ≥ 1. Then, for every q ∈ 1, ∞, the immersion HT1 T →
Lq T is compact.
Proof. The proving is similar to the way as in proving of 32, Corollary 3.11, and we omit it
here.
The following two Lemmas are an immediate consequence of the 23, Proposition 3.6
see also 23, Corollary 3.9.
Lemma 2.2. let {um }m∈N ⊂ HT1 T and u ∈ HT1 T. If {um }m∈N converges weakly in HT1 T to u,
then {um }m∈N converges uniformly to u on 0, T T .
Lemma 2.3. If u ∈ HT1 T, then
u∞ ≤ c1 u.
In particular, if
T
0
2.6
utΔ 0, then
u∞ ≤ T 1/2 uΔ t
L2 T
.
2.7
Lemma 2.4. Let L : 0, T T × Rn × Rn → R, t, x, y → Lt, x, y be measurable in t for each
x, y ∈ Rn × Rn and continuously differentiable in x, y for Δ-a.e. t ∈ 0, T T . If there exist a ∈
CR , R , b1 ∈ L1 0, T T , R , and c ∈ L2 0, T T , R , such that, for Δ-a.e. t ∈ 0, T T and each
x, y ∈ Rn × Rn , one has
L t, x, y ≤ a|x| b1 t y2 ,
Dx L t, x, y ≤ a|x| b1 t y2 ,
2.8
Dy L t, x, y ≤ a|x| ct y ,
then the functional ϕ defined by ϕu and
T
0
Lt, ut, uΔ tΔ is continuously differential on HT1 T
T
Dx L t, ut, uΔ t · vt Dy L t, ut, uΔ t · vΔ t Δ.
ϕ u, v 2.9
0
∗
Proof. In the following, we will prove that ϕ has a directional derivative ϕ u ∈ HT1 T
given by 2.9 and that the mapping
∗
ϕ : HT1 T −→ HT1 T
, u −→ ϕ u
2.10
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is continuous.
i It follows easily from 2.8 that ϕ is everywhere finite on HT1 T. Fixing u and v in
1
HT T, we define
Fλ, t L t, ut λvt, uΔ t λvΔ t for
ψλ ϕu λv T
Fλ, tΔ T
0
t ∈ 0, T T , λ ∈ −1, 1,
Fλ, t ◦ Edt 0
2.11
T
Fλ, Etdt.
0
We will apply Leibniz formula of differentiation under integral sign to ψ. According
to assumption 2.8, one obtains
|Dλ Fλ, Et|
≤ Dx L t, u λv, u λv ◦ E · v ◦ E Dy L t, u λv, u λv ◦ E · v ◦ E 2 ≤ a|uEt λvEt| b1 t u λv |v| ct u λv v ◦ E
≤ a0
2 b1 t u u |v| ct u v v ◦ E,
2.12
where a0 maxλ,t∈−1,1×0,T a|uEt λvEt|.
2
It is obvious that b1 ◦ E ∈ L1 0, T , R , |uΔ ◦ E| |vΔ ◦ E| ∈ L1 0, T , R , c ◦ E ∈
2
L 0, T , R . v ∈ HT1 T implies that vΔ ◦ E ∈ L2 0, T , R and v ◦ E ∈ L1 0, T , R hold,
hence we have
|Dλ Fλ, Et| ≤ d ◦ E ∈ L1 0, T , R .
2.13
In view of Leibniz formula and 1.6, we get
ψ 0 ϕ u, v T
T
Dλ F0, Etdt
0
Dx L Et, uEt, uΔ Et · vEtdt
0
T
0
Dy L Et, uEt, uΔ Et · vΔ Etdt
T
Dx L t, ut, uΔ t · vt Dy L t, ut, uΔ t · vΔ t Δ.
0
2.14
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Moreover
2 Δ Dx L t, u, u ≤ a|u| b1 t uΔ ∈ L1 0, T T , R ,
Dy L t, u, uΔ ≤ a|u| ct uΔ ∈ L2 0, T T , R .
2.15
Thus, from Lemma 2.3,
T
T
Dx L t, u, uΔ · v Dy L t, u, uΔ · vΔ Δ
Dλ F0, tΔ 0
2.16
0
≤ c1 v∞ c2 vΔ L2 T
≤ c3 v,
∗
and ϕ has a directional derivative ϕ u ∈ HT1 T , given by 2.9.
ii According to the theorem of Krasnosel’skii 33, assumption 2.8 implies that the
mapping from HT1 T into L1 ×L2 defined by u → Dx Lt, u, uΔ , Dy Lt, u, uΔ is continuous,
∗
thus, ϕ is continuous from HT1 T into HT1 T , and the proof is completed.
We also need the following theorem, which was the generalized mountain pass
theorem.
Lemma 2.5. [34] Let E be a real Hilbert space with E E1 ⊕ E2 and E2 E1⊥ . Suppose I ∈ C1 E, R,
satisfies (PS), and
I1 Iu 1/2Au, u mu, where Au A1 P1 u A2 P2 u and Ai : Ei → Ei is bounded
and self-adjoint, i 1, 2,
I2 m u is compact, and
I3 there exist a subspace E ⊂ E and sets S ⊂ E, Q ⊂ E and constants α > ω such that
i
S ⊂ E1 and I|S ≥ α;
ii
Q is bounded and I|∂Q ≤ ω;
iii
S and ∂Q link.
Then I possesses a critical value c ≥ α.
3. Existence Results
In this section, by using the minimax methods in critical theory, we establish the existence
of at least one nonzero periodic solution for second-order Hamiltonian system 1.1 on time
scales.
Throughout this section, the following is assumed.
H1 Suppose that there exist t1 , t2 ∈ 0, T T and b ∈ C0, T T , R satisfying bt ≥
T
1/2bt0 > 0 for all t ∈ t1 , t2 T , where t0 ∈ t1 , t2 T . In addition, 0 btΔ 0.
T
H2 0 Ht, xΔ ≥ 0 for all x ∈ Rn and Δ-a.e. t ∈ 0, T T .
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H3 Assume that there exists g ∈ L1 0, T T , R such that |∇Ht, x| ≤ gt for all x ∈ Rn
and Δ-a.e. t ∈ 0, T T .
H4 Assume that there exist α0 ∈ 0, 1/2T 2 and r0 > 0 such that |Ht, x| ≤ α0 |x|2 for all
x ≤ r0 and Δ-a.e. t ∈ 0, T T .
H5 Assume that σρt t.
If
1
2
L t, x, y L t, ut, uΔ t |uΔ t| − bt|ut|μ − Ht, ut,
2
3.1
then by Lemma 2.4, the functional ϕ is given by
ϕu 1
2
T
T
T
Δ 2
u t Δ − bt|ut|μ Δ − Ht, utΔ,
0
0
3.2
0
which is continuously differentiable on HT1 T. Moreover
ϕ u, v T
uΔ t · vΔ tΔ − μ
0
−
T
T
bt|ut|μ−2 ut · vtΔ
0
∇Ht, ut · vtΔ
3.3
∀u, v ∈
0
HT1 T.
That is, for all u, v ∈ HT1 T, we get
ϕ u, v −
T
uΔΔ ρt · vtΔ − μ
0
−
T
bt|ut|μ−2 ut · vtΔ −
T
0
u
ΔΔ
ρt μbt|ut|
T
∇Ht, ut.vtΔ
0
μ−2
3.4
ut ∇Ht, ut · vtΔ.
0
Hence , u ∈ HT1 T is a solution of problem 1.1 if and only if u is a critical point of ϕ.
Lemma 3.1. Let a sequence {un t} ⊂ HT1 T be such that ϕ un t → 0 and let {un t} be
bounded in HT1 T, then {un t} has a convergent subsequence in HT1 T.
Proof. Since {un t} is bounded in HT1 T, it follows from 30, Theorem 4.12 that there
exists a subsequence still denoted by {un t} which weakly converges to u0 ∈ HT1 T. By
Lemma 2.2, we have
un −→ u0
in 0, T T .
Hence, for t ∈ 0, T T , there exists an M > 0 such that |un t| ≤ M, n 1, 2, . . . .
3.5
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Lemma 2.1 leads to
HT1 T → L2 0, T T , Rn is compact.
3.6
Hence
un −→ u0
in L2 0, T T , Rn .
3.7
Since
ϕ un t − ϕ um t, un t − um t
T
Δ
Δ
Δ
uΔ
n t − um t · un t − um t Δ
0
−μ
T
bt |un t|μ−2 un t − |um t|μ−2 um t · un t − um tΔ
3.8
0
−
T
∇Ht, un t − ∇Ht, um t · un t − um tΔ,
0
in view of H1 and H3, one has
T
2
Δ
un t − uΔ
m t Δ ≤ ϕ un t − ϕ um t un t − um t
0
2μMμ−1 un t − um t∞
2un t − um t∞
T
T
3.9
|bt|Δ
0
gtΔ −→ 0,
as n, m −→ ∞.
0
Consequently
2
un t − um t T
0
|uΔ
n tΔ
−
2
uΔ
m t| Δ
T
|un t − um t|2 Δ −→ 0
as n, m −→ ∞, 3.10
0
which implies that {un } is a Cauchy sequence in HT1 T. By the completeness of HT1 T, we
obtain that {un } is a convergent sequence in HT1 T; the proof is completed.
Now, we list our main result.
Theorem 3.2. Suppose that μ > 2, (H1), (H2), (H3), (H4), and (H5) hold, then the problem 1.1 has
at least one nonzero solution.
Proof. It suffices to show that all the conditions of Lemma 2.5 hold with respect to ϕ.
First, we show that ϕ satisfies the PS condition.
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From Lemma 3.1, we only need to prove that {un } is bounded. Otherwise, there exists
a subsequence still denoted by un such that un → ∞. Let vn un /un , then {vn } is
bounded; it has a subsequence we still denote {vn } which weakly converges to v0 . In view
of Lemma 2.2, {vn } uniformly converges to v0 in 0, T T . Since vn 1 for all n ∈ N, one has
vn / 0.
According to un → ∞ as n → ∞, μ > 2 and H3, for all vn , v ∈ HT1 T, we get
T
T
μ−2
1−μ 2−μ Δ
Δ
−ϕ un , v un vn t · v tΔ
μ bt|vn t| vn t · vtΔ ≤ un 0
0
T
1−μ un − ∇Ht, un t · vtΔ
0
3.11
≤ un 1−μ ϕ un v C1 un 2−μ vn v
T
1−μ
un v∞ gtΔ −→ 0 as n −→ ∞.
0
Thus, it follows from Lebesgue dominated convergence theorem on time scales 28 that
T
μ−2
bt|v0 t| v0 t · vtΔ 0
0
∀ v ∈ HT1 T.
3.12
By the arbitrariness of v, one has
bt 0
for Δ-a.e. t ∈ 0, T T ,
3.13
which contradicts the condition H1. Hence ϕ satisfies the PS condition.
Second, if
Au ut,
T
T
1
mu − bt|ut| Δ − Ht, utΔ −
2
0
0
μ
T
u2 tΔ,
0
then it is easy to verify that I1 and I2 hold.
Third, we will prove that ϕ satisfies the condition I3 in Lemma 2.5.
3.14
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11
1 T with u ≤ r0 , H4 and 2.7 imply
For arbitrary u ∈ H
T
ϕu T
1
2
2
|uΔ t| Δ −
T
0
T
1
≥
2
bt|ut|μ Δ −
0
2
Δ
|u t| Δ −
μ
ut∞
0
T
T
Ht, utΔ
0
|bt|Δ − α0
0
T
|ut|2 Δ
0
3.15
T
1
μ
≥
u2∞ − ut∞ |bt|Δ − α0 T ut2∞
2T
0
T
1
μ
− α0 T u2∞ − ut∞ |bt|Δ.
≥
2T
0
Choose ρ > 0 small enough such that
α
T
1
2
μ
− α0 T ρ − ρ
|bt|Δ > 0,
2T
0
3.16
thus
ϕu ≥ α >
1 T with u ρ.
∀u∈H
T
3.17
If
1 T, u ρ ,
S u∈H
T
3.18
then ϕ|s ≥ α > 0; this implies that the condition i of Lemma 2.5 holds.
It is known that H1 and H2 lead to
ϕx −
T
bt|x|μ Δ −
T
0
Ht, xΔ ≤ 0
∀ x ∈ Rn .
3.19
0
1 T such that et 0 for all t ∈ 0, T T \ t1 , t2 T and
Choose 0 /
e ∈ H
T
T
btetΔ 0
t2
3.20
btetΔ 0.
t1
For arbitrary u ∈ HT1 T, let
1
ϕ1 u 2
T
0
Δ
2
|u t| Δ,
ϕ2 u −
T
0
μ
bt|ut| Δ,
ϕ3 u −
T
Ht, utΔ,
0
3.21
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then
ϕu ϕ1 u ϕ2 u ϕ3 u.
3.22
For all x ∈ Rn and r ≥ 0, in terms of H3, one has
T0
ϕ3 x re − ϕ3 x 0
≤r
T
0
∇Ht, x sret · ret dsΔ
1
3.23
gt|et|Δ ≤ re∞ g L1 T .
Since −ϕ3 x ≥ 0 for all x ∈ Rn , we have
ϕ3 x re ≤ re∞ g L1 T
∀ x ∈ Rn , r ≥ 0.
3.24
In terms of 3.20 and Hölder’s inequality on time scales, one obtains
t2
2
2
− bt |x| r |et| Δ 2
t2
t1
− bt|x| r|et|2 Δ
t1
≤
t
2
2/μ
μ
− bt|x| r|et| Δ
3.25
t1
×
t
2
μ−2/μ
− btΔ
∀ x ∈ Rn , r ≥ 0.
t1
T
Thus, by using 0 bt|x|μ Δ 0, 3.25, and Hölder’s inequality on time scales again, for all
x ∈ Rn and r ≥ 0, we obtain
ϕ2 x re −
T
bt |x ret|μ − |x|μ Δ
0
−
t1
μ
bt|x ret| Δ −
0
−
μ
bt|x ret| Δ −
bt|x ret|μ Δ t2
2
t1
bt|x ret| Δ t2
T
bt|x|μ Δ
0
bt|x|μ Δ
2
2
μ/2 t
− bt|x| r |et| Δ
2
t1
t2
μ
t1
t
≤−
T
t1
t2
t1
≤
t2
bt|x| r|et|μ Δ 2
2−μ/2
− btΔ
t1
t2
t1
t2
bt|x|μ Δ
t1
bt|x|μ Δ ≤ 0.
3.26
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3.24 and 3.26 lead to
ϕx re ϕ1 x re ϕ2 x re ϕ3 x re
T
1
2
≤ r 2 |eΔ | Δ re∞ g L1 T ∀ x ∈ Rn , r ≥ 0,
2
0
3.27
which means that there exists r3 > 0 such that
α
2
ϕx re ≤
∀x ∈ Rn , r ∈ 0, r3 .
3.28
For all R ≥ 0 and r ≥ 0, let
hR, r t
2
2
μ/2 t
2
2
− bt|R| r |et| Δ
2
t1
−
t2
t1
− btΔ
t2
t1
t
2
2−μ/2
2
bt|R|2 r 2 |et|2 Δ
bt|R|μ Δ
t1
μ/2 t
btΔ
t1
In view of
2−μ/2
t2
t1
3.29
bt|R|μ Δ.
t1
bt > 0, for all R ≥ 0 and r ≥ 0, we get
∂h
R, r −rμ
∂r
≤ −rμ
t2
2
t
2
bt|et| Δ
t1
t2
μ/2−1 t
2
2
btΔ
t1
2
t1
t
2
bt|et| Δ
μ/2−1 t
2
2
R btΔ
t1
t1
≤ −rμRμ−2
t2
2−μ/2
2
btR r |et| Δ
2
2−μ/2
btΔ
3.30
t1
bt|et|2 Δ.
t1
3.30 and hR, 0 0 imply that
1
hR, r ≤ − r 2 μRμ−2
2
t2
bt|et|2 Δ ∀ R ≥ 0,
r ≥ 0.
3.31
t1
Note that there exists R1 > ρ > 0 such that
T
0
2
|eΔ t| Δ − μRμ−2
t2
t1
bt|et|2 Δ ≤ −
2
e∞ g L1 T
r3
∀ R ≥ R1 .
3.32
14
Discrete Dynamics in Nature and Society
Therefore, by using 3.24, 3.26, 3.31, and 3.32, for x R, r ∈ r3 , R and R ≥ R1 , one has
ϕx re ϕ1 x re ϕ2 x re ϕ3 x re
1 2 T Δ 2
|e t| Δ hR, r re∞ g L1 T
≤ r
2
0
T
t2
1
1
2
≤ r 2 |eΔ t| Δ − r 2 μRμ−2 bt|et|2 Δ re∞ g L1
2
2
0
t1
≤−
3.33
r
r − r3 e∞ g L1 T ≤ 0.
r3
Thus, 3.28 and 3.33 can lead to
ϕx re ≤
α
,
2
∀ |x| R, r ∈ 0, R,
R ≥ R1 .
3.34
For all x ∈ Rn and R ≥ 0, denote
fx, R −
t
2
2
2
μ/2 t
2−μ/2
2
bt|x| R |et| Δ
2
btΔ
t1
t1
t2
bt|x|μ Δ.
3.35
t1
By using the similar way to inequality 3.26, one has
ϕ2 x Re ≤ fx, R
R ≥ 0,
∀ x ∈ Rn ,
3.36
for all x ∈ Rn and R ≥ 0, since
∂f
x, R −Rμ
∂R
t2
t
2
bt|et| Δ
t1
≤ −μRμ−1
2
2
μ/2−1 t
2
btΔ
t1
t
2
2−μ/2
2
bt|x| R |et| Δ
2
t1
μ/2 t
bt|et|2 Δ
btΔ
t1
3.37
2−μ/2
2
.
t1
In view of 3.37 and fx, 0 0, for all x ∈ Rn and R ≥ 0, we get
fx, R ≤ −R
μ
t
2
t1
2
μ/2 t
bt|et| Δ
2
2−μ/2
btΔ
t1
.
3.38
Discrete Dynamics in Nature and Society
15
From 3.24, 3.36, and 3.38, for all x ∈ Rn and R ≥ 0, we obtain
ϕx Re ϕ1 x Re ϕ2 x Re ϕ3 x Re
1
≤ R2
2
μ/2 t
2−μ/2
t
T
2
2
Δ 2
2
μ
bt|et| Δ
btΔ
Re∞ g L1 T ,
e t Δ − R
0
t1
t1
3.39
which implies that there exists R2 > R1 , such that
ϕx Re ≤ 0
∀x ∈ Rn ,
R ≥ R2 .
3.40
Now let
Q {x re | x ∈ Rn , x R, r ∈ 0, R, R ≥ R2 }.
3.41
Thus, 3.19, 3.34, and 3.40 lead to ϕ|∂Q ≤ 1/2α, which means that the condition ii of
Lemma 2.5 is satisfied.
It is easy to see that S and ∂Q link. Hence, all the conditions of the generalized
mountain pass theorem are satisfied. By Lemma 2.5, the problem 1.1 has at least one nonzero
solution.
4. An Example
In this section, we present a simple example to illustrate our result.
Let
T 0, 0.3 ∪ {0.4, 0.45, 0.5, 0.55, 0.6} ∪ 0.7, 1,
T 1.
4.1
Consider the following second-order Hamiltonian system on time scales T of the form
uΔΔ ρt μbt|ut|μ−2 ut ∇Ht, ut 0, Δ-a.e. t ∈ 0, 1T ,
u0 − u1 uΔ 0 − uΔ 1 0,
4.2
where μ > 2 is a constant; let ε > 0 is arbitrary small, Ht, u εu2 t for Δ-a.e. t ∈ 0, 1T
and
⎧
⎪
t,
⎪
⎪
⎨
bt −1.5t 0.75,
⎪
⎪
⎪
⎩
t − 1,
t ∈ 0, 0.3T ,
t ∈ 0.3, 0.7T ,
4.3
t ∈ 0.7, 1T .
It is easy to verify that all the conditions of Theorem 3.2 are satisfied. By Theorem 3.2, we see
that the problem 4.2 has at least one nonzero solution.
16
Discrete Dynamics in Nature and Society
Acknowledgment
This work was supported by the NSF of China 10571078 and the grant of XZIT
XKY2008311.
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