Volume 2009, Article ID 316812, 17 pagesdoi:10.1155/2009/316812 Research Article Variational Method to the Impulsive Equation with Neumann Boundary Conditions Juntao Sun and Haibo Chen D
Trang 1Volume 2009, Article ID 316812, 17 pages
doi:10.1155/2009/316812
Research Article
Variational Method to the Impulsive Equation with Neumann Boundary Conditions
Juntao Sun and Haibo Chen
Department of Mathematics, Central South University, Changsha, 410075 Hunan, China
Correspondence should be addressed to Juntao Sun,sunjuntao2008@163.com
Received 28 August 2009; Accepted 28 September 2009
Recommended by Pavel Dr´abek
We study the existence and multiplicity of classical solutions for second-order impulsive Sturm-Liouville equation with Neumann boundary conditions By using the variational method and critical point theory, we give some new criteria to guarantee that the impulsive problem has at least one solution, two solutions, and infinitely many solutions under some different conditions, respectively Some examples are also given in this paper to illustrate the main results
Copyrightq 2009 J Sun and H Chen 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
In this paper, we consider the boundary value problem of second-order Sturm-Liouville equation with impulsive effects
−p tut rtut qtut gt, ut, t / t k , a.e t ∈ 0, 1,
−Δp t k ut k I k ut k , k 1, 2, , p − 1,
u0 u
1−
0,
1.1
where 0 t0 < t1 < t2 < · · · < t p−1 < t p 1, p ∈ C10, 1, r, q ∈ C0, 1 with p and q
continuous,−Δpt k ut k −pt k ut
k −ut−
k , ut
k and ut−
k denote the right and the
left limits, respectively, of ut at t t k , u0 is the right limit of u0, and u1− is the left
limit of u1
In the recent years, a great deal of work has been done in the study of the
Trang 2of chemotherapy, population dynamics, optimal control, ecology, industrial robotics, and physics phenomena are described For the general aspects of impulsive differential equations,
tools or techniques have been used to study such problems in the literature These classical
On the other hand, in the last two years, some researchers have used variational methods to study the existence of solutions for impulsive boundary value problems Variational method has become a new powerful tool to study impulsive differential
following equation with impulsive effects:
−ρ tφ p
ut stφ p ut ft, ut, t / t j , a.e t ∈ a, b,
−Δρ
t j
φ p
u
t j
I j
u
t j
, j 1, 2, , l,
1.2
where f : a, b × 0, ∞ → 0, ∞ is continuous, I j:0, ∞ → 0, ∞, j 1, 2, , l, are continuous, and α, β, γ, σ > 0 They essentially proved that IBVP1.2 has at least two positive
theory, Nieto and O’Regan studied the existence of solutions of the following equation:
−ut λut ft, ut, t / t j , a.e t ∈ 0, T,
Δu
t j
I j
u
t j
, j 1, 2, , l,
u 0 uT 0,
1.3
−ut gtut ft, ut, t / t j , a.e t ∈ 0, T,
Δu
t j
I j
u
t j
, j 1, 2, , p,
u 0 uT 0,
1.4
variational method and critical point theorem
Motivated by the above facts, in this paper, our aim is to study the variational structure
Trang 3is no paper concerned impulsive differential equation with Neumann boundary conditions
effects are not involved
In this paper, we will need the following conditions
H1 There is constants β > 2, M > 0 such that for every t ∈ 0, 1 and u ∈ R with |u| ≥ M,
0 < βGt, u ≤ ugt, u, 0 < β
u
0
where Gt, u u
0g t, sds.
H2 limu→ 0gt, u/u 0 uniformly for t ∈ 0, 1, and lim u→ 0I k u/u 0.
H3 There exist numbers h1, h2> 0 and p1 > 1 such that
H4 There exist numbers a k , b k > 0 and γ k ∈ 0, 1 such that
H5 There exist numbers r1, r2> 0 and μ ∈ 0, 1 such that
k , b k > 0 and γ k ∈ 1, ∞ such that
I k u ≤ a
k b
examples are presented in this section to illustrate our main results in the last section
2 Preliminaries
following equivalent form:
−e −Lt p tut e −Lt q tut e −Lt g t, ut, t / t k , a.e t ∈ 0, 1,
−Δe −Lt kp t k ut k e −Lt kI k ut k , k 1, 2, , p − 1,
u0 u
1−
0.
2.1
Trang 4Obviously, the solutions of IBVP2.1 are solutions of IBVP 1.1 So it suffices to consider
In this section, the following theorem will be needed in our argument Suppose that E
Theorem 2.1 22, Theorem 38.A For the functional F : M ⊆ X → −∞, ∞ with
M / ∅, min u ∈M F u α has a solution for which the following hold:
i X is a real reflexive Banach space;
ii M is bounded and weakly sequentially closed;
iii F is weakly sequentially lower semicontinuous on M; that is, by definition, for each
sequence {u n } in M such that u n u as n → ∞, one has Fu ≤ lim inf n→ ∞F u n
holds.
Theorem 2.2 16, Theorem 2.2 Let E be a real Banach space and let ϕ ∈ C1E, R satisfy the
Palais-Smale condition Assume there exist u0, u1∈ E and a bounded open neighborhood Ω of u0such that u1∈ E \ Ω and
ϕ u0, ϕu1 < inf
Let
Γ {h | h : 0, 1 −→ E is continuous and h0 u0, h 1 u1},
c inf
h∈Γmax
s ∈0,1 ϕ hs. 2.3
Then c is a critical value of ϕ; that is, there exists u∗∈ E such that ϕu∗ Θ and ϕu∗ c, where
c > max {ϕu0, ϕu1}.
Theorem 2.3 23 Let E be a real Banach space, and let ϕ ∈ C1E, R be even satisfying the
Palais-Smale condition and ϕ 0 0 If E V ⊕ Y, where V is finite dimensional, and ϕ satisfies that
A1 there exist constants ρ, α > 0 such that ϕ| ∂Br ∩Y ≥ α,
A2 for each finite dimensional subspace W ⊂ E, there is R RW such that ϕu ≤ 0 for all
Then ϕ possesses an unbounded sequence of critical values.
consider the inner product
u, v
1
0
utvtdt
1
0
Trang 5which induces the usual norm
0
ut 2
dt
1
0
|ut|2dt
1/2
We also consider the inner product
u, v X
1
0
e −Lt p tutvtdt
1
0
e −Lt q tutvtdt, 2.6
and the norm
0
e −Lt p t ut 2
dt
1
0
e −Lt q t|ut|2dt
1/2
then the norm · Xis equivalent to the usual norm · in W 1,2 0, 1 Hence, X is reflexive.
We define the norm in C0, 1, L20, 1 as u∞ maxt ∈0,1 |ut| and u2 1
0|u|2dt1/2, respectively
For u ∈ W 2,2 0, 1, we have that u, uare absolutely continuous, and u ∈ L20, 1,
hence −Δe −Lt kp t k ut k −e −Lt kp t k ut
k − ut−
X, then u is absolutely continuous and u ∈ L20, 1 In this case, the one-side derivatives
u0, u1−, ut
k , ut−
introduce a different concept of solution We say that u ∈ C0, 1 is a classical solution of IBVP2.1 if it satisfies the equation in IBVP 2.1 a.e on 0, 1, the limits ut
k , ut−
k , k
1, 2, , p−1 exist and impulsive conditions in IBVP 2.1 hold, u0, u1− exist and u0
u1− 0 Moreover, for every k 0, 1, , p − 1, u k u| t k,tk1 satisfy u k ∈ W 2,2 t k , t k1
2u2
X−
p−1
k1
e −Lt ku t k
0
I k sds −
1
0
e −Lt G t, udt. 2.8
It is clear that ϕ is differentiable at any u ∈ X and
ϕuv
1
0
e −Lt p tutvt e −Lt q tutvtdt
−
p−1
k1
e −Lt kI k ut k vt k −
1
0
e −Lt g t, utvtdt
2.9
Lemma 2.4 If u ∈ X is a critical point of the functional ϕ, then u is a classical solution of IBVP
2.1
Trang 6Proof Let u ∈ X be a critical point of the functional ϕ It shows that
1
0
e −Lt p tutvte −Lt q tutvtdt
−
p−1
k1
e −Lt kI k ut k vt k −
1
0
e −Lt g t, utvtdt0
2.10
t ∈ t k , t k1 for k / j Equation 2.10 implies
tj1
tj
e −Lt p tutvt e −Lt q tutvt − e −Lt g t, utvtdt 0. 2.11
0 t j , t j1,
tj1
tj
e −Lt p tu
j twt e −Lt q tu j twt − e −Lt g
t, u j tw tdt 0, 2.12
where u j u| t j,tj1 Thus u jis a weak solution of the following equation:
−e −Lt p tut e −Lt q tut e −Lt g t, ut t ∈t j , t j1
0 t j , t j1 ⊂ Ct j , t j1 Let ht : e −Lt gt, u − qu, then 2.13
becomes the following form:
−e −Lt p tut ht ont j , t j1
u j t C1 C2
t
tj
e Ls−ln ps ds−
t
tj
e Ls−ln ps
s
tj
h r
p r e ln pr dr
ds t∈t j , t j1
where C1 and C2 are two constants Then uj ∈ Ct j , t j1 and u
j ∈ Ct j , t j1 Therefore, u j
Trang 7previous equation, we can easily get that the limits ut
j , ut−
j , j 1, 2, , p − 1, ut
0 and
ut−
p exist By integrating 2.10, one has
1
0
e −Lt p tutvt e −Lt q tutvtdt
−
p−1
k1
e −Lt kI k ut k vt k −
1
0
e −Lt g t, utvtdt
−
p−1
k1
Δe −Lt kp t k ut kv t k e −L1 p 1u
1−
v1
− e −L0 p 0u0v0 −
p−1
k1
e −Lt kI k ut k vt k
1
0
−e −Lt p tut e −Lt q tut − e −Lt g t, ut
v tdt
−
p−1
k1
Δe −Lt kp t k ut k e −Lt kI k ut kv t k
e −L1 p 1u
1−
v 1 − e −L0 p 0u0v0
1
0
−e −Lt p tut e −Lt q tut − e −Lt g t, ut
v tdt 0,
2.16
−
p−1
k1
Δe −Lt kp t k ut ke −Lt kI k ut kv t k
e −L1 p 1u
1−
v 1 − e −L0 p 0u0v00.
2.17
of generality, we assume that there exists i ∈ {1, 2, , p − 1} such that
e −Lt iI i ut i Δe −Lt ip t i ut i/ 0. 2.18
Let
v t
p
k 0, k / i
Trang 8Obviously, v ∈ X Substituting them into 2.17, we get
Δe −Lt ip t i ut i e −Lt iI i ut iv t i 0 2.20
becomes the following form:
e −L1 p 1u
1−
e −L0 p 0u0 0, and it implies u1− u0 0 Therefore, u is a classical solution
Lemma 2.5 Let u ∈ X Then u∞≤ M1u X , where
M1 21/2max
1
mint ∈0,1 e −Lt p t1/2 , 1
mint ∈0,1 e −Lt q t1/2
Proof By using the same methods of15, Lemma 2.6, we easily obtain the above result, and
we omit it here
3 Main Results
In this section, we will show our main results and prove them
Theorem 3.1 Assume that (H1) and (H2) hold Moreover, gt, u and the impulsive functions I k u
are odd about u, then IBVP1.1 has infinitely many classical solutions
Proof Obviously, ϕ is an even functional and ϕ0 0 We divide our proof into three parts
that
ϕ u n ≤ C3, ϕu n
Trang 9By2.8, 2.9, 3.1, and H1, we have
u n2
X β
2u n2
X − u n2
X
βϕu n − ϕu n u n β
p−1
k1
e −Lt kun t k
0
I k sds β
1
0
e −Lt G t, u n dt
−
p−1
k1
e −Lt kI k u n t k u n t k −
1
0
e −Lt g t, u n u n dt
p−1
k1
e −Lt k β
un t k
0
I k sds − I k u n t k u n t k
1
0
e −Lt
βG t, u n − gt, u n u n
dt βϕu n − ϕu n u n
1C3u nX
1
0
e −Lt dt max
t ∈0,1, u n t∈−M,M βG t, u n − gt, u n u n
p−1
k1
e −Lt k max
un t k ∈−M,M
β
un t k
0
I k sds − I k u n t k u n t k
.
3.2
ϕu n − ϕuu n − u
u n − u2
X
−
p−1
k1
e −Lt kI k u n t k − I k ut n u n t k − ut k
−
1
0
e −Lt
g t, u n t − gt, utu n t − utdt.
3.3
By u n u in X, we see that {u n } uniformly converges to u in C0, 1 So
1
0
e −Lt
g t, u n t − gt, utu n t − utdt −→ 0,
p−1
k1
e −Lt kI k u n t k − I k ut k u n t k − ut k −→ 0,
ϕun − ϕuu n − u −→ 0 as n −→ ∞.
3.4
Trang 10By3.3, 3.4, we obtain un − u X → 0 as n → ∞ That is, {u n } strongly converges to u in
X, which means the that P S condition holds for ϕ.
1
1
1
0e −Lt dt, 1/8M2
1
p−1
k1e −Lt k} > 0, there exists an δ > 0 such that for every u
with|u| < δ,
G t, u ≤ ε|u|2,
u
0
2u2
X−
p−1
k1
e −Lt ku t k
0
I k sds −
1
0
e −Lt G t, udt
2u2
X−
p−1
k1
e −Lt kε |u k t k|2−
1
0
e −Lt ε |ut|2dt
2u2
X − εM2 1
p−1
k1
e −Lt ku2
X − εM2 1
1
0
e −Lt dt u2
X
2u2
X−1
8u2
X−1
8u2
X
4u2
X
3.6
1, ρ δ/M1, then ϕu ≥ α, ∀u ∈ Y ∩ ∂B ρ
G t, u
u β
u
u β g t, u − βu β−1G t, u
Hence
G t, u
M β ≥ M −βmin
t ∈0,1 G t, M C> 0 3.8
for all t ∈ 0, 1 and u ≥ M > 0 This implies that Gt, u ≥ Cu β for all t ∈ 0, 1 and u ≥ M > 0 Similarly, we can prove that there is a constant C> 0 such that G t, u ≥ C|u| β for all t ∈ 0, 1 and u ≤ −M Since Gt, u − C4|u| βis continuous on0, 1 × −M, M, there exists C5> 0 such
Trang 11that Gt, u − C4|u| β > −C5on0, T × −M, M Thus, we have
where C4 min{C, C}
Similarly, there exist constants C6, C7> 0 such that
u
0
following inequality:
ϕ ξu ≤ 1
2ξu2
X−
p−1
k1
e −Lt k
C6|ξut k|β − C7
−
1
0
e −Lt
C4|ξu| β − C5
dt
≤ ξ2
2u2
X −C6|ξ| βp−1
k1
e −Lt k|ut k|β C7
p−1
k1
e −Lt k−C4|ξ| β
1
0
e −Lt |ut| β dt C5
1
0
e −Lt dt
3.11
holds Take w ∈ W such that w X 1, since β > 2, 3.11 implies that there exists ξ> 0 such
thatξw X > ρ and ϕ ξw < 0 for ξ ≥ ξ > 0 Since W is a finite dimensional subspace, there
critical points; that is, IBVP1.1 has infinite many classical solutions
Theorem 3.2 Assume that (H1) and the first equality in (H2) hold Moreover, gt, u is odd about
u and the impulsive functions I k u are odd and nonincreasing Then IBVP 1.1 has infinitely many
classical solutions.
Proof We only verifyA1 inTheorem 2.3 Since Ik u are odd and nonincreasing continuous
0I k sds < 0 So we havep−1
k1e −Lt kun t k
0 I k sds < 0 Take
1
1
0e −Lt dt > 0, α 3δ2/8M2
1, ρ δ/M1, like in3.6 we can obtain the result
Theorem 3.3 Suppose that the first inequalities in (H1), (H3), and (H4) hold Furthermore, one
assumes that g t, u and the impulsive functions I k u are odd about u and we have the following.
H7 There exists A0> 0 such that
A0
2 > M1
p−1
k1
e −Lt k
b k M γk1 A γk0 a k
h2M p1
1 A p1
0 h1
1 0
e −Lt dt. 3.12
Then IBVP1.1 has infinitely many classical solutions
Trang 12Proof Obviously, ϕ is an even functional and ϕ 0 0 Firstly, we will show that ϕ satisfies
u n2
X β
2u n2
X − u n2
X
βϕu n − ϕu n u n β
p−1
k1
e −Lt kun t k
0
I k sds β
1
0
e −Lt G t, u n dt
−
p−1
k1
e −Lt kI k u n t k u n t k −
1
0
e −Lt g t, u n u n dt
βϕu n − ϕu n u n
1
0
e −Lt
βG t, u n − gt, u n u n
dt
β
p−1
k1
e −Lt kun t k
0
I k sds −
p−1
k1
e −Lt kI k u n t k u n t k
1C3u nX
1
0
e −Lt dt max
t ∈0,1, u n t∈−M,M βG t, u n − gt, u n u n
β 1
p−1
k1
e −Lt k
a k M1u nX b k M1γk1u nγk1
X
.
3.13
satisfied
2u2
X−
p−1
k1
e −Lt ku t k
0
I k sds −
1
0
e −Lt G t, udt
2u2
X−
p−1
k1
e −Lt k
a k M1u X b k M γk11u γk1
X
−
1
0
e −Lt dt
h1M1u X h2M p1 1
1 u p1 1
X
2u2
X−
p−1
k1
e −Lt kb k M γk11u γk1
X − h2M p1 1
1
1
0
e −Lt dt u p1 1
X
− M1u X p−1
k1
e −Lt ka k h1
1
0
e −Lt dt
.
3.14
... we can obtain the resultTheorem 3.3 Suppose that the first inequalities in (H1), (H3), and (H4) hold Furthermore, one
assumes that g t, u and the impulsive functions... results and prove them
Theorem 3.1 Assume that (H1) and (H2) hold Moreover, gt, u and the impulsive functions I k u
are odd about u, then IBVP1.1...
Theorem 3.2 Assume that (H1) and the first equality in (H2) hold Moreover, gt, u is odd about
u and the impulsive functions I k u are odd and nonincreasing Then