Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2012, Article ID 295209, 12 pages
doi:10.1155/2012/295209
Research Article
Multiple Positive Solutions for Singular
Semipositone Periodic Boundary Value Problems
with Derivative Dependence
Huiqin Lu
School of Mathematical Sciences, Shandong Normal University, Shandong, Jinan 250014, China
Correspondence should be addressed to Huiqin Lu, lhy@sdu.edu.cn
Received 29 March 2012; Accepted 19 May 2012
Academic Editor: Yansheng Liu
Copyright q 2012 Huiqin Lu. 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.
By constructing a special cone in C1 0, 2π and the fixed point theorem, this paper investigates
second-order singular semipositone periodic boundary value problems with dependence on the
first-order derivative and obtains the existence of multiple positive solutions. Further, an example
is given to demonstrate the applications of our main results.
1. Introduction
In this paper, we are concerned with the existence of multiple positive solutions for the
second-order singular semipositone periodic boundary value problems PBVP, for short:
u t atut f t, ut, u t ,
u0 u2π,
t ∈ 0, 2π,
u 0 u 2π,
1.1
where a ∈ C0, 2π, the nonlinear term ft, u, v may be singular at t 0, t 2π, and u 0,
also may be negative for some value of t, u, and v.
In recent years, second-order singular periodic boundary value problems have
been studied extensively because they can be used to model many systems in celestial
mechanics such as the N-body problem see 1–11 and references therein. By applying the
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Krasnosel’skii’s fixed point theorem, Jiang 5 proves the existence of one positive solution
for the second-order PBVP
u t m2 u ft, u,
t ∈ 0, 2π,
1.2
u 0 u 2π,
u0 u2π,
where m ∈ 0, 1/2 is a constant and f ∈ C0, 2π × 0, ∞, 0, ∞. Zhang and Wang
6 used the same fixed point theorem to prove the existence of multiple positive solutions
for PBVP 1.2 when ft, u is nonnegative and singular at u 0, not singular at t 0,
t 2π. Lin et al. 7 only obtained the existence of one positive solution to PBVP 1.1 when
ft, u, v ft, u, f is semipositone and singular only at u 0. All the above works were
done under the assumption that the first-order derivative u is not involved explicitly in the
nonlinear term f.
Motivated by the works of 5–7, the present paper investigates the existence of
multiple positive solutions to PBVP 1.1. PBVP 1.1 has two special features. The first one
is that the nonlinearity f may depend on the first-order derivative of the unknown function
u, and the second one is that the nonlinearity ft, u, v is semipositone and singular at t 0,
t 2π, and u 0. We first construct a special cone different from that in 5–7 and then
deduce the existence of multiple positive solutions by employing the fixed point theorem on
a cone. Our results improve and generalize some related results obtained in 5–7.
A map u ∈ C1 0, 2π ∩ C2 0, 2π is said to be a positive solution to PBVP1.1 if and
only if u satisfies PBVP 1.1 and ut > 0 for t ∈ 0, 2π.
The contents of this paper are distributed as follows. In Section 2, we introduce some
lemmas and construct a special cone, which will be used in Section 3. We state and prove the
existence of at least two positive solutions to PBVP 1.1 in Section 3. Finally, an example is
worked out to demonstrate our main results.
2. Some Preliminaries and Lemmas
Define the set functions
Λ
⎧
⎨
a ∈ C0, 2π : a 0, t ∈ 0, 2π,
⎩
2π
0
1/p
ap dt
⎫
⎬
≤ K 2q for some p ≥ 1 ,
⎭
2.1
where q is the conjugate exponent of p,
K q ⎧
2
1−2/q ⎪
Γ 1/q
⎪
2
1
⎪
⎪
,
⎪
⎨ q2π2/q 2 q
Γ 1/2 1/q
⎪
⎪
⎪
⎪
⎪
⎩2,
π
where Γ is the Gamma function.
1 ≤ q < ∞,
2.2
q ∞,
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3
Given a ∈ Λ, let Gt, s be the Green function for the equation
u atut 0,
t ∈ 0, 2π,
2.3
u 0 u 2π.
u0 u2π,
Now, the following Lemma follows immediately from the paper 7.
Lemma 2.1. Gt, s has the following properties:
G1 Gt, s is continuous in t and s for all t, s ∈ 0, 2π;
G2 Gt, s > 0 for all t, s ∈ 0, 2π × 0, 2π, G0, s G2π, s and ∂G/∂t|0,s ∂G/∂t|2π,s ;
G3 denote l1 min0≤t,s≤2π Gt, s and l2 max0≤t,s≤2π Gt, s, then l2 > l1 > 0;
G4 there exist functions h, H ∈ C2 0, 2π such that
Gt, s ⎧
⎨α 1Hths β − 1 htHs cHtHs dhths,
0 ≤ s ≤ t ≤ 2π,
⎩αHths βhtHs cHtHs dhths,
0 ≤ t ≤ s ≤ 2π,
2.4
where α, β, c, d are constants, H, h are independent solutions of the linear differential equation
u atut 0, and H tht − h tHt 1;
G5 Gt t, s is bounded on 0, 2π × 0, 2π.
Denote l3 max0≤t,s≤2π |Gt t, s|, then l3 > 0.
Remark 2.2. Using paper 5, we can get Gt, s when at ≡ m2 and m ∈ 0, 1/2, obtaining
Gt, s Gt t, s
l1 ⎧ sin mt − s sin m2π − t s
⎪
⎪
,
⎪
⎨
2m1 − cos 2mπ
⎪
⎪
⎪
⎩ sin ms − t sin m2π − s t ,
2m1 − cos 2mπ
⎧ cos mt − s − cos m2π − t s
⎪
⎪
,
⎪
⎨
21 − cos 2mπ
⎪
⎪
⎪
⎩ − cos ms − t cos m2π − s t ,
21 − cos 2mπ
sin 2mπ
,
2m1 − cos 2mπ
l2 0 ≤ s ≤ t ≤ 2π,
0 ≤ t ≤ s ≤ 2π,
0 ≤ s ≤ t ≤ 2π,
2.5
0 ≤ t < s ≤ 2π,
sin mπ
,
m1 − cos 2mπ
l3 1
.
2
Let E {u ∈ C1 0, 2π : u0 u2π, u 0 u 2π} with norm u
max{
u
0 , u 0 }, where u
0 maxt∈0,2π |ut|. Then E, · is a Banach space. Let
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σ : min{l1 /l2 , l1 /l3 }, L : l3 /l1 , from Lemma 2.1, we know that σ, L are both constants
and 0 < σ < 1, L > 0.
Define
K u ∈ E : ut ≥ σ
u
, u t ≤ L
u
, ∀t ∈ 0, 2π ,
Ωr {u ∈ E : u
< r},
2.6
∀r > 0.
It is easy to conclude that K is a cone of E and Ωr is an open set of E.
Lemma 2.3 see 12. Let E be a Banach space and P a cone in E. Suppose Ω1 and Ω2 are bounded
open sets of E such that θ ∈ Ω1 ⊂ Ω1 ⊂ Ω2 and suppose that A : P ∩ Ω2 \ Ω1 → P is a completely
continuous operator such that
1 infu∈P ∩∂Ω1 Au
> 0 and u / λAu for u ∈ P ∩ ∂Ω1 , λ ≥ 1; u / λAu for u ∈ P ∩ ∂Ω2 , 0 <
λ ≤ 1, or
λAu for u ∈ P ∩ ∂Ω2 , λ ≥ 1; u /
λAu for u ∈ P ∩ ∂Ω1 , 0 <
2 infu∈P ∩∂Ω2 Au
> 0 and u /
λ ≤ 1.
Then A has a fixed point in P ∩ Ω2 \ Ω1 .
For convenience, let us list some conditions for later use.
H0 at ∈ Λ, f : 0, 2π × 0, ∞ × R → R is continuous and there exists a constant M > 0
such that
0 ≤ ft, u, v M ≤ gthu, v,
∀t, u, v ∈ 0, 2π × 0, ∞ × R,
where g ∈ C0, 2π, R , h ∈ C0, ∞ × R, R , and 0 <
2π
0
H1 there exist r1 > σ −1 2πMl2 and at ∈ L0, 2π with
M ft, u, v ≥ >at,
2.7
gtdt < ∞;
2π
0
atdt > ≥r1 l1−1 such that
∀t ∈ 0, 2π, u ∈ 0, r1 , v ∈ −Lr1 2πMl3 , Lr1 2πMl3 ;
2.8
H2 there exists R1 > r1 such that
max{l2 , l3 }
2π
0
gtdt < R1 M0−1 ,
2.9
where M0 : max{hu, v : u ∈ σR1 − 2πMl2 , R1 , v ∈ −LR1 2πMl3 , LR1 2πMl3 };
H3 there exists α∗ , β∗ ⊂ 0, 2π such that
ft, u, v
∞ uniformly with respect to t ∈ α∗ , β∗ , v ∈ R.
u → ∞
u
lim
2.10
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3. Main Results
Theorem 3.1. Assume that conditions (H0 )–(H3 ) are satisfied, then PBVP 1.1 has at least two
positive solutions u1 , u2 ∈ C1 0, 2π ∩ C2 0, 2π such that r1 < u1 Mω
< R1 < u2 Mω
,
2π
where ωt : 0 Gt, sds.
Proof. We consider the following PBVP:
u t atut f t, ut − Mωt, u t − Mω t M,
u0 u2π,
t ∈ 0, 2π,
u 0 u 2π.
3.1
It is easy to see that if u ∈ C1 0, 2π ∩ C2 0, 2π and r1 < u
< R1 is a positive solution
of PBVP 3.1 with ut > Mωt for t ∈ 0, 2π, then xt ut − Mωt is a positive solution
of PBVP 1.1 and r1 < x Mω
< R1 .
As a result, we will only concentrate our study on PBVP 3.1.
Define an operator T : K \ {θ} → E by
T ut :
2π
Gt, s f s, us − Mωs, u s − Mω s M ds,
∀t ∈ 0, 2π,
3.2
0
where Gt, s is the Green function to problem 2.3.
1 We first show that T : K ∩ ΩR \ Ωr1 → K is completely continuous for any
R > r1 .
For any u ∈ K ∩ ΩR \ Ωr1 , from H1 , we have ut − Mωt ≥ σr1 − 2πMl2 > 0. So,
by Lemma 2.1 and 3.2,
T u0 T u2π,
T ut 2π
0
≥
l1
l2
l2
3.3
Gt, s f s, us − Mωs, u s − Mω s M ds
2π
Gt, s f s, us − Mωs, u s − Mω s M ds
0
l1
max
≥
l2 τ∈0,2π
T u 0 T u 2π,
2π
Gτ, s f s, us − Mωs, u s − Mω s M ds
0
l1
T u
0 ≥ σ
T u
0 ,
l2
∀t ∈ 0, 2π,
3.4
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2π
T u t Gt t, s f s, us − Mωs, u s − Mω s M ds
0
≤
2π
G t, s f s, us − Mωs, u s − Mω s M ds
0
t
2π
l3
l1
f s, us − Mωs, u s − Mω s M ds
l1
0
2π
l3
Gτ, s f s, us − Mωs, u s − Mω s M ds
≤
l1 0
≤
l3
T uτ,
l1
3.5
∀t, τ ∈ 0, 2π.
From 3.5, we have T ut ≥ l1 /l3 maxτ∈0,2π |T u τ| ≥ σ
T u 0 . Therefore,
T ut ≥ σ
T u
, |T u t| ≤ L
T u
, for all t ∈ 0, 2π, that is, T : K ∩ ΩR \ Ωr1 → K.
Assume that un , u∗ ∈ K ∩ ΩR \ Ωr1 with un − u∗ → 0, n → ∞. Thus, from H1 ,
we have
lim f t, un t − Mωt, un t − Mω t
n → ∞
f t, u∗ t − Mωt, u∗ t − Mω t , t ∈ 0, 2π,
f t, un t − Mωt, un t − Mω t ≤ M M1 gt, t ∈ 0, 2π,
M M1 gt ∈ L0, 2π,
3.6
where M1 : max{hu, v : u ∈ σr1 − 2πMl2 , R, v ∈ −LR 2πMl3 , LR 2πMl3 }.
Lemma 2.1 and Lebesgue-dominated convergence theorem guarantee that
T un − T u∗ ≤ max{l2 , l3 }
2π
0
f t, un t − Mωt, un t − Mω t
−f t, u∗ t − Mωt, u∗ t − Mω t dt −→ 0,
n −→ ∞.
3.7
So, T : K ∩ ΩR \ Ωr1 → K is continuous.
For any bounded D ⊂ K ∩ ΩR \ Ωr1 , From Lemma 2.1 and H1 , for any u ∈ D, we
have
T u
≤ max{l2 , l3 }
2π
f s, us − Mωs, u s − Mω s M ds
0
≤ max{l2 , l3 }
2π
0
≤ max{l2 , l3 }M1
gsh us − Mωs, u s − Mω s ds
2π
gsds,
0
3.8
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which means the functions belonging to {T Dt} and the functions belonging to {T D t}
are uniformly bounded on 0, 2π. Notice that
T u t ≤ l3 M1
2π
gsds,
3.9
t ∈ 0, 2π, u ∈ D,
0
which implies that the functions belonging to {T Dt} are equicontinuous on 0, 2π. From
Lemma 2.1, we have
Gt t, s ⎧
⎨α 1H ths β − 1 h tHs cH tHs dh ths,
⎩αH ths βh tHs cH tHs dh ths,
0 ≤ s ≤ t ≤ 2π,
0 ≤ t < s ≤ 2π,
3.10
where α, β, c, d are constants, h, H ∈ C2 0, 2π are independent solutions of the linear
differential equation u atut 0, and H tht − h tHt 1.
It is easy to see that Gt t, s is continuous in t and s for 0 ≤ s ≤ t ≤ 2π and 0 ≤ t < s ≤ 2π.
So, for any t1 , t2 ∈ 0, 2π, t1 < t2 , we have
t1
t2
t1
G t1 , s − G t2 , sgsds −→ 0,
t
0
G t1 , s − G t2 , sgsds ≤ 2l3
t
t
2π
t2
as t1 −→ t−2 , or t2 −→ t1 ,
t
t2
gsds −→ 0,
t1
G t1 , s − G t2 , sgs ds −→ 0,
t
t
as t1 −→ t−2 , or t2 −→ t1 ,
3.11
as t1 −→ t−2 , or t2 −→ t1 .
Therefore,
T u t1 − T u t2 2π Gt t1 , s − Gt t2 , s f s, us − Mωs, u s − Mω s M ds
0
≤ M1
2π
G t1 , s − G t2 , sgsds
t
0
M1
t
1
t
G t1 , s − G t2 , sgsds t
0
2π
t2
t
3.12
t2
t1
G t1 , s − G t2 , sgsds
t
G t1 , s − G t2 , sgsds −→ 0,
t
t
t
as t1 −→ t−2 or t2 −→ t1 .
Thus, the functions belonging to {T D t} are equicontinuous on 0, 2π. By Arzela-Ascoli
theorem, T D is relatively compact in C1 0, 2π.
Hence, T : K ∩ ΩR \ Ωr1 → K is completely continuous for any R > r1 .
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Journal of Applied Mathematics
2 We now show that
inf
u∈K∩∂Ωr1
u
/ λT u,
T u
> 0,
∀u ∈ K ∩ ∂Ωr1 , λ ≥ 1.
3.13
For any u ∈ K ∩ ∂Ωr1 , we have
0 < σr1 − 2πMl2 ≤ ut − Mωt ≤ r1 ,
u t − Mω t ≤ u t Mω t ≤ Lr1 2πMl3 ,
∀t ∈ 0, 2π.
3.14
From H1 and 3.2,
T ut 2π
Gt, s f s, us − Mωs, u s − Mω s M ds
0
≥ l1
3.15
2π
asds >
0
l1 r1 l1−1
r1 > 0.
Suppose that there exist λ0 ≥ 1 and u0 ∈ K ∩ ∂Ωr1 such that u0 λ0 T u0 , that is, for
t ∈ 0, 2π,
u0 t ≥ T u0 t 2π
0
≥ l1
Gt, s f s, u0 s − Mωs, u0 s − Mω s M ds
3.16
2π
asds >
0
l1 r1 l1−1
r1 .
This is in contradiction with u0 ∈ K ∩ ∂Ωr1 and 3.13 holds.
3 Next, we show that
u
/ λT u
∀u ∈ K ∩ ∂ΩR1 , 0 < λ ≤ 1.
3.17
Suppose this is false, then there exist λ0 ∈ 0, 1 and u0 ∈ K ∩ ∂ΩR1 with u0 λ0 T u0 ,
that is, for t ∈ 0, 2π, we have
u0 t ≤ T u0 t 2π
0
u t λ0 T u0 t ≤
0
Gt, s f s, u0 s − Mωs, u0 s − Mω s M ds,
2π
0
G t, s f s, us − Mωs, u s − Mω s M ds.
3.18
t
From H2 , we have
0 < σR1 − 2πMl2 ≤ u0 t − Mωt ≤ R1 ,
u t − Mω t ≤ u t Mω t ≤ LR1 2πMl3 ,
0
0
∀t ∈ 0, 2π.
3.19
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Therefore, by 3.18, 3.19, and H2 , it follows that
u0 t ≤ l2
2π
0
≤ l2 M0
u t ≤ l3
0
2π
0
≤ l3 M0
gsh u0 s − Mωs, u0 s − Mω s ds
2π
gsds < R1 ,
∀t ∈ 0, 2π,
0
gsh u0 s − Mωs, u0 s − Mω s ds
2π
gsds < R1 ,
3.20
∀t ∈ 0, 2π.
0
Thus, u
< R1 . This is in contradiction with u0 ∈ K ∩ ∂ΩR1 and 3.17 holds.
4 Choose N ∗ 1 2πMl2 σl1 β∗ − α∗ −1 1. From H3 , there exists R2 >
max{R1 , 1} such that
ft, u, v ≥ N ∗ u,
∀u ≥ R2 , v ∈ R, t ∈ α∗ , β∗ .
3.21
Now, we show that
inf
u∈K∩∂ΩR
T u
> 0,
u/
λT u,
∀u ∈ K ∩ ∂ΩR , λ ≥ 1,
3.22
where R R2 2πMl2 σ −1 .
For any u ∈ K ∩ ∂ΩR , we have
ut − Mωt ≥ σR − 2πMl2 R2 ,
∀t ∈ 0, 2π.
3.23
This and 3.21 together with 3.2 imply
T ut 2π
Gt, s f s, us − Mωs, u s − Mω s M ds
0
≥ l1
β∗
α∗
f s, us − Mωs, u s − Mω s ds
≥ l1 N ∗ R2 β∗ − α∗ > 0.
3.24
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Suppose that there exist λ0 ≥ 1 and u0 ∈ K ∩ ∂ΩR such that u0 λ0 T u0 , then, for
t ∈ α∗ , β∗ , we have
u0 t ≥ T u0 t 2π
0
≥ l1
Gt, s f s, u0 s − Mωs, u0 s − Mω s M ds
β∗
α∗
f s, us − Mωs, u s − Mω s ds
3.25
≥ l1 N ∗ R2 β∗ − α∗ > R2 2πMl2 σ −1 R.
This is in contradiction with u0 ∈ K ∩ ∂ΩR and 3.22 holds.
Now, 3.13, 3.17, 3.22, and Lemma 2.3 guarantee that T has two fixed points u1 ∈
K ∩ ΩR1 \ Ωr1 , u2 ∈ K ∩ ΩR \ ΩR1 with r1 < u1 1 < R1 < u2 1 < R. Clear, PBVP 3.1 has
at least two positive solutions u1 , u2 ∈ C1 0, 2π ∩ C2 0, 2π.
Remark 3.2. From the proof of Theorem 3.1, when ft, u, v is nonnegative i.e., M 0 in
H0 , Theorem 3.1 still holds.
Corollary 3.3. Assume that (H0 )–(H2 ) hold, then PBVP 1.1 has at least one positive solution ut
2π
such that r1 < u Mω
< R1 , where ωt : 0 Gt, sds.
Corollary 3.4. Assume that (H0 ) and (H3 ) hold, and
(H4 ) there exist R1 > σ −1 2πMl2 such that
max{l2 , l3 }
2π
0
gtdt < R1 M0−1 ,
3.26
where M0 : max{hu, v : u ∈ σR1 − 2πMl2 , R1 and v ∈ −LR1 2πMl3 , LR1 2πMl3 }.
Then PBVP 1.1 has at least one positive solution ut such that u Mω
> R1 , where ωt :
2π
Gt, sds.
0
Example 3.5. Consider the following second-order singular semipositone PBVP:
√
u9/4 u 2 1
t
3
1
cos ,
−
u u
16
12
8πu t2π − t 30π
u0 u2π,
t ∈ 0, 2π,
3.27
u 0 u 2π.
4. Conclusion
PBVP 3.27 has at least two positive solutions u1 , u2 ∈ C1 0, 2π ∩ C2 0, 2π and u1 t, u2 t >
0 for t ∈ 0, 2π.
To
u9/4 v2 m 1/4, ft, u, v 9/4
see this, we√will apply Theorem 3.1 with
1/8πu t2π − t − 3/30π cost/12, gt 1/ t2π − t, hu, v u v2 1/8πu,
M 1/20π.
√
√
From Remark 2.2, it is easy to see that l1 2, l2 2 2, l3 1/2, σ 2/2, and L 1/4.
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11
2π
By simple computation, we easily get 0 ≤ ft, u, v M ≤ gthu, v and 0 gtdt π. So H0 holds.
√
√
Taking r1 1/2, at 1/4π t2π − t, then σ −1 2πMl2 2 · 2π · 1/20π · 2 2 2π
2/5 < r1 , 0 atdt 1/4 r1 l1−1 and for any t ∈ 0, 2π, u ∈ 0, 1/2, v ∈ −7/40, 7/40,
√
√
t
1
1/29/4 1
1
3
3
u9/4 v2 1
cos
≥
−
−
12 20π 4π t2π − t 30π 20π
8πu t2π − t 30π
√
1/29/4
1
1
3
≥
−
30π 20π 4π t2π − t
4π2
>
4π
1
t2π − t
4.1
.
Thus, H1 holds.
√
Taking R1 4, then for u ∈ 9/5 2, 4, |v| ≤ 21/20, we have
5
M0 ≤ √
72 2π
So, M0 max{l2 , l3 }
2π
0
9/2
2
21
20
2
1
<
5
65
√ 23 2 1 √ .
72 2π
36 2π
4.2
√
gtdt 2 2πM0 < 65/18 < 4 R1 . That is, H2 holds.
Let α∗ , β∗ π/2, π, then it is easy to check that H3 holds.
Thus all the conditions of Theorem 3.1 are satisfied, so PBVP 3.27 has at least two
positive solutions u1 , u2 ∈ C1 0, 2π ∩ C2 0, 2π and u1 t, u2 t > 0 for t ∈ 0, 2π.
Acknowledgments
Research supported by the Project of Shandong Province Higher Educational Science and
Technology Program J09LA08 and Reward Fund for Excellent Young and Middle-Aged
Scientists of Shandong Province BS2011SF022, China.
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