Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2012, Article ID 653675, 17 pages
doi:10.1155/2012/653675
Research Article
Existence of 2m − 1 Positive Solutions for
Sturm-Liouville Boundary Value Problems with
Linear Functional Boundary Conditions on
the Half-Line
Yanmei Sun1 and Zengqin Zhao2
1
Department of Mathematics and Information Sciences, Weifang University, Shandong,
Weifang 261061, China
2
Department of Mathematics, Qufu Normal University, Shandong,
Qufu 273165, China
Correspondence should be addressed to Yanmei Sun, sunyanmei2009@126.com
Received 11 January 2012; Accepted 5 March 2012
Academic Editor: Giuseppe Marino
Copyright q 2012 Y. Sun and Z. Zhao. 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 using the Leggett-Williams fixed theorem, we establish the existence of multiple positive
solutions for second-order nonhomogeneous Sturm-Liouville boundary value problems with
linear functional boundary conditions. One explicit example with singularity is presented to
demonstrate the application of our main results.
1. Introduction
In this paper, we consider the following Sturm-Liouville boundary value problems on the
half-line
ptu t Φtf t, ut, u t 0,
0 < t < ∞,
α1 u0 − β1 lim ptu t Tu,
t→0
1.1
α2 lim ut β2 lim ptu t Ku,
t → ∞
t → ∞
≡ 0 on any subinterval of R , here
where f : R × R × R → R is a continuous function, f /
R 0, ∞; Φ : R → R is a Lebesgue integrable function and may be singular at
2
Journal of Applied Mathematics
∞
t 0; p ∈ CR , R C1 R , 0 ds/ps < ∞; αi , βi ≥ 0 i 1, 2 with ρ α1 β2 ∞
α2 β1 α1 α2 0 ds/ps; T, K are linear positive functionals on CR T, K which are called
positive if Tu, Ku ≥ 0 for u ∈ CR .
The theory of nonlocal boundary value problems for ordinary differential equations
arises in different areas of applied mathematics and physics. There are many studies for
nonlocal, including three-point, m-point, and integral boundary value problems on finite
interval by applying different methods 1–3. It is well known that boundary value problems
on infinite interval arise in the study of radial solutions of nonlinear elliptic equations and
models of gas pressure in a semi-infinite porous medium 4–6. But the theory of SturmLiouville nonhomogeneous boundary value problems on infinite interval is yet rare.
The linear functional boundary conditions cover some nonlocal three-point, m-point,
and integral boundary conditions. In 7, Zhao and Li investigated some nonlinear singular
differential equations with linear functional boundary conditions. However, the differential
equations were defined only in a finite interval. Recently, Liu et al. 6 studied multiple
positive solutions for Sturm-Liouville boundary value problems on the half-line
ptu t mtft, ut 0,
0 ≤ t < ∞,
α1 u0 − β1 lim ptu t 0,
1.2
t→0
α2 lim ut β2 lim ptu t 0.
t → ∞
t → ∞
However, the authors did not consider the case when Sturm-Liouville boundary value
problems are nonhomogeneous. Therefore BVP1.1 is the direct extension of 7. So it is
worthwhile to investigate BVP1.1.
We denote
t
ds
,
ps
∞
ds
,
ps
0
t
ut ρ−1 1 atbty1 t,
u t y2 t,
f t, u, u Ψ t, y1 , y2 ,
∞
ds
,
a∞ lim at β1 α1
a0 lim at β1 ,
t → ∞
t→0
ps
0
∞
ds
b0 lim bt β2 α2
,
b∞ lim bt β2 .
t → ∞
t→0
ps
0
at β1 α1
bt β2 α2
1.3
1.4
In this paper, we always assume that the following conditions hold.
H1 Ψt,
∞ y1 , y2 ≤ qtQy1 , y2 , qt ∈ CR , R , Qy1 , y2 ∈ CR × R, R and
Φsqsds < ∞.
0
H2 For any constant τ ∈ 0, ∞, 0 < Taτ < ρ, 0 < Kbτ < ρ and
ρ − Tbτ Taτ Δ
> 0.
Kbτ ρ − Kaτ
1.5
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3
Motivated and inspired by 5–9, we are concerned with the existence of multiple positive solutions for BVP1.1 by applying Leggett-Williams fixed theorem. The main new features presented in this paper are as follows. Firstly, Sturm-Liouville nonhomogeneous boundary value problems with linear functional boundary conditions are seldom researched, it
brings about many difficulties when we imply the integral equations of BVP1.1. To solve the
problem, we use a new method of undetermined coefficient to obtain the integral equations of
boundary value problems with nonhomogeneous boundary conditions. Secondly, we discuss
the existence of triple positive solutions and 2m − 1 positive solutions of BVP1.1. Finally, the
methods used in this paper are different from 1, 6, 7 and the results obtained in this paper
generalize and involve some results in 5.
The rest of paper is organized as follows. In Section 2, we present some preliminaries
and lemmas. We state and prove the main results in Section 3. Finally, in Section 4, one example with a singular nonlinearity is presented to demonstrate the application of Theorem 3.1.
2. Preliminary
In order to discuss the main results, we need the following lemmas.
∞
Lemma 2.1. Under the condition 0 ds/ps < ∞ and ρ > 0, the boundary value problem
ptu t yt 0,
0 < t < ∞,
α1 u0 − β1 lim ptu t Tu,
t→0
2.1
α2 lim ut β2 lim ptu t Ku,
t → ∞
t → ∞
has a unique solution for any y ∈ L0, ∞. Moreover, this unique solution can be expressed in the
form
ut ∞
Gt, sysds A y at B y bt,
2.2
0
where Gt, s, Ay, and By are defined by
Gt, s ρ
−1
atbs,
0 ≤ t ≤ s < ∞,
asbt,
0 ≤ s < t < ∞,
∞
T
Gτ,
sysds
ρ
−
Tbτ
1
0
A y ∞
,
Δ −K
Gτ,
sysds
Kbτ
2.3
a
0
∞
K
Gτ,
sysds
ρ
−
Kbτ
1
0
B y ∞
.
Δ Gτ, sysds
Tbτ −T
0
b
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Proof. at and bt in 1.3 are two linear independent solutions of the equation ptu t 0, so the general solutions for the equation ptu t yt 0 can be expressed in the form
ut ∞
Gt, sysds Cat Dbt,
2.4
0
where C, D are undetermined constants. Through verifying directly, when C and D satisfy
a and b separately, ut in 2.4 is a solution of BVP2.1.
Now we need to prove that when ut in 2.4 is a solution of BVP2.1, C and D satisfy
a and b separately.
∞
Let ut 0 Gt, sysds Cat Dbt be a solution of BVP2.1, then
∞
1
1
asbtysds atbsysds Cat Dbt,
ρ
ρ
0
t
a t ∞
Cα1 Dα2
b t t
−
u t asysds bsysds ρ
ρ
pt pt
0
t
1
−α2 t
α1 ∞
asysds bsysds Cα1 − Dα2 ,
pt
ρ 0
ρ t
ut t
2.5
−α2
α1
α2 at α1 bt
ptu t atyt − btyt −
yt −yt.
ρ
ρ
ρ
That is, ptu t yt 0.
By 2.4, we have
β1
u0 ρ
∞
∞
ds
,
ps
0
0
∞
α1
1
u 0 bsysds Cα1 − Dα2 ,
p0 ρ 0
∞
∞
β2
ds
,
u∞ asysds Cβ1 Dβ2 Cα1
ρ 0
ps
0
α2 ∞
1
u ∞ −
asysds Cα1 − Dα2 ,
p∞
ρ 0
bsysds Cβ1 Dβ2 Dα2
2.6
then
∞
Dρ T
Gτ, sysds CTaτ DTbτ,
0
∞
Cρ K
Gτ, sysds CKaτ DKbτ.
2.7
0
From 2.7, we obtain that C and D satisfy a and b separately. The proof is completed.
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5
Remark 2.2. Assume that H2 holds. Then 0 ≤ Ay < ∞, 0 ≤ By < ∞ for any y ≥ 0 and
any solution ut of BVP2.1 is nonnegative.
Lemma 2.3. From 1.3 and 2.3, it is easy to get the following properties.
1 Gt, s/ρ−1 1 atbt ≤ 1, at/1 atbt < 1/bt ≤ 1/β2 , bt/1 atbt <
1/at ≤ 1/β1 .
2 Gs limt → ∞ Gt, s β2 /ρas < ∞.
3 Gt, s ≤ Gs, s ≤ asbs/ρ < ∞.
Lemma 2.4. For any constant 0 < a∗ < b∗ < ∞, there exists 0 < c∗ < 1, such that, for τ, s ∈ 0, ∞,
Gt, s/ρ−1 1 atbt ≥ c∗ Gτ, s/ρ−1 1 aτbτ, at/ρ−1 1 atbt ≥ c∗ aτ/ρ−1 1 aτbτ, bt/ρ−1 1 atbt ≥ c∗ bτ/ρ−1 1 aτbτ, t ∈ a∗ , b∗ .
Proof. By 1.3, it is obvious that at is increasing, and bt is decreasing on t ∈ 0, ∞;
therefore, by 2.3, we have
⎧
aa∗ β2
atbs
⎪
⎪
≥
, t ≤ s,
⎪
⎨
1 atbt 1 ab∗ ba∗ Gt, s
⎪ asbt
ρ−1 1 atbt ⎪
bb∗ β1
⎪
⎩
≥
, s ≤ t.
1 atbt 1 ab∗ ba∗ 2.8
We take c∗ min{aa∗ β2 /1 ab∗ ba∗ , bb∗ β1 /1 ab∗ ba∗ }, then 0 < c∗ < 1; this is
because that
aa∗ β2
ab∗ ba∗ ≤
< 1,
∗
∗
1 ab ba 1 ab∗ ba∗ bb∗ β1
ab∗ ba∗ ≤
< 1.
∗
∗
1 ab ba 1 ab∗ ba∗ 2.9
By Lemma 2.31, we have Gτ, s/ρ−1 1 aτbτ ≤ 1, then
Gt, s
Gτ, s
≥ c∗ ≥ c∗ −1
,
ρ 1 aτbτ
atbt
ρ−1 1
aa∗ β2
aa∗ at
1
aτ
1
≥
> c∗
.
≥ c∗
1 atbt 1 ab∗ ba∗ 1 ab∗ ba∗ β2
bt
1 aτbτ
2.10
Similarly, we can obtain that bt/1 atbt ≥ c∗ bτ/1 aτbτ. The proof is
completed.
In this paper, we use the space
E
|ut|
u ∈ C R : sup −1
< ∞, sup u t < ∞
t∈0,∞ ρ 1 atbt
t∈0,∞
1
2.11
with the norm u
max{
u
1 , u ∞ }, where u
1 supt∈0,∞ |ut|/ρ−1 1 atbt and
u
∞ supt∈0,∞ |ut|, then E, u
is a Banach space.
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Journal of Applied Mathematics
Let
P
u ∈ E : ut ≥ 0, mint∈a∗ ,b∗ ut
uτ
∗
.
≥
c
,
τ
∈
R
ρ−1 1 atbt
ρ−1 1 aτbτ
2.12
Clearly P is a cone of E.
Lemma 2.5 see 10. Let M ⊆ Cl R , R {x ∈ CR , R | limt → ∞ xt exists}, then M is
precompact if the following conditions hold:
a M is bounded in Cl ;
b the functions belonging to M are locally equicontinuous on any interval of R ;
c the functions from M are equiconvergent; that is, given ε > 0, there corresponds Tε > 0
such that |xt − x∞| < ε for any t ≥ Tε and x ∈ M.
We shall consider nonnegative continuous and concave functional α on P ; that is, α :
P → 0, ∞ is continuous and satisfies
α tx 1 − ty ≥ tαx 1 − tα y ,
∀x, y ∈ P, 0 ≤ t ≤ 1.
2.13
We denote the set {x ∈ P | a ≤ αx, x
≤ b}b > a > 0 by P α, a, b and
Pr {x ∈ P | x
< r}.
2.14
The key tool in our approach is the following Leggett-Williams fixed point theorem.
Theorem 2.6 see 11. Let T : Pc → Pc be completely continuous and α a nonnegative continuous
concave functional on P with αx ≤ x
for any x ∈ Pc . Suppose that there exist 0 < a < b < d ≤ c
such that
c1 {x ∈ P α, b, d | αx > b} /
φ, and αT x > b, for x ∈ P α, b, d;
c2 T x
< a, for x ∈ Pa ;
c3 αT x > b for x ∈ P α, b, c with T x
> d.
Then T has at least three fixed points x1 , x2 , x3 , with
x1 < a,
b < αx2 ,
x3 > a,
2.15
αx3 < b.
3. Existence Results
Define the operator T : P → P by
T ut ∞
Gt, sΦsf s, us, u s ds A Φf at B Φf bt,
0 < t < ∞.
0
Then ut is a fixed point of operator T if and only if ut is a solution of BVP1.1.
3.1
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7
For convenience, we denote δ, αx by
aa∗ bb∗ 0<δ≤
1 ab∗ ba∗ b∗
a∗
Φsds,
αu min
∗ ∗
t∈a ,b ρ−1 1
ut
,
atbt
∀u ∈ P.
3.2
Theorem 3.1. Suppose that H1 , H2 hold, and assume there exist 0 < r1 < b1 < l1 < r2 with
l1 max{b1 /c∗ , supt∈0,∞ b1 /c∗ pt}, such that
∞
H3 Qy1 , y2 ≤
min{r2 / 0 Φsqsds AΦq/ρ−1 β2 BΦq/ρ−1 β1 , r2 /
∞
supt∈0,∞ 1/pt 0 Φsqsds AΦqα1 BΦqα2 }, 0 ≤ y1 ≤ r2 , |y2 | ≤ r2 ,
H4 Ψt, y1 , y2 > b1 /δ, t ∈ a∗ , b∗ , b1 ≤ y1 ≤ r2 , |y2 | ≤ r2 ,
∞
H5 Qy1 , y2 < min{r1 / 0 Φsqsds AΦq/ρ−1 β2 BΦq/ρ−1 β1 , r1 /
∞
supt∈0,∞ 1/pt 0 Φsqsds AΦqα1 BΦqα2 }, 0 ≤ y1 ≤ r1 , |y2 | ≤ r1 .
Then BVP1.1 has at least three positive solutions u1 , u2 , and u3 with
u1 < r1 ,
b1 < αu2 ,
u3 > r1 ,
αu3 < b1 .
3.3
Proof. Firstly we prove that T : P → P is continuous.
We will show that T : P → P is well defined and T P ⊂ P . For all ut ∈ P , by H2 ,
Φt and f are nonnegative functions, and we have T ut ≥ 0. From H1 , H2 , we obtain
∞
T
Gτ, sΦsf s, us, u s ds
ρ − Tbτ
1
A Φf 0 ∞
Δ Gτ, sΦsf s, us, u s ds
Kbτ −K
0
∞
T
Gτ, sΦsqsds
ρ − Tbτ
maxy1 ∈0,
u
,|y2 |≤
u
Q y1 , y2 0
≤
∞
Δ
−K
Gτ,
sΦsqsds
Kbτ
A Φq
max
y1 ∈0,
u
,|y2 |≤
u
A
0
Q y1 , y2 .
In the same way, we have
Q y1 , y2 .
max
y1 ∈0,
u
,|y2 |≤
u
B Φf ≤ B Φq
B
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Journal of Applied Mathematics
By Lemma 2.31, A, B, and H1 , for all ut ∈ P , we have
A Φf at
Gt, s
Φsf s, us, u s ds −1
ρ−1 1 atbt
ρ 1 atbt
0
B Φf bt
−1
ρ 1 atbt
∞
B Φf
A Φf
≤
Φsf s, us, u s ds −1
−1
ρ β2
ρ β1
0
∞
A Φq
B Φq
≤
max
Q y1 , y2
Φsqsds −1
−1
y1 ∈0,
u
,|y2 |≤
u
ρ β2
ρ β1
0
T ut
−1
ρ 1 atbt
∞
3.4
< ∞,
t
T u t 1 −α2 as Φsf s, us, u s ds
pt 0
ρ
∞
α1 bs
Φsf s, us, u s ds A Φf α1 − B Φf α2 ρ
t
1
max
Q y1 , y2
t∈0,∞ pt y1 ∈0,
u
,|y2 |≤
u
∞
×
Φsqsds A Φq α1 B Φq α2
≤ sup
0
< ∞.
3.5
Hence, T : P → P is well defined. By 3.1, H1 , the Lebesgue dominated convergence
theorem and the continuity of pt, for any u ∈ P, t1 , t2 ∈ R , we have
t1
T u t1 − T u t2 ≤ α2 a∞ 1 − 1 Φsf s, us, u s ds
ρ
pt1 pt2 0
α2 a∞ t2
Φsf s, xs, x s ds
ρpt2 t1
1 ∞
α1 b0 1
−
Φsf s, us, u s ds
ρ
pt1 pt2 0
3.6
t2
α1 b0
Φsf s, xs, x s ds
ρpt2 t1
1
1 −
A Φf α1 B Φf α2 pt1 pt2 −→ 0,
as t1 −→ t2 .
That is, T ut ∈ C1 R0 ; therefore, T ut ∈ E.
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9
By Lemma 2.4, we have
min
∗ ∗
t∈a ,b
T ut
min
ρ−1 1 atbt
t∈a∗ ,b∗ ≥c
∗
c∗
∞
Gt, s
Φsf s, us, u s ds
atbt
0
btB Φf
atA Φf
−1
ρ 1 atbt ρ−1 1 atbt
ρ−1 1
∞
Gτ, s
Φsf s, us, u s ds
−1 1 aτbτ
ρ
0
aτA Φf
bτB Φf
−1
ρ 1 atbt ρ−1 1 atbt
3.7
T uτ
,
ρ−1 1 aτbτ
therefore T : P → P .
We show that T : P → P is continuous. In fact suppose {um } ⊆ P, u0 ∈ P and um →
u0 m → ∞, then there exists M > 0, such that um ≤ M. By H1 , we have
∞
0
Φsf s, um s, um s − f s, u0 s, u0 s ds ≤ 2
∞
Φsf s, us, u s ds
0
≤2
max
y1 ∈0,M,|y2 |≤M
×
∞
Q y1 , y2
3.8
Φsqsds
0
< ∞.
Therefore, by Lemma 2.31, the continuity of f and the Lebesgue dominated convergence
theorem imply that
T um t − T u0 t ρ−1 1 atbt ≤
∞
Gt, s
−1 1 atbt
ρ
0
× Φs f s, um s, um s − f s, u0 s, u0 s ds
∞
0
Φsf s, um s, um s − f s, u0 s, u0 s ds −→ 0,
m −→ ∞,
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Journal of Applied Mathematics
T um t − T u0 t ≤ sup
1
t∈0,∞ pt
−→ 0,
∞
0
Φsf s, um s, um s − f s, u0 s, u0 s ds
m −→ ∞.
3.9
Thus, T um − T u0 → 0m → ∞. Therefore T : P → P is continuous.
Secondly we show that T : P → P is compact operator.
For any bounded set B ⊂ P , there exists a constant L > 0 such that u
≤ L, for all
u ∈ B. By Lemma 2.31, A, B, and H1 , we have
T ut
atbt
∞
Gt, s
≤ ρ−1 1 a∞b0
Φsf s, us, u s ds
−1
ρ 1 atbt
0
B Φf bt
A Φf at
−1
ρ 1 atbt ρ−1 1 atbt
∞
B Φf
A Φf
−1
Φsf s, us, u s ds −1
−1
≤ ρ 1 a∞b0
ρ β2
ρ β1
0
∞
A Φq
B Φq
−1
Φsqsds −1
−1
≤ ρ 1 a∞b0 max Q y1 , y2
y1 ∈0,L,|y2 |≤L
ρ β2
ρ β1
0
T ut ρ−1 1 atbt
ρ−1 1
< ∞,
∞
GsΦsf s, us, u s ds A Φf a∞ B Φf b∞
u∞
T
0
β2
ρ
≤
∞
asΦsf s, us, u s ds A Φf a∞ B Φf b∞
0
β2 a∞ ∞
Q y1 , y2
Φsqsds A Φq a∞ B Φq b∞
ρ
y1 ∈0,L,|y2 |≤L
0
max
< ∞.
3.10
Therefore, T ut ⊆ Cl R , R.
By 3.4 and 3.5, we have
T ut
atbt
t∈0,∞
∞
A Φq
B Φq
≤
max Q y1 , y2
Φsqsds −1
−1
y1 ∈0,L,|y2 |≤L
ρ β2
ρ β1
0
T u
1 sup
< ∞,
ρ−1 1
Journal of Applied Mathematics
11
T u max T u t
∞
t∈0,∞
∞
1
max
Q y1 , y2
t∈0,∞ pt y1 ∈0,L,|y2 |≤L
≤ sup
Φsqsds A Φq α1 B Φq α2
0
< ∞,
3.11
so T B is bounded.
Given T > 0, t1 , t2 ∈ 0, T , by H1 and Lemma 2.31, we have
Gt1 , s
Gt2 , s
Q y1 , y2
max
ρ−1 1 at bt − ρ−1 1 at bt Φsf s, us, u s ≤ 2
y1 ∈0,L,|y2 |≤L
1
1
2
2
× Φsqs.
3.12
Therefore for any u ∈ B, by 3.1, the Lebesgue dominated convergence theorem and the
continuity of Gt, s, at, and bt, we have
T ut1 T xt2 −
ρ−1 1 at bt ρ−1 1 at bt 1
1
2
2
∞ Gt1 , s
Gt2 , s
≤
ρ−1 1 at bt − ρ−1 1 at bt 1
1
2
2
0
× Φsf s, us, u s ds
at1 at2 A Φf −1
− −1
ρ 1 at1 bt1 ρ 1 at2 bt2 bt1 bt2 B Φf −1
− −1
ρ 1 at1 bt1 ρ 1 at2 bt2 −→ 0,
3.13
as t1 −→ t2 .
By a similar proof as 3.6, we obtain |T u t1 − T u t2 | → 0, as t1 → t2 . Thus, T B is
equicontinuous on 0, T . Since T > 0 is arbitrary, T B is locally equicontinuous on 0, ∞.
By Lemma 2.32, H2 and the Lebesgue dominated convergence theorem, we obtain
T ut
lim −1
t → ∞ ρ 1 atbt ∞
1
β
asΦsf
s,
xs,
x
ds
A
Φf
a∞
B
Φf
b∞
s
2
−1
ρ 1 a∞b∞ 0
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Journal of Applied Mathematics
∞
maxy1 ∈0,L,|y2 |≤L Q y1 , y2
Φsqsds A Φq a∞ B Φq b∞
β2 a∞
≤
ρ−1 1 β1 β2
0
< ∞,
T ut
T u∞
−
ρ−1 1 atbt ρ−1 1 a∞b∞ ≤
t
0
asbt
t
0
1
1
Φsf s, xs, x s ds
−
1 atbt 1 a∞b∞ as
bt − β2 Φsf s, xs, x s ds
1 a∞b∞
∞
|at − as|
Φsf s, xs, x s ds
1 atbt
t
∞
1
1
Φsf s, xs, x s ds
−
asbs
1 atbt 1 a∞b∞
t
∞
as
bs − β2 Φsf s, xs, x s ds A Φf |at − a∞|
1
a∞b∞
ρ−1 1 atbt
t
bs
|bt − b∞|
B Φf −1
A Φf a∞ B Φf b∞
ρ 1 atbt
1
1
× −1
− −1
ρ 1 atbt ρ 1 a∞b∞ ≤
max
Q y1 , y2
y1 ∈0,M,|y2 |≤M
t
1
1
Φsqsds
× b0a∞ −
1
atbt
1
a∞b∞
0
t
∞
a∞
bt − β2 Φsqsds b0
|at − as|Φsqsds
1 a∞b∞ 0
1 β1 β2 t
∞
1
1
a∞b0
1 atbt − 1 a∞b∞ Φsqsds
t
∞
a∞
bs − β2 Φsqsds A Φq |at − a∞|
1 a∞b∞ t
ρ−1 1 atbt
|bt − b∞|
B Φq −1
A Φq a∞ B Φq b∞
ρ 1 atbt
1
1
−→ 0, as t −→ ∞.
× −1
− −1
ρ 1 atbt ρ 1 a∞b∞ 3.14
Journal of Applied Mathematics
13
By 3.5, we know that limt → ∞ |T u t| < ∞, then
T u t − T u ∞
∞
1 t −α as
α1 bs
1
2
Φsf s, us, u s ds Φsf s, us, u s ds
pt 0
ρ
pt t
ρ
t
α2 as
1
1 1
A Φf α1 −
B Φf α2 Φsf s, us, u s ds
pt
pt
p∞ 0 ρ
∞
α2 as
1
1
1
Φsf s, us, u s ds −
A Φf α1 B Φf α2 p∞ t
ρ
p∞
p∞
1
1 ≤
max
−
Q y1 , y2 pt p∞ y1 ∈0,L,|y2 |≤L
t
∞
1
1
α2 as
α1 bs
×
Φsqsds Φsqsds ρ
pt
ρ
p∞
0
t
∞
1
1 α2 as
−→ 0,
Φsqsds A Φq α1 B Φq α2 −
ρ
pt p∞ t
as t −→ ∞.
3.15
Therefore, T B is equiconvergent at ∞. By Lemma 2.5, T B is completely continuous.
Finally we will show that all conditions of Theorem 2.6 hold.
From the definition of α, we can get αu ≤ u
for all u ∈ P . For all u ∈ Pr2 , we have
u
≤ r2 ; therefore 0 ≤ y1 ≤ r2 , |y2 | ≤ r2 . By 3.4, 3.5, and H3 , we have
|T ut|
≤
−1
ρ 1 atbt
max
y1 ∈0,r2 ,|y2 |≤r2
Q y1 , y2
∞
0
A Φq
B Φq
Φsqsds −1
−1
ρ β2
ρ β1
≤ r2 ,
T u t ≤ sup
1
max
Q y1 , y2
t∈0,∞ pt y1 ∈0,r2 ,|y2 |≤r2
×
∞
3.16
Φsqsds A Φq α1 B Φq α2
0
≤ r2 ,
that is, T u
≤ r2 for u ∈ Pr2 . Thus T : Pr2 → Pr2 .
Similarly for any u ∈ Pr1 , we have T u
< r1 , which means that condition c2 of
Theorem 2.6 holds.
14
Journal of Applied Mathematics
In order to apply condition c1 of Theorem 2.6, we choose ut b1 ρ−1 1 atbt/
c , t ∈ R0 , then u
≤ l1 ; this is because
∗
u
1 b1
≤ l1 ,
c∗
3.17
−1 u sup u t sup b1 ρ a tbt atb t ≤ sup 1 b1 ≤ l1 ,
∞
t∈0,∞ pt c∗
c∗
t∈0,∞
t∈0,∞
and αu mint∈a∗ ,b∗ ut/ρ−1 1 atbt b1 /c∗ > b1 , which means that {u ∈
φ. For all u ∈ P α, b1 , l1 , we have αu ≥ b1 and u
≤ l1 , thus
P α, b1 , l1 |αu > b1 } /
b1 ≤ ut/ρ−1 1 atbt ≤ l1 , |u t| ≤ l1 , that is, b1 ≤ y1 ≤ l1 , |y2 | ≤ l1 . By H4 , we can
get
1
T ut
≥ min
−1
∗
∗
t∈a
,b
ρ 1 atbt
atbt
αT ut min
∗ ∗
t∈a ,b ρ−1 1
×
a∗
0
asbt
Φsf s, us, u s ds
ρ
t
b∗
asbt
atbs
Φsf s, us, u s ds Φsf s, us, u s ds
ρ
ρ
a∗
t
∞
atbs
Φsf s, us, u s ds
ρ
∗
b
aa∗ bb∗ >
1 ab∗ ba∗ >
aa∗ bb∗ 1 ab∗ ba∗ b∗
a∗
b∗
a∗
Φsf s, us, u s ds
Φsds
b1
δ
≥ b1 .
3.18
Consequently condition c1 of Theorem 2.6 holds.
We will prove that condition c3 of Theorem 2.6 holds. If u ∈ P α, b1 , r2 , and T ut
>
l1 , by H4 , we have
aa∗ bb∗ T ut
αT ut min
>
t∈a∗ ,b∗ ρ−1 1 atbt
1 ab∗ ba∗ b∗
a∗
Φsds
b1
≥ b1 .
δ
3.19
Therefore, condition c3 of Theorem 2.6 is satisfied. Then we can complete the proof of this
theorem by Leggett-Williams fixed point theorem.
Theorem 3.2. Suppose that H1 , H2 hold, and assume there exist 0 < r1 < b1 < l1 < r2 < b2 <
l2 < r3 < · · · < rm with li max{bi /c∗ , supt∈0,∞ bi /c∗ pt}, such that
Journal of Applied Mathematics
15
∞
<
min{ri / 0 Φsqsds AΦq/ρ−1 β2 BΦq/ρ−1 β1 , ri /
H6 Qy1 , y2 ∞
supt∈0,∞ 1/pt 0 Φsqsds AΦqα1 BΦqα2 }, 0 ≤ y1 ≤ ri , |y2 | ≤ ri ,
1 ≤ i ≤ m,
H7 Ψt, y1 , y2 > bi /δ, t ∈ a∗ , b∗ , bi ≤ y1 ≤ ri1 , |y2 | ≤ ri1 , 1 ≤ i ≤ m − 1.
Then BVP1.1 has at least 2m − 1 positive solutions.
Proof. When m 1, it follows from H6 that T has at least one positive solution by the
Schauder fixed point theorem. When m 2, it is clear that Theorem 3.1 holds. Then we
can obtain three positive solutions. In this way, we can finish the proof by the method of
induction.
4. Example
Consider the following singular Sturm-Liouville singular boundary value problems for
second-order differential equation on the half-line
e−t 1 t f t, ut, u t 0, 0 < t < ∞,
√
t
m−2
1 i
u0 − lim ptu t uξi , 0 < ξi < ∞,
t→0
3
i1
∞
1 −s
e 1 susds,
lim ut lim ptu t t → ∞
t → ∞
3
0
1 t2 u t
4.1
where
⎧
1 ⎪
⎪
y2 ,
y14 ⎨
550
f t, ut, u t Ψ t, y1 , y2 ⎪
⎪
⎩1 1 y2 ,
550
y1 ≤ 1,
4.2
y1 ≥ 1,
√
Φt e−t 1t/ t
pt 1t2 , α1 α2 β1 β2 1, at 2−1/1t, bt 11/1t,
m−2
∞
which is singular at t 0, ρ 3, Tu i1 1/3i uξi , Ku 0 1/3e−s 1 susds.
Set qt 1 and
⎧
1 ⎪
⎪
y2 ,
y14 ⎨
550
Q y1 , y2 ⎪
⎪
⎩1 1 y2 ,
550
y1 ≤ 1,
4.3
y1 ≥ 1,
∞
then 0 Φsqsds < 3, a0 1, a∞ 2, b0 2, b∞ 1, 1/2 < Taτ < 1, 1/2 <
Tbτ < 1, Kaτ Kbτ 1, Δ > 3, AΦq < 26/9, BΦq < 20/9.
16
Journal of Applied Mathematics
Choose r1 1/3, b1 7/5, r2 19. When a∗ 1, b∗ 2, by the definition of δ, we
may choose δ 8/5. By direct calculations, we imply that
min ∞
0
r1
,
Φsqsds A Φq /ρ−1 β2 B Φq /ρ−1 β1
supt∈0,∞ 1/pt
min ∞
0
∞
0
r2
,
Φsqsds A Φq /ρ−1 β2 B Φq /ρ−1 β1
supt∈0,∞ 1/pt
Q y1 , y2 ≤
⎫
⎬
3r1
⎭ > 55 ,
Φsqsds A Φq α1 B Φq α2
r1
r2
⎫
⎬
3r2
⎭ > 55 ,
∞
Φsqsds A Φq α1 B Φq α2
0
4
1
1
3r1
1
1
× <
,
3
550 3 55
55
3 × 19 3r2
19
Q y1 , y2 ≤ 1 <
,
550
55
55
7/5 b1
Ψ t, y1 , y2 ≥ 1 >
,
8/5
δ
for 0 ≤ y1 ≤
4.4
1 1
, y2 ≤ ,
3
3
for 0 ≤ y1 ≤ 19, y2 ≤ 19,
for t ∈ 1, 2, 7/5 ≤ y1 ≤ 19, y2 ≤ 19.
Therefore, the conditions H1 –H5 hold. Applying Theorem 3.1 we conclude that
BVP4.1 has at least three positive solutions.
Acknowledgments
The research was supported by the National Natural Science Foundation of China 10871116
and the Natural Science Foundation of Shandong Province of China ZR2010AM005.
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