Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2009, Article ID 609143, 11 pages
doi:10.1155/2009/609143
Research Article
Existence of Solutions for m-point Boundary Value
Problems on a Half-Line
Changlong Yu, Yanping Guo, and Yude Ji
College of Sciences, Hebei University of Science and Technology, Shijiazhuang 050018, Hebei, China
Correspondence should be addressed to Changlong Yu, changlongyu@126.com
Received 5 April 2009; Accepted 6 July 2009
Recommended by Paul Eloe
By using the Leray-Schauder continuation theorem, we establish the existence of solutions for
m-point boundary value problems on a half-line x t ft, xt, x t 0, 0 < t < ∞, x0 m−2
m−2
1 and 0 < η1 < η2 < · · · < ηm−2 < ∞ are
i1 αi xηi , limt → ∞ x t 0, where αi ∈ R,
i1 αi /
given.
Copyright q 2009 Changlong Yu et al. 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
Multipoint boundary value problems BVPs for second-order differential equations in a
finite interval have been studied extensively and many results for the existence of solutions,
positive solutions, multiple solutions are obtained by use of the Leray-Schauder continuation
theorem, Guo-Krasnosel’skii fixed point theorem, and so on; for details see 1–4 and the
references therein.
In the last several years, boundary value problems in an infinite interval have been
arisen in many applications and received much attention; see 5, 6. Due to the fact that
an infinite interval is noncompact, the discussion about BVPs on the half-line is more
complicated, see 5–14 and the references therein. Recently, in 15, Lian and Ge studied
the following three-point boundary value problem:
x t f t, xt, x t 0,
x0 αx η ,
0 < t < ∞,
lim x t 0,
t → ∞
1.1
2
Advances in Difference Equations
where α ∈ R, α /
1, and η ∈ 0, ∞ are given. In this paper, we will study the following
m-point boundary value problems:
x t f t, xt, x t 0,
x0 m−2
αi x ηi ,
0 < t < ∞,
1.2
lim x t 0,
t → ∞
i1
1, αi have the same signal, and 0 < η1 < η2 < · · · < ηm−2 < ∞ are
where αi ∈ R, m−2
i1 αi /
given. We first present the Green function for second-order multipoint BVPs on the half-line
and then give the existence results for 1.2 using the properties of this Green function and
the Leray-Schauder continuation theorem.
1
0, ∞ {x ∈ C1 0, ∞, limt → ∞ xt exists, limt → ∞ x t exists}
We use the space C∞
with the norm x max{x∞ , x ∞ }, where · ∞ is supremum norm on the half-line, and
1
L
∞0, ∞ {x : 0, ∞ → R is absolutely integrable on 0, ∞} with the norm xL1 |xt|dt.
0
We set
P
∞
psds,
P1 0
∞
spsds,
Q
∞
0
qsdt,
1.3
0
and we suppose αi , i 1, 2, . . . , m − 2 are the same signal in this paper and we always assume
α m−2
i1 αi .
2. Preliminary Results
In this section, we present some definitions and lemmas, which will be needed in the proof
of the main results.
Definition 2.1 see 15. It holds that f : 0, ∞ × R2 −→ R is called an S-Carathéodory
function if and only if
i for each u, v ∈ R2 , t → ft, u, v is measurable on 0, ∞,
ii for almost every t ∈ 0, ∞, u, v → ft, u, v is continuous on R2 ,
iii for each r > 0, there exists ϕr t ∈ L1 0, ∞ with tϕr t ∈ L1 0, ∞, ϕr t > 0 on
0, ∞ such that max{|u|, |v|} ≤ r implies |ft, u, v| ≤ ϕr t, for a.e. t ∈ 0, ∞.
Lemma 2.2. Suppose
BVP,
m−2
i1
αi /
1, if for any vt ∈ L1 0, ∞ with tvt ∈ L1 0, ∞, then the
x t vt 0,
x0 m−2
αi x ηi ,
i1
0 < t < ∞,
lim x t 0,
t → ∞
2.1
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3
has a unique solution. Moreover, this unique solution can be expressed in the form
xt ∞
2.2
Gt, svsds,
0
where Gt, s is defined by
⎧m−2
⎪
⎪
⎪
αi s Λs,
⎪
⎪
⎪
⎪
i1
⎪
⎪
⎪
⎪
m−2
⎪
⎪
⎪
⎪
αi s Λt,
⎪
⎪
⎪
⎪
i1
⎪
⎪
⎪ i
⎪
m−2
⎪
⎪
⎪
⎪
αk ηk αk s Λs,
⎪
1⎨
k1
Gt, s s ≤ η1 , s ≤ t,
s ≤ η1 , t ≤ s,
0 < ηi ≤ s ≤ ηi1 , s ≤ t, i 1, 2, . . . , m − 3,
ki1
2.3
i
m−2
Λ⎪
⎪
⎪
⎪
α
η
αk s Λt,
⎪
k
k
⎪
⎪
⎪
k1
ki1
⎪
⎪
⎪
⎪
m−2
⎪
⎪
⎪
⎪ αi ηi Λs,
⎪
⎪
⎪
⎪
i1
⎪
⎪
⎪
⎪
m−2
⎪
⎪
⎪
⎪
⎩ αi ηi Λt,
0 < ηi ≤ s ≤ ηi1 , t ≤ s, i 1, 2, . . . , m − 3,
s ≥ ηm−2 , s ≤ t,
s ≥ ηm−2 , t ≤ s,
i1
here note Λ 1 −
m−2
i1
αi .
Proof. Integrate the differential equation from t to ∞, noticing that vt, tvt ∈ L1 0, ∞,
then from 0 to t and one has
xt x0 t ∞
Since x0 xt m−2
1−
i1
2.4
vτdτ ds.
0
s
αi xηi , from 2.4, it holds that
1
m−2
i1
m−2
∞
i1
ηi
αi ηi
αi
vsds m−2
ηi
i1
0
αi
∞
svsds t
vsds t
t
svsds.
0
2.5
For 0 ≤ t ≤ η1 , the unique solution of 2.1 can be stated by
xt t m−2
i1
1−
0
m−2
m−3
ηi1
i1
ηi
αi s
i1
αi
i
k1
s vsds η1 m−2
i1
t
1−
αi s
m−2
i1
αi
t vsds
∞ m−2
αk ηk m−2
ki1 αk s Λt
i1 αi ηi
vsds t vsds.
1 − m−2
1 − m−2
ηm−2
i1 αi
i1 αi
2.6
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If ηi ≤ t ≤ ηi1 , 1 ≤ i ≤ m − 3, the unique solution of 2.1 can be stated by
xt η1 m−2
1−
0
i−1 ηj1
j1
αi s
i1
m−2
i1
⎛ j
⎝
αi
k1
s vsds
αk ηk 1−
ηj
m−2
kj1
m−2
i1
αk s Λs
αi
⎞
⎠vsds
t i
αk ηk m−2
ki1 αk s Λs
vsds
1 − m−2
ηi
i1 αi
m−2
ηi1 i
k1 αk ηk ki1 αk s Λt
vsds
1 − m−2
t
i1 αi
⎛ j
⎞
m−2
m−3
α
η
α
s
Λt
ηj1
k
k
k
kj1
⎝ k1
⎠vsds
m−2
1
−
η
j
ji1
i1 αi
k1
∞ m−2
i1
ηm−2
1−
αi ηi
m−2
i1
αi
2.7
t vsds.
If ηm−2 ≤ t < ∞, the unique solution of 2.1 can be stated by
η1 m−2
αk ηk m−2
ki1 αk s Λs
vsds
xt s vsds 1 − m−2
1 − m−2
0
i1 ηi
i1 αi
i1 αi
2.8
t m−2
∞ m−2
α
η
α
η
i i
i i
i1
i1
s vsds m−2 t vsds.
1 − m−2
α
1
−
ηm−2
t
i
i1
i1 αi
i1
We note Λ 1 −
Gt, s αi s
m−2
i1
m−3
ηi1
k1
αi , then
⎧m−2
⎪
⎪
⎪
αi s Λs,
⎪
⎪
⎪
i1
⎪
⎪
⎪
⎪
m−2
⎪
⎪
⎪
⎪ αi s Λt,
⎪
⎪
⎪
i1
⎪
⎪
⎪
⎪
i
m−2
⎪
⎪
⎪
⎪
αk ηk αk s Λs,
⎪
⎪
ki1
1 ⎨ k1
i
m−2
Λ⎪
⎪
⎪
⎪
αk ηk αk s Λt,
⎪
⎪
⎪
⎪
k1
ki1
⎪
⎪
⎪
⎪
m−2
⎪
⎪
⎪
⎪ αi ηi Λs,
⎪
⎪
⎪
i1
⎪
⎪
⎪
⎪
m−2
⎪
⎪
⎪ α η Λt,
⎪
⎩
i i
i1
i
s ≤ η1 , s ≤ t,
s ≤ η1 , t ≤ s,
0 < ηi ≤ s ≤ ηi1 , s ≤ t, i 1, 2, . . . , m − 3,
2.9
0 < ηi ≤ s ≤ ηi1 , t ≤ s, i 1, 2, . . . , m − 3,
s ≥ ηm−2 , s ≤ t,
s ≥ ηm−2 , t ≤ s.
Advances in Difference Equations
5
Therefore, the unique solution of 2.1 is xt proof.
∞
0
Gt, svsds, which completes the
Remark of Lemma 2.2.Obviously Gt, s satisfies the properties of a Green function, so we call
Gt, s the Green function of the corresponding homogeneous multipoint BVP of 2.1 on the
half-line.
Lemma 2.3. For all t, s ∈ 0, ∞, it holds that
⎧
m−2
⎪
⎪
⎪
αi < 0,
s,
⎪
⎪
⎪
⎪
i1
⎪
⎪
m−2
⎪
⎨s
0≤
αi < 1,
|Gt, s| ≤ Λ ,
⎪
i1
⎪
⎫
⎧ m−2
⎪
m−2
⎪
⎪
⎬ m−2
⎨
αs
αη
⎪
⎪
i1 i
i1 i m−2
⎪
⎪
,
,
max
α > 1.
⎪
⎩
⎭ i1 i
⎩ −Λ
−Λ
2.10
Proof. For each s ∈ 0, ∞, Gt, s is nondecreasing in t. Immediately, we have
m−2
min
i1
i
αi s
,
Λ
αk ηk Λ
k1
m−2
ki1
αk s
m−2
i1
,
αi ηi
Λ
≤ Gt, s ≤ Gs, s
⎧
⎪
s,
s ≤ η1 ,
⎪
⎪
⎪
⎪
⎪
i
m−2
⎪
⎪
αk Λ s, ηi ≤ s ≤ ηi1 < ∞, i 1, 2, . . . , m − 3,
1 ⎨ αk ηk k1
ki1
Λ⎪
⎪
⎪
⎪
⎪
⎪m−2
⎪
⎪
s ≥ ηm−2 .
⎩ αi ηi Λs,
2.11
i1
Further, we have
m−2
i1
Λ
m−2
0 < min
i1
Λ
m−2
min
αi s
i1
Therefore, we get the result.
Λ
αi s
αi s
≤ Gt, s ≤ s,
m−2
,
αi η1
Λ
m−2
,
i1
αi ηm−2
Λ
αi < 0,
i1
i1
m−2
≤ Gt, s ≤
s
,
Λ
≤ Gt, s ≤ s,
m−2
0≤
αi < 1,
i1
m−2
αi > 1.
i1
2.12
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Advances in Difference Equations
Lemma 2.4. For the Green function Gt, s, it holds that
lim Gt, s Gs
t → ∞
⎧
⎪
s,
⎪
⎪
⎪
⎪
⎪
⎪
i
m−2
⎪
⎪
⎨
αk ηk αk Λ s,
1
s ≤ η1 ,
ηi ≤ s ≤ ηi1 < ∞, i 1, 2, . . . , m − 3 2.13
k1
ki1
Λ⎪
⎪
⎪
⎪
m−2
⎪
⎪
⎪
⎪
⎪
⎩ αi ηi Λs,
s ≥ ηm−2 .
i1
Lemma 2.5. For the function x ∈ C1 0, ∞, it is satisfied that
x0 m−2
αi x ηi
2.14
i1
and αi i 1, 2, . . . , m − 2 have the same signal, 0 < η1 < η2 < · · · < ηm−2 < ∞, then there exists
η ∈ η1 , ηm−2 satisfying
x0 αx η ,
where α m−2
i1
2.15
αi .
Proof. Let αi i 1, 2, . . . , m − 2 are positive, and note M∗ max{xt | t ∈ η1 , ηm−2 }, m∗ min{xt | t ∈ η1 , ηm−2 }, then for every i i 1, 2, . . . , m − 2, we have m∗ ≤ xηi ≤ M∗ ,
m−2
m−2
m−2
m−2
∗
∗
∗
so m∗ m−2
i1 αi ≤
i1 αi xηi ≤ M
i1 αi , that is, m ≤
i1 αi xηi /
i1 αi x ≤ M .
Because xt is continuous on the interval η1 , ηm−2 , there exists η ∈ η1 , ηm−2 satisfying
x0 αxη, where α m−2
i1 αi .
Theorem 2.6 see 5. Let M ⊂ C∞ 0, ∞ {x ∈ C0, ∞, limt → ∞ xt exists}. Then M is
relatively compact in X if the following conditions hold:
a M is uniformly bounded in C∞ 0, ∞;
b the functions from M are equicontinuous on any compact interval of 0, ∞;
c the functions from M are equiconvergent, that is, for any given > 0, there exists a T T > 0 such that |ft − f∞| < , for any t > T, f ∈ M.
3. Main Results
1
Consider the space X {x ∈ C∞
0, ∞, x0 the operator T : X × 0, 1 → X by
∞
T x, λt λ
m−2
i1
αi xηi , limt → ∞ x t 0} and define
Gt, sf s, xs, x s ds,
0
The main result of this paper is following.
0 ≤ t < ∞.
3.1
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7
Theorem 3.1. Let f : 0, ∞ × R2 → R be an S-Carathéodory function. Suppose further that there
exists functions pt, qtrt ∈ L1 0, ∞ with tpt, tqttrt ∈ L1 0, ∞ such that
ft, u, v ≤ pt|u| qt|v| rt
3.2
for almost every t ∈ 0, ∞ and all u, v ∈ R2 . Then 1.2 has at least one solution provided:
ηm−2 P P1 Q < 1,
max
αηm−2
P P1 Q < 1,
1−α
α < 0,
0 ≤ α < 1,
αηm−2
αηm−2 P
αP1
P P1 Q,
α−1
α−1
α−1
3.3
< 1,
α > 1.
Lemma 3.2. Let f : 0, ∞ × R2 → R be an S-Carathéodory function. Then, for each λ ∈
0, 1, T x, λ is completely continuous in X.
Proof. First we show T is well defined. Let x ∈ X; then there exists r > 0 such that x ≤ r. For
each λ ∈ 0, 1, it holds that
∞
T x, λt λ
Gt, sf s, xs, x s ds
0
≤
∞
3.4
|Gt, s|ϕr sds < ∞,
∀t ∈ 0, ∞.
0
Further, Gt, s is continuous in t so the Lebesgue dominated convergence theorem implies
that
|T x, λt1 − T x, λt2 | ≤ λ
∞
|Gt1 , s − Gt2 , s|f s, xs, x s ds
0
∞
≤λ
|Gt1 , s − Gt2 , s|ϕr sds
3.5
0
−→ 0, as t1 −→ t2 ,
t2
T x, λ t1 − T x, λ t2 ≤ λ f s, xs, x s ds
≤
t2
t1
ϕr sds −→ 0
t1
where 0 ≤ t1 , t2 < ∞. Thus, T x ∈ C1 0, ∞.
3.6
as t1 −→ t2 ,
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Obviously, T x, λ0 m−2
i1
αi T x, ληi . Notice that
lim T x, λ t lim
t → ∞
∞
t → ∞ t
f s, xs, x s ds 0,
3.7
so we can get T x, λt ∈ X.
We claim that T x, λ is completely continuous in X, that is, for each λ ∈ 0, 1, T x, λ
is continuous in X and maps a bounded subset of X into a relatively compact set.
Let xn → x as n → ∞ in X. Next we prove that for each λ ∈ 0, 1, T xn , λ → T x, λ
as n → ∞ in X. Because f is a S-Carathéodory function and
∞ ∞
Gs f s, xn s, xn s − f s, xs, x s ds ≤ 2
Gsϕr0 sds < ∞,
0
3.8
0
where r0 > 0 is a real number such that max{maxn∈N\{0} xn , x} ≤ r0 , we have
∞ |T xn , λ∞ − T x, λ∞| ≤ λ
Gsf s, xn s, xn s − f s, xs, x s ds
0
−→ 0,
3.9
as n −→ ∞.
Also, we can get
|T xn , λt − T xn , λ∞| ≤ λ
∞ Gt, s − Gsf s, xn s, xn s ds
0
∞ ≤
Gt, s − Gsϕr0 sds
3.10
0
−→ 0, as t −→ ∞,
∞
T xn , λ t − T xn , λ ∞ ≤
f s, xn s, xn s ds
t
≤
∞
3.11
ϕr0 sds −→ 0,
as t −→ ∞.
t
Similarly, we have
|T x, λt − T x, λ∞| −→ 0, as t −→ ∞,
T x, λ t − T x, λ ∞ −→ 0, as t −→ ∞.
3.12
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For any positive number T0 < ∞, when t ∈ 0, T0 , we have
|T xn , λt − T x, λt| ≤
∞
0
|Gt, s|f s, xn s, xn s − f s, xs, x s ds
−→ 0, as n −→ ∞,
∞
T xn , λ t − T x, λ t ≤
f s, xn s, xn s − f s, xs, x s ds
3.13
t
−→ 0,
as n −→ ∞.
Combining 3.9–3.13, we can see that T ·, λ is continuous. Let B ⊂ X be a bounded
subset; it is easy to prove that T B is uniformly bounded. In the same way, we can prove
3.5,3.6, and 3.12, we can also show that T B is equicontinuous and equiconvergent. Thus,
by Theorem 2.6, T ·, λ : X×0, 1 → X is completely continuous. The proof is completed.
Proof of Theorem 3.1. In view of Lemma 2.2, it is clear that x ∈ X is a solution of the BVP 1.2 if
and only if x is a fixed point of T ·, 1. Clearly, T x, 0 0 for each x ∈ X. If for each λ ∈ 0, 1
the fixed points T ·, λ in X belong to a closed ball of X independent of λ, then the LeraySchauder continuation theorem completes the proof. We have known T ·, λ is completely
continuous by Lemma 3.2. Next we show that the fixed point of T ·, λ has a priori bound M
independently of λ. Assume x T x, λ and set
Q1 ∞
sqsds,
0
R
∞
rsds,
R1 0
∞
srsdt.
3.14
0
According to Lemma 2.5, we know that for any x ∈ X, there exists η ∈ η1 , ηm−2 satisfying
x0 αxη. Hence, there are three cases as follow.
Case 1 α < 0. For any x ∈ X, x0xη ≤ 0 holds and, therefore, there exists a t0 ∈ 0, η such
that xt0 0. Then, we have
t
|xt| x sds ≤ t η x ∞ ≤ t ηm−2 x ∞ ,
t0
t ∈ 0, ∞,
3.15
and so it holds that
x ≤ λft, x, x 1 ≤ ft, x, x 1
∞
L
L
≤ pt|xt| qt|x t| rtL1
≤ ηm−2 P P1 Q x ∞ R,
3.16
therefore,
x ≤
∞
R
M1 .
1 − ηm−2 P − P1 − Q
3.17
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At the same time, we have
∞
|xt| ≤ λ Gt, sf s, xs, x s ds
≤
∞
0
sf s, xs, x s ds
3.18
0
≤ P1 x∞ Q1 M1 R1 ,
t ∈ 0, ∞,
and so
x∞ ≤
Q1 M1 R1
M1 .
1 − P1
3.19
Set M max{M1 , M1 }, which is independent of λ.
Case 2 0 ≤ α < 1. For any x ∈ X, we have
t
|xt| αx η x sds ≤ αx η tx ∞ ,
0
t ∈ 0, ∞,
3.20
which implies that |xt| ≤ αη/1 − α tx ∞ ≤ αηm−2 /1 − α tx ∞ for all t ∈ 0, ∞.
In the same way as for Case 1, we can get
x ≤
∞
1 − αR
M2 ,
1 − α1 − P1 − Q − αηm−2 P
Q1 M2 R1
M2 .
x∞ ≤
1 − α − P1
3.21
Set M max{M2 , M2 }, which is independent of λ and is what we need.
Case 3 α > 1. For x ∈ X, we have
t
1
|xt| x η x sds ≤ |x0| t − ηx ∞ ,
α
η
t ∈ 0, ∞,
3.22
and so |xt| ≤ αη/α − 1 tx ∞ ≤ αηm−2 /α − 1 tx ∞ for all t ∈ 0, ∞.
Similarly, we obtain
x ≤
∞
α − 1R
M3 ,
−
11
−
P1 − Q − αηm−2 P
α
α Q1 M3 R1 αηm−2 QM3 R
M3 .
x∞ ≤
α − 1 − αηm−2 P
Set M max{M3 , M3 } and which is we need. So 1.2 has at least one solution.
3.23
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Acknowledgment
The Natural Science Foundation of Hebei Province A2009000664 and the Foundation of
Hebei University of Science and Technology XL200759 are acknowledged.
References
1 C. P. Gupta, “A note on a second order three-point boundary value problem,” Journal of Mathematical
Analysis and Applications, vol. 186, no. 1, pp. 277–281, 1994.
2 C. P. Gupta and S. I. Trofimchuk, “A sharper condition for the solvability of a three-point second
order boundary value problem,” Journal of Mathematical Analysis and Applications, vol. 205, no. 2, pp.
586–597, 1997.
3 R. Ma, “Positive solutions for second-order three-point boundary value problems,” Applied
Mathematics Letters, vol. 14, no. 1, pp. 1–5, 2001.
4 Y. Guo and W. Ge, “Positive solutions for three-point boundary value problems with dependence on
the first order derivative,” Journal of Mathematical Analysis and Applications, vol. 290, no. 1, pp. 291–301,
2004.
5 D. O’Regan, Theory of Singular Boundary Value Problems, World Scientific, River Edge, NJ, USA, 1994.
6 R. P. Agarwal and D. O’Regan, Infinite Interval Problems for Differential, Difference and Integral Equations,
Kluwer Academic Publishers, Dordrecht, The Netherlands, 2001.
7 J. V. Baxley, “Existence and uniqueness for nonlinear boundary value problems on infinite intervals,”
Journal of Mathematical Analysis and Applications, vol. 147, no. 1, pp. 122–133, 1990.
8 D. Jiang and R. P. Agarwal, “A uniqueness and existence theorem for a singular third-order boundary
value problem on 0, ∞,” Applied Mathematics Letters, vol. 15, no. 4, pp. 445–451, 2002.
9 R. Ma, “Existence of positive solution for second-order boundary value problems on infinite
intervals,” Applied Mathematics Letters, vol. 16, pp. 33–39, 2003.
10 C. Bai and J. Fang, “On positive solutions of boundary value problems for second-order functional
differential equations on infinite intervals,” Journal of Mathematical Analysis and Applications, vol. 282,
no. 2, pp. 711–731, 2003.
11 B. Yan and Y. Liu, “Unbounded solutions of the singular boundary value problems for second order
differential equations on the half-line,” Applied Mathematics and Computation, vol. 147, no. 3, pp. 629–
644, 2004.
12 Y. Tian and W. Ge, “Positive solutions for multi-point boundary value problem on the half-line,”
Journal of Mathematical Analysis and Applications, vol. 325, no. 2, pp. 1339–1349, 2007.
13 Y. Tian, W. Ge, and W. Shan, “Positive solutions for three-point boundary value problem on the halfline,” Computers & Mathematics with Applications, vol. 53, no. 7, pp. 1029–1039, 2007.
14 M. Zima, “On positive solutions of boundary value problems on the half-line,” Journal of Mathematical
Analysis and Applications, vol. 259, no. 1, pp. 127–136, 2001.
15 H. Lian and W. Ge, “Solvability for second-order three-point boundary value problems on a half-line,”
Applied Mathematics Letters, vol. 19, no. 10, pp. 1000–1006, 2006.