POSITIVE SOLUTIONS FOR SECOND-ORDER
BOUNDARY VALUE PROBLEM WITH INTEGRAL
BOUNDARY CONDITIONS AT RESONANCE ON A
HALF-LINE
Positive Solutions for BVP at
Resonance on a Half-Line
Aijun Yang and Weigao Ge
vol. 10, iss. 1, art. 9, 2009
AIJUN YANG AND WEIGAO GE
Title Page
Department of Applied Mathematics
Beijing Institute of Technology
Beijing, 100081, P. R. China.
Contents
EMail: yangaij2004@163.com
gew@bit.edu.cn
Received:
03 February, 2009
Accepted:
25 February, 2009
Communicated by:
R.P. Agarwal
2000 AMS Sub. Class.:
34B10; 34B15; 34B45
Key words:
Boundary value problem; Resonance; Cone; Positive solution; Coincidence.
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Abstract:
This paper deals with the second order boundary value problem with integral boundary conditions on a half-line:
(p(t)x0 (t))0 + g(t)f (t, x(t)) = 0, a.e. in (0, ∞),
Z ∞
x(0) =
x(s)g(s)ds,
lim p(t)x0 (t) = p(0)x0 (0).
0
t→∞
A new result on the existence of positive solutions is obtained. The interesting points are: firstly, the boundary value problem involved in the
integral boundary condition on unbounded domains; secondly, we employ
a new tool – the recent Leggett-Williams norm-type theorem for coincidences and obtain positive solutions. Also, an example is constructed to
illustrate that our result here is valid.
Positive Solutions for BVP at
Resonance on a Half-Line
Aijun Yang and Weigao Ge
vol. 10, iss. 1, art. 9, 2009
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Contents
1
Introduction
4
2
Related Lemmas
6
3
Main Result
9
Positive Solutions for BVP at
Resonance on a Half-Line
Aijun Yang and Weigao Ge
vol. 10, iss. 1, art. 9, 2009
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1.
Introduction
In this paper, we study the existence of positive solutions to the following boundary
value problem at resonance:
(1.1)
(p(t)x0 (t))0 + g(t)f (t, x(t)) = 0,
a.e.in (0, ∞),
Positive Solutions for BVP at
Resonance on a Half-Line
Z
(1.2)
x(0) =
∞
x(s)g(s)ds,
0
lim p(t)x0 (t) = p(0)x0 (0),
t→∞
Aijun Yang and Weigao Ge
vol. 10, iss. 1, art. 9, 2009
R∞
where g ∈ L1 [0, ∞) withRg(t) > 0 on [0, ∞) and 0 g(s)ds = 1, p ∈ C[0, ∞) ∩
∞ 1
dt ≤ 1 and p(t) > 0 on [0, ∞).
C 1 (0, ∞), p1 ∈ L1 [0, ∞), 0 p(t)
Second-order boundary value problems (in short: BVPs) on infinite intervals,
arising from the study of radially symmetric solutions of nonlinear elliptic equations
and models of gas pressure in a semi-infinite porous medium [10], have received
much attention, to identify a few, we refer the readers to [9] – [11] and references
therein. For example, in [9], Lian and Ge studied the following second-order BVPs
on a half-line
(1.3)
(1.4)
x00 (t) = f (t, x(t), x0 (t)),
x(0) = x(η), lim x0 (t) = 0
0 < t < ∞,
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and
(1.5)
(1.6)
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x00 (t) = f (t, x(t), x0 (t)) + e(t),
x(0) = x(η), lim x0 (t) = 0,
0 < t < ∞,
t→∞
By using Mawhin’s continuity theorem, they obtained the existence results.
N. Kosmanov in [11] considered the second-order nonlinear differential equation
at resonance
(1.7)
(p(t)u0 (t))0 = f (t, u(t), u0 (t)),
a.e. in (0, ∞)
with two sets of boundary conditions:
(1.8)
u0 (0) = 0,
n
X
i=1
κi ui (Ti ) = lim u(t)
t→∞
Aijun Yang and Weigao Ge
vol. 10, iss. 1, art. 9, 2009
and
(1.9)
Positive Solutions for BVP at
Resonance on a Half-Line
u(0) = 0,
n
X
i=1
κi ui (Ti ) = lim u(t).
Title Page
t→∞
The author established existenceP
theorems by the coincidence degree theorem of
Mawhin under the condition that ni=1 κi = 1.
Although the existing literature on solutions of BVPs is quite wide, to the best
of our knowledge, only a few papers deal with the existence of positive solutions
to BVPs at resonance. In particular, there has been no work done for the boundary
value problems with integral boundary conditions on a half-line, such as the BVP
(1.1) – (1.2). Moreover, our main approach is different from the existing ones and
our main ingredient is the Leggett-Williams norm-type theorem for coincidences
obtained by O’Regan and Zima [4], which is a new tool used to study the existence
of positive solutions for nonlocal BVPs at resonance. An example is constructed to
illustrate that our result here is valid and almost sharp.
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2.
Related Lemmas
For the convenience of the reader, we review some standard facts on Fredholm operators and cones in Banach spaces. Let X, Y be real Banach spaces. Consider a linear
mapping L : dom L ⊂ X → Y and a nonlinear operator N : X → Y . Assume that
1◦ L is a Fredholm operator of index zero, i.e., Im L is closed and dim Ker L =
codim Im L < ∞.
The assumption 1◦ implies that there exist continuous projections P : X → X
and Q : Y → Y such that Im P = Ker L and Ker Q = Im L. Moreover, since
dim Im Q = codim Im L, there exists an isomorphism J : Im Q → Ker L. Denote
by Lp the restriction of L to Ker P ∩ dom L. Clearly, Lp is an isomorphism from
Ker P ∩ dom L to Im L, we denote its inverse by Kp : Im L → Ker P ∩ dom L. It
is known (see [3]) that the coincidence equation Lx = N x is equivalent to
Positive Solutions for BVP at
Resonance on a Half-Line
Aijun Yang and Weigao Ge
vol. 10, iss. 1, art. 9, 2009
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x = (P + JQN )x + KP (I − Q)N x.
Let C be a cone in X such that
(i) µx ∈ C for all x ∈ C and µ ≥ 0,
(ii) x, −x ∈ C implies x = θ.
It is well known that C induces a partial order in X by
xy
if and only if y − x ∈ C.
The following property is valid for every cone in a Banach space X.
Lemma 2.1 ([7]). Let C be a cone in X. Then for every u ∈ C \ {0} there exists a
positive number σ(u) such that
||x + u|| ≥ σ(u)||x|| for all
x ∈ C.
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Let γ : X → C be a retraction, that is, a continuous mapping such that γ(x) = x
for all x ∈ C. Set
Ψ := P + JQN + Kp (I − Q)N
and
Ψγ := Ψ ◦ γ.
In order to prove the existence result, we present here a definition.
Definition 2.2. f : [0, ∞) × R → R is called a g-Carathéodory function if
(A1) for each u ∈ R, the mapping t 7→ f (t, u) is Lebesgue measurable on [0, ∞),
(A2) for a.e. t ∈ [0, ∞), the mapping u 7→ f (t, u) is continuous on R,
(A3) Rfor each l > 0 and g ∈ L1 [0, ∞), there exists αl : [0, ∞) → [0, ∞) satisfying
∞
g(s)αl (s)ds < ∞ such that
0
|u| ≤ l
implies
|f (t, u)| ≤ αl (t) for a.e.
Positive Solutions for BVP at
Resonance on a Half-Line
Aijun Yang and Weigao Ge
vol. 10, iss. 1, art. 9, 2009
Title Page
Contents
t ∈ [0, ∞).
We make use of the following result due to O’Regan and Zima.
Theorem 2.3 ([4]). Let C be a cone in X and let Ω1 , Ω2 be open bounded subsets
of X with Ω1 ⊂ Ω2 and C ∩ (Ω2 \ Ω1 ) 6= ∅. Assume that 1◦ and the following
conditions hold.
2◦ N is L-compact, that is, QN : X → Y is continuous and bounded and Kp (I −
Q)N : X → X is compact on every bounded subset of X,
◦
3 Lx 6= λN x for all x ∈ C ∩ ∂Ω2 ∩ Im L and λ ∈ (0, 1),
4◦ γ maps subsets of Ω2 into bounded subsets of C,
5◦ degB {[I − (P + JQN )γ]|Ker L , Ker L ∩ Ω2 , 0} 6= 0, where degB denotes the
Brouwer degree,
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6◦ there exists u0 ∈ C \ {0} such that ||x|| ≤ σ(u0 )||Ψx|| for x ∈ C(u0 ) ∩ ∂Ω1 ,
where C(u0 ) = {x ∈ C : µu0 x for some µ > 0} and σ(u0 ) such that
||x + u0 || ≥ σ(u0 )||x|| for every x ∈ C,
7◦ (P + JQN )γ(∂Ω2 ) ⊂ C,
8◦ Ψγ (Ω2 \ Ω1 ) ⊂ C.
Then the equation Lx = N x has a solution in the set C ∩ (Ω2 \ Ω1 ).
For simplicity of notation, we set
Z ∞ Z
(2.1)
ω :=
0
0
Aijun Yang and Weigao Ge
s
1
dτ
p(τ )
vol. 10, iss. 1, art. 9, 2009
g(s)ds
and
G(t, s) =
Positive Solutions for BVP at
Resonance on a Half-Line
Title Page



























1
ω
1
ω
Rt
∞ 1
g(r)drdτ
s p(τ ) τ
i
R∞ 1 R∞
Rτ
− 0 p(τ ) τ g(r)dr 0 g(r)drdτ
Rt
Rτ
Rt
+1 + 0 p(τ1 ) 0 g(r)drdτ − s p(τ1 ) dτ,
1
dτ
0 p(τ )
Rt
hR
R∞
0 ≤ s < t < ∞,
hR
∞ 1 R∞
g(r)drdτ
s p(τ ) τ
i
Rτ
R∞ 1 R∞
− 0 p(τ ) τ g(r)dr 0 g(r)drdτ
Rt
Rτ
+1 + 0 p(τ1 ) 0 g(r)drdτ,
1
dτ
0 p(τ )
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Note that G(t, s) ≥ 0 for t, s ∈ [0, 1], and set






1
(2.2)
0 < κ ≤ min 1,
.

sup G(t, s) 


t,s∈[0,∞)
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0 ≤ t ≤ s < ∞.
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3.
Main Result
We work in the Banach spaces
n
o
(3.1)
X = x ∈ C[0, ∞) : lim x(t) exists
t→∞
and
(3.2)
∞
Z
Y = y : [0, ∞) → R :
g(t)|y(t)|dt < ∞
0
with the norms ||x||X = sup |x(t)| and ||y||Y =
Positive Solutions for BVP at
Resonance on a Half-Line
Aijun Yang and Weigao Ge
vol. 10, iss. 1, art. 9, 2009
R∞
t∈[0,∞)
0
g(t)|y(t)|dt, respectively.
Define the linear operator L : dom L ⊂ X → Y and the nonlinear operator
N : X → Y with
n
(3.3) dom L = x ∈ X : lim p(t)x0 (t) exists, x, px0 ∈ AC[0, ∞)
t→∞
Z ∞
0 0
1
and gx, (px ) ∈ L [0, ∞), x(0) =
x(s)g(s)ds
0
o
and lim p(t)x0 (t) = p(0)x0 (0)
t→∞
1
by Lx(t) = − g(t)
(p(t)x0 (t))0 and N x(t) = f (t, x(t)), t ∈ [0, ∞), respectively. Then
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Ker L = {x ∈ dom L : x(t) ≡ c
and
Z
Im L = y ∈ Y :
0
∞
on [0, ∞)}
g(s)y(s)ds = 0 .
Next, define the projections P : X → X by (P x)(t) =
Y → Y by
Z ∞
(Qy)(t) =
g(s)y(s)ds.
R∞
0
g(s)x(s)ds and Q :
0
Clearly, Im P = Ker L and Ker Q = Im L. So dim Ker L = 1 = dim Im Q =
codim Im L. Notice that Im L is closed, L is a Fredholm operator of index zero.
Note that the inverse Kp : Im L → dom L ∩ Ker P of Lp is given by
Z ∞
(Kp y)(t) =
k(t, s)g(s)y(s)ds,
Positive Solutions for BVP at
Resonance on a Half-Line
Aijun Yang and Weigao Ge
vol. 10, iss. 1, art. 9, 2009
0
where
(3.4)
1
dτ
p(τ )
R ∞R τ
1
dτ
0 p(τ )
R ∞R τ
 Rt
ω1 0
k(t, s) :=
1 R t
ω
s
s
1
drg(τ )dτ
s p(r)
1
drg(τ )dτ,
s p(r)
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−
Rt
1
dτ,
s p(τ )
0 ≤ s < t < ∞,
0 ≤ t ≤ s < ∞.
R∞
1
It is easy to see that |k(t, s)| ≤ 2 0 p(s)
ds.
In order to apply Theorem 2.3, we have to prove that N is L-compact, that is, QN
is continuous and bounded and Kp (I − Q)N is compact on every bounded subset of
X. Since the Arzelà-Ascoli theorem fails in the noncompact interval case, we will
use the following criterion.
n
o
Theorem 3.1 ([10]). Let M ⊂ x ∈ C[0, ∞) : lim x(t) exists . Then M is relat→∞
tively compact if the following conditions hold:
(B1) all functions from M are uniformly bounded,
(B2) all functions from M are equicontinuous on any compact interval of [0, ∞),
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(B3) all functions from M are equiconvergent at infinity, that is, for any given ε > 0,
there exists a T = T (ε) > 0 such that |f (t) − f (∞)| < ε for all t > T and
f ∈ M.
Lemma 3.2. If f : [0, ∞) × R → R is a g-Carathéodory function, then N is Lcompact.
Proof. Suppose that Ω ⊂ X is a bounded set. Then there exists l > 0 such that
function, there exists αl ∈
||x||X ≤ l for x ∈ Ω. Since f is a g-Carathéodory
R∞
1
L [0, ∞) satisfying αl (t) > 0, t ∈ (0, ∞) and 0 g(s)αl (s)ds < ∞ such that for
a.e. t ∈ [0, ∞), |f (t, x(t))| ≤ αl (t) for x ∈ Ω. Then for x ∈ Ω,
Z ∞
Z ∞
Z ∞
||QN x||Y =
g(t) g(s)f (s, x(s))ds dt ≤
g(s)αl (s)ds < ∞,
0
0
Positive Solutions for BVP at
Resonance on a Half-Line
Aijun Yang and Weigao Ge
vol. 10, iss. 1, art. 9, 2009
Title Page
0
which implies that QN is bounded on Ω.
Next, we show that Kp (I − Q)N is compact, i.e., Kp (I − Q)N maps bounded
sets into relatively compact ones. Furthermore, denote KP,Q = KP (I − Q)N (see
[9], [11]). For x ∈ Ω, one gets
Z ∞
Z ∞
k(t, s)g(s) f (s, x(s)) −
ds
|(KP,Q x)(t)| ≤
g(τ
)f
(τ,
x(τ
))dτ
0
0
Z ∞
Z ∞
1
≤2
dτ
g(s)|f (s, x(s))|ds
p(τ )
0
0
Z ∞
Z ∞
+
g(s)
g(τ )|f (τ, x(τ ))|dτ ds
0
0
Z ∞
Z ∞
1
dτ
g(s)αl (s)ds < ∞,
≤4
p(τ )
0
0
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that is, KP,Q (Ω) is uniformly bounded. Meanwhile, for any t1 , t2 ∈ [0, T ] with T a
positive constant,
|(KP,Q x)(t1 ) − (KP,Q x)(t2 )|
Z ∞Z ∞ Z τ
1
1
= drg(τ )dτ g(s) f (s, x(s))
ω 0
s
s p(r)
Z t1
Z ∞
1
−
g(τ )f (τ, x(τ ))dτ ds
dτ
0
t2 p(τ )
Z t1 Z t1
Z ∞
1
−
dτ g(s) f (s, x(s)) −
g(τ )f (τ, x(τ ))dτ ds
p(τ )
0
s
0
Z t2 Z t2
Z ∞
1
−
dτ g(s) f (s, x(s)) −
g(τ )f (τ, x(τ ))dτ ds p(τ )
0
s
0
Z ∞
Z s
Z τ
Z ∞
1
1
≤
g(s)
g(r)|f (r, x(r))|drdτ ds +
g(τ )|f (τ, x(τ ))|dτ
ω 0
0 p(τ ) 0
0
Z t1
Z τ
Z ∞
Z s
1
1
g(r)drdτ ds] · dτ ·
g(s)
p(τ )
0 p(τ ) 0
t
0Z t1
Z s
Z2 s
Z ∞
1
g(τ )|f (τ, x(τ ))|dτ +
g(τ )
+ g(r)|f (r, x(r))|drdτ ds
p(s) 0
0
0
t
Z ∞ 2
Z t1
Z ∞
Z ∞
1
dτ ≤
g(r)|f (r, x(r))|dr +
g(τ )|f (τ, x(τ ))|dτ ·
g(r)dr 0
0
0
t2 p(τ )
Z t1
Z ∞
1
+2
g(r)|f (r, x(r))|dr dτ 0
t2 p(τ )
Z t1
Z ∞
1
≤4
g(s)αl (s)ds · dτ → 0,
uniformly as |t1 − t2 | → 0,
0
t2 p(τ )
Positive Solutions for BVP at
Resonance on a Half-Line
Aijun Yang and Weigao Ge
vol. 10, iss. 1, art. 9, 2009
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which means that KP,Q (Ω) is equicontinuous. In addition, we claim that KP,Q (Ω) is
equiconvergent at infinity. In fact,
|(KP,Q x)(∞) − (KP,Q x)(t)|
Z ∞
Z Z Z
Z ∞
1 ∞ ∞ τ 1
1
≤
drg(τ )dτ g(s) |f (s, x(s))|+
g(τ )|f (τ, x(τ ))|dτ ds·
dτ
ω 0 s s p(r)
0
t p(τ )
Z s
Z ∞
Z s
Z ∞
1
+
ds
g(τ )|f (τ, x(τ )|dτ +
g(τ )
g(r)|f (r, x(r))|drdτ ds
p(s)
t
0
0
0
Z ∞
Z ∞
1
≤4
g(s)αl (s)ds ·
dτ → 0, uniformly as t → ∞.
p(τ )
0
t
Hence, Theorem 3.1 implies that Kp (I − Q)N (Ω) is relatively compact. Furthermore, since f satisfies g-Carathéodory conditions, the continuity of QN and Kp (I −
Q)N on Ω follows from the Lebesgue dominated convergence theorem. This completes the proof.
Now, we state our main result on the existence of positive solutions for the BVP
(1.1) – (1.2).
Theorem 3.3. Assume that
(H1) f : [0, ∞) × R → R is a g-Carathéodory function,
(H2) there exist positive constants b1 , b2 , b3 , c1 , c2 , B with
Z ∞
c2
b2 c 2 b3
1
(3.5)
B>
+2
+
ds
c1
b1 c 1 b1
p(s)
0
Positive Solutions for BVP at
Resonance on a Half-Line
Aijun Yang and Weigao Ge
vol. 10, iss. 1, art. 9, 2009
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such that
−κx ≤ f (t, x),
f (t, x) ≤ −c1 x + c2 ,
f (t, x) ≤ −b1 |f (t, x)| + b2 x + b3
for t ∈ [0, ∞), x ∈ [0, B],
(H3) there exist b ∈ (0, B), t0 ∈ [0, ∞), ρ ∈ (0, 1] and δ ∈ (0, 1). For each
t ∈ [0, ∞), f (t,x)
is non-increasing on x ∈ (0, b] with
xρ
Z ∞
f (s, b)
1−δ
(3.6)
G(t0 , s)g(s)
ds ≥
.
b
δρ
0
Positive Solutions for BVP at
Resonance on a Half-Line
Aijun Yang and Weigao Ge
vol. 10, iss. 1, art. 9, 2009
Title Page
Then the BVP (1.1) – (1.2) has at least one positive solution on [0, ∞).
Proof. Consider the cone
C = {x ∈ X : x(t) ≥ 0 on
[0, ∞)}.
Let
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Ω1 = {x ∈ X : δ||x||X < |x(t)| < b
on
[0, ∞)}
and
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Ω2 = {x ∈ X : ||x||X < B}.
Clearly, Ω1 and Ω2 are bounded and open sets and
Ω1 = {x ∈ X : δ||x||X ≤ |x(t)| ≤ b
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on
[0, ∞)} ⊂ Ω2
(see [4]). Moreover, C ∩ (Ω2 \ Ω1 ) 6= ∅. Let J = I and (γx)(t) = |x(t)| for x ∈ X.
Then γ is a retraction and maps subsets of Ω2 into bounded subsets of C, which
means that 4◦ holds.
In order to prove 3◦ , suppose that there exist x0 ∈ ∂Ω2 ∩ C ∩ dom L and λ0 ∈
(0, 1) such that Lx0 = λ0 N x0 , then (p(t)x00 (t))0 + λ0 g(t)f (t, x0 (t)) = 0 for all
t ∈ [0, ∞). In view of (H2), we have
−
1
1
(p(t)x00 (t))0 = f (t, x0 (t)) ≤ −b1
|(p(t)x00 (t))0 | + b2 x0 (t) + b3 .
λ0 g(t)
λ0 g(t)
Hence,
(3.7)
−(p(t)x00 (t))0 ≤ −b1 |(p(t)x00 (t))0 | + λ0 b2 g(t)x0 (t) + λ0 b3 g(t).
Integrating both sides of (3.7) from 0 to ∞, one gets
Z ∞
0=−
(p(t)x00 (t))0 dt
0
Z ∞
Z ∞
Z
0
0
≤ −b1
|(p(t)x0 (t)) |dt + λ0 b2
g(t)x0 (t)dt + λ0 b3
0
0
∞
|(p(t)x00 (t))0 |dt
(3.8)
0
b2
<
b1
Title Page
∞
0
Z
∞
g(t)x0 (t)dt +
0
b3
.
b1
Similarly, from (H2), we also obtain
Z ∞
c2
(3.9)
g(t)x0 (t)dt ≤ .
c1
0
∞
Z
x0 (t) =
∞
Z
k(t, s)(p(s)x00 (s))0 ds
g(t)x0 (t)dt +
Z0 ∞
(3.11)
g(t)dt,
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On the other hand,
(3.10)
Aijun Yang and Weigao Ge
vol. 10, iss. 1, art. 9, 2009
which gives
Z
Positive Solutions for BVP at
Resonance on a Half-Line
≤
Z0 ∞
g(t)x0 (t)dt +
0
0
|k(t, s)| · |(p(s)x00 (s))0 |ds.
Then, (3.8)-(3.9) yield
c2
B = ||x0 ||X ≤
+2
c1
b2 c 2 b3
+
b1 c 1 b1
Z
0
∞
1
ds,
p(s)
which contradicts (3.5).
To prove 5◦ , consider x ∈ Ker L ∩ Ω2 . Then x(t) ≡ c on [0, ∞). Let
Z ∞
H(c, λ) = c − λ|c| − λ
g(s)f (s, |c|)ds
0
for c ∈ [−B, B] and λ ∈ [0, 1]. It is easy to show that 0 = H(c, λ) implies c ≥ 0.
Suppose 0 = H(B, λ) for some λ ∈ (0, 1]. Then, (3.5) leads to
Z ∞
0 ≤ B(1 − λ) = λ
g(s)f (s, B)ds ≤ λ(−c1 B + c2 ) < 0,
0
which is a contradiction. In addition, if λ = 0, then B = 0, which is impossible.
Thus, H(x, λ) 6= 0 for x ∈ Ker L ∩ ∂Ω2 and λ ∈ [0, 1]. As a result,
degB {H(·, 1), Ker L ∩ Ω2 , 0} = degB {H(·, 0), Ker L ∩ Ω2 , 0}.
Positive Solutions for BVP at
Resonance on a Half-Line
Aijun Yang and Weigao Ge
vol. 10, iss. 1, art. 9, 2009
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However,
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degB {H(·, 0), Ker L ∩ Ω2 , 0} = degB {I, Ker L ∩ Ω2 , 0} = 1.
Then,
degB {[I − (P + JQN )γ]Ker L , Ker L ∩ Ω2 , 0} = degB {H(·, 1), Ker L ∩ Ω2 , 0} =
6 0.
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Next, we prove 8◦ . Let x ∈ Ω2 \ Ω1 and t ∈ [0, ∞),
Z ∞
Z ∞
(Ψγ x)(t) =
g(s)|x(s)|ds +
g(s)f (s, |x(s)|)ds
0
0
Z ∞
Z ∞
+
k(t, s)g(s) f (s, |x(s)|) −
g(τ )f (τ, |x(τ )|)dτ ds
0
0
Z ∞
Z ∞
=
g(s)|x(s)|ds +
G(t, s)g(s)f (s, |x(s)|)ds
0
0
Z ∞
≥
(1 − κG(t, s))g(s)|x(s)|ds ≥ 0.
Positive Solutions for BVP at
Resonance on a Half-Line
Aijun Yang and Weigao Ge
vol. 10, iss. 1, art. 9, 2009
0
Hence, Ψγ (Ω2 \ Ω1 ) ⊂ C, i.e. 8◦ holds.
Since for x ∈ ∂Ω2 ,
Z ∞
Z ∞
(P + JQN )γx =
g(s)|x(s)|ds +
g(s)f (s, |x(s)|)ds
0
Z0 ∞
≥
(1 − κ)g(s)|x(s)|ds ≥ 0,
0
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◦
then, (P + JQN )γx ⊂ C for x ∈ ∂Ω2 , and 7 holds.
It remains to verify 6◦ . Let u0 (t) ≡ 1 on [0, ∞). Then u0 ∈ C \ {0}, C(u0 ) =
{x ∈ C : x(t) > 0 on [0, ∞)} and we can take σ(u0 ) = 1. Let x ∈ C(u0 ) ∩ ∂Ω1 .
Then x(t) > 0 on [0, ∞), 0 < ||x||X ≤ b and x(t) ≥ δ||x||X on [0, ∞). For every
x ∈ C(u0 ) ∩ ∂Ω1 , by (H3)
Z ∞
Z ∞
(Ψx)(t0 ) =
g(s)x(s)ds +
G(t0 , s)g(s)f (s, x(s))ds
0
0
Z ∞
f (s, x(s)) ρ
≥ δ||x||X +
G(t0 , s)g(s)
x (s)ds
xρ (s)
0
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Z
∞
f (s, b)
G(t0 , s)g(s) ρ ds
b
0
1−ρ Z ∞
b
f (s, b)
= δ||x||X + δ ρ ||x||X
G(t0 , s)g(s)
ds
1−ρ
b
||x||X
0
Z ∞
f (s, b)
ρ
≥ δ||x||X + δ ||x||X
G(t0 , s)g(s)
ds ≥ ||x||X .
b
0
≥ δ||x||X + δ
ρ
||x||ρX
Thus, ||x||X ≤ σ(u0 )||Ψx||X for all x ∈ C(u0 ) ∩ ∂Ω1 .
In addition, 1◦ holds and Lemma 3.2 yields 2◦ . Then, by Theorem 2.3, the BVP
(1.1) – (1.2) has at least one positive solution x∗ on [0, ∞) with b ≤ ||x∗ ||X ≤ B.
This completes the proof of Theorem 3.3.
Remark 1. Note that with the projection P (x) = x(0), Conditions 7◦ and 8◦ of
Theorem 2.3 are no longer satisfied.
To illustrate how our main result can be used in practice, we present here an
example.
Example 3.1. Consider the following BVP

 2(et x0 (t))0 + e−t f (t, x(t)) = 0, a.e. in (0, ∞),
(3.12)
R
 x0 (0) = lim et x0 (t), x(0) = ∞ e−s x(s)ds.
0
Positive Solutions for BVP at
Resonance on a Half-Line
Aijun Yang and Weigao Ge
vol. 10, iss. 1, art. 9, 2009
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t→∞
Corresponding to the BVP (1.1) – (1.2), p(t) = 2et , g(t) = e−t and f (t, x) =
(t − 12 )e−2t x + e−t x2 . We can get ω = 14 and
(3.13) G(t, s)
( 13 1 −t
+ 6 (e − 3e−s ) + 14 (e−2t + 2e−2s ) − 12 e−(t+2s) , 0 ≤ s ≤ t < ∞,
12
=
13
− 13 e−t + 14 (e−2t + 2e−2s ) − 21 e−(t+2s) ,
0 ≤ t ≤ s < ∞.
12
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Obviously, G(t, s) ≥ 0 for t, s ∈ [0, +∞). Choose κ = 12 , B = 5, c1 = 25 ,
3
3
c2 = 12 e− 2 , b1 = 12 , b2 = 32 and b3 = 32 e− 2 such that (H2) holds, and take b = 54 ,
t0 = 0, ρ = 1 and δ = 94 such that (H3) is satisfied. Then thanks to Theorem 3.3, the
BVP (3.12) has a positive solution on [0, ∞).
Positive Solutions for BVP at
Resonance on a Half-Line
Aijun Yang and Weigao Ge
vol. 10, iss. 1, art. 9, 2009
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References
[1] K. DEIMLING, Nonlinear Functional Analysis, New York, 1985.
[2] D. GUO AND V. LAKSHMIKANTHAM, Nonlinear Problems in Abstract
Cones, New York, 1988.
[3] J. MAWHIN, Topological degree methods in nonlinear boundary value problems, in: NSFCBMS Regional Conference Series in Mathematics, American
Mathematical Society, Providence, RI, 1979.
[4] D. O’REGAN AND M. ZIMA, Leggett-Williams norm-type theorems for coincidences, Arch. Math., 87 (2006), 233–244.
[5] W. GE, Boundary value problems for ordinary nonlinear differential equations,
Science Press, Beijing, 2007 (in Chinese).
[6] G. INFANTE AND M. ZIMA, Positive solutions of multi-point boundary value
problems at resonance, Nonlinear Anal., 69 (2008), 2458–2465.
[7] W.V. PETRYSHYN, On the solvability of x ∈ T x + λF x in quasinormal cones
with T and F k-set contractive, Nonlinear Anal., 5 (1981), 585–591.
[8] R.E. GAINES AND J. SANTANILLA, A coincidence theorem in convex sets
with applications to periodic solutions of ordinary differential equations, Rocky
Mountain. J. Math., 12 (1982), 669–678.
[9] H. LIAN, H. PANG AND W. GE, Solvability for second-order three-point
boundary value problems at resonance on a half-line, J. Math. Anal. Appl., 337
(2008), 1171–1181.
[10] R.P. AGARWAL AND D. O’REGAN, Infinite Interval Problems for Differential, Difference and Integral Equations, Kluwer Academic, 2001.
[11] N. KOSMATOV, Multi-point boundary value problems on an unbounded domain at resonance, Nonlinear Anal., 68 (2008), 2158–2171.
Positive Solutions for BVP at
Resonance on a Half-Line
Aijun Yang and Weigao Ge
vol. 10, iss. 1, art. 9, 2009
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