# Nonlinear Sum Operator Equations with a Parameter and Application to Second-Order Three-Point BVPs

Authors: Wen-Xia Wang; Xi-Lan Liu; Piao-Piao Shi
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

A class of nonlinear sum operator equations with a parameter on order Banach spaces were considered. The existence and uniqueness of positive solutions for this kind of operator equations and the dependence of solutions on the parameter have been obtained by using the properties of cone and nonlinear analysis methods. The critical value of the parameter was estimated. Further, the application to some nonlinear three-point boundary value problems was given to show the significance of the discussion.


---

## Body

## 1. Introduction and Preliminaries

The aim of this paper is to investigate the existence and uniqueness of positive solution for the following operator equations:(1)
L
(
λ
,
x
)
=
x
,
where L
(
λ
,
x
)
=
A
x
+
λ
B
x
,
A is an operator with concavity, B is a pseudo subhomogeneous operator, and λ is a parameter. In addition, by applying our results to the second-order three-point boundary value problem (BVP),
(2)
-
x
′′
(
t
)
=
λ
f
(
t
,
x
(
t
)
)
+
g
(
t
)
,
t
∈
(
0,1
)
,
x
(
0
)
=
τ
x
′
(
0
)
,
x
(
1
)
-
σ
x
(
ξ
)
=
∫
0
1
‍
h
(
x
(
s
)
)
d
s
,
where f
∈
C
(
[
0,1
]
×
R
)
,
g
∈
C
[
0,1
]
,
h
∈
C
(
R
)
,
τ
≥
0
,
0
<
ξ
<
1, and 0
<
σ
≤
1; we obtain that there exists a λ
*
>
0, such that BVP(2) has a unique positive solution and no positive solution for 0
<
λ
<
λ
* and λ
≥
λ
*, respectively. In particular, such a positive solution x
λ
(
t
) of BVP(2) is increasing and continuous in λ for λ
∈
[
0
,
λ
*
) and lim
⁡
λ
→
λ
*
-
0
∥
x
λ
∥
c
=
+
∞. Further, we estimate the critical value λ
*.In recent years, many authors focus on multipoint boundary value problems for differential equations, since these problems arise in a variety of different areas of applied mathematics and physics (see [1]). For example, by using degree-theoretic arguments, Gupta [2] obtained the existence and uniqueness theorems for the following three-point boundary value problem:
(3)
x
′′
(
t
)
=
f
(
t
,
x
(
t
)
,
x
′
(
t
)
)
-
e
(
t
)
,
t
∈
(
0,1
)
,
x
(
0
)
=
0
,
x
(
ξ
)
=
x
(
1
)
.
Since then, the existence of solutions for nonlinear multipoint boundary value problems has been studied by many authors (see [3–8] and their references). The cases with special boundary value conditions in BVP(2) were discussed by [2–7], where τ
=
0 and h
(
x
)
=
0 or τ
=
0
,
σ
=
1 and h
(
x
)
=
0. However, to the best of our knowledge, little has been done for the multipoint boundary value problem (2) with parameter, perturbed loading force, and nonlinear boundary conditions, especially on the existence and uniqueness of positive solution and the dependence of solutions on the parameter λ.It is well known that fixed point theory is an effective tool in the treatment of existence results of boundary value problems for nonlinear differential equations. Many researchers were concerned with the existence and uniqueness of fixed point for concave operators and sum operators. For example, [9–15] investigated eigenvalue problems of concave operators, the existence and uniqueness of positive fixed point for concave operators, and the existence and uniqueness of positive fixed point for sum operator, respectively. However, to our knowledge, few of the results in literature can be applied to BVP(2) successfully. The above reasons stimulate us to do this work.First, we consider the existence and property of positive solutions for nonlinear operator equations (1) on order Banach space E.To be clear, some definitions, notations, and lemmas are presented as follows.LetE be a real Banach space which is partially ordered by a cone P of E, that is, x
≤
y if and only if y
-
x
∈
P, and let θ be the zero element of E. If x
≤
y and x
≠
y, then we denote x
<
y or y
>
x. A nonempty closed convex set P
⊂
E is a cone if it satisfies (i) x
∈
P
,
r
≥
0
⇒
r
x
∈
P; (ii) x
∈
P
,
-
x
∈
P
⇒
x
=
θ. A cone P is said to be normal if there exists a positive number N, called the normal constant of P, such that, for all x
,
y
∈
E
,
θ
≤
x
≤
y implies ∥
x
∥
≤
N
∥
y
∥.LetD
⊆
E. An operator T
:
D
→
E is said to be increasing if, for x
1
,
x
2
∈
D
,
x
1
≤
x
2
⇒
T
x
1
≤
T
x
2. An element x
*
∈
D is called a fixed point of T if T
x
*
=
x
*. For details on cone theory, see [16, 17].Givene
>
θ, let
(4)
P
e
¯
=
{
x
∈
E
∣
there
exist
l
=
l
(
x
)
>
0
such
that
0
≤
x
≤
l
e
}
,
(5)
P
e
=
{
x
∈
E
∣
there
exist
l
1
=
l
1
(
x
)
>
0
,
l
2
=
l
2
(
x
)
>
0
such
that
l
1
e
≤
x
≤
l
2
e
}
.
Then P
e
⊂
P
e
¯
⊂
P and
(6)
∀
x
,
y
∈
P
e
,
∃
0
<
μ
0
<
1
<
l
0
<
+
∞
such
that
μ
0
y
≤
x
≤
l
0
y
.Lemma 1.
LetP be a normal cone in E and T
:
P
→
P an increasing operator. Suppose that(H1)
T
(
P
e
)
⊂
P
e;
(H2)
for anyr
∈
(
0,1
) and [
y
,
z
]
⊂
P
e there exists η
(
r
,
y
,
z
)
>
0 such that
(7)
T
(
r
x
)
≥
r
(
1
+
η
(
r
,
y
,
z
)
)
T
x
,
∀
x
∈
[
y
,
z
]
,
r
∈
(
0,1
)
.Then T has a unique fixed point x
* in P
e if and only if there exist u
,
v
∈
P
e such that u
≤
T
u
≤
T
v
≤
v. Moreover, for any initial value u
0
∈
P
e and a sequence u
n
=
T
u
n
-
1
(
n
=
1,2
,
…
), one has lim
⁡
n
→
+
∞
∥
u
n
-
x
*
∥
=
0.The proof of Lemma1 is standard; we omit it here.Definition 2 (see [14]).
An operatorT
:
P
→
P is said to be generalized α-concave if it satisfies (H1) and the following condition:(H3)
there exists anα
:
(
0,1
)
→
(
0,1
) such that
(8)
T
(
r
x
)
≥
r
α
(
r
)
T
x
,
∀
x
∈
P
e
,
r
∈
(
0,1
)
.
From Lemma1 it is easy to show that the following lemma holds.Lemma 3 (see [15]).
LetP be a normal cone of E and T
:
P
→
P increasing generalized α-concave. Then T has a unique fixed point x
* in P
e. Moreover, for any initial value u
0
∈
P
e and a sequence u
n
=
T
u
n
-
1
(
n
=
1,2
,
…
), one has lim
⁡
n
→
+
∞
∥
u
n
-
x
*
∥
=
0.In what follows, we introduce definitions of pseudo subhomogeneous operator and pseudo generalizedα-concave operator.Definition 4.
An operatorT
:
P
→
P is said to be pseudo subhomogeneous if it satisfies(H4)
T
(
P
e
)
⊂
P
e
¯;
(H5)
T
(
r
x
)
≥
r
T
x
,
∀
x
∈
P
e
,
r
∈
(
0,1
).Definition 5.
An operatorT
:
P
→
P is said to be pseudo generalized α-concave if it satisfies (H3) and (H4).Remark 6.
An increasing pseudo subhomogeneous operator and an increasing pseudo generalizedα-concave operator may have no fixed point in P
e. For example, Let E
=
C
[
0,1
]
,
P
=
{
x
∈
E
∣
x
(
t
)
≥
0
,
t
∈
[
0,1
]
}
,
e
(
t
)
≡
1, and T
1
x
(
t
)
=
(
1
-
t
)
x
(
t
)
/
(
1
+
x
(
t
)
). Obviously, T
1
:
P
→
P is increasing pseudo subhomogeneous, but T
1 has no fixed point in P
e. Let
(9)
T
2
x
(
t
)
=
∫
0
1
‍
t
(
1
-
s
)
x
α
(
s
)
d
s
,
x
∈
P
,
0
<
α
<
1. Clearly, T
2
:
P
→
P is increasing pseudo generalized α-concave. Since T
2
x
(
0
)
=
0 for x
∈
P
e, then T
2
x
∉
P
e for all x
∈
P
e. Therefore, T
2 is not generalized α-concave and T
2 has no fixed point in P
e.Remark 7.
From Definitions4 and 5, it is clear that a pseudo generalized α-concave operator is a pseudo subhomogeneous operator.Remark 8.
It is easy to show that (8) is equivalent to
(10)
T
(
s
x
)
≤
s
α
(
1
/
s
)
T
x
,
x
∈
P
e
,
s
>
1
,
and (H5) is equivalent to
(11)
T
(
s
x
)
≤
s
T
x
,
x
∈
P
e
,
s
>
1
.


## 2. Positive Solutions of Operator Equation

In this section, we assume thatE is a real Banach space, P is a normal cone in E with the normal constant N, e
>
θ, and A
,
B
:
P
→
P are increasing operators.Theorem 9.
Assume thatA is a generalized α-concave operator and B is a pseudo subhomogeneous operator. Then the following four results are true.(i)
There exists aλ
*
>
0 such that (1) has a unique solution x
λ in P
e for λ
∈
[
0
,
λ
*
). For any initial value u
0
∈
P
e, set u
n
=
L
(
λ
,
u
n
-
1
)
(
n
=
1,2
,
…
); then lim
⁡
n
→
+
∞
∥
u
n
-
x
λ
∥
=
0.
(ii)
Equation (1) has no solution in P
e for λ
≥
λ
*.
(iii)
x
λ is increasing in λ for λ
∈
[
0
,
λ
*
).
(iv)
x
λ is continuous with respect to λ for λ
∈
[
0
,
λ
*
).Proof.
By Lemma3, A has a unique fixed point x
0
∈
P
e and
(12)
L
(
λ
,
x
0
)
=
A
x
0
+
λ
B
x
0
≥
A
x
0
=
x
0
,
λ
≥
0
.
IfB
|
P
e
≡
θ
,
L
(
λ
,
·
)
|
P
e
=
A for any λ
≥
0. Set λ
*
=
+
∞; it is obvious that conclusions (i)–(iv) hold.
IfB
|
P
e
≢
θ, there exists x
-
∈
P
e such that B
x
-
>
θ. By (6), for any x
∈
P
e, there exists μ
0
∈
(
0,1
) such that μ
0
x
-
≤
x. Moreover, Definition 4 shows that there exists l
0
∈
(
1
,
+
∞
) such that
(13)
θ
<
μ
0
B
x
-
≤
B
x
≤
l
0
x
0
,
x
∈
P
e
.
The facts that A
(
P
e
)
⊂
P
e and B
(
P
e
)
⊂
P
e
¯ imply that
(14)
L
(
λ
,
P
e
)
⊂
P
e
,
λ
≥
0
.
Next, we prove all statements by five steps.
Step  1. Existence of the critical value λ
*. Set
(15)
Λ
=
{
λ
≥
0
∣
there
exists
y
λ
∈
P
e
such
that
x
0
≤
y
λ
,
L
(
λ
,
y
λ
)
≤
y
λ
}
,
and λ
*
=
sup
⁡
Λ. Now we show that Λ
=
[
0
,
λ
*
).
Define a mappingρ
:
P
e
→
[
0
,
+
∞
) by
(16)
ρ
(
x
)
=
inf
⁡
{
τ
>
0
∣
B
x
≤
τ
x
0
}
,
x
∈
P
e
.
By (13) it is obvious that 0
<
ρ
(
x
)
<
+
∞. In addition, for any x
1
,
x
2
∈
P
e, x
1
≤
x
2, we have B
x
1
≤
B
x
2
≤
ρ
(
x
2
)
x
0, which implies that ρ
(
x
1
)
≤
ρ
(
x
2
). That is, ρ
(
x
) is increasing in x for x
∈
P
e.
For a givens
0
>
1, set y
0
=
s
0
x
0. Then y
0
∈
P
e and x
0
≤
y
0. It follows from Remark 8 that
(17)
L
(
λ
,
y
0
)
≤
s
0
(
s
0
α
(
1
/
s
0
)
-
1
+
λ
ρ
(
x
0
)
)
x
0
≤
y
0
,
λ
∈
[
0
,
1
ρ
(
x
0
)
(
1
-
s
0
α
(
1
/
s
0
)
-
1
)
]
.
This means that
(18)
[
0
,
1
ρ
(
x
0
)
(
1
-
s
0
α
(
1
/
s
0
)
-
1
)
]
⊂
Λ
,
λ
*
≥
1
ρ
(
x
0
)
(
1
-
s
0
α
(
1
/
s
0
)
-
1
)
>
0
.
We assert that λ
*
∉
Λ. If λ
*
=
+
∞, from (15) it is obvious that λ
*
∉
Λ. Suppose that λ
*
<
+
∞ and λ
*
∈
Λ. Then again by (15) there exists y
λ
*
∈
P
e with x
0
≤
y
λ
* such that L
(
λ
*
,
y
λ
*
)
≤
y
λ
*. Set v
0
=
s
1
y
λ
* for a given s
1
>
1. Then v
0
∈
P
e and x
0
≤
v
0. Note that (
1
/
ρ
(
y
λ
*
)
)
(
1
-
s
1
α
(
1
/
s
1
)
-
1
)
>
0; we can take a number δ
>
0 sufficiently small such that
(19)
δ
<
1
ρ
(
y
λ
*
)
(
1
-
s
1
α
(
1
/
s
1
)
-
1
)
.
From (16)–(19) and the fact that x
0
=
A
x
0
≤
A
y
λ
*, we obtain
(20)
L
(
λ
*
+
δ
,
v
0
)
=
A
(
s
1
y
λ
*
)
+
(
λ
*
+
δ
)
B
(
s
1
y
λ
*
)
≤
s
1
(
A
y
λ
*
+
λ
*
B
y
λ
*
)
+
(
s
1
α
(
1
/
s
1
)
-
s
1
)
A
y
λ
*
+
δ
s
1
B
y
λ
*
≤
s
1
L
(
λ
*
,
y
λ
*
)
-
(
s
1
-
s
1
α
(
1
/
s
1
)
)
A
x
0
+
δ
s
1
ρ
(
y
λ
*
)
x
0
≤
s
1
y
λ
*
+
(
δ
s
1
ρ
(
y
λ
*
)
-
(
s
1
-
s
1
α
(
1
/
s
1
)
)
)
x
0
≤
s
1
y
λ
*
=
v
0
,
which means that λ
*
+
δ
∈
Λ. This contradicts the definition of λ
*. Therefore, we conclude that
(21)
Λ
=
[
0
,
λ
*
)
.
Step  2. Conclusion (i) holds. For given λ
∈
[
0
,
λ
*
), consider (1). Since A and B are increasing, L
(
λ
,
x
) is increasing in x. Moreover, combining (12) and (15) gives
(22)
x
0
≤
L
(
λ
,
x
0
)
≤
L
(
λ
,
y
λ
)
≤
y
λ
.
Besides, by (H2) and (H4) we obtain
(23)
L
(
λ
,
r
x
)
≥
r
α
(
r
)
A
x
+
λ
r
B
x
≥
r
L
(
λ
,
x
)
+
(
r
α
(
r
)
-
r
)
A
x
≥
r
(
1
+
r
α
(
r
)
-
r
r
φ
(
x
,
λ
)
)
L
(
λ
,
x
)
,
x
∈
P
e
,
r
∈
(
0,1
)
,
where
(24)
φ
(
λ
,
x
)
=
sup
⁡
{
τ
>
0
∣
A
x
≥
τ
L
(
λ
,
x
)
}
,
x
∈
P
e
.
For any u
,
v
∈
P
e with u
≤
v, (H1) implies that A
u
,
A
v
∈
P
e. Further, there exists 0
<
ϵ
≤
1 such that ϵ
A
v
≤
A
u. Hence,
(25)
A
x
≥
A
u
≥
ϵ
A
v
≥
ϵ
φ
(
λ
,
v
)
L
(
λ
,
v
)
≥
ϵ
φ
(
λ
,
v
)
L
(
λ
,
x
)
,
x
∈
[
u
,
v
]
.
Evidently,
(26)
φ
(
λ
,
x
)
≥
ϵ
φ
(
λ
,
v
)
>
0
,
∀
x
∈
[
u
,
v
]
.
Therefore, let η
(
r
,
u
,
v
)
=
(
ϵ
(
r
α
(
r
)
-
r
)
/
r
)
φ
(
λ
,
v
); then it follows from (23) and (26) that
(27)
L
(
λ
,
r
x
)
≥
r
(
1
+
η
(
r
,
u
,
v
)
)
L
(
λ
,
x
)
,
x
∈
[
u
,
v
]
,
r
∈
(
0,1
)
.
The application of Lemma 1 concludes the proof of (i).
Step  3. Conclusion (ii) holds. Suppose that there exists λ
′
≥
λ
* such that L
(
λ
′
,
x
)
=
x has a solution x
λ
′ in P
e. Since x
0
,
x
λ
′
∈
P
e, by (6) there exists s
′
>
1 such that x
0
≤
s
′
x
λ
′. Set y
λ
′
=
s
′
x
λ
′. Then
(28)
L
(
λ
′
,
y
λ
′
)
=
A
(
s
′
x
λ
′
)
+
λ
′
B
(
s
′
x
λ
′
)
≤
s
′
(
(
s
′
)
α
(
1
/
s
′
)
-
1
A
x
λ
′
+
λ
′
B
x
λ
′
)
≤
s
′
x
λ
′
=
y
λ
′
,
which means that λ
′
∈
Λ. Equation (21) implies that λ
′
<
λ
*, which is a contradiction to the hypothesis λ
′
≥
λ
*.
Step  4. Conclusion (iii) holds. Let λ
1
,
λ
2
∈
[
0
,
λ
*
) with λ
1
≤
λ
2. Then L
(
λ
1
,
x
λ
2
)
≤
L
(
λ
2
,
x
λ
2
)
=
x
λ
2, and, further, x
0
≤
L
(
λ
1
,
x
0
)
≤
L
(
λ
1
,
x
λ
2
)
≤
x
λ
2. By the proof of conclusion (i), x
=
L
(
λ
1
,
x
) has a unique solution x
λ
1
*
∈
[
x
0
,
x
λ
2
] in P
e, which implies that x
λ
1
=
x
λ
1
*. Thus, x
λ
1
≤
x
λ
2.
Step  5. Conclusion (iv) holds. Let λ
0
∈
(
0
,
λ
*
). By conclusion (iii) we have
(29)
x
λ
≤
x
λ
0
,
∀
0
<
λ
<
λ
0
.
Let
(30)
τ
λ
=
sup
⁡
{
τ
>
0
x
λ
≥
τ
x
λ
0
}
,
0
<
λ
<
λ
0
.
Then τ
λ is nondecreasing in λ for λ
∈
[
0
,
λ
0
), and
(31)
0
<
τ
λ
≤
1
,
x
λ
≥
τ
λ
x
λ
0
,
0
<
λ
<
λ
0
.
We assert that
(32)
lim
⁡
λ
→
λ
0
-
τ
λ
=
1
,if, otherwise, there exists a sequence {
λ
n
}
(33)
0
<
λ
1
≤
λ
2
≤
⋯
≤
λ
n
≤
⋯
≤
λ
0
,
lim
⁡
n
→
+
∞
λ
n
=
λ
0
such that
(34)
lim
⁡
n
→
+
∞
τ
λ
n
=
τ
0
,
τ
λ
n
≤
τ
0
<
1
.
By (23), (24), and (30), we obtain
(35)
x
λ
n
=
L
(
λ
n
,
x
λ
n
)
≥
L
(
λ
n
,
τ
λ
n
x
λ
0
)
=
L
(
λ
n
,
τ
λ
n
τ
0
τ
0
x
λ
0
)
≥
τ
λ
n
τ
0
L
(
λ
n
,
τ
0
x
λ
0
)
≥
λ
n
λ
0
τ
λ
n
τ
0
L
(
λ
0
,
τ
0
x
λ
0
)
≥
λ
n
λ
0
τ
λ
n
(
1
+
τ
0
α
(
τ
0
)
-
τ
0
τ
0
φ
(
x
λ
0
,
λ
0
)
)
x
λ
0
.
Therefore,
(36)
τ
λ
n
≥
λ
n
λ
0
τ
λ
n
(
1
+
τ
0
α
(
τ
0
)
-
τ
0
τ
0
φ
(
x
λ
0
,
λ
0
)
)
.
That is,
(37)
λ
n
λ
0
(
1
+
τ
0
α
(
τ
0
)
-
τ
0
τ
0
φ
(
x
λ
0
,
λ
0
)
)
≤
1
.
Taking the limit n
→
+
∞, we get
(38)
1
+
τ
0
α
(
τ
0
)
-
τ
0
τ
0
φ
(
x
λ
0
,
λ
0
)
≤
1
,
which is a contradiction. So, (32) holds.
From (29), (31), and (32), we have
(39)
∥
x
λ
0
-
x
λ
∥
≤
N
(
1
-
τ
λ
)
∥
x
λ
0
∥
,
0
<
λ
<
λ
0
,
which implies that ∥
x
λ
0
-
x
λ
∥
→
0
as
λ
→
λ
0
-.
A similar argument shows that, for anyλ
0
∈
[
0
,
λ
*
)
,
∥
x
λ
-
x
λ
0
∥
→
0 as λ
→
λ
0
+. Thus, conclusion (iv) holds. The proof of Theorem 9 is complete.Noting (18) and (21), we can easily obtain the following result.Theorem 10.
Assume that the hypotheses in Theorem9 hold. If lim
⁡
r
→
0
+
r
1
-
α
(
r
)
=
0, then λ
* in Theorem 9 satisfies
(40)
λ
*
≥
1
ρ
(
x
0
)
,
where ρ
(
x
0
) are defined by (16).Corollary 11.
Assume thatA is a generalized α
1-concave operator and B is a pseudo generalized α
2-concave operator. Then(i)
(1) has a unique solution x
λ
∈
P
e for λ
∈
[
0
,
+
∞
). Moreover, for any u
0
∈
P
e, set u
n
=
L
(
λ
,
u
n
-
1
)
(
n
=
1,2
,
…
); then lim
⁡
n
→
+
∞
∥
u
n
-
x
λ
∥
=
0;
(ii)
x
λ is increasing in λ for λ
∈
[
0
,
+
∞
);
(iii)
x
λ is continuous with respect to λ for λ
∈
[
0
,
+
∞
);
(iv)
eitherx
λ
=
x
0 (the unique fixed point of A in P
e) or lim
⁡
λ
→
+
∞
∥
x
λ
∥
=
+
∞.Proof.
From Definitions2 and 4, it is clear that L
(
λ
,
P
e
)
⊂
P
e. Since A and B are increasing, L
(
λ
,
x
) is increasing in x for any λ
≥
0. Let α
(
r
)
=
max
⁡
{
α
1
(
r
)
,
α
2
(
r
)
}; then
(41)
L
(
λ
,
r
x
)
=
A
(
r
x
)
+
λ
B
(
r
x
)
≥
r
α
1
(
r
)
A
x
+
λ
r
α
2
(
r
)
B
x
≥
r
α
(
r
)
L
(
λ
,
x
)
,
x
∈
P
e
,
r
∈
(
0,1
)
.
Thus, conclusion (i) follows from Lemma 3. Similar to the proofs of Theorem 9, the proofs of (ii) and (iii) can be completed.
Note that(42)
x
λ
=
A
x
λ
=
x
0
,
if
B
|
P
e
≡
θ
,
x
λ
=
A
x
λ
+
λ
B
x
λ
≥
λ
B
x
λ
≥
λ
B
x
0
,
if
B
|
P
e
≢
θ
,
and, therefore, normality of P implies that conclusion (iv) holds. This ends the proof.Theorem 12.
Assume thatA is a pseudo generalized α
1-concave operator and B is a generalized α
2-concave operator. Then,(i)
(1) has a unique solution x
λ
∈
P
e for λ
∈
(
0
,
+
∞
). Moreover, for any u
0
∈
P
e, set u
n
=
L
(
λ
,
u
n
-
1
)
(
n
=
1,2
,
…
); then lim
⁡
n
→
+
∞
∥
u
n
-
x
λ
∥
=
0;
(ii)
x
λ is increasing with respect to λ for λ
∈
(
0
,
+
∞
);
(iii)
x
λ is continuous with respect to λ for λ
∈
(
0
,
+
∞
);
(iv)
lim
⁡
λ
→
0
∥
x
λ
-
A
x
λ
∥
=
0
;
lim
⁡
λ
→
+
∞
∥
x
λ
∥
=
+
∞.Proof.
Similar to the proof of Corollary11, the proofs of (i), (ii), and (iii) can be completed. To prove (iv), let x
1
∈
P
e be the unique solution of (1) with λ
=
1. From conclusion (ii) of this theorem, we obtain
(43)
θ
<
x
λ
-
A
x
λ
=
λ
B
x
λ
≤
λ
B
x
1
,
0
<
λ
<
1
,
x
λ
=
A
x
λ
+
λ
B
x
λ
≥
λ
B
x
λ
≥
λ
B
x
1
>
θ
,
λ
>
1
.
Therefore,
(44)
∥
x
λ
-
A
x
λ
∥
≤
N
λ
∥
B
x
1
∥
,
0
<
λ
<
1
,
∥
x
λ
∥
≥
λ
N
∥
B
x
1
∥
,
λ
>
1
,
which implies that lim
⁡
λ
→
0
∥
x
λ
-
A
x
λ
∥
=
0 and lim
⁡
λ
→
+
∞
∥
x
λ
∥
=
+
∞. This ends the proof.Next, we discuss the case of (1) with L
(
λ
,
x
)
=
x
0
+
λ
B
x
,
x
0
∈
E; that is,
(45)
x
0
+
λ
B
x
=
x
which can be widely applied to various problems for differential equations.Theorem 13.
Assume thatx
0
∈
P
e and B is a pseudo subhomogeneous operator. Then(i)
there exists aλ
*
≥
(
1
/
ρ
0
)
>
0 such that (45) has a unique solution x
λ
∈
P
e for λ
∈
[
0
,
λ
*
). Moreover, for any u
0
∈
P
e and a sequence u
n
=
x
0
+
λ
B
u
n
-
1
(
n
=
1,2
,
…
), one has lim
⁡
n
→
+
∞
∥
u
n
-
x
λ
∥
=
0;
(ii)
(45) has no solution in P
e for λ
≥
λ
*;
(iii)
x
λ is increasing in λ for λ
∈
[
0
,
λ
*
);
(iv)
x
λ is continuous with respect to λ for λ
∈
[
0
,
λ
*
); moreover, lim
⁡
λ
→
0
∥
x
λ
-
x
0
∥
=
0;
(v)
further, ifB
x
∈
P
e
¯ for x
≥
x
0 and B is completely continuous, then lim
⁡
λ
→
λ
*
-
0
∥
x
λ
∥
=
+
∞.
Hereρ
0
=
inf
⁡
{
τ
>
0
∣
B
x
0
≤
τ
x
0
}.Proof.
Define an operatorA by A
x
=
x
0 for x
∈
P. Then A
:
P
→
P
e is increasing and x
0 is the unique fixed point of A in P
e.
Conclusions (i)–(iv) can be proved similarly to the proof of Theorem9. We only prove conclusion (v) by considering the following two cases.
Case  1  (
λ
*
=
+
∞
). Note that x
λ
=
x
0
+
λ
B
x
λ
≥
λ
B
x
λ
≥
λ
B
x
0 for λ
∈
[
0
,
+
∞
); it is clear that lim
⁡
λ
→
λ
*
-
0
∥
x
λ
∥
=
+
∞.
Case  2  (
λ
*
<
+
∞
). In this case, suppose, to the contrary, that lim
⁡
λ
→
λ
*
-
0
∥
x
λ
∥
≠
+

∞. Then, there exists a nondecreasing sequence {
λ
n
}
⊂
[
0
,
λ
*
) and a constant M
>
0 such that lim
⁡
n
→
+
∞
λ
n
=
λ
* and ∥
x
λ
n
∥
≤
M.
SinceB is completely continuous, there exist x
*
∈
P and subsequence {
λ
n
i
}
⊂
{
λ
n
} such that lim
⁡
i
→
+
∞
B
x
λ
n
i
=
x
*. Taking the limitation i
→
+
∞ to both sides of
(46)
x
λ
n
i
=
x
0
+
λ
n
i
B
x
λ
n
i
,
we have lim
⁡
i
→
+
∞
x
λ
n
i
=
x
0
+
λ
*
x
*. Equation (46) implies that
(47)
x
0
+
λ
*
x
*
=
x
0
+
λ
*
B
(
x
0
+
λ
*
x
*
)
.
The relation x
0
+
λ
*
x
*
>
x
0 shows that B
(
x
0
+
λ
*
x
*
)
∈
P
e
¯. Further, x
0
+
λ
*
B
(
x
0
+
λ
*
x
*
)
∈
P
e. This means that x
0
+
λ
*
x
*
∈
P
e. Therefore, (47) gives rise to the contradiction λ
*
∈
[
0
,
λ
*
). This finishes the proof.Corollary 14.
Assume thatx
0
∈
P
e and B is pseudo generalized α-concave. Then,(i)
(45) has a unique solution x
λ
∈
P
e for λ
∈
[
0
,
+
∞
). Moreover, for any u
0
∈
P
e, set u
n
=
x
0
+
λ
B
u
n
-
1
(
n
=
1,2
,
…
); then lim
⁡
n
→
+
∞
∥
u
n
-
x
λ
∥
=
0;
(ii)
x
λ is increasing with respect to λ for λ
∈
[
0
,
+
∞
);
(iii)
x
λ is continuous with respect to λ for λ
∈
[
0
,
+
∞
);
(iv)
eitherx
λ
=
x
0
,
λ
≥
0 or lim
⁡
λ
→
+
∞
∥
x
λ
∥
=
+
∞.Proof.
It is obvious thatL
(
λ
,
x
)
=
x
0
+
λ
B
x is increasing in x and L
(
λ
,
P
e
)
⊂
P
e for any λ
≥
0. In addition,
(48)
L
(
λ
,
r
x
)
=
x
0
+
λ
B
(
r
x
)
≥
x
0
+
λ
r
α
(
r
)
B
x
≥
r
α
(
r
)
L
(
λ
,
x
)
,
x
∈
P
e
,
r
∈
(
0,1
)
.
This means that L
(
λ
,
·
) is generalized α-concave. Thus, conclusion (i) follows from Lemma 3. Similarly to the proof of Corollary 11, the proofs of (ii), (iii), and (iv) can be completed.From the proofs of Theorem12, Theorem 13, and Corollary 14, we easily prove the following results.Theorem 15.
Assume thatx
0
∈
P
e
¯ and B is a generalized α-concave operator. Then,(i)
(45) has a unique solution x
λ
∈
P
e for λ
∈
(
0
,
+
∞
). Moreover, for any u
0
∈
P
e, set u
n
=
x
0
+
λ
B
u
n
-
1
(
n
=
1,2
,
…
); then lim
⁡
n
→
+
∞
∥
u
n
-
x
λ
∥
=
0;
(ii)
x
λ is increasing in λ and x
λ
≥
x
0 for λ
∈
(
0
,
+
∞
);
(iii)
x
λ is continuous with respect to λ for λ
∈
(
0
,
+
∞
);
(iv)
lim
⁡
λ
→
0
∥
x
λ
-
x
0
∥
=
0 and lim
⁡
λ
→
+
∞
∥
x
λ
∥
=
+
∞.Remark 16.
Different from Theorem9 and Corollary 11, even if (1) satisfies the conditions in Theorem 12, (1) may not have a fixed point in P
e when λ
=
0.


## 3. Three-Point Nonlinear Boundary Value Problem

In this section, based on the discussion of the previous section, we study the existence and uniqueness of positive solutions for the three-point BVP(2) and the dependence of solutions on the parameter λ.In what follows, setE
=
C
[
0,1
], the Banach space of continuous functions on [
0,1
] with the norm ∥
x
∥
=
max
⁡
t
∈
[
0,1
]
|
x
(
t
)
|. Consider that P
=
{
x
∈
C
[
0,1
]
|
x
(
t
)
≥
0
,
t
∈
[
0,1
]
}. It is clear that P is a normal cone with the normality constant 1; P
e is given as in (5) with e
(
t
)
=
τ
+
t. Let
(49)
G
(
t
,
s
)
=
1
1
+
τ
-
σ
(
ξ
+
τ
)
×
{
(
1
-
σ
ξ
-
(
1
-
σ
)
t
)
(
τ
+
s
)
,
0
≤
s
≤
min
⁡
{
ξ
,
t
}
,
(
1
-
t
)
(
τ
+
s
)
+
σ
(
ξ
+
τ
)
(
t
-
s
)
,
ξ
≤
s
≤
t
≤
1
,
(
τ
+
t
)
(
1
-
s
-
σ
(
ξ
-
s
)
)
,
0
≤
t
≤
s
≤
ξ
,
(
τ
+
t
)
(
1
-
s
)
,
max
⁡
{
ξ
,
t
}
≤
s
≤
1
.
It is easy to prove that
(50)
0
≤
G
(
t
,
s
)
≤
1
+
σ
1
+
τ
-
σ
(
ξ
+
τ
)
(
τ
+
t
)
(
1
-
s
)
,
t
,
s
∈
[
0,1
]
.
Define two operators A
:
P
→
C
[
0,1
] and B
:
P
→
C
[
0,1
] by
(51)
A
x
(
t
)
=
τ
+
t
1
+
τ
-
σ
(
ξ
+
τ
)
∫
0
1
‍
h
(
x
(
s
)
)
d
s
+
∫
0
1
‍
G
(
t
,
s
)
g
(
s
)
d
s
.
(52)
B
x
(
t
)
=
∫
0
1
‍
G
(
t
,
s
)
f
(
s
,
x
(
s
)
)
d
s
.
It is clear that a positive solution of BVP(2) is equivalent to nontrivial solution of (1) in P.The following lemma can be proved easily by the Ascoli-Arzela theorem.Lemma 17.
B
:
P
→
C
[
0,1
] is completely continuous.The following hypotheses are needed in this section.(L1)
f
:
[
0,1
]
×
[
0
,
+
∞
)
→
[
0
,
+
∞
) is continuous.
(L2)
f
(
t
,
x
) is increasing in x
∈
[
0
,
+
∞
) for fixed t
∈
[
0,1
] and
(53)
f
(
t
,
r
x
)
≥
r
f
(
t
,
x
)
,
∀
t
∈
[
0,1
]
,
r
∈
(
0,1
)
,
x
∈
[
0
,
+
∞
)
.
(L3)
h
:
[
0
,
+
∞
)
→
[
0
,
+
∞
) is continuous and h
(
x
)
≢
0.
(L4)
h
(
x
) is increasing in x
∈
[
0
,
+
∞
) and there exists a function α
(
r
)
∈
(
0,1
) such that
(54)
h
(
r
x
)
≥
r
α
(
r
)
h
(
x
)
,
∀
r
∈
(
0,1
)
,
x
∈
(
0
,
+
∞
)
.
(L5)
g
:
[
0,1
]
→
[
0
,
+
∞
) is continuous.
(L6)
∫
0
1
‍
(
τ
+
s
)
(
1
-
s
)
g
(
s
)
d
s
>
0.Lemma 18.
Suppose that (L1) and (L2) hold. ThenB
:
P
→
P is increasing and satisfies(i)
B
(
P
∖
{
θ
}
)
⊂
P
e
¯;
(ii)
B
(
r
x
)
≥
r
B
x
,
x
∈
P
e
,
r
∈
(
0,1
).Proof.
From (50), (52), and (L1), it is clear that B
(
P
)
⊂
P. By (L2) we obtain that B is an increasing operator and, for any x
∈
P
∖
{
θ
},
(55)
B
x
(
t
)
=
∫
0
1
‍
G
(
t
,
s
)
f
(
s
,
x
(
s
)
)
d
s
≤
(
1
+
σ
)
∫
0
1
‍
(
1
-
s
)
f
(
s
,
x
(
s
)
)
d
s
1
+
τ
-
σ
(
ξ
+
τ
)
(
τ
+
t
)
≤
(
1
+
σ
)
∫
0
1
‍
(
1
-
s
)
f
(
s
,
∥
x
∥
)
d
s
1
+
τ
-
σ
(
ξ
+
τ
)
e
(
t
)
≤
l
(
x
)
e
(
t
)
,
where l
(
x
)
≥
max
⁡
{
(
(
1
+
σ
)
∫
0
1
‍
(
1
-
s
)
f
(
s
,
∥
x
∥
)
d
s
)
/
(
1
+
τ
-
σ
(
ξ
+
τ
)
)
,
1
}, and
(56)
B
(
r
x
)
(
t
)
=
∫
0
1
‍
G
(
t
,
s
)
f
(
s
,
r
x
(
s
)
)
d
s
≥
r
∫
0
1
‍
G
(
t
,
s
)
f
(
s
,
x
(
s
)
)
d
s
=
r
B
x
(
t
)
,
r
∈
(
0,1
)
.
The proof is complete.Theorem 19.
Suppose that (L1)–(L5) hold. Then there existsλ
*
>
0 such that BVP(2) has a unique positive solution x
λ
(
t
) in P
e for λ
∈
[
0
,
λ
*
) and has no solution in P
e for λ
≥
λ
*. Furthermore such a solution x
λ
(
t
) satisfies the following properties:(i)
for anyu
0
∈
P
e, set
(57)
u
n
(
t
)
=
τ
+
t
1
+
τ
-
σ
(
ξ
+
τ
)
∫
0
1
‍
h
(
u
n
-
1
(
s
)
)
d
s
+
∫
0
1
‍
G
(
t
,
s
)
g
(
s
)
d
s
+
λ
∫
0
1
‍
G
(
t
,
s
)
f
(
s
,
u
n
-
1
(
s
)
)
d
s
,
and then u
n
(
t
) uniformly converges to x
λ
(
t
) on [
0,1
];
(ii)
x
λ
(
t
) is nondecreasing in λ for λ
∈
[
0
,
λ
*
);
(iii)
x
λ
(
t
) is continuous with respect to λ for λ
∈
[
0
,
λ
*
).Proof.
ConsiderA and B defined by (51) and (52). By (L3)–(L5), A
:
P
→
P is increasing. For any x
∈
P
e, from (50) we have
(58)
∫
0
1
‍
h
(
x
(
s
)
)
d
s
1
+
τ
-
σ
(
ξ
+
τ
)
(
τ
+
t
)
≤
A
x
(
t
)
≤
∫
0
1
‍
h
(
x
(
s
)
)
d
s
+
(
1
+
σ
)
∫
0
1
‍
(
1
-
s
)
g
(
s
)
d
s
1
+
τ
-
σ
(
ξ
+
τ
)
(
τ
+
t
)
,
which means that A
(
P
e
)
⊂
P
e. In addition, for any r
∈
(
0,1
) and x
∈
P
e, by (L4) we obtain
(59)
A
(
r
x
)
(
t
)
=
τ
+
t
1
+
τ
-
σ
(
ξ
+
τ
)
×
∫
0
1
‍
h
(
r
x
(
s
)
)
d
s
+
∫
0
1
‍
G
(
t
,
s
)
g
(
s
)
d
s
≥
r
α
(
r
)
(
τ
+
t
1
+
τ
-
σ
(
ξ
+
τ
)
∫
0
1
‍
h
(
x
(
s
)
)
d
s
+
r
-
α
(
r
)
∫
0
1
‍
G
(
t
,
s
)
g
(
s
)
d
s
)
≥
r
α
(
r
)
A
x
(
t
)
.
So, the operator A is generalized α-concave. Lemma 18 implies that B is pseudo subhomogeneous. The conclusion follows from Theorem 9. The proof is complete.In the following, we consider three special cases of BVP(2).Case 1.
BVP(2) has no perturbation; that is, g
(
t
)
≡
0.
From the proof of Theorem19, we have the following result.Theorem 20.
Suppose that (L1)–(L4) hold. Then there existsλ
*
>
0 such that
(60)
-
x
′′
(
t
)
=
λ
f
(
t
,
x
(
t
)
)
,
t
∈
(
0,1
)
,
x
(
0
)
=
τ
x
′
(
0
)
,
x
(
1
)
-
σ
x
(
ξ
)
=
∫
0
1
‍
h
(
x
(
s
)
)
d
shave a unique positive solution x
λ
(
t
)
∈
P
e for λ
∈
[
0
,
λ
*
) and no solution in P
e for λ
≥
λ
*. Furthermore such a solution x
λ
(
t
) satisfies the following properties:(i)
for anyu
0
∈
P
e, set
(61)
u
n
(
t
)
=
τ
+
t
1
+
τ
-
σ
(
ξ
+
τ
)
∫
0
1
‍
h
(
u
n
-
1
(
s
)
)
d
s
+
λ
∫
0
1
‍
G
(
t
,
s
)
f
(
s
,
u
n
-
1
(
s
)
)
d
s
,
and then u
n
(
t
) uniformly converges to x
λ
(
t
) on [
0,1
];
(ii)
x
λ
(
t
) is nondecreasing in λ for λ
∈
[
0
,
λ
*
);
(iii)
x
λ
(
t
) is continuous with respect to λ for λ
∈
[
0
,
λ
*
).Case 2.
The nonlinear boundary value control functionh
(
x
) in BVP(2) reduces to the linear one h
0
(
>
0
).Theorem 21.
Suppose that (L1), (L2), and (L5) hold. Then, there existsλ
*
>
0 such that
(62)
-
x
′′
(
t
)
=
λ
f
(
t
,
x
(
t
)
)
+
g
(
t
)
,
t
∈
(
0,1
)
,
x
(
0
)
=
τ
x
′
(
0
)
,
x
(
1
)
-
σ
x
(
ξ
)
=
h
0have a unique positive solution x
λ
(
t
)
∈
P
e for λ
∈
[
0
,
λ
*
) and no solution in P
e for λ
≥
λ
*; furthermore such a solution x
λ
(
t
) satisfies the following properties:(i)
for anyu
0
∈
P
e, set
(63)
u
n
(
t
)
=
τ
+
t
1
+
τ
-
σ
(
ξ
+
τ
)
h
0
+
∫
0
1
‍
G
(
t
,
s
)
g
(
s
)
d
s
+
λ
∫
0
1
‍
G
(
t
,
s
)
f
(
s
,
u
n
-
1
(
s
)
)
d
s
,
and then u
n
(
t
) uniformly converges to x
λ
(
t
) on [
0,1
];
(ii)
x
λ
(
t
) is nondecreasing in λ for λ
∈
[
0
,
λ
*
);
(iii)
x
λ
(
t
) is continuous with respect to λ for λ
∈
[
0
,
λ
*
);
(iv)
lim
⁡
λ
→
λ
*
-
0
∥
x
λ
∥
=
+
∞.Proof.
Define operatorB as (52) and let
(64)
x
0
(
t
)
=
τ
+
t
1
+
τ
-
σ
(
ξ
+
τ
)
h
0
+
∫
0
1
‍
G
(
t
,
s
)
g
(
s
)
d
s
.
Then x
0
∈
P
e. By Lemmas 17 and 18, B
:
P
→
P is completely continuous and B is pseudo subhomogeneous, and B
x
∈
P
e
¯ for x
≥
x
0. The application of Theorem 13 completes the proof.Case 3.
The nonlinear boundary value control functionh
(
x
) in BVP(2) vanishes; that is, h
(
x
)
≡
0 for t
∈
[
0,1
].Theorem 22.
Suppose that (L1), (L2), (L5), and (L6) hold. Then there existsλ
*
>
0 such that
(65)
-
x
′′
(
t
)
=
λ
f
(
t
,
x
(
t
)
)
+
g
(
t
)
,
t
∈
(
0,1
)
,
x
(
0
)
=
τ
x
′
(
0
)
,
x
(
1
)
-
σ
x
(
ξ
)
=
0have a unique positive solution x
λ
(
t
)
∈
P
e for λ
∈
[
0
,
λ
*
) and no solution in P
e for λ
≥
λ
*; furthermore such a solution x
λ
(
t
) satisfies the following properties:(i)
for anyu
0
∈
P
e, set
(66)
u
n
(
t
)
=
∫
0
1
‍
G
(
t
,
s
)
g
(
s
)
d
s
+
λ
∫
0
1
‍
G
(
t
,
s
)
f
(
s
,
u
n
-
1
(
s
)
)
d
s
,
and then u
n
(
t
) uniformly converges to x
λ
(
t
) on [
0,1
];
(ii)
x
λ
(
t
) is nondecreasing in λ for λ
∈
[
0
,
λ
*
);
(iii)
x
λ
(
t
) is continuous with respect to λ for λ
∈
[
0
,
λ
*
);
(iv)
lim
⁡
λ
→
λ
*
-
0
∥
x
λ
∥
=
+
∞.Proof.
Define operatorB as (52) and let
(67)
x
0
(
t
)
=
∫
0
1
‍
G
(
t
,
s
)
g
(
s
)
d
s
.
Note the monotonicity of G
(
t
,
s
) in t; it follows that
(68)
G
(
t
,
s
)
≥
σ
(
1
-
ξ
)
(
τ
+
ξ
)
(
1
+
τ
-
σ
(
ξ
+
τ
)
)
(
τ
+
1
)
2
×
(
τ
+
s
)
(
1
-
s
)
(
τ
+
t
)
,
t
,
s
∈
[
0,1
]
,
which together with (50), (L5), and (L6) leads to x
0
∈
P
e. By Lemma 18, B
:
P
→
P is pseudo subhomogeneous and B
x
∈
P
e
¯ for x
≥
x
0. The application of Theorem 13 finishes the proof.



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*Source: 259016-2014-03-16.xml*