# Delta-Nabla Type Maximum Principles for Second-Order Dynamic Equations on Time Scales and Applications

Authors: Jiang Zhu; Dongmei Liu
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

Some delta-nabla type maximum principles for second-order dynamic equations on time scales are proved. By using these maximum principles, the uniqueness theorems of the solutions, the approximation theorems of the solutions, the existence theorem, and construction techniques of the lower and upper solutions for second-order linear and nonlinear initial value problems and boundary value problems on time scales are proved, the oscillation of second-order mixed delat-nabla differential equations is discussed and, some maximum principles for second order mixed forward and backward difference dynamic system are proved.


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## Body

## 1. Introduction

Maximum principles are a well known tool for studying differential equations, which can be used to receive prior information about solutions of differential inequalities and to obtain lower and upper solutions of differential equations and so on. Maximum principles include continuous maximum principles and discrete maximum principles. It is well known that there are many results and applications for continuous and discrete maximum principles. For example, about these theories and applications, we can refer to [1–15] and the references therein. On the other hand, Hilger [16] established the theory of time scales calculus to unify the continuous and discrete calculus in 1990. After that, ordinary dynamic equations and partial dynamic equations on time scales have been extensively studied by some authors. For example, about these, we can refer to [17–23] and the references therein. However, the study on the maximum principles on time scales is very little, about these, we can refer to Stehik and Thompson’s recent works [24, 25].Inspired by the above works, we will be devoted to study delta-nabla type maximum principles for second-order dynamic equations on one-dimensional time scales and the applications of these maximum principles.This paper is organized as follows. In Section2, we state and prove some basic notations and results on time scales. In Section 3, we will first prove some delta-nabla type maximum principles for second-order dynamic equations on time scales; then, by using these maximum principles, we get some maximum principles for second-order mixed forward and backward difference dynamic system and discuss the oscillation of second-order mixed delta-nabla differential equations. In Section 4, we apply the maximum principles proved in Section 3 to obtain uniqueness of the solutions, the approximating techniques of the solutions, the existence theorem, and construction techniques of the lower and upper solutions for second-order linear initial value problems. In Section 5, we apply the maximum principles proved in Section 3 to obtain uniqueness of the solutions, the approximating techniques of the solutions, the existence theorem, and construction techniques of the lower and upper solutions for second-order linear boundary value problems. Finally, in Section 6, we extended the results of linear operator established in Sections 4 and 5 to nonlinear operators.


## 2. Preliminaries

Definition 1 (see [22]).
A time scale𝕋 is a nonempty closed subset of the real numbers. Throughout this paper, 𝕋 denotes a time scale.Definition 2 (see [22]).
Let𝕋 be a time scale. For t
∈
𝕋 one defines the forward jump operator σ
:
𝕋
→
𝕋 by σ
(
t
)
:
=
inf
⁡
⁡
{
s
∈
𝕋
:
s
>
t
}, while the backward jump operator ρ
:
𝕋
→
𝕋 is defined by ρ
(
t
)
:
=
sup
⁡
⁡
{
s
∈
𝕋
:
s
<
t
}. If σ
(
t
)
>
t, one says that t is right-scattered, while if ρ
(
t
)
<
t, we say that t is left-scattered. Points that are right-scattered and left-scattered at the same time are called isolated. Also, if t
<
sup
⁡
𝕋 and σ
(
t
)
=
t, then t is called right-dense, and if t
>
inf
⁡
𝕋 and ρ
(
t
)
=
t, then t is called left-dense. Finally, the graininess function μ
(
t
)
:
𝕋
→
[
0
,
∞
) is defined by
(1)
μ
(
t
)
∶
=
σ
(
t
)
-
t
.Definition 3 (see [22]).
If𝕋 has a left-scattered maximum M, then one defines 𝕋
k
=
𝕋
-
{
M
}; otherwise 𝕋
k
=
𝕋. Assume f
:
𝕋
→
R is a function and let t
∈
𝕋
k. Then one defines f
Δ
(
t
) to be the number (provided it exists) with the property that, given any ɛ
>
0, there is a neighborhood U of t (i.e., U
=
(
t
-
δ
,
t
+
δ
)
∩
𝕋 for some δ
>
0) such that
(2)
|
[
f
(
σ
(
t
)
)
-
f
(
s
)
]
-
f
Δ
(
t
)
[
σ
(
t
)
-
s
]
|
≤
ɛ
|
σ
(
t
)
-
s
|
∀
s
∈
U
.
We call f
Δ
(
t
) thedelta derivative of f at t.Definition 4 (see [22]).
If𝕋 has a right-scattered minimum m, then one defines 𝕋
k
=
𝕋
-
{
m
}; otherwise 𝕋
k
=
𝕋. The backwards graininess function υ
(
t
)
:
𝕋
→
[
0
,
∞
) is defined by
(3)
ν
(
t
)
∶
=
t
-
ρ
(
t
)
.
Assume f
:
𝕋
→
R is a function and let t
∈
𝕋
k. Then we define f
∇
(
t
) to be the number (provided it exists) with the property that, given any ɛ
>
0, there is a neighborhood U of t (i.e., U
=
(
t
-
δ
,
t
+
δ
)
∩
𝕋 for some δ
>
0) such that
(4)
|
[
f
(
ρ
(
t
)
)
-
f
(
s
)
]
-
f
∇
(
t
)
[
ρ
(
t
)
-
s
]
|
≤
ɛ
|
ρ
(
t
)
-
s
|
∀
s
∈
U
.
We call f
∇
(
t
) thenabla derivative of f at t. Define the second derivative by u
Δ
∇
=
(
u
Δ
)
∇.Definition 5 (see [21]).
Letf
:
𝕋
→
ℝ. Define and denote f
∈
C
rd
⁡
(
𝕋
;
ℝ
) as right-dense continuous if for each t
∈
𝕋
(5)
lim
⁡
s
→
t
+
f
(
s
)
=
f
(
t
)
,
if
t
∈
𝕋
is
right-dense
,
lim
⁡
s
→
t
-
f
(
s
)
exists
and
is
finite
if
t
∈
𝕋
is
left-dense
.Definition 6 (see[21]).
Letf
:
𝕋
→
ℝ. Define and denote f
∈
C
ld
⁡
(
𝕋
;
ℝ
) as left-dense continuous if for each t
∈
𝕋
(6)
lim
⁡
s
→
t
-
f
(
s
)
=
f
(
t
)
,
if
t
∈
𝕋
is
left-dense
,
lim
⁡
s
→
t
+
f
(
s
)
exists
and
is
finite
if
t
∈
𝕋
is
right-dense
.Theorem 7 (see [21]).
Assume thatf
:
f
:
𝕋
→
ℝ and let t
∈
𝕋
k.(i)
Iff is Δ
-differentiable at t then f is continuous at t.
(ii)
Iff is continuous at t and t is right-scattered then f is Δ
-differentiable at t with
(7)
f
Δ
(
t
)
=
f
(
σ
(
t
)
)
-
f
(
t
)
σ
(
t
)
-
t
.
(iii)
Ift is right-dense, then f is differentiable at t if and only if the limit
(8)
lim
⁡
s
→
t
f
(
t
)
-
f
(
s
)
t
-
s
exists as a finite number. In this case(9)
f
Δ
(
t
)
=
lim
⁡
s
→
t
f
(
t
)
-
f
(
s
)
t
-
s
.
(iv)
Iff is Δ
-differentiable at t, then
(10)
f
(
σ
(
t
)
)
=
f
(
t
)
+
μ
(
t
)
f
Δ
(
t
)
.Theorem 8 (see [22]).
Assume thatf
:
f
:
𝕋
→
ℝ and let t
∈
𝕋
k.(i)
Iff is nabla differentiable at t then f is continuous at t.
(ii)
Iff is continuous at t and t is left-scattered then f is nabla differentiable at t with
(11)
f
∇
(
t
)
=
f
(
ρ
(
t
)
)
-
f
(
t
)
ρ
(
t
)
-
t
.
(iii)
Ift is left-dense, then f is nabla differentiable at t if and only if the limit
(12)
lim
⁡
s
→
t
f
(
t
)
-
f
(
s
)
t
-
s
exists as a finite number. In this case(13)
f
∇
(
t
)
=
lim
⁡
s
→
t
f
(
t
)
-
f
(
s
)
t
-
s
.
(iv)
Iff is nabla differentiable at t, then
(14)
f
(
ρ
(
t
)
)
=
f
(
t
)
-
ν
(
t
)
f
∇
(
t
)
.Theorem 9 (see [22]).
Iff
:
𝕋
→
ℝ is Δ
-differentiable and f
Δ is right-dense continuous on 𝕋
k, then f is ∇
-differentiable, and
(15)
f
∇
(
t
)
=
{
f
Δ
(
ρ
(
t
)
)
t
∈
𝕋
A
lim
⁡
s
→
t
-
f
Δ
(
s
)
t
∈
A
.
If g
:
𝕋
→
ℝ is ∇
-differentiable and g
∇ is left-dense continuous on 𝕋
k, then f is Δ
-differentiable, and
(16)
g
Δ
(
t
)
{
g
∇
(
σ
(
t
)
)
t
∈
𝕋
B
lim
⁡
s
→
t
+
g
∇
(
s
)
t
∈
B
,
where
(17)
A
:
=
{
t
∈
𝕋
:
t
i
s
l
e
f
t
-
d
e
n
s
e
a
n
d
r
i
g
h
t
-
s
c
a
t
t
e
r
e
d
}
,
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
l
l
𝕋
A
:
=
𝕋
∖
A
,
B
≔
{
t
∈
𝕋
:
t
i
s
l
e
f
t
-
s
c
a
t
t
e
r
e
d
a
n
d
r
i
g
h
t
-
d
e
n
s
e
}
,
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
l
l
𝕋
B
:
=
𝕋
∖
B
.Corollary 10 (see [22]).
Iff
:
𝕋
→
ℝ is Δ
-differentiable and f
Δ is continuous on 𝕋
k, g
:
𝕋
→
ℝ is ∇
-differentiable, and g
∇ is continuous on 𝕋
k, then
(18)
f
∇
(
t
)
=
f
Δ
(
ρ
(
t
)
)
f
o
r
t
∈
𝕋
k
,
g
Δ
(
t
)
=
g
∇
(
σ
(
t
)
)
f
o
r
t
∈
𝕋
k
.Theorem 11 (see [21]).
Assumef
,
g
:
𝕋
→
ℝ are differentiable at t
∈
𝕋
k. Then
(
i
) the sum f
+
g
:
𝕋
→
ℝ is differentiable at t with
(19)
(
f
+
g
)
Δ
(
t
)
=
f
Δ
(
t
)
+
g
Δ
(
t
)
;
(
i
i
) for any constant α
,
α
f: 𝕋
→
ℝ is differentiable at t with
(20)
(
α
f
)
Δ
(
t
)
=
α
f
Δ
(
t
)
;
(
i
i
i
) the product f
g
:
𝕋
→
ℝ is differentiable at t with
(21)
(
f
g
)
Δ
(
t
)
=
f
Δ
(
t
)
g
(
t
)
+
f
(
σ
(
t
)
)
g
Δ
(
t
)
=
f
(
t
)
g
Δ
(
t
)
+
f
Δ
(
t
)
g
(
σ
(
t
)
)
;
(
i
v
) if g
(
t
)
g
(
σ
(
t
)
)
≠
0, then f
/
g is differentiable at t with
(22)
(
f
g
)
Δ
(
t
)
=
f
Δ
(
t
)
g
(
t
)
-
f
(
t
)
g
Δ
(
t
)
g
(
t
)
g
(
σ
(
t
)
)
.Theorem 12 (see [22]).
Assumef
,
g
:
𝕋
→
ℝ are nabla differentiable at t
∈
𝕋
k. Then
(
i
) the sum f
+
g: 𝕋
→
ℝ is nabla differentiable at t with
(23)
(
f
+
g
)
∇
(
t
)
=
f
∇
(
t
)
+
g
∇
(
t
)
;
(
i
i
) for any constant α
,
α
f: 𝕋
→
ℝ is nabla differentiable at t with
(24)
(
α
f
)
∇
(
t
)
=
α
f
∇
(
t
)
;
(
i
i
i
)   the product f
g
:
𝕋
→
ℝ is nabla differentiable at t with
(25)
(
f
g
)
∇
(
t
)
=
f
∇
(
t
)
g
(
t
)
+
f
(
ρ
(
t
)
)
g
∇
(
t
)
=
f
(
t
)
g
∇
(
t
)
+
f
∇
(
t
)
g
(
ρ
(
t
)
)
;
(
i
v
) if g
(
t
)
g
(
ρ
(
t
)
)
≠
0, then f
/
g is nabla differentiable at t with
(26)
(
f
g
)
∇
(
t
)
=
f
∇
(
t
)
g
(
t
)
-
f
(
t
)
g
∇
(
t
)
g
(
t
)
g
(
ρ
(
t
)
)
.Theorem 13 (see [22]).
Iff, f
Δ, and f
∇ are continuous, then(
i
)
[
∫
a
t
‍
f
(
t
,
s
)
Δ
s
]
Δ
=
∫
a
t
‍
f
Δ
(
t
,
s
)
Δ
s
+
f
(
σ
(
t
)
,
t
);
(
i
i
)
[
∫
a
t
‍
f
(
t
,
s
)
Δ
s
]
∇
=
∫
a
t
‍
f
∇
(
t
,
s
)
Δ
s
+
f
(
ρ
(
t
)
,
ρ
(
t
)
);
(
i
i
i
)
[
∫
a
t
‍
f
(
t
,
s
)
∇
s
]
Δ
=
∫
a
t
‍
f
Δ
(
t
,
s
)
∇
s
+
f
(
σ
(
t
)
,
σ
(
t
)
);
(
i
v
)
[
∫
a
t
‍
f
(
t
,
s
)
∇
s
]
∇
=
∫
a
t
‍
f
∇
(
t
,
s
)
∇
s
+
f
(
ρ
(
t
)
,
t
).Definition 14 (see [21]).
One says that a functionp
:
𝕋
→
ℝ is regressive provided 1
+
μ
(
t
)
p
(
t
)
≠
0 for all t
∈
𝕋
k holds. The set of all regressive and rd-continuous functions f
:
𝕋
→
ℝ will be denoted by ℜ
=
ℜ
(
𝕋
)
=
ℜ
(
𝕋
,
ℝ
).Definition 15 (see [21]).
One definesξ
h
(
z
)
=
(
1
/
h
)
log
⁡
(
1
+
z
h
)
(
ξ
h
:
ℂ
h
→
ℤ
h
), where h
>
0. If p
∈
ℜ, then one defines the exponential function by
(27)
e
p
(
t
,
s
)
=
exp
⁡
(
∫
s
t
‍
ξ
μ
(
τ
)
(
p
(
τ
)
)
Δ
τ
)
for
s
,
t
∈
𝕋
.
If p
∈
ℜ, then the first-order linear dynamic equation
(28)
y
Δ
=
p
(
t
)
y
is called regressive.Theorem 16 (see [21]).
Suppose (28) is regressive and fix t
0
∈
𝕋. Then e
p
(
·
,
t
0
) is a solution of the initial value problem
(29)
y
Δ
=
p
(
t
)
y
,
y
(
t
0
)
=
1
on 𝕋.Theorem 17 (see [21]).
Suppose (28) is regressive; then the only solution of (29) is given by e
p
(
·
,
t
0
).Theorem 18 (see [21]).
Ifp
∈
ℜ, then(
i
)
e
0
(
t
,
s
)
=
1 and e
p
(
t
,
t
)
=
1;
(
i
i
)
e
p
(
σ
(
t
)
,
s
)
=
(
1
+
μ
(
t
)
p
(
t
)
)
e
p
(
t
,
s
);
(
i
i
i
)
e
p
(
t
,
s
)
=
1
/
e
p
(
s
,
t
);
(
i
v
)
(
1
/
e
p
(
·
,
s
)
)
Δ
=
-
p
(
t
)
/
(
e
p
σ
(
·
,
s
).According to the above theorems and definitions, we can obtain the following corollary.Corollary 19.
Suppose (28) is regressive and fix x
0
∈
𝕋
k
k, and if one chooses p
(
t
)
=
α, where α is a positive constant, then the following equality holds on 𝕋
k
k.(
a
)
e
α
∇
(
x
,
x
0
)
=
α
e
α
(
ρ
(
x
)
,
x
0
)
=
α
·
(
e
α
(
x
,
x
0
)
/
(
1
+
α
ν
(
x
)
)
).
(
b
)
e
α
Δ
∇
(
x
,
x
0
)
=
α
2
·
(
e
α
(
x
,
x
0
)
/
(
1
+
α
ν
(
x
)
)
).Proof.
(a) Since(30)
e
α
∇
(
x
,
x
0
)
=
e
α
Δ
(
ρ
(
x
)
,
x
0
)
=
α
e
α
(
ρ
(
x
)
,
x
0
)
,
e
α
(
ρ
(
x
)
,
x
0
)
=
e
α
(
x
,
x
0
)
-
ν
(
x
)
e
α
∇
(
x
,
x
0
)
,
we have
(31)
e
α
(
ρ
(
x
)
,
x
0
)
=
e
α
(
x
,
x
0
)
1
+
α
ν
(
x
)
,
and thus
(32)
e
α
∇
(
x
,
x
0
)
=
α
·
e
α
(
x
,
x
0
)
1
+
α
ν
(
x
)
.
(b) Obviously,
(33)
e
α
Δ
∇
(
x
,
x
0
)
=
(
α
·
e
α
(
x
,
x
0
)
)
∇
=
α
2
e
α
(
ρ
(
x
)
,
x
0
)
=
α
2
·
e
α
(
x
,
x
0
)
1
+
α
ν
(
x
)
.Definition 20 (see [22]).
One definesξ
^
h
(
z
)
=
-
(
1
/
h
)
log
⁡
(
1
-
z
h
)
(
ξ
^
h
:
ℂ
h
→
ℤ
h
), where h
>
0. If p
∈
ℜ
ν, then one defines the exponential function by
(34)
e
^
p
(
t
,
s
)
=
exp
⁡
(
∫
s
t
‍
ξ
^
ν
(
τ
)
(
p
(
τ
)
)
∇
τ
)
for
s
,
t
∈
𝕋
.
If p
∈
ℜ
ν, then the first-order linear dynamic equation
(35)
y
∇
=
p
(
t
)
y
is called regressive.Theorem 21 (see [22]).
Suppose (35) is regressive and fix t
0
∈
𝕋. Then e
^
p
(
·
,
t
0
) is a solution of the initial value problem
(36)
y
∇
=
p
(
t
)
y
,
y
(
t
0
)
=
1
on 𝕋.Theorem 22 (see [22]).
Suppose (35) is regressive; then the only solution of (36) is given by e
^
p
(
·
,
t
0
).Theorem 23 (see [22]).
Ifp
∈
ℜ
ν, then(i)
e
^
0
(
t
,
s
)
=
1 and e
^
p
(
t
,
t
)
=
1;
(ii)
e
^
p
(
ρ
(
t
)
,
s
)
=
(
1
-
ν
(
t
)
p
(
t
)
)
e
^
p
(
t
,
s
);
(iii)
e
^
p
(
t
,
s
)
=
1
/
e
^
p
(
s
,
t
);
(iv)
e
^
p
(
t
,
u
)
e
^
p
(
u
,
s
)
=
e
^
p
(
t
,
s
);
(v)
(
1
/
e
^
p
(
·
,
s
)
)
∇
=
-
p
(
t
)
/
e
p
ρ
(
·
,
s
).Definition 24 (see [22]).
One defines the setℜ
ν
+ of all positively ν
-regressive elements of ℜ
ν by
(37)
ℜ
ν
+
=
{
p
∈
ℜ
ν
:
(
1
-
ν
(
t
)
p
(
t
)
)
>
0
∀
t
∈
𝕋
}
.Corollary 25 (see [22]).
Ifp
∈
ℜ
ν
+ and t
0
∈
𝕋, then e
^
p
(
t
,
t
0
)
>
0.According to the above theorems and definitions, we can obtain the following corollary.Corollary 26.
Suppose (35) is regressive and fix x
0
∈
𝕋
k
k, and if one chooses p
(
t
)
=
α, where α is a negative constant, then the following equality holds on 𝕋
k
k.(
a
)
e
^
α
Δ
(
x
,
x
0
)
=
α
e
^
α
(
σ
(
x
)
,
x
0
)
=
α
(
e
^
α
(
x
,
x
0
)
/
(
1
-
α
μ
(
x
)
).
(
b
)
e
^
α
∇
(
x
0
,
x
)
=
-
α
e
^
α
(
x
0
,
ρ
(
x
)
)
=
-
α
(
e
^
α
(
x
0
,
x
)
/
(
1
-
α
ν
(
x
)
)
).
(
c
)
e
^
α
Δ
(
x
0
,
x
)
=
-
α
·
e
^
α
(
x
0
,
x
).
(
d
)
e
^
α
Δ
∇
(
x
0
,
x
)
=
α
2
e
^
α
(
x
0
,
ρ
(
x
)
)
=
α
2
(
e
^
α
(
x
0
,
x
)
/
(
1
-
α
ν
(
x
)
)
).Proof.
(a) It is easy to see that(38)
e
^
α
Δ
(
x
,
x
0
)
=
e
^
α
∇
(
σ
(
x
)
,
x
0
)
=
α
e
^
α
(
σ
(
x
)
,
x
0
)
,
and we have
(39)
e
^
α
(
σ
(
x
)
,
x
0
)
=
e
^
α
(
x
,
x
0
)
+
μ
(
x
)
e
^
α
Δ
(
x
,
x
0
)
=
e
^
α
(
x
,
x
0
)
+
μ
(
x
)
α
e
^
α
(
σ
(
x
)
,
x
0
)
,
which can obtain
(40)
e
^
α
(
σ
(
x
)
,
x
0
)
=
e
^
α
(
x
,
x
0
)
1
-
α
μ
(
x
)
,
and thus
(41)
e
^
α
Δ
(
x
,
x
0
)
=
α
e
^
α
(
σ
(
x
)
,
x
0
)
=
α
e
^
α
(
x
,
x
0
)
1
-
α
μ
(
x
)
.
(b) Obviously,(42)
e
^
α
∇
(
x
0
,
x
)
=
(
1
e
^
α
(
x
,
x
0
)
)
∇
=
-
α
e
^
α
(
ρ
(
x
)
,
x
0
)
=
-
α
e
^
α
(
x
0
,
ρ
(
x
)
)
,
and we have
(43)
e
^
α
(
x
0
,
ρ
(
x
)
)
=
e
^
α
(
x
0
,
x
)
-
ν
(
x
)
e
^
α
∇
(
x
0
,
x
)
=
e
^
α
(
x
0
,
x
)
-
ν
(
x
)
[
-
α
e
^
α
(
x
0
,
ρ
(
x
)
)
]
,
which can obtain
(44)
e
^
α
(
x
0
,
ρ
(
x
)
)
=
e
^
α
(
x
0
,
x
)
1
-
α
ν
(
x
)
,
and therefore, we get
(45)
e
^
α
∇
(
x
0
,
x
)
=
-
α
e
^
α
(
x
0
,
x
)
1
-
α
ν
(
x
)
.
(c) We have(46)
e
^
α
Δ
(
x
0
,
x
)
=
(
1
e
^
α
(
x
,
x
0
)
)
Δ
=
-
e
^
α
Δ
(
x
,
x
0
)
e
^
α
(
σ
(
x
)
,
x
0
)
e
^
α
(
x
,
x
0
)
,
and then
(47)
e
^
α
Δ
(
x
0
,
x
)
=
-
α
e
^
α
(
σ
(
x
)
,
x
0
)
e
^
α
(
σ
(
x
)
,
x
0
)
e
^
α
(
x
,
x
0
)
=
-
α
e
^
α
(
x
0
,
x
)
.
(d) Obviously,(48)
e
^
α
Δ
∇
(
x
0
,
x
)
=
(
e
^
α
Δ
(
x
0
,
x
)
)
∇
=
(
-
α
e
^
α
(
x
0
,
x
)
)
∇
=
α
2
e
^
α
(
x
0
,
ρ
(
x
)
)
.
And hence, we get
(49)
e
^
α
Δ
∇
(
x
0
,
x
)
=
α
2
e
^
α
(
x
0
,
ρ
(
x
)
)
=
α
2
e
^
α
(
x
0
,
x
)
1
-
α
ν
(
x
)
.Theorem 27 (see [22]).
Letf be a continuous function on [
a
,
b
]
𝕋, that is, delta differentiable on [
a
,
b
)
𝕋. Then f is increasing, decreasing, nondecreasing, and nonincreasing on [
a
,
b
)
𝕋 if f
Δ
>
0
,
f
Δ
<
0
,
f
Δ
≥
0
,
f
Δ
≤
0 for all t
∈
[
a
,
b
)
𝕋, respectively.Definition 28.
One says that a functionf
:
𝕋
→
ℝ is left-increasing at t
0
∈
𝕋
k provided(i)
ift
0 is left-scattered, then f
(
t
0
)
>
f
(
ρ
(
t
0
)
);
(ii)
ift
0 is left-dense, then there is a neighbourhood U of t
0 such that f
(
t
0
)
>
f
(
t
) for all t
∈
U with t
0
>
t.
Similarly, we say thatf is left-decreasing if in the above (i) f
(
t
0
)
<
f
(
ρ
(
t
0
)
) and in (ii) f
(
t
0
)
<
f
(
t
).Theorem 29.
Supposef
:
𝕋
→
ℝ is nabla differentiable at t
0
∈
𝕋
k. If f
∇
(
t
0
)
>
0, then f is left-increasing. If f
∇
(
t
0
)
<
0, then f is left-decreasing.Proof.
We only showf
∇
(
t
0
)
>
0 as the second statement can be shown similarly. If t
0 is left-scattered, then
(50)
f
∇
(
t
0
)
=
f
(
ρ
(
t
0
)
)
-
f
(
t
0
)
ρ
(
t
0
)
-
t
0
and hence f
(
ρ
(
t
0
)
)
<
f
(
t
0
); that is, f is left-increasing. Let now t
0 be left-dense. Then
(51)
f
∇
(
t
0
)
=
lim
⁡
t
→
t
0
f
(
t
0
)
-
f
(
t
)
t
0
-
t
and therefore for ɛ
=
f
∇
(
t
0
) there is a neighbourhood U of t
0 such that
(52)
|
f
(
t
0
)
-
f
(
t
)
t
0
-
t
-
f
∇
(
t
0
)
|
<
ɛ
for all t
∈
U with t
≠
t
0. Hence
(53)
0
<
f
(
t
0
)
-
f
(
t
)
t
0
-
t
<
2
f
∇
(
t
0
)
.
Therefore, f
(
t
)
<
f
(
t
0
) for all t
∈
U with t
<
t
0. Combining what we have proved, we can get that if f
∇
(
t
0
)
>
0, then f is left-increasing.Definition 30.
We say that a functionf
:
𝕋
→
ℝ assumes its local left-minimum at t
0
∈
𝕋
k provided(i)
ift
0 is left-scattered, then f
(
t
0
)
≤
f
(
ρ
(
t
0
)
);
(ii)
ift
0 is left-dense, then there is a neighbourhood U of t
0 such that f
(
t
0
)
≤
f
(
t
) for all t
∈
U with t
0
>
t.
Similarly, we say thatf assumes its local left-maximum if in the above (i) f
(
t
0
)
≥
f
(
ρ
(
t
0
)
) and in (ii) f
(
t
0
)
≥
f
(
t
).Theorem 31.
Supposef
:
𝕋
→
ℝ is nabla differentiable at t
0
∈
𝕋
k. If f attains its local left-minimum at t
0, then f
∇
(
t
0
)
≤
0. If f attains its local left-maximum at t
0, then f
∇
(
t
0
)
≥
0.Proof.
Suppose thatf attains its local left-minimum at t
0. To show that f
∇
(
t
0
)
≤
0, we assume the opposite, that is, f
∇
(
t
0
)
>
0. Then f is left-increasing by Theorem 29, contrary to the assumption that f attains its local left-minimum at t
0. Thus, we must have f
∇
(
t
0
)
≤
0. The second statement can be shown similarly.Theorem 32.
Letf be a continuous function on [
a
,
b
]
𝕋, that is, nabla differentiable on (
a
,
b
]
𝕋 (the differentiability at b is understood as left-sided) and satisfies f
(
a
)
=
f
(
b
). Then, there exists ξ
,
τ
∈
(
a
,
b
]
𝕋 such that
(54)
f
∇
(
ξ
)
≤
0
≤
f
∇
(
τ
)
.Proof.
Sincef is continuous function on [
a
,
b
]
𝕋,f attains its minimum m and its maximum M. Therefore, here exists ξ
,
τ
∈
[
a
,
b
]
𝕋 such that m
=
f
(
ξ
)
,
M
=
f
(
τ
). Since f
(
a
)
=
f
(
b
), we may assume that ξ
,
τ
∈
(
a
,
b
]
𝕋. Clearly f attains its local left-minimum at ξ and its local left-maximum at τ. Then, by Theorem 31 we have f
∇
(
ξ
)
≤
0 and f
∇
(
τ
)
≥
0.Theorem 33.
Letf be a continuous function on [
a
,
b
]
𝕋, that is, nabla differentiable on (
a
,
b
]
𝕋. Then f is increasing, decreasing, nondecreasing, and nonincreasing on (
a
,
b
]
𝕋 if f
∇
>
0, f
∇
<
0, f
∇
≥
0, f
∇
≤
0 for all t
∈
(
a
,
b
]
𝕋, respectively.Proof.
Suppose the functionφ
(
t
) defined on [
a
,
c
]
𝕋
∈
[
a
,
b
]
𝕋 by
(55)
φ
(
t
)
=
f
(
t
)
-
f
(
a
)
-
f
(
c
)
-
f
(
a
)
c
-
a
(
t
-
a
)
.
Clearly φ
(
t
) is continuous on [
a
,
c
]
𝕋 and nabla differentiable on (
a
,
c
]
𝕋. Also φ
(
a
)
=
φ
(
c
)
=
0, and so
(56)
φ
∇
(
ξ
)
≤
0
≤
φ
∇
(
τ
)
for some ξ
,
τ
∈
(
a
,
c
]
𝕋 by Theorem 32. Hence, taking into account that
(57)
φ
∇
(
t
)
=
f
∇
(
t
)
-
f
(
c
)
-
f
(
a
)
c
-
a
,
then we have
(58)
f
∇
(
ξ
)
≤
f
(
c
)
-
f
(
a
)
c
-
a
≤
f
∇
(
τ
)
for some ξ
,
τ
∈
(
a
,
b
]
𝕋.
Iff
∇
>
0, f
∇
<
0, f
∇
≥
0, f
∇
≤
0 for all t
∈
(
a
,
c
]
𝕋, then f
(
c
)
-
f
(
a
)
>
0, f
(
c
)
-
f
(
a
)
<
0, f
(
c
)
-
f
(
a
)
≥
0, f
(
c
)
-
f
(
a
)
≤
0, respectively. Considering the arbitrary of c, we arrive at the statement of the theorem.


## 3. Delta-Nabla Type Maximum Principles

In this paper, we denoteΛ
∶
=
[
a
,
b
]
𝕋 as an interval on time scales. We study those functions defined on Λ which belong to 𝒟
(
Λ
), where 𝒟
(
Λ
) is the set of all functions u
:
Λ
→
ℝ, such that u
Δ is continuous on [
a
,
b
)
𝕋, u
∇ is continuous on (
a
,
b
]
𝕋, and u
Δ
∇ exists in (
a
,
b
)
𝕋.First we give a necessary condition thatu
(
x
)
∈
𝒟
(
Λ
) attains its maximum at some point x
0
∈
(
a
,
b
)
𝕋.Lemma 34.
Ifu
(
x
)
∈
𝒟
(
Λ
) attains a maximum at a point x
0
∈
(
a
,
b
)
𝕋, then
(59)
u
Δ
∇
(
x
0
)
≤
0
,
u
∇
(
x
0
)
≥
0
,
u
Δ
(
x
0
)
≤
0
.
The strict inequality in the last two inequalities can occur only at left-scattered points.Proof.
Let us divide our proof into three parts.
(i) Ifx
0 is left-scattered, then the maximality of u at x
0 implies that u
∇
(
x
0
)
≥
0 and u
Δ
(
x
0
)
≤
0 and consequently
(60)
u
Δ
∇
(
x
0
)
=
u
Δ
(
x
0
)
-
u
Δ
(
ρ
(
x
0
)
)
ν
(
x
0
)
=
u
Δ
(
x
0
)
-
u
∇
(
x
0
)
ν
(
x
0
)
≤
0
.
(ii) Ifx
0 is left-dense and right-scattered, in this case, we have u
Δ
(
x
0
)
≤
0. If there is no positive number sequence {
h
n
} such that lim
⁡
n
→
∞
h
n
=
0 and u
Δ
(
x
0
-
h
n
)
≥
0, then there exists a δ
>
0 such that u
Δ
(
x
)
<
0 for each x
∈
[
x
0
-
δ
,
x
0
)
𝕋; by Theorem 27, a contraction with u attains its maximum at interior point x
0 of (
a
,
b
)
𝕋. Thus, there exists {
h
n
} such that lim
⁡
n
→
∞
h
n
=
0 and u
Δ
(
x
0
-
h
n
)
≥
0. This yields
(61)
u
Δ
∇
(
x
0
)
=
lim
⁡
h
→
0
u
Δ
(
x
0
)
-
u
Δ
(
x
0
-
h
)
h
≤
0
.
Furthermore, the continuity of the delta derivative u
Δ
(
x
) implies that
(62)
0
≤
lim
⁡
h
→
0
u
Δ
(
x
0
-
h
)
=
u
Δ
(
x
0
)
≤
0
,
and consequently u
Δ
(
x
0
)
=
0. Then by using Corollary 10 we have that
(63)
u
∇
(
x
0
)
=
u
Δ
(
ρ
(
x
0
)
)
=
u
Δ
(
x
0
)
=
0
.
(iii) Ifx
0 is left-dense and right-dense, in this case the maximality of u at x
0 and standard continuous necessary conditions imply that
(64)
u
Δ
∇
(
x
0
)
≤
0
,
u
∇
(
x
0
)
=
u
Δ
(
x
0
)
=
0
.According to Lemma34, we can obtain the first simple maximum principle for the time scale.Corollary 35.
Assuming thatu
(
x
)
∈
𝒟
(
Λ
), if u
Δ
∇
(
x
0
)
>
0 at some point x
0
∈
(
a
,
b
)
𝕋, then u cannot attain its maximum at x
0. Moreover, if u
Δ
∇
(
x
)
>
0 in (
a
,
b
)
𝕋, then u cannot attain its maximum in (
a
,
b
)
𝕋.We give a variant of Corollary35 where we weaken the condition u
Δ
∇
>
0.Theorem 36.
Letu
(
x
)
∈
𝒟
(
Λ
). If u
Δ
∇
(
x
)
≥
0 in (
a
,
b
)
𝕋, then u cannot attain its maximum in (
a
,
b
)
𝕋, unless u
≡
M.Proof.
We suppose that the result is false. Then there arex
0
,
x
1
∈
(
a
,
b
)
𝕋, such that u
(
x
0
)
=
M and u
(
x
1
)
<
M. Let us assume first that x
0
<
x
1 and let us define a function z
(
x
)
∈
𝒟
(
Λ
) by
(65)
z
(
x
)
∶
=
e
α
(
x
,
x
0
)
-
1
,
where α
>
0 and e
α
(
x
,
x
0
) is an exponential function on 𝕋 (see Section 2), and then
(66)
z
(
x
)
{
>
0
,
x
>
x
0
=
0
,
x
=
x
0
<
0
.
x
<
x
0
.
Considering α
>
0 and the positivity of e
α, we obtain
(67)
z
Δ
∇
(
x
)
=
α
2
e
α
(
x
,
x
0
)
1
+
α
ν
>
0
.
Let us define a function w
(
x
)
∈
𝒟
(
Λ
) by
(68)
w
(
x
)
∶
=
u
(
x
)
+
ɛ
z
(
x
)
,
where ɛ
>
0 is chosen so that
(69)
ɛ
<
M
-
u
(
x
1
)
z
(
x
1
)
.
Since e
α
(
a
,
x
0
)
<
1, we have
(70)
w
(
a
)
=
u
(
a
)
+
ɛ
z
(
a
)
<
u
(
a
)
≤
M
.
Furthermore, the definition of ɛ yields that
(71)
w
(
x
1
)
=
u
(
x
1
)
+
ɛ
z
(
x
1
)
<
u
(
x
1
)
+
M
-
u
(
x
1
)
=
M
.
Finally, e
α
(
x
0
,
x
0
)
=
1 derives
(72)
w
(
x
0
)
=
M
.
It shows that w attains its maximum in (
a
,
x
1
)
𝕋.
However,(73)
w
Δ
∇
=
u
Δ
∇
+
ɛ
z
Δ
∇
>
0
,
which contradicts the statement of Corollary 35. If x
0
>
x
1,
(74)
z
(
x
)
∶
=
∫
x
b
‍
(
∫
s
b
‍
∇
τ
)
Δ
s
-
∫
x
0
b
‍
(
∫
s
b
‍
∇
τ
)
Δ
s
.
Then we have that(75)
z
(
x
)
{
<
0
,
x
>
x
0
=
0
,
x
=
x
0
>
0
.
x
<
x
0
,
z
Δ
∇
(
x
)
=
1
>
0
.
Let us define a function w
(
x
)
∈
𝒟
(
Λ
) by
(76)
w
(
x
)
∶
=
u
(
x
)
+
ɛ
z
(
x
)
,
where ɛ
>
0 is chosen so that
(77)
ɛ
<
M
-
u
(
x
1
)
z
(
x
1
)
.
Since z
(
b
)
∶
=
-
∫
x
0
b
‍
(
∫
s
b
‍
∇
τ
)
Δ
s
<
0, we have
(78)
w
(
b
)
=
u
(
b
)
+
ɛ
z
(
b
)
<
u
(
b
)
≤
M
.
Furthermore, the definition of ɛ yields that
(79)
w
(
x
1
)
=
u
(
x
1
)
+
ɛ
z
(
x
1
)
<
u
(
x
1
)
+
M
-
u
(
x
1
)
=
M
.
Finally, since z
(
x
0
)
=
0, we derive
(80)
w
(
x
0
)
=
M
.
It shows that w attains its maximum in (
x
1
,
b
)
𝕋.
However,(81)
w
Δ
∇
=
u
Δ
∇
+
ɛ
z
Δ
∇
>
0
,
which is a contradiction with Corollary 35. The proof is completed.As a natural extension of the above simple maximum principle, we consider the operator of the following type:(82)
L
[
u
]
∶
=
u
∇
Δ
+
g
1
u
Δ
+
g
2
u
∇
.
By the above results, we can obtain Theorem 37.Theorem 37.
Assume that the functionsg
1
,
g
2
:
[
a
,
b
]
𝕋
→
R satisfy
(83)
1
+
ν
g
1
≥
0
,
o
n
(
a
,
b
)
𝕋
,
-
1
+
ν
g
2
≤
0
,
o
n
(
a
,
b
)
𝕋
.
Letting u
(
x
)
∈
𝒟
(
Λ
), if L
[
u
]
(
x
0
)
>
0 at some point x
0
∈
(
a
,
b
)
𝕋, then u cannot attain its maximum M at x
0. Moreover, if L
[
u
]
∶
=
u
Δ
∇
+
g
1
u
Δ
+
g
2
u
∇
>
0, for each x
∈
(
a
,
b
)
𝕋, then u cannot attain its maximum in (
a
,
b
)
𝕋.Proof.
We suppose thatL
[
u
]
(
x
0
)
>
0 at some point x
0
∈
(
a
,
b
)
𝕋 and u attains its maximum at a point x
0. We divide our proof into two parts.
(i) Ifx
0 is left-scattered, in this case, we have
(84)
u
Δ
∇
(
x
0
)
=
u
Δ
(
x
0
)
-
u
Δ
(
ρ
(
x
0
)
)
ν
(
x
0
)
=
u
Δ
(
x
0
)
-
u
∇
(
x
0
)
ν
(
x
0
)
.
Multiplying L
[
u
]
(
x
0
) by ν
(
x
0
), we obtain
(85)
ν
(
x
0
)
L
[
u
]
(
x
0
)
=
ν
(
x
0
)
(
u
Δ
∇
(
x
0
)
+
g
1
u
Δ
(
x
0
)
+
g
2
u
∇
(
x
0
)
)
=
(
1
+
ν
(
x
0
)
g
1
)
u
Δ
(
x
0
)
+
(
-
1
+
ν
(
x
0
)
g
2
)
u
∇
(
x
0
)
>
0
.
However, it follows from Lemma34 and the conditions
(86)
1
+
ν
g
1
≥
0
,
on
⁡
(
a
,
b
)
𝕋
,
-
1
+
ν
g
2
≤
0
,
on
⁡
(
a
,
b
)
𝕋
,
that ν
(
x
0
)
L
[
u
]
(
x
0
)
≤
0, which is a contradiction.
(ii) Ifx
0 is left-dense, then by Lemma 34 we know that
(87)
u
Δ
(
x
0
)
=
u
∇
(
x
0
)
=
0
.
Therefore,L
[
u
]
(
x
0
) reduces to
(88)
L
[
u
(
x
0
)
]
=
u
Δ
∇
(
x
0
)
>
0
,
which is a contradiction with Lemma 34. Combining the proof of (i) and (ii), we get that u cannot attain its maximum at x
0. The proof is completed.Next, we weaken the condition(89)
L
[
u
]
∶
=
u
Δ
∇
+
g
1
u
Δ
+
g
2
u
∇
>
0
to
(90)
L
[
u
]
∶
=
u
Δ
∇
+
g
1
u
Δ
+
g
2
u
∇
≥
0
.Theorem 38.
Assume that the functionsg
1
,
g
2
:
[
a
,
b
]
𝕋
→
R satisfy
(91)
1
+
ν
g
1
>
0
,
o
n
(
a
,
b
)
𝕋
,
(92)
-
1
+
ν
g
2
<
0
,
o
n
(
a
,
b
)
𝕋
,
(93)
g
1
+
g
2
1
+
g
1
ν
i
s
b
o
u
n
d
e
d
o
n
a
n
y
c
l
o
s
e
d
s
u
b
i
n
t
e
r
v
a
l
o
f
(
a
,
b
)
𝕋
.
Let u
(
x
)
∈
𝒟
(
Λ
) satisfy L
[
u
]
∶
=
u
Δ
∇
+
g
1
u
Δ
+
g
2
u
∇
≥
0, for each x
∈
(
a
,
b
)
𝕋. Then u cannot attain its maximum in (
a
,
b
)
𝕋, unless u
≡
M.Proof.
Assume thatu attains its maximum at a point x
0 in (
a
,
b
)
𝕋 but does not identically equal M. That is, u
(
x
0
)
=
M, and there exists x
p
∈
(
a
,
b
)
𝕋 such that u
(
x
p
)
<
M. Let us assume first that x
0
<
x
p and let us define a function z
(
x
)
∈
𝒟
(
Λ
) by
(94)
z
(
x
)
=
e
α
(
x
,
x
0
)
-
1
.
Therefore, we have
(95)
L
[
z
]
=
z
Δ
∇
+
g
1
z
Δ
+
g
2
z
∇
=
[
(
1
+
ν
g
1
)
α
2
+
α
·
(
g
1
+
g
2
)
]
e
α
(
x
,
x
0
)
1
+
α
ν
=
α
[
α
+
g
1
+
g
2
1
+
ν
g
1
]
e
α
(
x
,
x
0
)
1
+
α
ν
(
1
+
ν
g
1
)
.
Thus, by (93) we can take arbitrary α
>
0, such that α
>
-
(
g
1
+
g
2
)
/
(
1
+
g
1
ν
) in [
a
1
,
x
p
]
𝕋, where a
1
=
σ
(
a
) if σ
(
a
)
>
a and a
1
∈
(
a
,
x
0
)
𝕋 if σ
(
a
)
=
a. Then we have L
[
z
]
>
0 in [
a
1
,
x
p
]
𝕋. Let us define a function w
(
x
)
∈
𝒟
(
Λ
) by
(96)
w
(
x
)
∶
=
u
(
x
)
+
ɛ
z
(
x
)
,
where ɛ
>
0 is chosen so that
(97)
ɛ
<
M
-
u
(
x
p
)
z
(
x
p
)
.
If a
1
=
σ
(
a
)
=
x
0, since e
α
(
a
,
x
0
)
<
1, we have that z
(
a
)
<
0 and
(98)
w
(
a
)
=
u
(
a
)
+
ɛ
z
(
a
)
<
u
(
a
)
≤
M
.
Moreover, the definition of ɛ yields that
(99)
w
(
x
p
)
=
u
(
x
p
)
+
ɛ
z
(
x
p
)
<
u
(
x
p
)
+
M
-
u
(
x
p
)
=
M
.
Finally, e
α
(
x
0
,
x
0
)
=
1 implies that w
(
x
0
)
=
M. It follows that w has a maximum in (
a
,
x
p
)
𝕋. However,
(100)
L
[
w
]
=
L
[
u
]
+
ɛ
L
[
z
]
>
0
,
which is a contradiction with Theorem 37. If a
1
<
x
0, then we have w
(
a
1
)
<
M. It follows that w has a maximum in (
a
1
,
x
p
)
𝕋. This is again a contradiction with Theorem 37. Thus, we have proved that if x
0
∈
(
a
,
b
)
𝕋 is a maximum point, then u
(
x
)
=
M for any x
≥
x
0. Let
(101)
m
=
inf
⁡
{
x
≥
a
:
u
(
t
)
=
M
,
∀
t
∈
[
x
,
x
0
]
𝕋
}
.
From this, we obtain that u
(
m
)
=
M and u
Δ
(
m
)
=
0. Then we have that x
0
≥
m
>
x
p and u
(
t
)
<
M for any x
0
∈
(
a
,
m
)
𝕋. If m is left-scattered, then
(102)
m
m
m
m
m
m
ρ
(
m
)
<
m
,
u
Δ
(
ρ
(
m
)
)
=
u
(
m
)
-
u
(
ρ
(
m
)
)
m
-
ρ
(
m
)
>
0
,
u
Δ
∇
(
m
)
=
u
Δ
(
ρ
(
m
)
)
-
u
Δ
(
m
)
ρ
(
m
)
-
m
m
m
m
m
=
u
Δ
(
ρ
(
m
)
)
ρ
(
m
)
-
m
.
Since L
[
u
]
(
m
)
≥
0, we multiply L
[
u
]
(
m
) by ν
(
m
) and get that
(103)
0
≤
L
[
u
]
(
m
)
·
ν
(
m
)
=
-
u
Δ
(
ρ
(
m
)
)
+
u
∇
(
m
)
g
2
(
m
)
ν
(
m
)
=
-
u
Δ
(
ρ
(
m
)
)
+
u
Δ
(
ρ
(
m
)
)
g
2
(
m
)
ν
(
m
)
=
u
Δ
(
ρ
(
m
)
)
[
g
2
(
m
)
ν
(
m
)
-
1
]
<
0
.
This is a contradiction. If m is left-dense, let
(104)
z
(
x
)
=
e
-
α
(
x
,
m
)
-
1
,
where α
>
0, such that α
>
(
g
1
+
g
2
)
/
(
1
+
g
1
ν
) in [
x
p
,
m
]
𝕋. We choose d closely enough to m, such that 1
-
α
μ
>
0, 1
-
α
ν
>
0 on [
d
,
m
]
𝕋, and
(105)
w
(
x
)
∶
=
u
(
x
)
+
ɛ
z
(
x
)
,
where ɛ
>
0 such that
(106)
0
<
ɛ
<
M
-
u
(
d
)
z
(
d
)
.
Therefore, we have
(107)
L
[
z
]
=
z
Δ
∇
+
g
1
z
Δ
+
g
2
z
∇
=
[
α
2
(
1
+
ν
g
1
)
-
α
(
g
1
+
g
2
)
]
e
-
α
(
x
,
m
)
1
-
α
ν
=
α
[
α
-
g
1
+
g
2
1
+
ν
g
1
]
e
-
α
(
x
,
m
)
1
-
α
ν
(
1
+
ν
g
1
)
>
0
.
Thus, L
[
w
]
=
L
[
u
]
+
ɛ
L
[
z
]
>
0 on [
d
,
m
]
𝕋. By Theorem 37 we know that w cannot attain its maximum in (
d
,
m
)
𝕋. Note that
(108)
w
(
d
)
=
u
(
d
)
+
ɛ
z
(
d
)
<
u
(
d
)
+
M
-
u
(
d
)
=
M
.
We get that w
(
m
)
=
u
(
m
)
=
M is the maximum of w on [
d
,
m
]
𝕋. Since u
(
x
)
=
M for any x
≥
m and z
(
x
) is increasing for x
≥
m, we have that w
Δ
(
m
)
≥
0; however, we also have that
(109)
w
Δ
(
m
)
=
u
Δ
(
m
)
+
ɛ
z
Δ
(
m
)
=
ɛ
z
Δ
(
m
)
=
-
α
ɛ
<
0
.
This is a contradiction. The proof is completed.In Theorem38, if we take 𝕋
=
ℝ, we have the following corollary which is the result that appeared in [3].Corollary 39.
Assuming that the functiong
:
[
a
,
b
]
→
ℝ is bounded on any closed subinterval of (
a
,
b
), if u
(
x
)
∈
𝒟
(
Λ
) satisfies (
L
+
h
)
[
u
]
∶
=
u
′′
+
g
u
′′
≥
0 in (
a
,
b
), then u cannot attain its maximum M in (
a
,
b
), unless u
≡
M.In Theorem38, if we take 𝕋
=
ℤ, where ℤ is the set of all integral numbers, we can obtain the following new maximum principle for second-order mixed Δ and ∇ difference dynamic system.Corollary 40.
Assume that the functionsg
1 and g
2
:
[
a
,
b
]
ℤ
→
ℝ satisfy
(110)
1
+
g
1
(
k
)
>
0
,
1
-
g
2
(
k
)
>
0
,
∀
k
∈
(
a
,
b
)
ℤ
,
and let u
(
x
)
∈
𝒟
(
Λ
); if
(111)
∇
Δ
u
(
k
)
+
g
1
(
k
)
Δ
u
(
k
)
+
g
2
(
k
)
∇
u
(
k
)
≥
0
,
∀
k
∈
(
a
,
b
)
ℤ
,
then u cannot attain its maximum M in (
a
,
b
)
ℤ, unless u
≡
M.To show that conditions (91), (92), and (93) are necessary for the validity of our results, we give the following examples.Example 41.
Let𝕋
=
{
q
n
:
n
∈
ℤ
}
∪
{
0
}, where ℤ is the set of all integral numbers and q
>
1, and u is defined by
(112)
u
(
x
)
=
{
x
,
if
x
≤
q
8
;
q
8
,
if
x
>
q
8
.
Then
(113)
u
Δ
(
x
)
=
{
1
,
if
x
<
q
8
;
0
,
if
x
>
q
8
,
0
,
if
x
=
q
8
,
u
∇
(
x
)
=
{
1
,
if
x
<
q
8
;
0
,
if
x
>
q
8
,
1
,
if
x
=
q
8
,
u
Δ
∇
(
x
)
=
{
0
,
if
x
<
q
8
;
0
,
if
x
>
q
8
,
1
q
7
(
1
-
q
)
,
if
x
=
q
8
.
Letting g
1
(
x
)
≡
1, g
2
(
x
)
≡
1
/
ν
(
x
), x
∈
[
a
,
b
]
𝕋
=
[
1
,
q
9
]
𝕋, then
(114)
u
Δ
∇
(
x
)
+
g
1
(
x
)
u
Δ
(
x
)
+
g
2
(
x
)
u
∇
(
x
)
=
u
Δ
∇
(
x
)
+
u
Δ
(
x
)
+
u
∇
(
x
)
≥
0
.
1
+
ν
(
x
)
g
1
(
x
)
=
1
+
1
>
0, 1
-
ν
(
x
)
g
2
(
x
)
=
1
-
1
=
0 for any x
∈
[
1
,
q
9
]
𝕋, and
(115)
g
1
(
x
)
+
g
2
(
x
)
1
+
g
1
(
x
)
ν
(
x
)
=
1
+
(
1
/
ν
(
x
)
)
1
+
ν
(
x
)
=
1
ν
(
x
)
is bounded on any closed subinterval of [
1
,
q
9
]
𝕋. Thus, conditions (91) and (93) hold, but (92) does not hold. The conclusion of Theorem 38 also does not hold, since u attains its maximum q
8 in (
1
,
q
9
)
ℤ, but u is not constant.Example 42.
Let𝕋
=
{
q
n
:
n
∈
ℤ
}
∪
{
0
}, where ℤ is the set of all integral numbers and q
>
1, and u is defined by
(116)
u
(
x
)
=
{
q
5
,
if
x
≤
q
5
;
2
q
5
-
x
,
if
x
>
q
5
.
Then(117)
u
Δ
(
x
)
=
{
0
,
if
x
<
q
5
;
-
1
,
if
x
>
q
5
,
-
1
,
if
x
=
q
5
,
u
∇
(
x
)
=
{
0
,
if
x
<
q
5
;
-
1
,
if
x
>
q
5
,
0
,
if
x
=
q
5
,
u
Δ
∇
(
x
)
=
{
0
,
if
x
<
q
5
;
0
,
if
x
>
q
5
,
1
q
4
(
1
-
q
)
,
if
x
=
q
5
.
Letting g
1
(
x
)
=
g
2
(
x
)
≡
-
1
/
ν
(
x
), [
a
,
b
]
𝕋
=
[
1
,
q
7
]
𝕋, then
(118)
u
Δ
∇
(
x
)
+
g
1
(
x
)
u
Δ
(
x
)
+
g
2
(
x
)
u
∇
(
x
)
=
u
Δ
∇
(
x
)
-
u
Δ
(
x
)
-
u
∇
(
x
)
≥
0
.
1
+
ν
(
x
)
g
1
(
x
)
=
1
-
1
=
0, 1
-
ν
(
x
)
g
2
(
x
)
=
1
+
1
>
0 for any x
∈
[
1
,
q
7
]
𝕋, and (
g
1
+
g
2
)
/
(
1
+
g
1
ν
)
=
1 is bounded on any closed subinterval of [
1
,
q
7
]
𝕋. Thus, condition (92) holds, but (91) does not hold; thus (93) does hold. The conclusion of Theorem 38 also does not hold, since u attains its maximum q
5 in (
1
,
q
7
)
ℤ, but u is not constant.Example 43.
Let𝕋
=
{
1
/
n
:
n
∈
ℤ
}
∪
{
0
}, u
(
x
)
=
-
e
α
(
x
,
0
), [
a
,
b
]
𝕋
=
[
-
1
/
2,1
/
2
]
𝕋, and 2
<
α
<
3. Then
(119)
u
Δ
(
x
)
=
-
α
e
α
(
x
,
0
)
,
u
∇
(
x
)
=
-
α
1
+
α
ν
(
x
)
e
α
(
x
,
0
)
,
u
Δ
∇
(
x
)
=
-
α
2
1
+
α
ν
(
x
)
e
α
(
x
,
0
)
.
Let
(120)
g
1
(
x
)
=
{
-
α
|
x
|
,
if
x
=
1
n
,
0
if
x
=
0
;
g
2
(
x
)
=
-
α
,
then
(121)
u
Δ
∇
(
x
)
+
g
1
(
x
)
u
Δ
(
x
)
+
g
2
(
x
)
u
∇
(
x
)
=
[
-
α
2
1
+
α
ν
(
x
)
-
α
g
1
(
x
)
+
α
2
1
+
α
ν
(
x
)
]
×
e
α
(
x
,
0
)
≥
0
,
1
+
ν
(
x
)
g
1
(
x
)
=
{
1
+
-
α
|
n
+
1
|
>
0
,
if
x
=
1
n
,
1
>
0
if
x
=
0
,
1
-
ν
(
x
)
g
2
(
x
)
=
{
1
+
α
n
(
n
+
1
)
>
0
,
if
x
=
1
n
,
1
>
0
if
x
=
0
,
g
1
(
x
)
+
g
2
(
x
)
1
+
ν
(
x
)
g
1
(
x
)
=
{
-
α
n
-
α
1
+
(
-
α
/
|
n
+
1
|
)
,
if
x
=
1
n
,
-
α
if
x
=
0
.
These show that conditions (91) and (92) hold, but (93) does not hold on [
-
1
/
2,1
/
2
]
𝕋. The conclusion of Theorem 38 also does not hold on [
-
1
/
2,1
/
2
]
𝕋, since u attains its maximum -
1 in (
-
1
/
2,1
/
2
)
𝕋, but u is not constant.Now, we establish a generalized maximum principle.Theorem 44.
Assume that the functionsg
1 and g
2
:
[
a
,
b
]
𝕋
→
ℝ satisfy (91), (92), and let u
(
x
)
∈
𝒟
(
Λ
) and
(122)
(
L
+
h
)
[
u
]
∶
=
u
Δ
∇
+
g
1
u
Δ
+
g
2
u
∇
+
h
u
,
where h
(
x
)
≤
0. If (
L
+
h
)
[
u
]
(
x
0
)
>
0 at some point x
0
∈
(
a
,
b
)
𝕋, then u cannot attain its maximum M at x
0. Moreover, if (
L
+
h
)
[
u
]
(
x
)
>
0, for each x
∈
(
a
,
b
)
𝕋, then u cannot attain its maximum in (
a
,
b
)
𝕋.Proof.
Assume thatu attains its nonnegative maximum M at a point x
0 in (
a
,
b
)
𝕋 and (
L
+
h
)
[
u
]
(
x
0
)
>
0. If x
0 is left-scattered, then by Lemma 34, we have that
(123)
ρ
(
x
0
)
<
x
0
,
u
∇
(
x
0
)
=
u
Δ
(
ρ
(
x
0
)
)
m
m
m
m
=
u
(
x
0
)
-
u
(
ρ
(
x
0
)
)
x
0
-
ρ
(
x
0
)
≥
0
,
u
Δ
∇
(
x
0
)
=
u
Δ
(
ρ
(
x
0
)
)
-
u
Δ
(
x
0
)
ρ
(
x
0
)
-
x
0
,
u
Δ
(
x
0
)
≤
0
,
and then
(124)
0
<
(
L
+
h
)
[
u
]
(
x
0
)
·
ν
(
x
0
)
=
-
u
Δ
(
ρ
(
x
0
)
)
+
u
Δ
(
x
0
)
+
u
Δ
(
x
0
)
g
1
(
x
0
)
ν
(
x
0
)
+
u
∇
(
x
0
)
g
2
(
x
0
)
ν
(
x
0
)
+
h
(
x
0
)
u
(
x
0
)
ν
(
x
0
)
=
-
u
∇
(
x
0
)
+
u
Δ
(
x
0
)
+
u
Δ
(
x
0
)
g
1
(
x
0
)
ν
(
x
0
)
+
u
∇
(
x
0
)
g
2
(
x
0
)
ν
(
x
0
)
+
h
(
x
0
)
u
(
x
0
)
ν
(
x
0
)
=
u
∇
(
x
0
)
[
g
2
(
x
0
)
ν
(
x
0
)
-
1
]
+
u
Δ
(
x
0
)
[
1
+
g
1
(
x
0
)
ν
(
x
0
)
]
+
h
(
x
0
)
u
(
x
0
)
ν
(
x
0
)
≤
0
.
This is a contradiction. If x
0 is left-dense, by Lemma 34 we have that u
Δ
(
x
0
)
=
u
∇
(
x
0
)
=
0 and u
Δ
∇
(
x
0
)
≤
0. Then
(125)
(
L
+
h
)
[
u
]
(
x
0
)
=
u
Δ
∇
(
x
0
)
+
h
(
x
0
)
u
(
x
0
)
>
0
,
and h
(
x
0
)
u
(
x
0
)
≤
0 imply that u
Δ
∇
(
x
0
)
>
0. This is also a contradiction with Corollary 35. Thus, we have that u cannot attain its maximum M at x
0. The proof is completed.In Theorem44, if we take 𝕋
=
ℝ, we have the following corollary which is an improvement for the result that appeared in [3].Corollary 45.
Letg
,
h
:
[
a
,
b
]
→
ℝ be functions, h
(
x
)
≤
0 on [
a
,
b
] and (
L
+
h
)
[
u
]
:
=
u
′′
+
g
u
′
+
h
u. If (
L
+
h
)
[
u
]
(
x
0
)
>
0 at some point x
0
∈
(
a
,
b
), then u cannot attain its maximum M at x
0. Moreover, if (
L
+
h
)
[
u
]
(
x
)
>
0, for each x
∈
(
a
,
b
), then u cannot attain its maximum in (
a
,
b
)
𝕋.In Theorem44, if we take 𝕋
=
ℤ, where ℤ is the set of all integral numbers, we can obtain the following new maximum principle for second-order mixed Δ and ∇ difference dynamic system.Corollary 46.
Assume that the functionsh, g
1, and g
2
:
[
a
,
b
]
ℤ
→
ℝ satisfy
(126)
1
+
g
1
(
k
)
>
0
,
1
-
g
2
(
k
)
>
0
,
∀
k
∈
(
a
,
b
)
ℤ
,
and h
(
k
)
≤
0 on [
a
,
b
]
ℤ. Let u
(
x
)
∈
𝒟
(
Λ
). If for some k
∈
(
a
,
b
)
ℤ
(127)
∇
Δ
u
(
k
)
+
g
1
(
k
)
Δ
u
(
k
)
+
g
2
(
k
)
∇
u
(
k
)
+
h
(
k
)
u
(
k
)
>
0
,
then u cannot attain its maximum M at k. If
(128)
∇
Δ
u
(
k
)
+
g
1
(
k
)
Δ
u
(
k
)
+
g
2
(
k
)
∇
u
(
k
)
+
h
(
k
)
u
(
k
)
>
0
,
∀
k
∈
(
a
,
b
)
ℤ
,
then u cannot attain its maximum M in (
a
,
b
)
ℤ.Theorem 47.
Assume that the functionsg
1 and g
2
:
[
a
,
b
]
𝕋
→
ℝ satisfy (91), (92), and (93), and let u
(
x
)
∈
𝒟
(
Λ
) satisfy
(129)
(
L
+
h
)
[
u
]
∶
=
u
Δ
∇
+
g
1
u
Δ
+
g
2
u
∇
+
h
u
≥
0
,
m
m
m
m
m
m
m
m
m
m
m
l
l
f
o
r
e
a
c
h
x
∈
(
a
,
b
)
𝕋
,
where h
(
x
)
≤
0 and
(130)
h
(
x
)
1
+
ν
g
1
i
s
b
o
u
n
d
e
d
o
n
a
n
y
c
l
o
s
e
d
s
u
b
i
n
t
e
r
v
a
l
o
f
(
a
,
b
)
𝕋
.
If u attains a nonnegative maximum M in (
a
,
b
)
𝕋, then u
≡
M.Proof.
Assume thatu attains its nonnegative maximum M at a point x
0 in (
a
,
b
)
𝕋 but does not identically equal M. Thus, we can choose x
0
,
x
p
∈
(
a
,
b
)
𝕋, such that u
(
x
0
)
=
M, u
(
x
p
)
<
M. If x
p
>
x
0, we define a function z
(
x
)
∈
𝒟
(
Λ
) by
(131)
z
(
x
)
=
e
α
(
x
,
x
0
)
-
1
.
Then(132)
(
L
+
h
)
[
z
]
=
z
Δ
∇
+
g
1
z
Δ
+
g
2
z
∇
+
h
z
=
[
(
1
+
ν
g
1
)
α
2
+
α
·
(
g
1
+
g
2
)
+
h
(
1
+
α
ν
(
x
)
)
m
m
m
×
(
1
-
e
α
(
x
0
,
x
)
)
(
1
+
ν
g
1
)
α
2
]
e
α
(
x
,
x
0
)
1
+
α
ν
=
[
α
2
+
α
·
g
1
+
g
2
1
+
ν
g
1
m
m
m
+
h
(
1
+
α
ν
(
x
)
)
(
1
-
e
α
(
x
0
,
x
)
)
1
+
ν
g
1
]
m
m
m
×
e
α
(
x
,
x
0
)
(
1
+
ν
g
1
)
1
+
α
ν
.
It is similar to the proof of Theorem 38; we choose sufficiently larger α such that
(133)
(
L
+
h
)
[
z
]
>
0
holds on (
a
1
,
x
p
)
𝕋, where x
0
∈
(
a
1
,
x
p
)
𝕋
⊂
(
a
,
x
p
)
𝕋. Let us define a function w
(
x
)
∈
𝒟
(
Λ
) by
(134)
w
(
x
)
∶
=
u
(
x
)
+
ɛ
z
(
x
)
,
where ɛ
>
0 is chosen so that
(135)
ɛ
<
M
-
u
(
x
p
)
z
(
x
p
)
.
Since e
α
(
a
,
x
0
)
<
1, we have
(136)
w
(
a
)
=
u
(
a
)
+
ɛ
z
(
a
)
<
u
(
a
)
≤
M
.
Moreover, the definition of ɛ yields that
(137)
w
(
x
p
)
=
u
(
x
p
)
+
ɛ
z
(
x
p
)
<
u
(
x
p
)
+
M
-
u
(
x
p
)
=
M
.
Finally, e
α
(
x
0
,
x
0
)
=
1 implies that w
(
x
0
)
=
M. It implies that w has a maximum in (
a
,
x
p
)
𝕋. However,
(138)
L
[
w
]
≥
(
L
+
h
)
[
w
]
=
(
L
+
h
)
[
u
]
+
ɛ
(
L
+
h
)
[
z
]
>
0
holds on (
a
1
,
x
p
)
𝕋. This is a contradiction with Theorem 37. Thus, we have proved that if x
0
∈
(
a
,
b
)
𝕋 is a maximum point, then u
(
x
)
=
M for any x
≥
x
0. Let
(139)
m
=
inf
⁡
{
x
≥
a
:
u
(
t
)
=
M
,
∀
t
∈
[
x
,
x
0
]
𝕋
}
.
From this, we obtain that u
(
m
)
=
M and u
Δ
(
m
)
=
0. Then we have that x
0
≥
m
>
x
p and u
(
t
)
<
M for any x
0
∈
(
a
,
m
)
𝕋. If m is left-scattered, then it is similar to the proof of Theorem 38; we have that
(140)
m
m
m
m
m
m
ρ
(
m
)
<
m
,
u
Δ
(
ρ
(
m
)
)
=
u
(
m
)
-
u
(
ρ
(
m
)
)
m
-
ρ
(
m
)
>
0
,
u
Δ
∇
(
m
)
=
u
Δ
(
ρ
(
m
)
)
-
u
Δ
(
m
)
ρ
(
m
)
-
m
m
m
m
m
=
u
Δ
(
ρ
(
m
)
)
ρ
(
m
)
-
m
,
(141)
0
≤
L
[
u
]
(
m
)
·
ν
(
m
)
=
-
u
Δ
(
ρ
(
m
)
)
+
u
∇
(
m
)
g
2
(
m
)
ν
(
m
)
+
h
(
m
)
u
(
m
)
ν
(
m
)
=
-
u
Δ
(
ρ
(
m
)
)
+
u
Δ
(
ρ
(
m
)
)
g
2
(
m
)
ν
(
m
)
+
h
(
m
)
u
(
m
)
ν
(
m
)
=
u
Δ
(
ρ
(
m
)
)
[
g
2
(
m
)
ν
(
m
)
-
1
]
+
h
(
m
)
u
(
m
)
ν
(
m
)
<
0
.
This is a contradiction. If m is left-dense, let
(142)
z
(
x
)
=
e
-
α
(
x
,
m
)
-
1
,
where α
>
0, such that
(143)
α
>
g
1
+
g
2
1
+
ν
g
1
-
h
1
+
ν
g
1
in [
x
p
,
m
]
𝕋. We choose d closely enough to m, such that 1
-
α
μ
>
0, 1
-
α
ν
>
0 on [
d
,
m
]
𝕋 and
(144)
w
(
x
)
∶
=
u
(
x
)
+
ɛ
z
(
x
)
,
where ɛ
>
0 such that
(145)
0
<
ɛ
<
M
-
u
(
d
)
z
(
d
)
.
Therefore, we have
(146)
L
[
z
]
=
z
Δ
∇
+
g
1
z
Δ
+
g
2
z
∇
+
h
z
=
[
α
2
(
1
+
ν
g
1
)
-
α
(
g
1
+
g
2
)
]
×
e
-
α
(
x
,
m
)
1
-
α
ν
+
h
(
e
-
α
(
x
,
m
)
-
1
)
=
α
{
α
-
g
1
+
g
2
1
+
ν
g
1
+
h
1
+
ν
g
1
(
1
-
α
ν
)
m
m
m
×
(
1
-
1
e
-
α
(
x
,
m
)
)
}
e
-
α
(
x
,
m
)
1
-
α
ν
(
1
+
ν
g
1
)
>
0
.
Thus, L
[
w
]
=
L
[
u
]
+
ɛ
L
[
z
]
>
0 on [
d
,
m
]
𝕋. By Theorem 38 we know that w cannot attain its maximum in (
d
,
m
)
𝕋. Note that
(147)
w
(
d
)
=
u
(
d
)
+
ɛ
z
(
d
)
<
u
(
d
)
+
M
-
u
(
d
)
=
M
;
we get that w
(
m
)
=
u
(
m
)
=
M is the maximum of w on [
d
,
m
]
𝕋. This implies that w
Δ
(
m
)
≥
0; however, we also have that
(148)
w
Δ
(
m
)
=
u
Δ
(
m
)
+
ɛ
z
Δ
(
m
)
=
ɛ
z
Δ
(
m
)
=
-
α
ɛ
<
0
.
This is a contradiction. The proof is completed.Corollary 48.
Assume thath
(
x
) is not always equal to 0 in Theorem 47; if u attains its nonnegative maximum M in (
a
,
b
)
𝕋, then the nonnegative maximum M
=
0. Especially, if u
(
a
)
≤
0, u
(
b
)
≤
0, then u
(
x
)
<
0 for x
∈
(
a
,
b
)
𝕋, unless u
(
x
)
≡
0.In Theorem47, if we take 𝕋
=
ℝ, we have the following corollary which is the result that appeared in [3].Corollary 49.
Assuming that the functionsg
,
h
:
[
a
,
b
]
→
ℝ onin (
a
,
b
) and h
(
x
)
≤
0 on [
a
,
b
], if u
(
x
)
∈
𝒟
(
Λ
) satisfies (
L
+
h
)
[
u
]
:
=
u
′′
+
g
u
′
+
h
u
≥
0 in (
a
,
b
), then u cannot attain its maximum M in (
a
,
b
), unless u
≡
M.In Theorem47, if we take 𝕋
=
ℤ, where ℤ is the set of all integral numbers, we can obtain the following new maximum principle for second-order mixed Δ and ∇ difference dynamic system.Corollary 50.
Assume that the functionsh, g
1, and g
2
:
[
a
,
b
]
ℤ
→
ℝ satisfy
(149)
1
+
g
1
(
k
)
>
0
,
1
-
g
2
(
k
)
>
0
,
∀
k
∈
(
a
,
b
)
ℤ
,
and h
(
k
)
≤
0 on [
a
,
b
]
ℤ. If u
(
x
)
∈
𝒟
(
Λ
) satisfies
(150)
∇
Δ
u
(
k
)
+
g
1
(
k
)
Δ
u
(
k
)
+
g
2
(
k
)
∇
u
(
k
)
+
h
(
k
)
u
(
k
)
≥
0
,
∀
k
∈
(
a
,
b
)
ℤ
,
then u cannot attain its maximum M in (
a
,
b
)
ℤ, unless u
≡
M.All of the above results investigate the behavior of functions inside the considered interval. Now, we will discuss the behavior of functions by providing the information about the boundary points.Theorem 51.
Letg
1, g
2, and h satisfy (91), (92), and h
(
x
)
≤
0. Assume that u
(
x
)
∈
𝒟
(
Λ
) is not constant, such that
(151)
(
L
+
h
)
[
u
]
∶
=
u
Δ
∇
+
g
1
u
Δ
+
g
2
u
∇
+
h
u
≥
0
,
for each x
∈
(
a
,
b
)
𝕋, u has unilateral derivative at points of a
,
b, and (93), (130) hold on (
a
,
b
)
𝕋.(1)
Ifu attains its nonnegative maximum at a point of a, then u
Δ
(
a
)
<
0;
(2)
Ifu attains its nonnegative maximum at a point of b, then u
∇
(
b
)
>
0.Proof.
We suppose thatu attains its nonnegative maximum M at a, that is, u
(
a
)
=
M, and there exists a point x
0
∈
[
a
,
b
]
𝕋, such that u
(
x
0
)
<
M; we define a function z
(
x
)
∈
𝒟
(
Λ
) by
(152)
z
(
x
)
=
e
α
(
x
,
a
)
-
1
,
where α
>
0. It is similar to the proof of Theorem 38; we can choose a larger enough α, such that
(153)
(
L
+
h
)
[
z
]
>
0
.
Moreover, we define a function w
(
x
)
∈
𝒟
(
Λ
) by
(154)
w
(
x
)
=
u
(
x
)
+
ɛ
z
(
x
)
,
where
(155)
0
<
ɛ
<
M
-
u
(
x
0
)
z
(
x
0
)
.
Thus
(156)
(
L
+
h
)
[
w
]
>
0
,
and by using Theorem 47 to w on [
a
,
x
0
]
𝕋, we get that w attains its maximum at a or x
0. Note that w
(
a
)
=
M
>
w
(
x
0
), and thus w attains its maximum at a. Therefore, unilateral derivative of w
(
a
) is not positive:
(157)
w
Δ
(
a
)
=
u
Δ
(
a
)
+
ɛ
z
Δ
(
a
)
≤
0
.
However,
(158)
z
Δ
(
a
)
=
α
>
0
,
and hence
(159)
u
Δ
(
a
)
<
0
.
If u
(
b
)
=
M, we can prove u
∇
(
b
)
>
0 as the similar way above. The proof is completed.In Theorem51, if we take 𝕋
=
ℝ, we have the following corollary which is the result that appeared in [3].Corollary 52.
Assuming that the functionsg
,
h
:
[
a
,
b
]
→
ℝ are bounded in (
a
,
b
) and h
(
x
)
≤
0 on [
a
,
b
], if u
(
x
)
∈
𝒟
(
Λ
) satisfies (
L
+
h
)
[
u
]
∶
=
u
′
′
+
g
u
′
+
h
u
≥
0 in (
a
,
b
), and u has unilateral derivative at points of a
,
b.(1)
Ifu attains its nonnegative maximum at a point of a, then u
′
(
a
)
<
0;
(2)
Ifu attains its nonnegative maximum at a point of b, then u
′
(
b
)
>
0.In Theorem51, if we take 𝕋
=
ℤ, where ℤ is the set of all integral numbers, we can obtain the following new maximum principle for second-order mixed Δ and ∇ difference dynamic system.Corollary 53.
Assume that the functionsh, g
1, and g
2
:
[
a
,
b
]
ℤ
→
ℝ satisfy
(160)
1
+
g
1
(
k
)
>
0
,
1
-
g
2
(
k
)
>
0
,
∀
k
∈
(
a
,
b
)
ℤ
,
and h
(
k
)
≤
0 on [
a
,
b
]
ℤ, and assume that u
(
x
)
∈
𝒟
(
Λ
) satisfies
(161)
∇
Δ
u
(
k
)
+
g
1
(
k
)
Δ
u
(
k
)
+
g
2
(
k
)
∇
u
(
k
)
+
h
(
k
)
u
(
k
)
≥
0
,
∀
k
∈
(
a
,
b
)
ℤ
.
(1)
Ifu attains its nonnegative maximum at a point of a, then u
(
a
+
1
)
<
u
(
a
);
(2)
Ifu attains its nonnegative maximum at a point of b, then u
(
b
-
1
)
>
u
(
b
).Next, we consider that(
L
+
h
)
[
u
]
≥
0, and h
(
x
) may take positive value. Let g
1, g
2, and h
(
x
) satisfy (91) and (92). Assume that we can find a function w
(
x
)
∈
𝒟
(
Λ
) which satisfies
(162)
m
m
m
w
(
x
)
>
0
for
each
x
∈
Λ
,
(
L
+
h
)
[
w
]
≤
0
for
each
x
∈
(
a
,
b
)
𝕋
.
Then there exists a function F which is predifferentiable with region of differentiation Λ such that
(163)
F
∇
(
x
)
=
w
(
x
)
∀
x
∈
Λ
,
and therefore, w
σ
∇
(
x
)
=
F
Δ
∇
(
x
) since w
σ
(
x
)
=
F
∇
(
σ
(
x
)
)
=
F
Δ
(
x
).We define a new functionv
(
x
)
∈
𝒟
(
Λ
) by
(164)
v
(
x
)
=
u
(
x
)
w
(
x
)
,
and then v
(
x
) satisfies
(165)
v
Δ
∇
+
F
Δ
∇
+
g
1
w
σ
w
σ
ρ
v
Δ
+
w
Δ
ρ
+
g
2
w
ρ
w
σ
ρ
v
∇
+
1
w
σ
ρ
(
L
+
h
)
[
w
]
v
≥
0
.Lemma 54.
Assume that (91), (92), and (93) hold, and a function w
(
x
) satisfies (162). Then, the following inequalities hold in (
a
,
b
)
𝕋;(
a
)
(
1
/
w
σ
ρ
)
(
L
+
h
)
[
w
]
≤
0,
(
b
)
1
+
ν
(
(
F
Δ
∇
+
g
1
w
σ
)
/
w
σ
ρ
)
>
0,
(
c
)
-
1
+
ν
(
(
w
Δ
ρ
+
g
2
w
ρ
)
/
w
σ
ρ
)
<
0.
Moreover, ifg
1, g
2, w
Δ
∇, and h
(
x
) are bounded on (
a
,
b
)
𝕋, and there exist t
1
>
0, such that
(166)
1
+
ν
g
1
≥
t
1
>
0
,
o
n
(
a
,
b
)
𝕋
,
then (
G
1
+
G
2
)
/
(
1
+
ν
G
1
) and H
/
(
1
+
ν
G
1
) are bounded on (
a
,
b
)
𝕋, where
(167)
G
1
=
F
Δ
∇
+
g
1
w
σ
w
σ
ρ
,
G
2
=
w
Δ
ρ
+
g
2
w
ρ
w
σ
ρ
,
m
m
m
m
m
m
H
=
1
w
σ
ρ
(
L
+
h
)
[
w
]
.Proof.
(a) Since(168)
(
L
+
h
)
[
w
]
≤
0
,
w
(
x
)
>
0
,
then
(169)
1
w
σ
ρ
(
L
+
h
)
[
w
]
≤
0
.
If ν
(
x
)
=
0, then (b) and (c) are satisfied at x, and so we suppose that ν
(
x
)
>
0.
(b) It is easy to see that(170)
ν
F
Δ
∇
(
x
)
=
ν
F
Δ
(
x
)
-
F
Δ
(
ρ
(
x
)
)
ν
=
F
Δ
(
x
)
-
F
∇
(
x
)
=
w
σ
(
x
)
-
w
(
x
)
.
Since ν
(
x
)
>
0, and hence
(171)
σ
(
ρ
(
x
)
)
=
x
,
then
(172)
w
(
x
)
=
w
σ
ρ
(
x
)
,
and, therefore,
(173)
1
+
ν
F
Δ
∇
+
g
1
w
σ
w
σ
ρ
=
1
+
ν
w
σ
∇
+
ν
g
1
w
σ
w
σ
ρ
=
1
+
w
σ
-
w
σ
ρ
+
ν
g
1
w
σ
w
σ
ρ
=
(
1
+
ν
g
1
)
w
σ
w
σ
ρ
>
0
.
(c) Finally,(174)
ν
w
Δ
ρ
=
[
x
-
ρ
(
x
)
]
w
σ
(
ρ
(
x
)
)
-
w
(
ρ
(
x
)
)
σ
(
ρ
(
x
)
)
-
ρ
(
x
)
.
Since ν
(
x
)
>
0, and hence
(175)
σ
(
ρ
(
x
)
)
=
x
,
thus
(176)
-
1
+
ν
w
Δ
ρ
+
g
2
w
ρ
w
σ
ρ
=
-
w
σ
ρ
+
w
σ
ρ
-
w
ρ
+
ν
g
2
w
ρ
w
σ
ρ
=
(
-
1
+
ν
g
2
)
w
ρ
w
σ
ρ
<
0
.
The boundness of (
G
1
+
G
2
)
/
(
1
+
ν
G
1
) and H
/
(
1
+
ν
G
1
) can be deduced by (166) and (173). The proof is completed.From Theorem47, Theorem 51, and Lemma 54 we obtain the following theorem.Theorem 55.
Assume thatu
(
x
)
∈
𝒟
(
Λ
) satisfies (
L
+
h
)
[
u
]
≥
0, and let g
1, g
2, w
Δ
∇, and h
(
x
) be bounded on (
a
,
b
)
𝕋, such that (92) and (166) hold. Assume that there exists a function w
(
x
)
∈
𝒟
(
Λ
) satisfying (162). If v
(
x
)
=
u
(
x
)
/
w
(
x
) attains its nonnegative maximum M in (
a
,
b
)
𝕋, then v
(
x
)
=
u
(
x
)
/
w
(
x
)
≡
M. If v
(
a
)
=
M, then v
Δ
(
a
)
<
0; if v
(
b
)
=
M, then v
∇
(
b
)
>
0.In Theorem55, if we take 𝕋
=
ℝ, we have the following corollary which is the result that appeared in [3].Corollary 56.
Assume that the functionsg
,
h
:
[
a
,
b
]
→
ℝ are bounded in (
a
,
b
), and u
(
x
)
∈
𝒟
(
Λ
) satisfies (
L
+
h
)
[
u
]
∶
=
u
′
′
+
g
u
′
+
h
u
≥
0 in (
a
,
b
). Assume that there exists a function w
(
x
)
∈
𝒟
(
Λ
) such that
(177)
m
m
m
w
(
x
)
>
0
f
o
r
e
a
c
h
x
∈
Λ
,
(
L
+
h
)
[
w
]
≤
0
f
o
r
e
a
c
h
x
∈
(
a
,
b
)
.
If v
(
x
)
=
u
(
x
)
/
w
(
x
) attains its nonnegative maximum M in (
a
,
b
), then v
(
x
)
=
u
(
x
)
/
w
(
x
)
≡
M. If v
(
a
)
=
M, then v
′
(
a
)
<
0; if v
(
b
)
=
M, then v
′
(
b
)
>
0.In Theorem55, if we take 𝕋
=
ℤ, where ℤ is the set of all integral numbers, we can obtain the following new maximum principle for second-order mixed Δ and ∇ difference inequality.Corollary 57.
Assume that the functionsh, w, g
1, and g
2
:
[
a
,
b
]
ℤ
→
ℝ satisfy w
(
k
)
>
0
,
∀
k
∈
[
a
,
b
]
ℤ,
(178)
1
+
g
1
(
k
)
>
0
,
1
-
g
2
(
k
)
>
0
,
∀
k
∈
(
a
,
b
)
ℤ
,
∇
Δ
w
(
k
)
+
g
1
(
k
)
Δ
w
(
k
)
+
g
2
(
k
)
∇
w
(
k
)
+
h
(
k
)
w
(
k
)
≤
0
,
∀
k
∈
(
a
,
b
)
ℤ
.
Let u
(
x
)
∈
𝒟
(
Λ
) satisfy
(179)
∇
Δ
u
(
k
)
+
g
1
(
k
)
Δ
u
(
k
)
+
g
2
(
k
)
∇
u
(
k
)
+
h
(
k
)
u
(
k
)
≥
0
,
∀
k
∈
(
a
,
b
)
ℤ
.
If v
(
x
)
=
u
(
x
)
/
w
(
x
) attains its nonnegative maximum M in (
a
,
b
)
ℤ, then v
(
x
)
=
u
(
x
)
/
w
(
x
)
≡
M. If v
(
a
)
=
M, then v
(
a
+
1
)
<
v
(
a
); if v
(
b
)
=
M, then v
(
b
-
1
)
>
v
(
b
).To show the value of Theorem55, we need the following definition.Definition 58.
One says thatx
0 is a change sign point of u, if there exist δ
>
0 and η
>
0, such that u
(
x
) has different sign on (
x
0
-
δ
,
x
0
)
𝕋 and [
x
0
,
x
0
+
η
)
𝕋, that is, either u
(
x
)
>
0 on (
x
0
-
δ
,
x
0
)
𝕋 and u
(
x
)
≤
0 on [
x
0
,
x
0
+
η
)
𝕋 or u
(
x
)
<
0 on (
x
0
-
δ
,
x
0
)
𝕋 and u
(
x
)
≥
0 on [
x
0
,
x
0
+
η
)
𝕋.Remark 59.
Theorem55 shows that a function u which satisfies (151) cannot oscillate too rapidly. In fact, assuming that u
>
0 between two of its change sign points x
=
a, x
=
b, then u
/
w must have a positive maximum between them. Hence, Theorem 55 will be violated. Thus, we have the following corollary.Corollary 60.
Assumingu
(
x
)
∈
𝒟
(
Λ
) satisfies (
L
+
h
)
[
u
]
≥
0, then u can have at most two change sign points (between which u is negative) in any interval (
a
,
b
)
𝕋 in which Theorem 55 holds.By applying the same reasoning to bothu and -
u, we can obtain the following corollary.Corollary 61.
Ifu
(
x
) is a solution of equation u
Δ
∇
+
g
1
u
Δ
+
g
2
u
∇
+
h
u
=
0, then u can have at most one change sign point in any interval (
a
,
b
)
𝕋 in which Theorem 55 holds.Theorem55 depends on the existence of the function w, and now, we discuss the existence of the function w.Lemma 62.
Assume thatg
1
(
x
), g
2
(
x
), and h
(
x
) satisfy the suppositions of Theorem 55, and there are positive numbers M
,
L, such that the following properties hold.(
a
)
2
M
<
1
(
b
-
a
)
2.
(
b
)
min
⁡
x
∈
[
a
,
b
]
𝕋
{
1
+
g
1
(
x
)
∫
σ
(
a
)
x
‍
∇
q
+
g
2
(
x
)
∫
σ
(
a
)
ρ
(
x
)
‍
∇
q
+
h
(
x
)
∫
a
x
‍
(
∫
σ
(
a
)
y
‍
∇
q
)
Δ
y
}
>
L
/
2.
(
c
)
|
h
(
x
)
|
≤
M.
Then there exists a functionw
(
x
) satisfying (162) and w
Δ
∇ is bounded in (
a
,
b
)
𝕋.Proof.
We can choose(180)
w
(
x
)
=
L
-
β
∫
a
x
‍
(
∫
σ
(
a
)
y
‍
∇
q
)
Δ
y
,
and then
(181)
w
∇
(
x
)
=
-
β
∫
σ
(
a
)
ρ
(
x
)
‍
∇
q
,
w
Δ
(
x
)
=
-
β
∫
σ
(
a
)
x
‍
∇
q
,
w
Δ
∇
(
x
)
=
-
β
.
Moreover,
(182)
(
L
+
h
)
[
w
]
=
-
β
{
1
+
g
1
∫
σ
(
a
)
x
‍
∇
q
+
g
2
∫
σ
(
a
)
ρ
(
x
)
‍
∇
q
m
m
m
m
+
h
∫
a
x
‍
(
∫
σ
(
a
)
y
‍
∇
q
)
Δ
y
}
+
h
L
.
Since |
h
(
x
)
|
≤
M,
(183)
min
⁡
x
∈
[
a
,
b
]
𝕋
{
1
+
g
1
∫
σ
(
a
)
x
‍
∇
q
+
g
2
∫
σ
(
a
)
ρ
(
x
)
‍
∇
q
m
m
m
m
+
h
∫
a
x
‍
(
∫
σ
(
a
)
y
‍
∇
q
)
Δ
y
}
>
L
2
.
Then ∀
β
:
2
M
<
β
<
(
L
/
(
b
-
a
)
2
); we have
(184)
(
L
+
h
)
[
w
]
≤
0
,
β
∫
a
x
‍
(
∫
σ
(
a
)
y
‍
Δ
q
)
∇
y
<
L
,
and hence
(185)
w
(
x
)
≥
L
-
β
(
b
-
a
)
2
>
0
.Lemma 63.
Letr
(
x
) be a solution of equation
(186)
r
Δ
∇
+
g
1
r
Δ
+
g
2
r
∇
+
h
r
=
0
,
where g
1, g
2, h, and w satisfy the conditions of Theorem 55. If r is not identically zero and
(187)
r
(
a
)
=
0
,
then r cannot vanish in some right neighbourhood of a.Proof.
Ifa is right-scattered, then r
(
σ
(
a
)
)
≠
0. Otherwise, we have that r
Δ
(
a
)
=
0; this shows that
(188)
v
(
a
)
=
0
,
v
Δ
(
a
)
=
r
Δ
(
a
)
w
(
a
)
-
r
(
a
)
w
Δ
(
a
)
w
(
a
)
w
(
σ
(
a
)
)
=
0
.
Then we can obtain v
(
x
)
≡
0. In fact, according to Theorem 55, v
=
r
/
w cannot attain its maximum nor minimum at a. If v attains its maximum in (
a
,
b
)
𝕋, then v
≡
0 since v
(
a
)
=
0. If v attains its maximum at b, hence -
v attains its maximum in (
a
,
b
)
𝕋. Next we apply Theorem 47 to -
v and obtain that -
v
(
x
) is constant; then v
(
x
)
≡
0 since -
v
(
a
)
=
0. Thus, in all cases we get that v
(
x
)
≡
0; this implies that r
(
x
)
≡
0 which is contradiction with the assumption.
Ifa is right-dense, we obtain that r cannot vanish in some right neighbourhood of a. In fact, if it is not so, then there exists a sequence t
n
→
a
+, and r
(
t
n
)
=
0; then r
Δ
(
a
)
=
lim
⁡
n
→
∞
(
r
(
a
)
-
r
(
t
n
)
)
/
(
a
-
t
n
)
=
0. Again we obtain that r
(
x
)
≡
0 by a similar proof of above, which is contradiction with the assumption. Thus, r cannot vanish in some right neighbourhood of a.Remark 64.
Under the conditions of Lemma63, if r has any change sign point at the right of a, we denote the first one by a
* and call it the conjugate change sign point of a. Thus, r does not change its sign in the interval (
a
,
a
*
)
𝕋. Without loss of the generality, we assume that
(189)
r
(
x
)
>
0
for
x
∈
(
a
,
a
*
)
𝕋
.
Then function r
/
w is positive in (
a
,
a
*
)
𝕋 and a
* is also a change sign point of r
/
w. By the definition of change sign point, we have that r
(
a
*
)
/
w
(
a
*
)
≤
0. Hence, r
/
w has a maximum in (
a
,
a
*
)
𝕋. Therefore by Theorem 55, w cannot satisfy (
L
+
h
)
[
w
]
≤
0. That is, under these cases, there is no function w satisfying the condition of Theorem 51.
On the other hand, ifb is any point in (
a
,
a
*
)
𝕋, a function w can be found so that r
/
w satisfies the maximum principle of Theorem 55. To see this, we observe first that r
(
x
) is bounded from below by a positive number on any subinterval [
c
,
b
]
𝕋 contained in (
a
,
a
*
)
𝕋. Consequently, for sufficiently small ɛ
>
0, the function w
(
x
)
=
r
(
x
)
+
ɛ
(
2
-
e
α
(
x
,
a
)
) is positive on [
a
,
b
]
𝕋. If α is selected so that (
L
+
h
)
[
2
-
e
α
(
x
,
a
)
]
≤
0 in (
a
,
b
)
𝕋, then w is a function for which Theorem 55 holds. Thus, we get the following result.Theorem 65.
Ifa
* is the conjugate change sign point of a, letting g
1, g
2, and h
(
x
) be bounded on (
a
,
b
)
𝕋, such that (92) and (166) hold, then there exists a w
(
x
)
>
0 such that Theorem 55 holds on the interval [
a
,
b
]
𝕋 if and only if b
<
a
*. If r
(
x
) (the solution of (186) which satisfies r
(
a
)
=
0) has no change sign point at the right of a, one sets a
*
=
∞, and Theorem 55 holds on every interval [
a
,
b
]
𝕋.In Theorem65, if we take 𝕋
=
ℝ, we have the following corollary which is the result that appeared in [3].Corollary 66.
Assume thata
* is the conjugate change sign point of a, and the functions g
,
h
:
[
a
,
b
]
→
ℝ are bounded in (
a
,
b
); then there exists a function w
(
x
)
>
0 such that Corollary 52 holds on the interval [
a
,
b
] if and only if b
<
a
*. If r
(
x
) (the solution of r
′′
+
g
r
′
+
h
r
=
0, which satisfies r
(
a
)
=
0) has no change sign point at the right of a, one sets a
*
=
∞, and Corollary 52 holds on every interval [
a
,
b
].In Theorem65, if we take 𝕋
=
ℤ, where ℤ is the set of all integral numbers, we can obtain the following new maximum principle for second-order mixed Δ and ∇ difference inequality.Corollary 67.
Assuming that the functionsh
,
w
,
g
1, and g
2
:
[
a
,
b
]
ℤ
→
ℝ satisfy
(190)
1
+
g
1
(
k
)
>
0
,
1
-
g
2
(
k
)
≥
0
,
∀
k
∈
(
a
,
b
)
ℤ
,
then there exists a function w
(
k
)
>
0, ∀
k
∈
[
a
,
b
]
ℤ, such that Corollary 53 holds on the interval [
a
,
b
]
ℤ if and only if b
<
a
*. If r
(
x
) (the solution of the equation
(191)
∇
Δ
r
(
k
)
+
g
1
(
k
)
Δ
r
(
k
)
+
g
2
(
k
)
∇
r
(
k
)
+
h
(
k
)
r
(
k
)
=
0
,
which satisfies r
(
a
)
=
0) has no change sign point at the right of a, one sets a
*
=
∞, and Corollary 53 holds on every interval [
a
,
b
]
ℤ.


## 4. Applications to Initial Value Problems

In this section, as an application of the maximum principles established in section three, firstly, we will prove some uniqueness theorem of the solution for initial value problem:(192)
u
Δ
∇
+
g
1
u
Δ
+
g
2
u
∇
+
h
u
=
f
(
x
)
on
⁡
(
a
,
b
)
𝕋
,
u
(
a
)
=
γ
1
,
u
Δ
(
a
)
=
γ
2
,
in 𝒟
(
Λ
). Secondly, we will discuss the existence of the lower and upper solutions of (192). Thirdly, we will give a general scheme for obtaining upper and lower solutions.Theorem 68.
Assume thatg
1, g
2, and h satisfy (91) and (92), and h
≤
0, (93), (130) hold on (
a
,
b
)
𝕋. If u
1 and u
2 are solutions of the initial value problem (192), then u
1
≡
u
2.Proof.
We define a functionv
(
x
)
∈
𝒟
(
Λ
) by
(193)
v
(
x
)
=
u
1
(
x
)
-
u
2
(
x
)
.
Since both u
1 and u
2 satisfy (192), the function v satisfies
(194)
v
Δ
∇
+
g
1
v
Δ
+
g
2
v
∇
+
h
v
=
0
,
v
(
a
)
=
0
,
v
Δ
(
a
)
=
0
.
According to Theorem 51, v cannot attain its maximum nor minimum at a. If v attains its maximum at an interior point of Λ
,
v
≡
0 since v
(
a
)
=
0. If v attains its maximum at b, hence -
v attains its maximum at an interior point of Λ. Next we apply Theorem 47 to -
v and obtain that -
v
(
x
) is constant; then v
(
x
)
≡
0 since -
v
(
a
)
=
0. The proof is completed.It follows from Theorem65; we get Theorem 69.Theorem 69.
Letg
1, g
2, and h
(
x
) be bounded on (
a
,
b
)
𝕋, such that (92) and (166) hold. Assuming that u
1 and u
2 are solutions of the initial value problem (192), if b
<
a
*, where a
* is the conjugate change sign point of a, then u
1
≡
u
2.More generally, we can prove the following theorem which shows that the conclusion of Theorem69 holds on any interval [
a
,
b
]
𝕋.Theorem 70.
Letg
1, g
2, and h
(
x
) be bounded on (
a
,
b
)
𝕋, such that (92) and (166) hold. Assuming that u
1 and u
2 are solutions of the initial value problem (192), then u
1
≡
u
2.Proof.
We define a functionu
(
x
)
∈
𝒟
(
Λ
) by
(195)
u
(
x
)
=
u
1
(
x
)
-
u
2
(
x
)
.
Since both u
1 and u
2 satisfy (192), the function u satisfies (194). We give our proof by two steps.
(1) Ifa is right-scattered, it follows from (194) that u
(
σ
(
a
)
)
=
0, u
∇
(
σ
(
a
)
)
=
u
Δ
(
a
)
=
0, and then we have that u
Δ
∇
(
σ
(
a
)
)
+
g
1
(
σ
(
a
)
)
u
Δ
(
σ
(
a
)
)
=
0. On the other hand,
(196)
u
Δ
∇
(
σ
(
a
)
)
=
u
Δ
(
a
)
-
u
Δ
(
σ
(
a
)
)
a
-
σ
(
a
)
,
this implies that
(197)
u
Δ
(
a
)
-
u
Δ
(
σ
(
a
)
)
a
-
σ
(
a
)
+
g
1
(
σ
(
a
)
)
u
Δ
(
σ
(
a
)
)
=
u
Δ
(
σ
(
a
)
)
[
1
+
ν
(
σ
(
a
)
)
g
1
(
σ
(
a
)
)
]
ν
(
σ
(
a
)
)
=
0
.
Note that (166); we know that 1
+
ν
(
σ
(
a
)
)
g
1
(
σ
(
a
)
)
>
0. This shows that u
Δ
(
σ
(
a
)
)
=
0. If a is right-dense, by using Lemma 62, there is a ɛ
>
0 enough small and a function w
(
x
)
>
0 on [
a
,
a
+
ɛ
]
𝕋. Let v
=
u
/
w; then
(198)
m
m
m
m
m
m
m
m
l
v
(
a
)
=
0
,
v
Δ
(
a
)
=
u
Δ
(
a
)
w
(
a
)
-
u
(
a
)
w
Δ
(
a
)
w
(
a
)
w
(
σ
(
a
)
)
=
0
,
v
Δ
∇
+
F
Δ
∇
+
g
1
w
σ
w
σ
ρ
v
Δ
+
w
Δ
ρ
+
g
2
w
ρ
w
σ
ρ
v
∇
+
1
w
σ
ρ
(
L
+
h
)
[
w
]
v
=
0
.
According to Theorem 55, v
=
r
/
w cannot attain its maximum nor minimum at a. If v attains its maximum in (
a
,
a
+
ɛ
)
𝕋, then v
≡
0 since v
(
a
)
=
0. If v attains its maximum at a
+
ɛ, hence -
v attains its maximum in (
a
,
a
+
ɛ
)
𝕋. Next we apply Theorem 55 to -
v and obtain that -
v
(
x
) is constant; then v
(
x
)
≡
0 since -
v
(
a
)
=
0. Thus, in all cases we get that v
(
x
)
≡
0; this implies that u
(
x
)
≡
0 on [
a
,
a
+
ɛ
]
𝕋. Thus, we can get that u
Δ
(
x
)
≡
0 on [
a
,
a
+
ɛ
)
𝕋. If a
+
ɛ is left-dense, by using the continuous u
Δ
(
x
), we have that u
Δ
(
a
+
ɛ
)
=
0. If a
+
ɛ is left-scattered, then u
Δ
(
ρ
(
a
+
ɛ
)
)
=
0, and ρ
(
a
+
ɛ
) is right-scattered; then similar to the above proof of u
Δ
(
σ
(
a
)
)
=
0, we have that u
Δ
(
a
+
ɛ
)
=
u
Δ
(
σ
(
ρ
(
a
+
ɛ
)
)
)
=
0. Synthesizing the above proof, we have proven that there exists ɛ
>
0 (if a is right-scattered, ɛ
=
μ
(
a
)), such that u
(
x
)
≡
0 and u
Δ
(
x
)
≡
0 on [
a
,
a
+
ɛ
]
𝕋.
(2) Let(199)
λ
=
sup
⁡
{
t
∈
(
a
,
b
]
𝕋
:
u
(
x
)
≡
0
,
u
Δ
(
x
)
≡
0
on
⁡
[
a
,
t
)
𝕋
}
.
If λ
=
b, then the conclusion of Theorem 70 will be proved. If λ
<
b, then u
(
x
)
≡
0 and u
Δ
(
x
)
≡
0 on [
a
,
λ
)
𝕋. If λ is left-dense, by using the continuous u
Δ
(
x
), we have that u
(
λ
)
=
0 and u
Δ
(
λ
)
=
0. If λ is left-scattered, then u
Δ
(
ρ
(
λ
)
)
=
0, and ρ
(
λ
) is right-scattered; then similar to the above proof of u
Δ
(
σ
(
a
)
)
=
0, we can prove that u
Δ
(
λ
)
=
u
Δ
(
σ
(
ρ
(
λ
)
)
)
=
0. By using the conclusion of step (1), we have that there exists ɛ
>
0 such that u
(
x
)
≡
0 and u
Δ
(
x
)
≡
0 on [
λ
,
λ
+
ɛ
]
𝕋. Then we get that u
(
x
)
≡
0 and u
Δ
(
x
)
≡
0 on [
a
,
λ
+
ɛ
]
𝕋. This is a contradiction with the definition of λ. Thus, λ
=
b; this shows that u
1
≡
u
2 on [
a
,
b
]
𝕋. The proof is completed.Remark 71.
Theorems68 and 69 show that (
L
+
h
)
[
u
]
=
u
Δ
∇
+
g
1
u
Δ
+
g
2
u
∇
+
h
u
=
f
(
x
), at most, has one solution satisfying u
(
a
)
=
γ
1, u
Δ
(
a
)
=
γ
2. On the other hand, in many cases, it is difficult to find a solution of the initial value problem directly, and therefore, it becomes important to find a lower and upper solution.Assume thatg
1, g
2, and h are bounded on (
a
,
b
)
𝕋, h
(
x
)
≤
0 on (
a
,
b
)
𝕋 and satisfy (91), (92), (93), and (130) for each x
∈
(
a
,
b
)
𝕋. If we can find a function z
1
(
x
)
∈
𝒟
(
Λ
) satisfying
(200)
(
L
+
h
)
[
z
1
]
≥
f
(
x
)
,
(201)
z
1
(
a
)
≥
γ
1
,
z
1
Δ
(
a
)
≥
γ
2
,
we define a function v
1
(
x
)
∈
𝒟
(
Λ
) by
(202)
v
1
(
x
)
≡
z
1
(
x
)
-
u
(
x
)
,
where u
(
x
) is the solution of (192). Thus,
(203)
(
L
+
h
)
[
v
1
]
≥
0
,
v
1
(
a
)
≥
0
,
v
1
Δ
(
a
)
≥
0
.
Since v
1
(
a
)
≥
0, v
1
(
x
) has a nonnegative maximum at any interval [
a
,
x
0
]
𝕋, and using Theorem 47, we know that the maximum point must be a or x
0. However, v
1
Δ
(
a
)
≥
0, and from Theorem 51 maximum point cannot be a unless v
1
(
x
)
≡ constant. Thus, we obtain
(204)
max
⁡
x
∈
[
a
,
x
0
]
𝕋
v
1
(
x
)
=
v
1
(
x
0
)
≥
v
1
(
a
)
,
x
0
∈
(
a
,
b
)
𝕋
.
Since x
0
∈
(
a
,
b
)
𝕋 is arbitrary, we can deduce that
(205)
v
1
Δ
(
x
)
≥
0
,
x
∈
(
a
,
b
)
𝕋
.
Using x to take the place of x
0, inequality (204) implies
(206)
u
(
x
)
≤
γ
1
+
z
1
(
x
)
-
z
1
(
a
)
,
x
≥
a
,
and inequality (205) implies
(207)
u
Δ
(
x
)
≤
z
1
Δ
(
x
)
,
x
∈
[
a
,
b
)
𝕋
.
Since
(208)
z
1
(
a
)
-
γ
1
≥
0
,
inequality (206) implies
(209)
u
(
x
)
≤
z
1
(
x
)
.
Similarly, assume that we can find a function z
2
(
x
)
∈
𝒟
(
Λ
) satisfying
(210)
(
L
+
h
)
[
z
2
]
≤
f
(
x
)
,
(211)
z
2
(
a
)
≤
γ
1
,
z
2
Δ
(
a
)
≤
γ
2
.
The same as the above statement, define
(212)
v
2
(
x
)
≡
u
(
x
)
-
z
2
(
x
)
,
and we obtain
(213)
u
(
x
)
≥
z
2
(
x
)
,
u
Δ
(
x
)
≥
z
2
Δ
(
x
)
.
Therefore, we have established the following theorem, which gives a sufficient condition for the lower and upper solutions.Theorem 72.
Assume thatg
1, g
2, and h are bounded on (
a
,
b
)
𝕋, h
(
x
)
≤
0 on (
a
,
b
)
𝕋 and satisfy (91), (92), (93), and (130) for each x
∈
(
a
,
b
)
𝕋. Let u
(
x
)
∈
𝒟
(
Λ
) be a solution of (192). Let z
1
(
x
) and z
2
(
x
) satisfy (200), (201) and (210), (211). Then z
2
(
x
)
≤
u
(
x
)
≤
z
1
(
x
), z
2
Δ
(
x
)
≤
u
Δ
(
x
)
≤
z
1
Δ
(
x
).In the following, we will discuss the existence of the lower and upper solutions.Theorem 73.
Assume thatg
1, g
2, and h are bounded on (
a
,
b
)
𝕋, are bounded on (
a
,
b
)
𝕋, and satisfy (166), (92), (93), and (130).(1)
Iff
(
x
) is continuous on Λ, then there exist functions z
1
(
x
) and z
2
(
x
) which satisfy (200), (201) and (210), (211), respectively.
(2)
Moreover, ifh
(
x
)
≤
0 on (
a
,
b
)
𝕋, then z
2
(
x
)
≤
u
(
x
)
≤
z
1
(
x
), z
2
Δ
(
x
)
≤
u
Δ
(
x
)
≤
z
1
Δ
(
x
).Proof.
It follows from (166) that
(214)
1
+
ν
g
1
≥
t
1
>
0
,
on
⁡
(
a
,
b
)
𝕋
,
so we can select α
>
0 large enough, such that k
>
0, where k is defined by
(215)
k
=
min
⁡
x
∈
[
a
,
b
]
𝕋
[
(
1
+
ν
g
1
)
α
2
+
α
(
g
1
+
g
2
)
+
h
(
1
+
α
ν
(
x
)
)
]
×
e
α
(
x
,
a
)
1
+
α
ν
.
Let
(216)
A
=
max
⁡
{
γ
1
,
γ
2
α
,
1
k
·
max
⁡
x
∈
[
a
,
b
]
𝕋
{
f
(
x
)
}
,
0
}
;
we show that, under the stated assumptions, the function
(217)
z
1
(
x
)
∶
=
A
e
α
(
x
,
a
)
satisfies (200) and (201). To see that (200) is satisfied, we note that
(218)
(
L
+
h
)
[
z
1
]
=
A
[
(
1
+
ν
g
1
)
α
2
+
α
(
g
1
+
g
2
)
+
h
(
1
+
α
ν
)
]
×
e
α
(
x
,
a
)
1
+
α
ν
≥
A
k
≥
1
k
·
max
⁡
x
∈
[
a
,
b
]
𝕋
{
f
(
x
)
}
·
k
≥
f
(
x
)
.
To see that (201) is satisfied, we note that
(219)
z
1
(
a
)
=
A
≥
γ
1
,
z
1
Δ
(
a
)
=
A
α
≥
γ
2
.
Similarly, we can choose
(220)
z
2
(
x
)
∶
=
B
e
α
(
x
,
a
)
,
where
(221)
B
=
min
⁡
{
γ
1
,
γ
2
α
,
1
k
·
{
min
⁡
x
∈
[
a
,
b
]
𝕋
f
(
x
)
}
,
0
}
.
To see that (210) is satisfied, we note that
(222)
(
L
+
h
)
[
z
2
]
=
B
[
(
1
+
ν
g
1
)
α
2
+
α
(
g
1
+
g
2
)
+
h
(
1
+
α
ν
(
x
)
)
]
×
e
α
(
x
,
a
)
1
+
α
ν
≤
B
k
≤
1
k
·
min
⁡
x
∈
[
a
,
b
]
𝕋
{
f
(
x
)
}
·
k
≤
f
(
x
)
.
To see that (211) is satisfied note that
(223)
z
2
(
a
)
=
B
≤
γ
1
,
z
2
Δ
(
a
)
=
B
α
≤
γ
2
.
Thus, conclusion (1) holds. Conclusion (2) can be deduced from Theorem 72.
The proof is completed.As we all know, the accuracy of the approximation will depend on how well we can choose the functionsz
1
(
x
) and z
2
(
x
). So we next search for the following general scheme for obtaining upper and lower bounds. Suppose we divide the interval [
a
,
b
]
𝕋 into N subintervals
(224)
a
=
x
0
<
x
1
<
⋯
<
x
N
-
1
<
x
N
=
b
.
On each subinterval, we will select z
1
(
x
) as the following form:
(225)
c
∫
t
x
‍
(
∫
t
y
‍
∇
q
)
Δ
y
+
d
∫
t
x
‍
∇
q
+
e
and choose the coefficients c
,
d
,
e so that z
1
(
a
)
=
γ
1, z
1
Δ
(
a
)
=
γ
2, and z
1
∈
𝒟
(
Λ
). Also, z
1 will be selected so that inequality (200) holds in each subinterval (
x
i
-
1
,
x
i
)
𝕋. We set
(226)
z
1
(
x
)
=
c
i
∫
x
i
x
‍
(
∫
x
i
y
‍
∇
q
)
Δ
y
+
d
i
∫
x
i
x
‍
∇
q
+
e
i
for
x
∈
[
x
i
,
x
i
+
1
)
𝕋
,
i
=
0,1
,
…
,
N
-
1
.
The constants c
i
,
d
i
,
e
i
,
i
=
0,1
,
2
,
…
,
N
-
1, and the number N of subintervals will be chosen so that all the required conditions are satisfied. We proceed in a step by step manner starting with the interval (
x
0
,
x
1
)
𝕋. The initial conditions
(227)
z
1
(
a
)
=
γ
1
,
z
1
Δ
(
a
)
=
γ
2
,
require that e
0
=
γ
1 and d
0
=
γ
2. Next, we divide our proof into three parts.(
i
) If x
0 is right-scattered and σ
(
x
0
) is also right-scattered, we let x
1
=
σ
(
σ
(
x
0
)
), and then we only have one point σ
(
x
0
) in (
x
0
,
x
1
)
𝕋, and hence, in this point, the inequality
(228)
(
L
+
h
)
[
z
1
]
≥
f
(
x
)
becomes
(229)
c
0
[
1
+
g
1
σ
(
x
0
)
ν
(
σ
(
x
0
)
)
]
+
γ
2
(
g
1
σ
(
x
0
)
+
g
2
σ
(
x
0
)
)
+
h
(
σ
(
x
0
)
)
(
γ
2
∫
x
0
σ
(
x
0
)
‍
∇
q
+
γ
1
)
≥
f
(
σ
(
x
0
)
)
.
If 1
+
g
1
ν
≥
t
1
>
0 and g
1
,
g
2
,
h, and f are bounded, then c
0 can be properly selected so that (229) is an equality. Thus, in this case, z
1
(
x
) is a solution of (192) in (
x
0
,
x
1
)
𝕋.(
ii
) If x
0 is right-scattered and σ
(
x
0
) is right-dense, we let x
1
>
σ
(
x
0
), and then the inequality
(230)
(
L
+
h
)
[
z
1
]
≥
f
(
x
)
,
x
∈
(
x
0
,
x
1
)
𝕋
,
becomes
(231)
c
0
[
1
+
g
1
(
x
)
(
∫
x
0
x
‍
∇
q
)
+
g
2
(
x
)
(
∫
x
0
ρ
(
x
)
‍
∇
q
)
+
h
(
x
)
∫
x
0
x
‍
(
∫
x
0
y
‍
∇
q
)
Δ
y
]
+
γ
2
(
g
1
(
x
)
+
g
2
(
x
)
)
+
h
(
x
)
(
γ
2
∫
x
0
x
‍
∇
q
+
γ
1
)
≥
f
(
x
)
,
x
∈
(
x
0
,
x
1
)
𝕋
.
If x
=
σ
(
x
0
), we have that
(232)
c
0
[
1
+
g
1
(
σ
(
x
0
)
)
(
∫
x
0
σ
(
x
0
)
‍
∇
q
)
+
g
2
(
σ
(
x
0
)
)
(
∫
x
0
ρ
(
σ
(
x
0
)
)
‍
∇
q
)
+
h
(
σ
(
x
0
)
)
∫
x
0
σ
(
x
0
)
‍
(
∫
x
0
y
‍
∇
q
)
Δ
y
]
+
γ
2
(
g
1
(
σ
(
x
0
)
)
+
g
2
(
σ
(
x
0
)
)
)
+
h
(
σ
(
x
0
)
)
(
γ
2
∫
x
0
σ
(
x
0
)
‍
∇
q
+
γ
1
)
=
c
0
[
1
+
g
1
(
σ
(
x
0
)
)
ν
(
σ
(
x
0
)
)
]
+
γ
2
(
g
1
(
σ
(
x
0
)
)
+
g
2
(
σ
(
x
0
)
)
)
+
h
(
σ
(
x
0
)
)
(
γ
2
ν
(
σ
(
x
0
)
)
+
γ
1
)
≥
f
(
σ
(
x
0
)
)
.
Thus, if g
1, g
2, f, and h are bounded, then x
1 can be selected so close to σ
(
x
0
), and c
0 can be taken so large that (231) holds for x
∈
(
x
0
,
x
1
)
𝕋. Moreover, when x
1 is sufficiently close to σ
(
x
0
), we can properly select c
0, such that (231) is close to an equality; then z
1
(
x
) is also close to the solution of (192) in (
x
0
,
x
1
)
𝕋.(
iii
) If x
0 is right-dense, the inequality
(233)
(
L
+
h
)
[
z
1
]
≥
f
(
x
)
,
x
∈
(
x
0
,
x
1
)
𝕋
,
becomes
(234)
c
0
[
1
+
g
1
(
x
)
(
∫
x
0
x
‍
∇
q
)
+
g
2
(
x
)
(
∫
x
0
ρ
(
x
)
‍
∇
q
)
+
h
(
x
)
∫
x
0
x
‍
(
∫
x
0
y
‍
∇
q
)
Δ
y
]
+
γ
2
(
g
1
+
g
2
)
+
h
(
γ
2
∫
x
0
x
‍
∇
q
+
γ
1
)
≥
f
(
x
)
,
x
∈
(
x
0
,
x
1
)
𝕋
.
If g
1, g
2, and h are bounded, then x
1 can be selected so close to x
0 that
(235)
[
1
+
g
1
(
x
)
(
∫
x
0
x
‍
∇
q
)
+
g
2
(
x
)
(
∫
x
0
ρ
(
x
)
‍
∇
q
)
+
h
(
x
)
∫
x
0
x
‍
(
∫
x
0
y
‍
∇
q
)
Δ
y
]
≥
l
>
0
,
for
x
∈
(
x
0
,
x
1
)
𝕋
,
where l
>
0 is a positive constant. If, in addition, f is bounded, then c
0 can be taken so large that (234) holds for all x in (
x
0
,
x
1
)
𝕋. Moreover, when x
1 is sufficiently close to x
0, we can properly select c
0, such that (234) is close to an equality; then z
1
(
x
) is also close to the solution of (192) in (
x
0
,
x
1
)
𝕋.Following all of the above proof, we have proved that there exists anx
1
>
x
0 and a large enough c
0, such that (200) holds for all x in (
x
0
,
x
1
)
𝕋 for
(236)
z
1
(
x
)
=
c
0
∫
x
0
x
‍
(
∫
x
0
y
‍
∇
q
)
Δ
y
m
m
m
m
+
γ
2
∫
x
0
x
‍
∇
q
+
γ
1
for
x
∈
(
x
0
,
x
1
)
𝕋
.We now turn to the interval(
x
1
,
x
2
)
𝕋, with z
1
(
x
) being defined by
(237)
z
1
(
x
)
=
c
1
∫
x
1
x
‍
(
∫
x
1
y
‍
∇
q
)
Δ
y
+
d
1
∫
x
1
x
‍
∇
q
+
e
1
for
x
∈
(
x
1
,
x
2
)
𝕋
.To insure the continuity ofz
1, z
1
Δ, and z
1
∇ at x
1, we choose
(238)
e
1
=
c
0
∫
x
0
x
1
‍
(
∫
x
0
y
‍
∇
q
)
Δ
y
+
γ
2
∫
x
0
x
1
‍
∇
q
+
γ
1
,
d
1
=
c
0
∫
x
0
x
1
‍
∇
q
+
γ
2
.
In fact, by computing we get that
(239)
lim
⁡
x
→
x
1
z
1
(
x
)
=
e
1
=
z
1
(
x
1
)
,
lim
⁡
x
→
x
1
z
1
Δ
(
x
)
=
d
1
=
c
0
∫
x
0
x
1
‍
∇
q
+
γ
2
=
z
1
Δ
(
x
1
)
,
lim
⁡
x
→
x
1
+
z
1
∇
(
x
)
=
d
1
=
c
0
∫
x
0
x
1
‍
∇
q
+
γ
2
,
lim
⁡
x
→
x
1
-
z
1
∇
(
x
)
=
c
0
∫
x
0
ρ
(
x
1
)
‍
∇
q
+
γ
2
=
z
1
∇
(
x
1
)
.
Thus, z
1
,
z
1
Δ are continuous at x
1, and z
1
∇ is left-dense continuous at x
1. In the interval (
x
1
,
x
2
)
𝕋, we apply the same reasoning of (
x
0
,
x
1
)
𝕋 to (
x
1
,
x
2
)
𝕋 and get that there exists an x
2
>
x
1 and a large enough c
1, such that (200) holds for all x in (
x
1
,
x
2
)
𝕋.Proceeding in this fashion, we determine eachd
i
,
e
i so that z
1 and z
1
Δ are continuous everywhere; z
1
∇ is left-dense continuous everywhere, and if x
i is a left-dense point, we always take interval (
x
i
,
x
i
+
1
)
𝕋 so small, such that the coefficient of c
i satisfies:
(240)
[
1
+
g
1
(
x
)
(
∫
x
i
x
‍
∇
q
)
+
g
2
(
x
)
(
∫
x
i
ρ
(
x
)
‍
∇
q
)
+
h
(
x
)
∫
x
i
x
‍
(
∫
x
i
y
‍
∇
q
)
Δ
y
]
≥
l
i
>
0
,
for
x
∈
(
x
0
,
x
1
)
𝕋
,
where l
i
>
0 is a positive constant. Also, we take the constant c
i to be large enough, so that (
L
+
h
)
[
z
1
]
≥
f
(
x
) holds on (
x
i
,
x
i
+
1
)
𝕋. In fact, the quantities d
i
,
e
i are determined by the recursion formulas
(241)
e
i
=
c
i
-
1
∫
x
i
-
1
x
i
‍
(
∫
x
i
-
1
y
‍
∇
q
)
Δ
y
+
d
i
-
1
∫
x
i
-
1
x
i
‍
∇
q
+
e
i
-
1
,
d
i
=
c
i
-
1
∫
x
i
-
1
x
‍
∇
q
+
d
i
-
1
.
In an actual computation to determine the c
i, it is convenient to replace f by its maximum in the ith subinterval and to replace g
1, g
2, and h by either their maximum or minimum, whichever may be appropriate for making (
L
+
h
)
[
z
1
]
≥
f
(
x
) throughout.In a similar manner we may construct lower bounds. The constantsd
i, e
i are selected in exactly the same way, and the quantities -
c
i are taken so large that (
L
+
h
)
[
z
2
]
≤
f
(
x
) holds everywhere.Iff, g
1, g
2, and h are continuous, by the above process, it can be shown that, as the maximum length of the subintervals, the upper and lower bounds both tend to the solution u. The above discussion leads to the following theorem.Theorem 74.
Assume thatg
1, g
2, and h are bounded and continuous on (
a
,
b
)
𝕋 and satisfy (166), (92), (93), and (130). If f
(
x
) is continuous on Λ and h
(
x
)
≤
0 on (
a
,
b
)
𝕋, then there exist the upper solution sequence {
z
1
(
n
)
(
x
)
} and lower solution {
z
2
(
n
)
(
x
)
}; they both tend to the solution u of (192).Thus far in this section, we have assumed thath
(
x
)
≤
0. We now take up the problem of approximating the solution of the equation
(242)
(
L
+
h
)
[
u
]
=
u
Δ
∇
+
g
1
u
Δ
+
g
2
u
∇
+
h
u
=
f
(
x
)
with initial conditions
(243)
u
(
a
)
=
γ
1
,
u
Δ
(
a
)
=
γ
2
when the function h
(
x
) may be positive. Under these circumstances we employ the generalized maximum principle (Theorem 51). To do so, we suppose that there is a function w which is positive on [
a
,
b
]
𝕋 and which has property that
(244)
(
L
+
h
)
[
w
]
≤
0
for
each
x
∈
[
a
,
b
]
𝕋
.
For example, we can take the function
(245)
w
(
x
)
=
L
-
β
∫
a
x
‍
(
∫
σ
(
a
)
y
‍
∇
q
)
Δ
y
,
defined in Lemma 62.We saw in Section3 that v
=
u
/
w satisfies an equation of the form
(246)
(
L
¯
+
H
)
[
v
]
=
v
Δ
∇
+
G
1
v
Δ
+
G
2
v
∇
+
H
v
=
f
w
σ
ρ
with G
1
=
(
F
Δ
∇
+
g
1
w
σ
)
/
w
σ
ρ, G
2
=
(
w
Δ
ρ
+
g
2
w
ρ
)
/
w
σ
ρ, H
=
(
1
/
w
σ
ρ
)
(
L
+
h
)
[
w
]
≤
0. Now, we define the comparison functions z
1
(
x
) and z
2
(
x
), so that z
1
/
w and z
2
/
w provide the bounds for u
/
w. First, we take z
1
(
x
) and z
2
(
x
), such that the inequalities
(247)
(
L
+
h
)
[
z
1
]
≥
f
(
x
)
,
z
1
(
a
)
≥
γ
1
,
z
1
Δ
(
a
)
w
(
a
)
-
z
1
(
a
)
w
Δ
(
a
)
≥
γ
2
w
(
a
)
-
γ
1
w
Δ
(
a
)
,
(
L
+
h
)
[
z
2
]
≤
f
(
x
)
,
z
2
(
a
)
≤
γ
1
,
z
2
Δ
(
a
)
w
(
a
)
-
z
2
(
a
)
w
Δ
(
a
)
≤
γ
2
w
(
a
)
-
γ
1
w
Δ
(
a
)
hold. Then, at x
=
a,
(248)
z
2
w
≤
u
w
≤
z
1
w
,
(
z
2
w
)
Δ
≤
(
u
w
)
Δ
≤
(
z
1
w
)
Δ
.
Moreover, it is easily seen by computing that
(249)
(
L
¯
+
H
)
[
z
2
w
]
≤
(
L
¯
+
H
)
[
u
w
]
≤
(
L
¯
+
H
)
[
z
1
w
]
.
Hence, if the conditions of Lemma 54 hold, by using Lemma 54 and Theorem 72, we know that, for x
∈
[
a
,
b
]
𝕋,
(250)
z
2
w
≤
u
w
≤
z
1
w
,
(
z
2
w
)
Δ
≤
(
u
w
)
Δ
≤
(
z
1
w
)
Δ
.
The first of these sets of inequalities gives the bounds
(251)
z
2
(
x
)
≤
u
(
x
)
≤
z
1
(
x
)
.The second set yields(252)
z
2
Δ
(
x
)
w
(
x
)
-
z
2
(
x
)
w
Δ
(
x
)
≤
u
Δ
(
x
)
w
(
x
)
-
u
(
x
)
w
Δ
(
x
)
≤
z
1
Δ
(
x
)
w
(
x
)
-
z
1
(
x
)
w
Δ
(
x
)
.
Since w is positive on [
a
,
b
]
𝕋, we find
(253)
z
2
Δ
(
x
)
+
w
Δ
(
x
)
w
(
x
)
[
u
(
x
)
-
z
2
(
x
)
]
≤
u
Δ
(
x
)
≤
z
1
Δ
(
x
)
-
w
Δ
(
x
)
w
(
x
)
×
[
z
1
(
x
)
-
u
(
x
)
]
.
If w
Δ
(
x
)
≤
0, we may substitute the upper bound of u
(
x
) as given in (251) into the left side of (253) and we may substitute the lower bound of u
(
x
) into the right side of (253). If w
Δ
(
x
)
≥
0, we use the lower bound of u
(
x
) on the left and the upper bound of u
(
x
) on the right. We thus find that
(254)
z
2
Δ
(
x
)
-
-
w
Δ
(
x
)
w
(
x
)
[
z
1
(
x
)
-
z
2
(
x
)
]
≤
u
Δ
(
x
)
≤
z
1
Δ
(
x
)
+
-
w
Δ
(
x
)
w
(
x
)
[
z
1
(
x
)
-
z
2
(
x
)
]
,
m
m
m
m
m
m
m
m
m
m
m
m
m
l
if
w
Δ
(
x
)
≤
0
,
z
2
Δ
(
x
)
≤
u
Δ
(
x
)
≤
z
1
Δ
(
x
)
,
m
m
m
m
l
l
l
l
if
w
Δ
(
x
)
≥
0
.
Inequalities (251) and (254) give the bounds for u
(
x
) and u
Δ
(
x
) which are more precise when z
1
(
x
)
-
z
2
(
x
) and z
1
Δ
(
x
)
-
z
2
Δ
(
x
) are smaller.It is always possible to find a positive functionw which satisfies (
L
+
h
)
[
w
]
≤
0 on a sufficiently small interval, but in general, there is no such function if the interval is too large. Once more we resort to breaking up the interval and piecing together functions defined on subintervals. Let w
>
0 and (
L
+
h
)
[
w
]
≤
0 on an interval [
a
,
x
*
]
𝕋, and let w
* be another positive function which satisfies (
L
+
h
)
[
w
*
]
≤
0 on an interval [
x
*
,
b
]
𝕋. We wish to find bounds for the solution u of the initial value problem (192), on the whole interval [
a
,
b
]
𝕋.Letz
1
(
x
) and z
2
(
x
) satisfy the conditions
(255)
(
L
+
h
)
[
z
2
]
≤
f
(
x
)
≤
(
L
+
h
)
[
z
1
]
on the interval [
a
,
x
*
]
𝕋, and
(256)
z
2
(
a
)
≤
γ
1
≤
z
1
(
a
)
,
z
2
Δ
(
a
)
w
(
a
)
-
z
2
(
a
)
w
Δ
(
a
)
≤
γ
2
w
(
a
)
-
γ
1
w
Δ
(
a
)
≤
z
1
Δ
(
a
)
w
(
a
)
-
z
1
(
a
)
w
Δ
(
a
)
.
Then
(257)
z
2
≤
u
≤
z
1
,
(
z
2
w
)
Δ
≤
(
u
w
)
Δ
≤
(
z
1
w
)
Δ
for
x
∈
[
a
,
x
*
]
𝕋
.
From these, we get the bounds for u
(
x
*
) and u
Δ
(
x
*
). In addition, if a function w
* which satisfies (
L
+
h
)
[
w
*
]
≤
0 on an interval [
x
*
,
b
]
𝕋 is given, we can then find bounds for u
/
w
* and (
u
/
w
*
)
Δ, as before. Let the functions z
1
* and z
2
* be defined on [
x
*
,
b
]
𝕋 and they satisfy
(258)
(
L
+
h
)
[
z
2
*
]
≤
f
(
x
)
≤
(
L
+
h
)
[
z
1
*
]
in
[
x
*
,
b
]
𝕋
,
z
2
*
(
x
*
)
≤
u
(
x
*
)
≤
z
1
*
(
x
*
)
,
(
z
2
*
w
*
)
Δ
≤
(
u
w
*
)
Δ
≤
(
z
1
*
w
*
)
Δ
at
x
=
x
*
.
Then we find, as we did previously, that
(259)
m
m
m
m
m
m
z
2
*
(
x
)
≤
u
(
x
)
≤
z
1
*
(
x
)
,
(
z
2
*
w
*
)
Δ
≤
(
u
w
*
)
Δ
≤
(
z
1
*
w
*
)
Δ
for
x
∈
[
x
*
,
b
]
𝕋
.
While we do not know u
(
x
*
) and (
u
/
w
*
)
Δ at x
*, but we know their bounds, therefore, we can give explicit conditions on the values of z
1
*, (
z
1
*
)
Δ, z
2
*, and (
z
2
*
)
Δ at x
* which assure that the above inequalities are satisfied.In fact, since atx
* we should have that
(260)
(
u
w
*
)
Δ
=
(
u
w
·
w
w
*
)
Δ
=
(
u
w
)
Δ
·
w
w
*
+
u
σ
w
σ
·
(
w
w
*
)
Δ
≤
(
z
1
w
)
Δ
·
w
w
*
+
u
σ
w
σ
·
(
w
w
*
)
Δ
=
(
z
1
w
)
Δ
·
w
w
*
-
u
σ
w
*
σ
w
w
σ
(
(
w
*
)
Δ
w
*
-
w
Δ
w
)
,
(
u
w
*
)
Δ
=
(
u
w
·
w
w
*
)
Δ
≥
(
z
2
w
)
Δ
·
w
w
*
-
u
σ
w
*
σ
w
w
σ
(
(
w
*
)
Δ
w
*
-
w
Δ
w
)
,
if (
w
*
)
Δ
/
w
*
≥
w
Δ
/
w, then we can give the conditions on the values of z
1
*, (
z
1
*
)
Δ, z
2
*, and (
z
2
*
)
Δ at x
* which assure that inequality (258) holds as follows:
(261)
z
1
*
≥
z
1
,
w
*
(
z
1
*
w
*
)
Δ
≥
w
(
z
1
w
)
Δ
-
w
*
w
*
σ
w
w
σ
m
m
m
m
m
m
l
l
×
(
(
w
*
)
Δ
w
*
-
w
Δ
w
)
z
2
*
σ
at
x
=
x
*
,
z
2
*
≤
z
2
,
w
*
(
z
2
*
w
*
)
Δ
≤
w
(
z
2
w
)
Δ
-
w
*
w
*
σ
w
w
σ
m
m
m
m
m
m
l
l
×
(
(
w
*
)
Δ
w
*
-
w
Δ
w
)
z
1
*
σ
at
x
=
x
*
.
If (
w
*
)
Δ
/
w
*
≤
w
Δ
/
w, we replace z
2 by z
1 in the coefficient of (
(
w
*
)
Δ
/
w
*
-
w
Δ
/
w
) in the first row of the inequalities and replace z
1 by z
2 in the second row. If these conditions are satisfied, we have the bounds
(262)
z
2
*
(
x
)
≤
u
(
x
)
≤
z
1
*
(
x
)
,
(
z
2
*
w
*
)
Δ
≤
(
u
w
*
)
Δ
≤
(
z
1
*
w
*
)
Δ
.We now considerw
* as an extension of w to the interval [
x
*
,
b
]
𝕋, z
1
* as an extension z
1 to the interval [
x
*
,
b
]
𝕋, and z
2
* as an extension of z
2 to the interval [
x
*
,
b
]
𝕋. Then these extended functions are, in general, discontinuous at x
*. However, the above inequalities relating w,  w
*,  z
1
*,  z
1,  z
2
*, and z
2 at x
=
x
* establish the relation between the right and left limits at discontinuous points. It may, of course, be necessary or desirable to divide the interval [
a
,
b
]
𝕋 into more than two subintervals.The above discussion leads to the following theorem.Theorem 75.
Letz
1
(
x
), z
2
(
x
), and w
(
x
) be piecewise continuous functions on the interval [
a
,
b
]
𝕋. Moreover, we assume that z
i
Δ
(
x
), z
i
Δ
∇
(
x
)
(
i
=
1,2
) and w
Δ
(
x
), w
Δ
∇
(
x
) are piecewise continuous on the interval [
a
,
b
]
𝕋, z
i
∇
(
x
) and w
∇
(
x
) are piecewise left-dense continuous on the interval [
a
,
b
]
𝕋, and g
1, g
2, and h satisfy the conditions of Lemma 54. If the following properties hold:(a)
w
>
0 on [
a
,
b
]
𝕋;
(b)
z
2
(
a
)
≤
γ
1
≤
z
1
(
a
);
(c)
z
2
Δ
(
a
)
w
(
a
)
-
z
2
(
a
)
w
Δ
(
a
)
≤
γ
2
w
(
a
)
-
γ
1
w
Δ
(
a
)
≤
z
1
Δ
(
a
)
w
(
a
)
-
z
1
(
a
)
w
Δ
(
a
);
(d)
(
L
+
h
)
[
w
]
≤
0, (
L
+
h
)
[
z
2
]
≤
f
(
x
)
≤
(
L
+
h
)
[
z
1
] hold at all points where the derivatives occurring in these formulas are continuous;
(e)
at each point of discontinuityx
* the functions z
1, -
z
2, and w
Δ
/
w have nonnegative jumps, the jump of w
(
z
1
/
w
)
Δ is at least -
(
(
w
*
/
w
*
σ
)
(
w
/
w
σ
)
)
z
2
*
σ times as many as jump of w
Δ
/
w, and the jump of w
(
z
2
/
w
)
Δ is at most -
(
(
w
*
/
w
*
σ
)
(
w
/
w
σ
)
)
z
1
*
σ times as many as jump of w
Δ
/
w,then
(263)
z
2
w
≤
u
w
≤
z
1
w
,
(
z
2
w
)
Δ
≤
(
u
w
)
Δ
≤
(
z
1
w
)
Δ
o
n
[
a
,
b
]
𝕋
.


## 5.  Applications to Boundary Value Problems

In this section, by using the maximum principles proved in Section3 to some general boundary value problems, the uniqueness of the solutions, the existence of the upper and lower solutions, and some necessary and sufficient conditions for the existence of the approximation solutions are discussed. First, we consider the following boundary value problems:
(264)
u
Δ
∇
+
g
1
u
Δ
+
g
2
u
∇
+
h
u
=
f
(
x
)
,
on
⁡
(
a
,
b
)
𝕋
,
(265)
u
(
a
)
=
γ
1
,
u
(
b
)
=
γ
2
.Theorem 76.
Assume thatg
1, g
2, and h are bounded on (
a
,
b
)
𝕋 such that (91), (92), (93), and (130) hold and h
(
x
)
≤
0 at each x
∈
(
a
,
b
)
𝕋. If u
1
,
u
2
∈
𝒟
(
Λ
) are solutions of (264) and satisfy the boundary value problem (265), then u
1
≡
u
2.Proof.
We define a functionv
(
x
)
∈
𝒟
(
Λ
) by
(266)
v
(
x
)
=
u
1
(
x
)
-
u
2
(
x
)
.
Since both u
1 and u
2 satisfy (264) and (265), the function v
(
x
) satisfies
(267)
v
Δ
∇
+
g
1
v
Δ
+
g
2
u
∇
+
h
v
=
0
,
v
(
a
)
=
0
,
v
(
b
)
=
0
.
It follows from Theorem 47 that v
(
x
)
≤
0, for each x
∈
(
a
,
b
)
𝕋. Since -
v
(
x
) satisfies the same boundary value problem, we have -
v
(
x
)
≤
0, for each x
∈
(
a
,
b
)
𝕋, and thus v
(
x
)
≡
0, for each x
∈
[
a
,
b
]
𝕋.Next we study general boundary value problems of the form(268)
u
Δ
∇
+
g
1
u
Δ
+
g
2
u
∇
+
h
u
=
f
(
x
)
,
(269)
-
u
Δ
(
a
)
cos
⁡
θ
+
u
(
a
)
sin
θ
=
γ
1
,
u
∇
(
b
)
cos
⁡
ϕ
+
u
(
b
)
sin
ϕ
=
γ
2
,
where θ
,
ϕ
,
γ
1
,
γ
2 are all constant, and 0
≤
θ
≤
π
/
2
,
0
≤
ϕ
≤
π
/
2. If θ
=
ϕ
=
π
/
2, (269) becomes (265).Theorem 77.
Assume thatg
1, g
2, and h are bounded on (
a
,
b
)
𝕋 such that (91), (92), (93), and (130) hold. If u
1 and u
2 are solutions of (268) and (269), then u
1
≡
u
2, except that u
1
(
x
)
-
u
2
(
x
) is a constant when h
(
x
)
≡
0, θ
=
ϕ
=
0.Proof.
We define a functionv
(
x
)
∈
𝒟
(
Λ
) by
(270)
v
(
x
)
=
u
1
(
x
)
-
u
2
(
x
)
.
Since both u
1 and u
2 satisfy (268) and (269), the function v
(
x
) satisfies
(271)
v
Δ
∇
+
g
1
v
Δ
+
g
2
u
∇
+
h
v
=
0
,
(272)
-
v
Δ
(
a
)
cos
⁡
θ
+
v
(
a
)
sin
θ
=
0
,
(273)
v
∇
(
b
)
cos
⁡
ϕ
+
v
(
b
)
sin
ϕ
=
0
.
It is clear that v
(
x
)
≡
C satisfies all the above conditions, if and only if h
(
x
)
≡
0
,
θ
=
ϕ
=
0. Then we assume first that v
(
x
)
>
0 at some point and v
(
x
) is not constant. Using Theorem 47 we know that v
(
x
) attains its maximum M at a or b. Suppose that v
(
a
)
=
M, and by using Theorem 51 we get v
Δ
(
a
)
<
0, which do not satisfy (272). Suppose that v
(
b
)
=
M, and by using Theorem 51 we get v
∇
(
b
)
>
0, which do not satisfy (273). Thus, we obtain v
(
x
)
≤
0. We can also prove that -
v
(
x
)
≤
0, and then v
(
x
)
≡
0, for each x
∈
[
a
,
b
]
𝕋.Similar to the initial value problems, in most cases it is impossible to find such a solution explicitly. But, it is frequently desirable to approximate a solution in such a way that an explicit bound for the error is known. Such an approximation is equivalent to the determination of both upper and lower bounds for the values of the solution. Thus, in the following, we will discuss the existence of the upper and lower solutions for boundary value problems.We will assume that the functionsf, g
1, g
2, h, are bounded and h
(
x
)
≤
0 in [
a
,
b
]
𝕋. Under these circumstances it is possible to use the maximum principle in Theorem 55 to obtain a bound for a solution u without any actual knowledge of u itself.Suppose we can find a functionz
1
(
x
)
∈
𝒟
(
Λ
) satisfying
(274)
(
L
+
h
)
[
z
1
]
≤
f
(
x
)
,
(275)
z
1
(
a
)
≥
γ
1
,
z
1
(
b
)
≥
γ
2
.
Then the function
(276)
v
1
(
x
)
≡
u
(
x
)
-
z
1
(
x
)
satisfies
(277)
(
L
+
h
)
[
v
1
]
≥
0
,
v
1
(
a
)
≤
0
,
v
1
(
b
)
≤
0
.
The maximum principles as given in Theorem 47 in Section 3 may be applied to v
1, and we conclude that v
1
(
x
)
≤
0 on [
a
,
b
]
𝕋. That is,
(278)
u
(
x
)
≤
z
1
(
x
)
for
x
∈
[
a
,
b
]
𝕋
.
The function z
1
(
x
) is an upper bound for u
(
x
).Similarly, a lower bound foru
(
x
) may be obtained by finding a function z
2
(
x
) with the properties
(279)
(
L
+
h
)
[
z
2
]
≥
f
(
x
)
,
(280)
z
2
(
a
)
≤
γ
1
,
z
2
(
b
)
≤
γ
2
.
By using the maximum principle (Theorem 51) to z
2
(
x
)
-
u
(
x
), we can get that
(281)
u
(
x
)
≥
z
2
(
x
)
for
x
∈
[
a
,
b
]
𝕋
.Functionsz
1
(
x
), z
2
(
x
) with the desired properties are easily constructed. For example, we may set
(282)
z
1
(
x
)
=
A
(
2
-
e
^
α
(
a
,
x
)
)
with α
<
0 and try to select A and α so that (274) and (275) are satisfied. In fact, if (166) holds, we can choose -
α to be so large that
(283)
(
L
+
h
)
[
e
^
α
(
a
,
x
)
]
=
[
α
2
(
1
+
ν
g
1
)
-
α
(
g
1
+
g
2
)
+
h
(
1
-
α
ν
(
x
)
)
]
×
e
^
α
(
a
,
x
)
1
-
α
ν
(
x
)
>
0
for x
∈
[
a
,
b
]
𝕋. Let
(284)
k
=
min
⁡
x
∈
[
a
,
b
]
𝕋
[
e
^
α
(
a
,
x
)
1
-
α
ν
(
x
)
[
α
2
(
1
+
ν
g
1
)
-
α
(
g
1
+
g
2
)
+
h
(
1
-
α
ν
(
x
)
)
]
m
m
m
m
m
m
×
e
^
α
(
a
,
x
)
1
-
α
ν
(
x
)
]
for x
∈
[
a
,
b
]
𝕋, and
(285)
A
=
max
⁡
{
γ
1
,
γ
2
,
1
k
max
⁡
x
∈
[
a
,
b
]
𝕋
{
-
f
(
x
)
}
,
0
}
.
For the selections of A and α as just described, the function z
1
(
x
)
=
A
(
l
-
e
^
α
(
a
,
x
)
) satisfies (274) and (275), where l
=
1
+
e
^
α
(
a
,
b
).To determine a lower bound, we choose(286)
z
2
(
x
)
=
B
(
l
-
e
^
α
(
a
,
x
)
)
,
with α
<
0, and try to select B and α such that (279) and (280) are satisfied. In fact, we choose -
α to be so large that (283) holds. Let
(287)
B
=
min
⁡
{
γ
1
,
γ
2
,
-
1
k
max
⁡
x
∈
[
a
,
b
]
𝕋
{
f
(
x
)
}
,
0
}
.
With the selections of B and α as just described, the function z
2
(
x
)
=
B
(
l
-
e
^
α
(
a
,
x
)
) satisfies (279) and (280). Then
(288)
B
(
l
-
e
^
α
(
a
,
x
)
)
≤
u
(
x
)
≤
A
(
l
-
e
^
α
(
a
,
x
)
)
.
In particular, we have
(289)
|
u
(
x
)
|
≤
l
max
⁡
[
|
γ
1
|
,
|
γ
2
|
,
1
k
max
⁡
x
∈
[
a
,
b
]
𝕋
|
f
(
x
)
|
,
0
]
.Ifu is a solution of (264) and (265) and u
¯ is a solution of the related problem
(290)
u
¯
Δ
∇
+
g
1
u
¯
Δ
+
g
2
u
¯
∇
+
h
u
¯
=
f
¯
(
x
)
,
u
¯
(
a
)
=
γ
1
¯
,
u
¯
(
b
)
=
γ
2
¯
,
then the difference u
-
u
¯ satisfies
(291)
(
u
-
u
¯
)
Δ
∇
+
g
1
(
u
-
u
¯
)
Δ
+
g
2
(
u
-
u
¯
)
∇
+
h
(
u
-
u
¯
)
=
f
-
f
¯
,
u
(
a
)
-
u
¯
(
a
)
=
γ
1
-
γ
1
¯
,
u
(
b
)
-
u
¯
(
b
)
=
γ
2
-
γ
2
¯
.
Inequality (289) shows that
(292)
|
u
(
x
)
-
u
¯
(
x
)
|
≤
l
max
⁡
[
1
k
max
⁡
x
∈
[
a
,
b
]
𝕋
|
f
(
x
)
-
f
¯
(
x
)
|
|
γ
1
-
γ
1
¯
|
,
|
γ
2
-
γ
2
¯
|
,
m
m
m
m
m
m
m
l
1
k
max
⁡
x
∈
[
a
,
b
]
𝕋
|
f
(
x
)
-
f
¯
(
x
)
|
]
.
Therefore, if the quantities
(293)
[
|
γ
1
-
γ
1
¯
|
,
|
γ
2
-
γ
2
¯
|
,
max
⁡
x
∈
[
a
,
b
]
𝕋
|
f
(
x
)
-
f
¯
(
x
)
|
]
are all small, then |
u
(
x
)
-
u
¯
(
x
)
| is small for all x in the interval [
a
,
b
]
𝕋. Under these circumstances, we say that the solution u of the problem (264), (265) depends continuously on f
(
x
) and the boundary values γ
1,γ
2.Combining the above discussions, we get the following result.Theorem 78.
Ifg
1, g
2, and h are bounded on (
a
,
b
)
𝕋 such that (92), (93), (130), and (166) hold and h
(
x
)
≤
0 in [
a
,
b
]
𝕋, then the following conclusions hold:(1)
there exist functionsz
1
(
x
)
,
z
2
(
x
)
∈
𝒟
(
Λ
) satisfying (274), (275) and (279), (280), respectively;
(2)
the solutionu of the problem (264), (265) satisfies z
1
(
x
)
≤
u
(
x
)
≤
z
2
(
x
), x
∈
[
a
,
b
]
𝕋;
(3)
the solutionu of the problem (264), (265) depends continuously on f
(
x
) and the boundary values γ
1, γ
2.Next, we consider the question of approximations of solutions of (268) and (269).If we can find a functionz
1
(
x
)
∈
𝒟
(
Λ
) satisfying
(294)
(
L
+
h
)
[
z
1
]
≤
f
(
x
)
,
(295)
-
z
1
Δ
(
a
)
cos
⁡
θ
+
z
1
(
a
)
sin
θ
≥
γ
1
,
z
1
∇
(
b
)
cos
⁡
ϕ
+
z
1
(
b
)
sin
ϕ
≥
γ
2
,
we define a function v
1
(
x
)
∈
𝒟
(
Λ
) by
(296)
v
1
(
x
)
≡
u
(
x
)
-
z
1
(
x
)
.
Thus,
(297)
(
L
+
h
)
[
v
1
]
≥
0
,
-
v
1
Δ
(
a
)
cos
⁡
θ
+
v
1
(
a
)
sin
θ
≤
0
,
v
1
∇
(
b
)
cos
⁡
ϕ
+
v
1
(
b
)
sin
ϕ
≤
0
.
It follows from Theorem 51, (297) we know that v
1
(
x
) cannot attain its maximum at a or b. If v
1
(
x
)
>
0 at some point x
∈
(
a
,
b
)
𝕋, then by using Theorem 47, we have that v
1
(
x
) is positive constant which implies that θ
=
ϕ
=
0 and h
(
x
)
≡
0. Otherwise, if h
(
x
)
≡
0, θ
=
0, ϕ
=
0 are not all hold, then we have that v
1
(
x
)
≤
0, that is u
(
x
)
≤
z
1
(
x
). Similarly, we assume that we can find a function z
2
(
x
)
∈
𝒟
(
Λ
) satisfying:
(298)
(
L
+
h
)
[
z
2
]
≥
f
(
x
)
,
(299)
-
z
2
Δ
(
a
)
cos
⁡
θ
+
z
2
(
a
)
sin
θ
≤
γ
1
,
z
2
∇
(
b
)
cos
⁡
ϕ
+
z
2
(
b
)
sin
ϕ
≤
γ
2
.
The same as the above statement, we define
(300)
v
2
(
x
)
≡
z
2
(
x
)
-
u
(
x
)
,
and we obtain
(301)
u
(
x
)
≥
z
2
(
x
)
.
Therefore, we establish an approximation theorem as in the following.Theorem 79.
Assume thatg
1, g
2, and h are bounded on (
a
,
b
)
𝕋 and satisfy (91), (92), (93), and (130) at each x
∈
(
a
,
b
)
𝕋. If u
(
x
) is a solution of (268) and (269), where h
(
x
)
≤
0, 0
≤
θ
≤
π
/
2, 0
≤
ϕ
≤
π
/
2. Let z
1
(
x
) and z
2
(
x
)
∈
𝒟
(
Λ
) satisfy (294), (295), (298), and (299). If h
(
x
)
≡
0, θ
=
0, ϕ
=
0 are not all hold, then z
2
(
x
)
≤
u
(
x
)
≤
z
1
(
x
).Now, we consider deleting the conditionsh
(
x
)
≤
0,0
≤
θ
≤
π
/
2, 0
≤
ϕ
≤
π
/
2 in Theorem 79. Without loss of generality, we can assume that -
π
/
2
≤
θ
≤
π
/
2, -
π
/
2
≤
ϕ
≤
π
/
2.In order to use the generalized maximum principle established in Section3, we suppose that there is a positive function w
(
x
) which satisfies the inequalities
(302)
(
L
+
h
)
[
w
]
≤
0
in
[
a
,
b
]
𝕋
,
(303)
-
w
Δ
(
a
)
cos
⁡
θ
+
w
(
a
)
sin
θ
≥
0
,
w
∇
(
b
)
cos
⁡
ϕ
+
w
(
b
)
sin
ϕ
≥
0
.
We set
(304)
v
=
u
w
,
where u
(
x
) is a solution of (268) and (269). Then v must satisfy
(305)
(
L
¯
+
H
)
[
v
]
=
v
Δ
∇
+
G
1
v
Δ
+
G
2
v
∇
+
H
v
=
f
w
σ
ρ
with G
1
=
(
F
Δ
∇
+
g
1
w
σ
)
/
w
σ
ρ, G
2
=
(
w
Δ
ρ
+
g
2
w
ρ
)
/
w
σ
ρ, H
=
(
1
/
w
σ
ρ
)
(
L
+
h
)
[
w
]
≤
0, and
(306)
-
[
w
Δ
(
a
)
v
(
a
)
+
w
(
σ
(
a
)
)
v
Δ
(
a
)
]
cos
⁡
θ
+
v
(
a
)
w
(
a
)
sin
θ
=
γ
1
,
[
w
∇
(
b
)
v
(
b
)
+
w
(
ρ
(
b
)
)
v
∇
(
b
)
]
cos
⁡
ϕ
+
v
(
b
)
w
(
b
)
sin
ϕ
=
γ
2
.
We can rewrite (306) to be
(307)
-
v
Δ
(
a
)
cos
⁡
θ
~
+
v
(
a
)
sin
θ
~
=
γ
1
~
,
v
∇
(
b
)
cos
⁡
ϕ
~
+
v
(
b
)
sin
ϕ
~
=
γ
2
~
,
where
(308)
tan
θ
~
=
-
w
Δ
(
a
)
cos
⁡
θ
+
w
(
a
)
sin
θ
w
(
σ
(
a
)
)
cos
⁡
θ
≥
0
,
tan
ϕ
~
=
w
∇
(
b
)
cos
⁡
ϕ
+
w
(
b
)
sin
ϕ
w
(
ρ
(
b
)
)
cos
⁡
ϕ
≥
0
,
γ
1
~
=
γ
1
[
w
(
a
)
sin
θ
-
w
Δ
(
a
)
cos
⁡
θ
]
2
+
[
w
(
σ
(
a
)
)
cos
⁡
θ
]
2
,
γ
2
~
=
γ
2
[
w
∇
(
b
)
cos
⁡
ϕ
+
w
(
b
)
sin
ϕ
]
2
+
[
w
(
ρ
(
b
)
)
cos
⁡
ϕ
]
2
.
By (303) we may choose θ
~ and ϕ
~ in the range 0
≤
θ
~
≤
π
/
2, 0
≤
ϕ
~
≤
π
/
2.Suppose that there is a functionw
(
x
), which is positive on [
a
,
b
]
𝕋 and satisfies conditions (302) and (303). If z
1
(
x
) and z
2
(
x
) satisfy (294), (295) and (298), (299), respectively, then the functions
(309)
z
1
w
,
z
2
w
satisfy the analogous conditions with respect to (305) with boundary (307); that is, the following inequalities hold:
(310)
(
L
¯
+
H
)
[
z
1
w
]
≤
f
w
σ
ρ
,
-
[
z
1
w
]
Δ
(
a
)
cos
⁡
θ
~
+
z
1
w
(
a
)
sin
θ
~
≥
γ
~
1
,
[
z
1
w
]
∇
(
b
)
cos
⁡
ϕ
~
+
z
1
w
(
b
)
sin
ϕ
~
≥
γ
~
2
,
m
m
m
m
(
L
¯
+
H
)
[
z
2
w
]
≥
f
w
σ
ρ
,
-
[
z
2
w
]
Δ
(
a
)
cos
⁡
θ
~
+
z
2
w
(
a
)
sin
θ
~
≤
γ
~
1
,
[
z
2
w
]
∇
(
b
)
cos
⁡
ϕ
~
+
z
2
w
(
b
)
sin
ϕ
≤
γ
~
2
.
Hence, by Theorem 79, we have the inequalities
(311)
z
2
w
≤
u
w
≤
z
1
w
unless θ
~
=
ϕ
~
=
0 and H
(
x
)
≡
0.Ifw
(
x
) satisfies (302) and (303) with equality rather than inequality, we may add a multiple of w to a solution u of (268) and (269) to obtain another solution. That is, the solution is not unique. Of course, there may be no solution at all, but if there is at least one, then there are many. Therefore, if there is a positive function w
(
x
) that satisfies (302) and (303) but such that not all the inequalities are equations, we obtain the bounds
(312)
z
2
(
x
)
≤
u
(
x
)
≤
z
1
(
x
)
as before.If inequality (312) holds for the solution of (268) and (269), then, particularly, the solution w of
(313)
(
L
+
h
)
[
w
]
=
0
in
(
a
,
b
)
𝕋
,
which satisfies the boundary conditions
(314)
-
w
Δ
(
a
)
cos
⁡
θ
+
w
(
a
)
sin
θ
=
1
,
w
∇
(
b
)
cos
⁡
ϕ
+
w
(
b
)
sin
ϕ
=
1
,
must be nonnegative.In fact, if we selectz
2
(
x
)
≡
0, then with respect to (313) and (314), z
2
(
x
) satisfies (298) and (299) for f
(
x
)
≡
0, γ
1
=
γ
2
=
1. Then we have that z
2
(
x
)
≤
w
(
x
), that is, w
(
x
)
≥
0. Moreover, if w
(
x
0
)
=
0
,
x
0 in (
a
,
b
)
𝕋, then -
w
(
x
0
)
=
0 is a maximum of -
w in (
a
,
b
)
𝕋; by using Lemma 34 to -
w we obtain
(315)
w
Δ
(
x
0
)
≥
0
,
w
∇
(
x
0
)
≤
0
.
If x
0 is left-scattered, since
(316)
(
L
+
h
)
[
w
]
=
0
we have
(317)
(
L
+
h
)
[
w
(
x
0
)
]
=
w
Δ
∇
(
x
0
)
+
g
1
w
Δ
(
x
0
)
+
g
2
w
∇
(
x
0
)
+
h
w
(
x
0
)
=
w
Δ
(
x
0
)
-
w
∇
(
x
0
)
ν
(
x
0
)
+
g
1
w
Δ
(
x
0
)
+
g
2
w
∇
(
x
0
)
=
0
,
and hence
(318)
(
1
+
ν
(
x
0
)
g
1
)
w
Δ
(
x
0
)
=
-
(
-
1
+
ν
(
x
0
)
g
2
)
w
∇
(
x
0
)
;
by (166), (92), and (315), we obtain w
Δ
(
x
0
)
=
0 (if -
1
+
ν
(
x
0
)
g
2
<
0, we can also have that w
∇
(
x
0
)
=
0). If x
0 is left-dense, by Lemma 34, we get that
(319)
w
Δ
(
x
0
)
=
0
,
w
∇
(
x
0
)
=
w
Δ
(
ρ
(
x
0
)
)
=
w
Δ
(
x
0
)
=
0
.
So we have proved that w
Δ
(
x
0
)
=
0 if w
(
x
0
)
=
0. The uniqueness theorem for the initial value problem implies that w
≡
0, which violates the boundary conditions (314). So w cannot vanish in (
a
,
b
)
𝕋. If w vanishes at endpoint, say at a, then the first condition in (314) becomes -
w
Δ
(
a
)
cos
⁡
θ
=
1, which is a contradiction. Therefore w
(
a
)
>
0 and, similarly, w
(
b
)
>
0. Hence w
>
0 on [
a
,
b
]
𝕋.Thus far, under the hypothesis that the problem (313), (314) has a solution, we have established the following result.Theorem 80.
Letu be a solution of (268), (269) with -
π
/
2
≤
θ
≤
π
/
2
,
-
π
/
2
≤
ϕ
≤
π
/
2. Let z
1
(
x
) and z
2
(
x
)
∈
𝒟
(
Λ
) satisfy (294), (295) and (298), (299), respectively; g
1, g
2, and h are bounded on (
a
,
b
)
𝕋 and satisfy (91), (92). If problem (313), (314) has a solution, then the bounds
(320)
z
2
(
x
)
≤
u
(
x
)
≤
z
1
(
x
)
hold in (
a
,
b
)
𝕋 if and only if there exists a positive function w on [
a
,
b
]
𝕋 which satisfies inequalities (302) and (303) in such a way that not all the inequalities in (302) and (303) are equalities.Remark 81.
Moreover, ifh
(
x
)
≤
0, 0
≤
θ
≤
π
/
2, and 0
≤
ϕ
≤
π
/
2, then the function w
≡
1 satisfies conditions (302) and (303); thus Theorem 79 can be deduced from Theorem 80.The functionw does not appear in the inequality (320). Therefore, it is of interest to obtain a theorem which eliminates w entirely and which provides conditions on z
1 and z
2 guaranteeing that they form the upper and lower bounds of the solution of (268), (269). The next result gives a necessary and sufficient condition for this case.Theorem 82.
Letu be a solution of (268), (269) with -
π
/
2
<
θ
<
π
/
2
,
-
π
/
2
<
ϕ
<
π
/
2. Suppose z
1
(
x
) and z
2
(
x
)
∈
𝒟
(
Λ
) satisfy (294), (295) and (298), (299), respectively, and in all these inequalities, at least there is one which is a strict inequality. Let g
1
,
g
2 be bounded and satisfy 1
+
ν
g
1
>
0, -
1
+
ν
g
2
<
0. Then the bounds
(321)
z
2
(
x
)
≤
u
(
x
)
≤
z
1
(
x
)
hold if and only if z
2
(
x
)
≤
z
1
(
x
) for x
∈
[
a
,
b
]
𝕋.Proof.
If (321) holds, then it is clear that z
2
(
x
)
≤
z
1
(
x
). We now assume that z
1
(
x
)
-
z
2
(
x
) is nonnegative, and we must show that (321) holds. If
(322)
q
(
x
)
≡
z
1
(
x
)
-
z
2
(
x
)
is strictly positive on [
a
,
b
]
𝕋, then we may select q as the function w of Theorem 80. All the requirements are fulfilled and Theorem 80 implies that (321) holds. Therefore, we need only to investigate the possibility that q has a zero point on [
a
,
b
]
𝕋.
According to (294), (295), (298), and (299), q satisfies the inequalities
(323)
(
L
+
h
)
[
q
]
≤
0
in
[
a
,
b
]
𝕋
,
(324)
-
q
Δ
(
a
)
cos
⁡
θ
+
q
(
a
)
sin
θ
≥
0
,
q
∇
(
b
)
cos
⁡
ϕ
+
q
(
b
)
sin
ϕ
≥
0
,
and, in view of the hypotheses, at least there is one which is a strict inequality in (323), (324).
First supposeq
(
c
)
=
0, c
∈
(
a
,
b
)
𝕋. Then q has a minimum at c, which shows that -
q has a maximum at c. So by Lemma 34 we obtain
(325)
q
Δ
(
c
)
≥
0
,
q
∇
(
c
)
≤
0
.
If c is left-scattered, since
(326)
(
L
+
h
)
[
q
]
≤
0
we have
(327)
(
L
+
h
)
[
q
(
c
)
]
=
q
Δ
∇
(
c
)
+
g
1
q
Δ
(
c
)
+
g
2
q
∇
(
c
)
+
h
q
(
c
)
=
q
Δ
(
c
)
-
q
∇
(
c
)
ν
(
c
)
+
g
1
q
Δ
(
c
)
+
g
2
q
∇
(
c
)
≤
0
,
and hence
(328)
0
≤
(
1
+
ν
g
1
)
q
Δ
(
c
)
≤
-
(
-
1
+
ν
(
x
)
g
2
)
q
∇
(
c
)
≤
0
.
Therefore
(329)
q
Δ
(
c
)
=
q
∇
(
c
)
=
0
.
If c is left-dense by Lemma 34, we have that
(330)
q
∇
(
c
)
=
q
Δ
(
c
)
=
0
.
And we conclude from Theorem 68 that q
≡
0. Then equality holds in all the conditions (323), (324) contrary to our hypotheses.
The only remaining possibility is thatq
>
0 in (
a
,
b
)
𝕋 but that q
=
0 at an endpoint, say at x
=
a. Then according to Theorem 55, q
Δ
(
a
)
>
0. But then the first inequality in (324) is violated unless θ
=
±
π
/
2, which is a contradiction with θ
≠
±
π
/
2. Similarly, if q vanishes at b, then ϕ
=
±
π
/
2, which is a contradiction with ϕ
≠
±
π
/
2. Therefore, we have proved that q is positive on [
a
,
b
]
𝕋, and it can be used as an auxiliary function in Theorem 80. The proof is completed.


## 6. Applications to Nonlinear Operator

In this section, we discuss nonlinear equations. We can extend the results of linear operator in Sections4 and 5 to nonlinear operator.Suppose thatu
(
x
)
∈
𝒟
(
Λ
) is a solution of
(331)
u
Δ
∇
+
H
(
x
,
u
,
u
Δ
,
u
∇
)
=
0
.
Assume that
(332)
H
(
x
,
y
,
z
,
p
)
,
∂
H
(
x
,
y
,
z
,
p
)
∂
y
,
∂
H
(
x
,
y
,
z
,
p
)
∂
z
,
∂
H
(
x
,
y
,
z
,
p
)
∂
p
are all continuous functions in their domain and satisfy
(333)
1
+
ν
∂
H
(
x
,
y
,
z
,
p
)
∂
z
>
0
,
-
1
+
ν
∂
H
(
x
,
y
,
z
,
p
)
∂
p
<
0
,
where H
(
x
,
y
,
z
,
p
) is defined on 𝕋
×
ℝ
×
ℝ
×
ℝ. Besides, we suppose that
(334)
H
(
x
,
y
1
,
z
,
p
)
≤
H
(
x
,
y
2
,
z
,
p
)
when
y
1
≥
y
2
for each x
,
z
,
p, or equivalently
(335)
∂
H
(
x
,
y
,
z
,
p
)
∂
y
≤
0
.Remark 83.
Note that if(336)
H
(
x
,
u
,
u
Δ
,
u
∇
)
=
g
1
u
Δ
+
g
2
u
∇
+
h
u
-
f
(
x
)
,
then (331) is the linear equation considered in the previous sections.Assume thatw
(
x
)
∈
𝒟
(
Λ
) satisfies
(337)
w
Δ
∇
+
H
(
x
,
w
,
w
Δ
,
w
∇
)
≥
0
.
We define a function v
(
x
)
∈
𝒟
(
Λ
) by
(338)
v
(
x
)
=
w
(
x
)
-
u
(
x
)
.
Subtracting (331) from (337), we derive that
(339)
v
Δ
∇
+
H
(
x
,
w
,
w
Δ
,
w
∇
)
-
H
(
x
,
u
,
u
Δ
,
u
∇
)
≥
0
.
Applying mean value theorem to H, we obtain
(340)
v
Δ
∇
+
∂
H
(
x
,
y
,
z
,
p
)
∂
z
v
Δ
+
∂
H
(
x
,
y
,
z
,
p
)
∂
p
v
∇
+
∂
H
(
x
,
y
,
z
,
p
)
∂
y
v
≥
0
,
where ∂
H
(
x
,
y
,
z
,
p
)
/
∂
y, ∂
H
(
x
,
y
,
z
,
p
)
/
∂
z, and ∂
H
(
x
,
y
,
z
,
p
)
/
∂
p are calculated at the point of (
x
,
u
+
α
(
w
-
u
), u
Δ
+
α
(
w
Δ
-
u
Δ
), u
∇
+
α
(
w
∇
-
u
∇
)
), and 0
<
α
<
1. Obviously, the function v
(
x
) satisfies a linear equation. We can use Theorems 47 and 72 to get the following results.Theorem 84.
Assume thatw
(
x
), u
(
x
)
∈
𝒟
(
Λ
) satisfy
(341)
w
Δ
∇
+
H
(
x
,
w
,
w
Δ
,
w
∇
)
≥
u
Δ
∇
+
H
(
x
,
u
,
u
Δ
,
u
∇
)
,
where H, ∂
H
/
∂
y,  ∂
H
/
∂
z,  ∂
H
/
∂
p, are all continuous functions such that (333) and (335) hold and
(342)
m
m
m
l
∂
H
(
x
,
y
,
z
,
p
)
/
∂
y
1
+
ν
(
∂
H
(
x
,
y
,
z
,
p
)
/
∂
z
)
,
∂
H
(
x
,
y
,
z
,
p
)
/
∂
y
+
∂
H
(
x
,
y
,
z
,
p
)
/
∂
p
1
+
ν
(
∂
H
(
x
,
y
,
z
,
p
)
/
∂
z
)
are bounded on Λ
×
w
(
Λ
)
×
w
Δ
(
Λ
)
×
w
∇
(
Λ
). If w
(
x
)
-
u
(
x
) attains its maximum M in (
a
,
b
)
𝕋, then w
(
x
)
-
u
(
x
)
≡
M.Theorem 85.
Assume thatu
(
x
)
∈
𝒟
(
Λ
) is a solution of
(343)
u
Δ
∇
+
H
(
x
,
u
,
u
Δ
,
u
∇
)
=
0
x
∈
(
a
,
b
)
𝕋
,
(344)
u
(
a
)
=
γ
1
,
u
Δ
(
a
)
=
γ
2
,
where H, ∂
H
/
∂
y, ∂
H
/
∂
z, and ∂
H
/
∂
p are all continuous functions such that (333) and (335) hold and
(345)
m
m
m
m
∂
H
(
x
,
y
,
z
,
p
)
/
∂
y
1
+
ν
(
∂
H
(
x
,
y
,
z
,
p
)
/
∂
z
)
,
∂
H
(
x
,
y
,
z
,
p
)
/
∂
y
+
∂
H
(
x
,
y
,
z
,
p
)
/
∂
p
1
+
ν
(
∂
H
(
x
,
y
,
z
,
p
)
/
∂
z
)
are bounded on Λ
×
w
(
Λ
)
×
w
Δ
(
Λ
)
×
w
∇
(
Λ
). If a function z
1
(
x
) satisfies
(346)
z
1
Δ
∇
+
H
(
x
,
z
1
,
z
1
Δ
,
z
1
∇
)
≥
0
,
z
1
(
a
)
≥
γ
1
,
z
1
Δ
(
a
)
≥
γ
2
,
and a function z
2
(
x
)
∈
𝒟
(
Λ
) satisfies
(347)
z
2
Δ
∇
+
H
(
x
,
z
2
,
z
2
Δ
,
z
2
∇
)
≤
0
,
z
2
(
a
)
≤
γ
1
,
z
2
Δ
(
a
)
≤
γ
2
,
then we have
(348)
z
2
(
x
)
≤
u
(
x
)
≤
z
1
(
x
)
,
z
2
Δ
(
x
)
≤
u
Δ
(
x
)
≤
z
1
Δ
(
x
)
.Remark 86.
Theorem85 implies that the solution of (343), (344) is unique. In fact, if u
1 and u
2 are solutions of (343), (344), we just need to let z
2
(
x
)
=
z
1
(
x
)
=
u
2, and then we can obtain u
1
≡
u
2.By using Theorem79, we have the following result.Theorem 87.
Assume thatu
(
x
)
∈
𝒟
(
Λ
) is a solution of
(349)
u
Δ
∇
+
H
(
x
,
u
,
u
Δ
,
u
∇
)
=
0
x
∈
(
a
,
b
)
𝕋
,
(350)
-
u
Δ
(
a
)
cos
⁡
θ
+
u
(
a
)
sin
θ
=
γ
1
,
u
∇
(
b
)
cos
⁡
ϕ
+
u
(
b
)
sin
ϕ
=
γ
2
,
where 0
≤
θ
≤
π
/
2, 0
≤
ϕ
≤
π
/
2, and θ
,
ϕ are not all equal to 0, H, ∂
H
/
∂
y, ∂
H
/
∂
z, and ∂
H
/
∂
p are all continuous functions such that (333) and (335) hold, and
(351)
∂
H
(
x
,
y
,
z
,
p
)
/
∂
y
1
+
ν
(
∂
H
(
x
,
y
,
z
,
p
)
/
∂
z
)
,
∂
H
(
x
,
y
,
z
,
p
)
/
∂
y
+
∂
H
(
x
,
y
,
z
,
p
)
/
∂
p
1
+
ν
(
∂
H
(
x
,
y
,
z
,
p
)
/
∂
z
)
are bounded on Λ
×
w
(
Λ
)
×
w
Δ
(
Λ
)
×
w
∇
(
Λ
). If a function z
1
(
x
)
∈
𝒟
(
Λ
) satisfies
(352)
z
1
Δ
∇
+
H
(
x
,
z
1
,
z
1
Δ
,
z
1
∇
)
≤
0
,
-
z
1
Δ
(
a
)
cos
⁡
θ
+
z
1
(
a
)
sin
θ
≥
γ
1
,
z
1
∇
(
b
)
cos
⁡
ϕ
+
z
1
(
b
)
sin
ϕ
≥
γ
2
,
and a function z
2
(
x
)
∈
𝒟
(
Λ
) satisfies
(353)
z
2
Δ
∇
+
H
(
x
,
z
2
,
z
2
Δ
,
z
2
∇
)
≥
0
,
-
z
2
Δ
(
a
)
cos
⁡
θ
+
z
2
(
a
)
sin
θ
≤
γ
1
,
z
2
∇
(
b
)
cos
⁡
ϕ
+
z
2
(
b
)
sin
ϕ
≤
γ
2
,
then we have
(354)
z
2
(
x
)
≤
u
(
x
)
≤
z
1
(
x
)
.Remark 88.
Theorem87 implies that the solution of (349), (350) is unique. In fact, if u
1 and u
2 are solutions of (349), (350), we just need to let z
2
(
x
)
=
z
1
(
x
)
=
u
2, and then we obtain u
1
≡
u
2.



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*Source: 165429-2014-05-11.xml*