EXISTENCE AND NONEXISTENCE OF POSITIVE SOLUTIONS FOR A SYSTEM OF EVEN ORDER DYNAMIC EQUATION ON TIME SCALES

Journal of Applied Mathematics & Informatics.
2015.
Sep,
33(5_6):
531-543

- Received : September 23, 2014
- Accepted : February 23, 2015
- Published : September 30, 2015

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We determine interval of two eigenvalues for which there existence and nonexistence of positive solution for a system of even-order dynamic equation on time scales subject to Sturm-Liouville boundary conditions.
AMS Mathematics Subject Classification : 34B15, 39B10, 34B18.
λ
and
μ
for which there exist and nonexist of positive solutions for the system of dynamic equations,
with the Sturm-Liouville boundary conditions,
for 0 ≤
i
≤
n
− 1,
n
≥ 1 with
a
∈ T
_{kn}
,
b
∈ T
^{kn}
for a time scale
T
and
σ^{n}
(
a
) <
ρ^{n}
(
b
). Our interest in this paper is to investigate the existence and nonexistence of eigenvalues
λ
and
μ
that yields positive and no positive solutions to the associated boundary value problems, (1.1)-(1.2).
We assume that:
The rest of this paper is organized as follows. In Section 2, we present some preliminaries and lemmas that will be used to prove our main results. In Section 3, we discuss the existence of a positive solution of the system (1.1)-(1.2). The intervals in which the parameters
λ
and
μ
can guarantee the existence of a solution are obtained. In Section 4, we will consider the conditions of the nonexistence of a positive solution.
^{n}u
^{(Δ∇)n}
(
t
) = 0,
t
∈ [
a
,
b
] satisfying boundary conditions (1.2). For 1 ≤
j
≤
n
, let
G_{j}
(
t
,
s
) be the Green’s function for the boundary value problems,
First, we need few results on the related second order homogeneous boundary value problem (2.1)-(2.2).
Lemma 2.1.
For
1 ≤
j
≤
n, let d_{j}
=
γ_{j}
β_{j}
+
α_{j}
δ_{j}
+
α_{j}
γ_{j}
(
b
−
a
).
The homogeneous boundary value problem
(2.1)-(2.2)
has only the trivial solution if and only if d_{j}
> 0.
Lemma 2.2.
For
1 ≤
j
≤
n, the Green's function G_{j}
(
t
,
s
)
for the homogeneous boundary value problem
(2.1)-(2.2),
is given by
Lemma 2.3.
Assume that condition
(
A
1)
is satisfied. Then, the Green's function G_{j}
(
t
,
s
)
satisfies the following inequality
where
for
1 ≤
j
≤
n
.
Proof
. It is straightforward to see that
this expression yields both inequalities in (2.4) for
g_{j}
as in (2.5). □
Lemma 2.4.
Assume that the condition
(
A
1)
is satisfied, and G_{j}
(
t
,
s
)
as in
(2.3).
Let us define H
_{1}
(
t
,
s
) =
G
_{1}
(
t
,
s
)
, and recursively define
for
2 ≤
j
≤
n
.
Then H_{n}
(
t
,
s
)
is the Green's function for the corresponding homogeneous problem
(1.1)-(1.2).
Let
ξ
and
ω
are chosen from
T
such that
a
<
ξ
<
ω
<
b
and also
for
g_{j}
as in (2.5).
Let
τ
∈ [
ξ
,
ω
] be defined by
Lemma 2.5.
Assume that the condition
(
A
1)
holds. If we define
then the Green's function H_{n}
(
t
,
s
)
in
Lemma 2.4
satisfies
and
where m_{n} is given in
(2.7),
Proof
. We using mathematical induction on
n
it is straightforward. □
By using Green’s function, our problem (1.1)-(1.2) can be written equivalently as the following nonlinear system of integral equations
We consider the Banach space
B
=
C
[
a
,
b
] ×
C
[
a
,
b
] with the norm
We define the cone
P
⊂
B
by
For
λ
,
μ
> 0, we introduce the operators
Q_{λ}
,
Q_{μ}
:
C
[
a
,
b
] ×
C
[
a
,
b
] →
C
[
a
,
b
] by
and an operator
Q
:
C
[
a
,
b
] ×
C
[
a
,
b
] →
C
[
a
,
b
] ×
C
[
a
,
b
] as
Then seeking solution to our BVP (1.1)-(1.2) is equivalent to looking for fixed points of the equation
Q
(
u
,
v
) = (
u
,
v
) in the Banach space
B
.
Lemma 2.6.
Q
:
P
→
P is completely continuous
.
Proof
. By using standard arguments, we can easily show that, under assumptions (
A
1) − (
A
2), the operator
Q
is completely continuous, we need only to prove
Q
(
P
) ⊆
P
. Choose some (
u
,
v
) ∈
P
. Then by Lemma 2.5 we have
and thus
which implies that
Q
(
P
) ⊆
P
for every (
u
,
v
) ∈
P
.
As
Q_{λ}
and
Q_{μ}
are integral operators, it is not difficult to see that using standard arguments we may conclude that both
Q_{λ}
and
Q_{μ}
are completely continuous, hence
Q
is completely continuous operator. □
Theorem 3.1
(
Krasnosel'skii
).
Let B be a Banach space, and let P
⊂
B be a cone in B
.
Assume that
Ω
_{1}
and
Ω
_{2}
are open subsets of B with
0 ∈ Ω
_{1}
⊂
⊂ Ω
_{2}
, and let T
:
P
∩(
╲Ω
_{1}
) →
P be a completely continuous operator such that either
Then, T has a fixed point in P
∩ (
╲Ω
_{1}
).
For our first result, define positive numbers
M
_{1}
and
M
_{2}
by
Theorem 3.2.
Assume that conditions
(
A
1) − (
A
4)
are satisfied. Then, for each λ
,
μ satisfying
there exists a pair
(
u
,
v
)
satisfying
(1.1)-(1.2)
such that u
(
t
) > 0
and v
(
t
) > 0
on
(
a
,
b
).
Proof
. Let
λ
,
μ
be as in (3.1). Let
ϵ
> 0 be chosen such that
Let
Q
be defined as in (2.10), then
Q
is a cone preserving, completely continuous operator. By the definitions of
f
_{0}
and
g
_{0}
, there exists
H
_{1}
> 0 such that
f
(
u
,
v
) ≤ (
f
_{0}
+
ϵ
)(
u
+
v
) for (
u
,
v
) ∈
P
with 0 < (
u
,
v
) ≤
H
_{1}
, and
g
(
u
,
v
) ≤ (
g
_{0}
+
ϵ
)(
u
+
v
) for (
u
,
v
) ∈
P
with 0 < (
u
,
v
) ≤
H
_{1}
. Set Ω
_{1}
= {(
u
,
v
) ∈
B
:║(
u
,
v
)║<
H
_{1}
}. Now let (
u
,
v
) ∈
P
∩ ∂Ω
_{1}
, i.e., let (
u
,
v
) ∈
P
with║(
u
,
v
)║=
H
_{1}
. Then, in view of the inequality (2.8) and choice of
ϵ
, for
a
≤
s
≤
b
, we have
and so,
Similarly, we prove that║
Q_{μ}
(
u
,
v
)║≤
║(
u
,
v
)║. Thus, for (
u
,
v
) ∈
P
∩ ∂Ω
_{1}
it follows that
that is,
Due to the definition of
f
_{∞}
and
g
_{∞}
, there exists an
> 0 such that
f
(
u
,
v
) ≥ (
f
_{∞}
−
ϵ
)(
u
+
v
) for all
u
,
v
≥
and
g
(
u
,
v
) ≥ (
g
_{∞}
−
ϵ
)(
u
+
v
) for all
u
,
v
≥
. Set
H
_{2}
=
and define Ω
_{2}
= {(
u
,
v
) ∈
P
:║(
u
,
v
)║<
H
_{2}
}. If (
u
,
v
) ∈
P
with║(
u
,
v
)║=
H
_{2}
then, min
_{t∈[ξ,ω]}
(
u
+
v
)(
t
) ≥
, by consequently, from (2.9) and choice of
ϵ
, for
a
≤
s
≤
b
, we have that
that is,
Q_{λ}
(
u
,
v
)(
t
) ≥
║(
u
,
v
)║for all
t
≥
τ
and so,
Q_{λ}
(
u
,
v
)(
t
) ≥
║(
u
,
v
)║. Similarly, we find that
Q_{μ}
(
u
,
v
) ≥
║(
u
,
v
)║. Thus, for (
u
,
v
) ∈
P
∩ ∂Ω
_{2}
it follows that
that is,
Applying Theorem 3.1 to (3.2) and (3.3), we obtain that
Q
has a fixed point in
P
∩ (
╲Ω
_{1}
) such that
H
_{1}
≤║(
u
,
v
)║≤
H
_{2}
, and so (1.1)-(1.2) has a positive solution. The proof is complete. □
For our next result we define the positive numbers
We are now ready to state and prove our main result.
Theorem 3.3.
Assume that conditions
(
A
1) − (
A
4)
are satisfied. Then, for each λ
,
μ satisfying
there exists a pair
(
u
,
v
)
satisfying
(1.1)-(1.2)
such that u
(
t
) > 0
and v
(
t
) > 0
on
(
a
,
b
).
Proof
. Let
λ
,
μ
be as in (3.4) and choose a sufficiently small
ϵ
> 0 such that
By the definition of
f
_{0}
and
g
_{0}
, there exists an
H
_{3}
> 0 such that
f
(
u
,
v
) ≥ (
f
_{0}
−
ϵ
)(
u
+
v
), for all (
u
,
v
) with 0 < (
u
,
v
) ≤
H
_{3}
and
g
(
u
,
v
) ≥ (
g
_{0}
−
ϵ
)(
u
+
v
), for all (
u
,
v
) with 0 < (
u
,
v
) ≤
H
_{3}
. Set Ω
_{3}
= {(
u
,
v
) ∈
P
:║(
u
,
v
)║<
H
_{3}
} and let (
u
,
v
) ∈
P
∩∂Ω
_{3}
. Thus we have, from (2.9) and choice of
ϵ
, for
a
≤
s
≤
b
,
that is,║
Q_{λ}
(
u
,
v
)║≥
║(
u
,
v
)║. In a similar manner,║
Q_{μ}
(
u
,
v
)║≥
║(
u
,
v
)║. Thus, for an arbitrary (
u
,
v
) ∈
P
∩ ∂Ω
_{3}
it follows that
and so,
Now let us define two functions
f
^{∗}
,
g
^{∗}
: [0,∞) → [0,∞) by
It follows that
f
(
u
,
v
) ≤
f
^{∗}
(
t
) and
g
(
u
,
v
) ≤
g
^{∗}
(
t
) for all (
u
,
v
) with 0 ≤
u
+
v
≤
t
. It is clear that the function
f
^{∗}
and
g
^{∗}
are nondecreasing. Also, there is no difficulty to see that
In view of the definitions of
f
_{∞}
and
g
_{∞}
, there exists an
such that
Set
H
_{4}
=
, and Ω
_{4}
= {(
u
,
v
) : (
u
,
v
) ∈
P
and║(
u
,
v
)║<
H
_{4}
}. Let (
u
,
v
) ∈
P
∩ ∂Ω
_{4}
and observe that, by the definition of
f
^{∗}
, it follows that for any
s
∈ [
a
,
b
], we have
In view of the observation and by the use of inequality (2.8),
which implies║
Q_{λ}
(
u
,
v
)║≤
║(
u
,
v
)║. In a similar manner, we can prove that║
Q_{μ}
(
u
,
v
)║≤
║(
u
,
v
)║. Thus, for (
u
,
v
) ∈
P
∩ ∂Ω
_{4}
, it follows that
and so,
Applying Theorem 3.1 to (3.5) and (3.6), we obtain that
Q
has a fixed point in
P
∩ (
╲Ω
_{3}
) such that
H
_{3}
≤║(
u
,
v
)║≤
H
_{4}
, and so (1.1)-(1.2) has a positive solution. The proof is complete. □
Theorem 4.1.
Assume that
(
A
1)−(
A
4)
hold. If f
_{0}
,
f
_{∞}
,
g
_{0}
,
g
_{∞}
< ∞,
then there exist positive constants λ
_{0}
,
μ
_{0}
such that for every λ
∈ (0,
λ
_{0}
)
and μ
∈ (0,
μ
_{0}
)
, the boundary value problem
(1.1)-(1.2)
has no positive solution
.
Proof
. Since
f
_{0}
,
f
_{∞}
< ∞, we deduce that there exist
such that
We consider
M
_{1}
=
. Then, we obtain
f
(
u
,
v
) ≤
M
_{1}
(
u
+
v
), ∀
u
,
v
≥ 0. Since
g
_{0}
,
g
_{∞}
< ∞, we deduce that there exist
such that
We consider
M
_{2}
=
Then, we obtain
g
(
u
,
v
) ≤
M
_{2}
(
u
+
v
), ∀
u
,
v
≥ 0. We define
, where
. We shall show that for every
λ
∈ (0,
λ
_{0}
) and
μ
∈ (0,
μ
_{0}
), the problem (1.1)-(1.2) has no positive solution.
Let
λ
∈ (0,
λ
_{0}
) and
μ
∈ (0,
μ
_{0}
). We suppose that (1.1)-(1.2) has a positive solution (
u
(
t
),
v
(
t
)),
t
∈ [
a
,
b
]. Then, we have
Therefore, we conclude
In a similar manner,
Therefore, we conclude
Hence,║(
u
,
v
)║=║
u
║+║
v
║<
║(
u
,
v
)║+
║(
u
,
v
)║=║(
u
,
v
)║, which is a contradiction. So, the boundary value problem (1.1)-(1.2) has no positive solution. □
Theorem 4.2.
Assume that
(
A
1) − (
A
4)
hold
.
(
i
)
If f
_{0}
,
f
_{∞}
> 0,
then there exists a positive constant
such that for every
λ
>
and μ
> 0,
the boundary value problem
(1.1)-(1.2)
has no positive solution
.
(
ii
)
If g
_{0}
,
g
_{∞}
> 0,
then there exists a positive constant
such that for every μ
>
and
λ
> 0,
the boundary value problem
(1.1)-(1.2)
has no positive solution
.
(
iii
)
If f
_{0}
,
f
_{∞}
,
g
_{0}
,
g
_{∞}
> 0,
then there exist positive constants
such that for every
,
the boundary value problem
(1.1)-(1.2)
has no positive solution
.
Proof
. (
i
) Since
f
_{0}
,
f
_{∞}
> 0, we deduce that there exist
such that
We introduce
m
_{1}
=
. Then, we obtain
f
(
u
,
v
) ≥
m
_{1}
(
u
+
v
), ∀
u
,
v
≥ 0. We define
. We shall show that for every
λ
>
and
μ
> 0 the problem (1.1)-(1.2) has no positive solution.
Let
λ
>
and
μ
> 0. We suppose that (1.1)-(1.2) has a positive solution (
u
(
t
),
v
(
t
)),
t
∈ [
a
,
b
]. Then, we obtain
Therefore, we deduce
and so,║(
u
,
v
)║=║
u
║+║
v
║≥║
u
║>║(
u
,
v
)║, which is a contradiction. Therefore, the boundary value problem (1.1)-(1.2) has no positive solution.
(
ii
) Since
g
_{0}
,
g
_{∞}
> 0, we deduce that there exist
such that
We introduce
m
_{2}
=
. Then, we obtain
g
(
u
,
v
) ≥
m
_{2}
(
u
+
v
), ∀
u
,
v
≥ 0. We define
. We shall show that for every
μ
>
and
λ
> 0 the problem (1.1)-(1.2) has no positive solution.
Let
μ
>
and
λ
> 0. We suppose that (1.1)-(1.2) has a positive solution (
u
(
t
),
v
(
t
)),
t
∈ [
a
,
b
]. Then, we obtain
Therefore, we deduce
and so,║(
u
,
v
)║=║
u
║+║
v
║≥║
v
║>║(
u
,
v
)║, which is a contradiction. Therefore, the boundary value problem (1.1)-(1.2) has no positive solution.
(
iii
) Because
f
_{0}
,
f
_{∞}
,
g
_{0}
,
g
_{∞}
> 0, we deduce as above, that there exist
m
_{1}
,
m
_{2}
> 0 such that
f
(
u
,
v
) ≥
m
_{1}
(
u
+
v
),
g
(
u
,
v
) ≥
m
_{2}
(
u
+
v
), ∀
u
,
v
≥ 0. We define
. Then for every
λ
>
and
μ
>
, the problem (1.1)-(1.2) has no positive solution.
Indeed, let
λ
>
and
μ
>
. We suppose that (1.1)-(1.2) has a positive solution (
u
(
t
),
v
(
t
)),
t
∈ [
a
,
b
]. Then in a similar manner as above, we deduce
and so,
which is a contradiction. Therefore, the boundary value problem (1.1)-(1.2) has no positive solution. □
S.N. Rao received M.Sc. from Andhra University and Ph.D at Andhra University, India. Since 2014 he has been at Jazan University, Kingdom of Saudi Arabia. His research focuses on boundary value problems.
Department of Mathematics, Jazan University, Jazan, Kingdom of Saudi Arabia.
e-mail: snrao@jazanu.edu.sa

System of equations
;
time scales
;
eigenvalue intervals
;
positive solutions
;
existence
;
nonexistence
;
cone

1. Introduction

The theory of time scales was introduced and developed by Hilger
[9]
to unify both continuous and discrete analysis. Time scales theory presents us with the tools necessary to understand and explain the mathematical structure underpinning the theories of discrete and continuous dynamic systems and allows us to connect them. The theory is widely applied to various situations like epidemic models, the stock market and mathematical modeling of physical and biological systems. Certain economically important phenomena contain processes that feature elements of both the continuous and discrete. The book on the subject of time scales by Bohner and Peterson
[4
,
5]
, summarizes and organizes much of the time scale calculus.
In recent years, the existence and nonexistence of positive solutions of the higher order boundary value problems (BVPs) on time scales have been studied extensively due to their striking applications to almost all area of science, engineering and technology, Anderson
[2
,
3]
, Chyan and Henderson
[6]
, Erbe and Peterson
[7]
, Kameswararao and Nageswararao
[14]
, Sun
[16]
.
We are concerned with determining values of
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- (A1)αj,βj,γj,δj≥ 0 anddj=γjβj+αjδj+αjγj(b−a) > 0;
- (A2)f,g∈C([0,∞) × [0,∞), [0,∞));
- (A3)p,q∈C([a,b], [0,∞)), and each does not vanish identically on any subinterval;
- (A4) All of

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2. Preliminary results

In this section, we state some lemmas that will be used to prove our results. Shortly we will be concerned with a completely continuous operator whose kernel is the Green’s function for the related homogeneous problem (−1)
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3. Existence results

In this section, we apply Krasnosel’skii fixed point theorem
[13]
to obtain the solutions in a cone (that is, positive solution) of (1.1)-(1.2).
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- (i)║Tu║≤║u║,u∈P∩ ∂Ω1, and║Tu║≥║u║,u∈P∩ ∂Ω2, or
- (ii)║Tu║≥║u║,u∈P∩ ∂Ω1, and║Tu║≤║u║,u∈P∩ ∂Ω2.

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4. Nonexistence results

In this section, we give some sufficient conditions for the nonexistence of positive solutions to the BVP (1.1)-(1.2).
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Citing 'EXISTENCE AND NONEXISTENCE OF POSITIVE SOLUTIONS FOR A SYSTEM OF EVEN ORDER DYNAMIC EQUATION ON TIME SCALES
'

@article{ E1MCA9_2015_v33n5_6_531}
,title={EXISTENCE AND NONEXISTENCE OF POSITIVE SOLUTIONS FOR A SYSTEM OF EVEN ORDER DYNAMIC EQUATION ON TIME SCALES}
,volume={5_6}
, url={http://dx.doi.org/10.14317/jami.2015.531}, DOI={10.14317/jami.2015.531}
, number= {5_6}
, journal={Journal of Applied Mathematics & Informatics}
, publisher={Korean Society of Computational and Applied Mathematics}
, author={RAO, SABBAVARAPU NAGESWARA}
, year={2015}
, month={Sep}