EXISTENCE OF MULTIPLE POSITIVE SOLUTIONS FOR THE SYSTEMS OF HIGHER ORDER BOUNDARY VALUE PROBLEMS ON TIME SCALES

Journal of Applied Mathematics & Informatics.
2015.
Jan,
33(1_2):
1-12

- Received : September 10, 2014
- Accepted : November 07, 2014
- Published : January 30, 2015

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This paper is concerned with boundary value problems for systems of
n
-th order dynamic equations on time scales. Under the suitable conditions, the existence and multiplicity of positive solutions are established by using abstract fixed-point theorems.
AMS Mathematics Subject Classification : 34B15, 39B10, 34B18.
be a time scale with
Given an interval
J
of ℝ, we will use the interval notation
In this paper we are concerned with the existence and multiplicity of positive solutions for dynamic equation on time scales
satisfying the boundary conditions
where
ƒ
(
t
, 0) ≡ 0,
g
(
t
, 0) ≡ 0.
Recently, existence and multiplicity of solutions for boundary value problems of dynamic equations have been of great interest in mathematics and its applications to engineering sciences. To our knowledge, most existing results on this topic are concerned with the single equation and simple boundary conditions.
It should be pointed out that Eloe and Henderson
[6]
discussed the boundary value problem as follows
By using a Krasnosel’skii fixed point theorem, the existence of solutions are obtained in the case when, either
ƒ
is superlinear, or
ƒ
is sublinear. Yang and Sun
[15]
considered the boundary value problem of the system of differential equations
By appealing to the degree theory, the existence of solutions are established. Note, that, there is only one differential equation in
[4]
and BVP in
[15]
contains the system of second order differential equations.
The arguments for establishing the existence of solutions of the BVP (1)-(2) involve properties of Green’s function that play a key role in defining some cones. A fixed point theorem due to Krasnosel’skii
[10]
is applied to yield the existence of positive solutions of the BVP (1)-(2). Another fixed point theorem about multiplicity is applied to obtain the multiplicity of positive solutions of BVP (1)-(2).
The rest of this paper is organized as follows. In Section 2, we shall provide some properties of certain Green’s functions and preliminaries which are needed later. For the sake of convenience, we also state Krasnosel’skii fixed point theorem in a cone. In Section 3, we establish the existence and multiplicity of positive solutions of the BVP (1)-(2). In Section 4, some examples are given to illustrate our main results.
G
(
t
,
s
) be the Green’s function for the boundary value problem
Using the Cauchy function concept
G
(
t
,
s
) is given by
Lemma 1.
For
we have
Proof
. For
t
≤
s
, we have
Similarly, for
σ
(
s
) ≤
t
and
σ^{i}
(
s
) ≤
t
,
i
= 1, · · · ,
n
− 1, we have
Thus, we have
□
Lemma 2.
Let
we have
Proof
. The Green’s function for the BVP (3)-(4) is given in (5) shows that
For
t
≤
s
,
t
∈
I
and
σ
^{i−1}
(
a
) ≤
σ
^{n−2}
(
a
),
i
= 1, 2, · · · ,
n
− 1, we have
For
σ
(
s
) ≤
t
,
t
∈
I
and
σ
^{i−1}
(
a
) ≤
σ
^{n−2}
(
a
),
i
= 1, 2, · · · ,
n
− 1, we have
Therefore
□
We note that a pair (
u
(
t
),
v
(
t
)) is a solution of the BVP (1)-(2) if and only if
Assume throughout that
is such that
both exist and satisfy
Next, let
be defined by
Finally, we define
and let
For our construction, let
with supremum norm
Then (
E
, ∥·∥) is a Banach space. Define
It is obivious that
P
is a positive cone in
E
. Define an integral operator
T
:
P
→
E
by
Lemma 3.
If the operator T is defined as
(10),
then T
:
P → P is completely continuous
.
Proof
. From the continuity of
ƒ
and
g
, and (8) that, for
u
∈
P
,
Tu
(
t
) ≥ 0 on
Also, for
u
∈
P
, we have from (6) that
so that
Next, if
u
∈
P
, we have from (7), (9), and (10) that
Therefore
T
:
P
→
P
. Since
G
(
t
,
s
),
ƒ
(
t
,
u
) and
g
(
t
,
u
) are continuous, it is easily known that
T
:
P
→
P
is completely continuous. The proof is complete. □
From above arguments, we know that the existence of positive solutions of (1)-(2) can be transferred to the existence of positive fixed points of the operator
T
.
Lemma 4
(
[3
,
4
,
10]
).
Let
(
E
, ∥ · ∥)
be a Banach space, and let P
⊂
E be a cone in E. Assume that
Ω
_{1}
and
Ω
_{2}
are open bounded subsets of E such that
0 ∈ Ω
_{1}
.
If
is a completely continuous operator such that either
then T has a fixed point in
Lemma 5
(
[3
,
4
,
10]
).
Let
(
E
, ∥ · ∥)
be a Banach space, and let P
⊂
E be a cone in E. Assume that
Ω
_{1}
, Ω
_{2}
and
Ω
_{3}
are open bounded subsets of E such that
If
is a completely continuous operator such that:
then T has at least two fixed points
and furthermore
(A2)
(A3)
(A4)
(A5)
ƒ
(
t
,
u
),
g
(
t
,
u
) are increasing functions with respect to
u
and, there is a number
N
> 0, such that
Theorem 1.
If
(
A
1)
and
(
A
2)
are satisfied, then
(1)-(2)
has at least one positive solution
satisfying u
(
t
) > 0,
v
(
t
) > 0.
Proof
. From (
A
1) there is a number
H
_{1}
∈ (0; 1) such that for each
one has
where
η
> 0 satisfies
For every
u
∈
P
and ∥
u
∥ =
H
_{1}
/2, note that
thus
So, ∥
Tu
∥ ≤ ∥
u
∥. If we set
then
On the other hand, from (
A
2) there exist four positive numbers
μ
,
μ
′,
C
_{1}
and
C
_{2}
such that
where
μ
and
μ
′ satisfy
For
u
∈
P
, we have
where
Therefore
from which it follows that ∥
Tu
∥ ≥
Tu
(
τ
) ≥ ∥
u
∥ as ∥
u
∥ → ∞.
Let Ω
_{2}
= {
u
∈
E
: ∥
u
∥ <
H
_{2}
}. Then for
u
∈
P
and ∥
u
∥ =
H
_{2}
> 0 sufficient by large, we have
Thus, from (11), (12) and Lemma 4, we know that the operator
T
has a fixed point in
The proof is complete. □
Theorem 2.
If
(
A
3)
and
(
A
4)
are satisfied, then
(1)-(2)
has at least one positive solution
satisfying u
(
t
) > 0,
v
(
t
) > 0.
Proof
. From (
A
3) there is a number
such that for each
one has
where
λ
> and
λ
′ satisfy
From
g
(
t
, 0) ≡ 0 and the continuity of
g
(
t
,
u
), we know that there exists a number
small enough such that
For every
u
∈
P
and ∥
u
∥ =
H
_{3}
, note that
thus
So, ∥
Tu
∥ ≥ ∥
u
∥. If we set
then
On the other hand, we know from (
A
4) that there exist three positive numbers
η
′,
C
_{4}
, and
C
_{5}
such that for every
where
Thus we have
where
from which it follows that
Tu
(
t
) ≤ ∥
u
∥ as ∥
u
∥ → ∞. Let Ω
_{4}
= {
u
∈
E
:∥
u
∥<
H
_{4}
}. For each
u
∈
P
and ∥
u
∥ =
H
_{4}
> 0 large enough, we have
From (13), (14) and Lemma 4, we know that the operator
T
has a fixed point in
The proof is complete. □
Theorem 3.
If
(
A
2), (
A
3)
and
(
A
5)
are satisfied, then
(1)-(2)
has at least two distinct positive solutions
satisfying u_{i}
(
t
) > 0,
v_{i}
(
t
) > 0 (
i
= 1, 2).
Proof
. Note that
Then from (
A
5), for every
we have
Thus
And from (
A
2) and (
A
3) we have
We can choose
H
_{2}
,
H
_{3}
and
N
such that
H
_{3}
≤
N
≤
H
_{2}
and (15)-(17) are satisfied. From Lemma 5,
T
has at least two fixed points in
respectively. The proof is complete. □
Example 1.
Consider the following dynamic equations
satisfying the boundary conditions
where
f
(
t
,
v
) =
v
^{3}
,
g
(
t
,
u
) =
u
^{2}
, then conditions of Theorem 1 are satisfied. From Theorem 1, the BVP (18)-(19) has at least one positive solution.
Example 2.
Consider the following dynamic equations
satisfying the boundary conditions
where
f
(
t
,
v
) =
v
^{2/3}
,
g
(
t
,
u
) =
u
^{3/4}
, then conditions of Theorem 2 are satisfied. From Theorem 2, the BVP (20)-(21) has at least one positive solution.
Example 3.
Consider the following system of boundary value problems
where
We can choose
N
= 100, then conditions of Theorem 3 are satisfied. From Theorem 3, the BVP (22)-(23) has at least two positive solutions.
A. K. Rao received M. Sc degree from Andhra University, Visakhapatnam. M. Phil degree from M. K. University, Madhurai and Ph. D from Andhra University under the esteemed guidence of Prof. K. R. Prasad. His research interest focuses on Boundary Value Problems.
Department of Mathematics, Gayatri Vidya Parishad College of Engineering for Women, Madhurawada, Visakhapatnam, 530 048, India.
e-mail: kamesh−1724@yahoo.com

Boundary value problems
;
dynamic equations
;
green’s function
;
positive solutions
;
multiplicity
;
cones

1. Introduction

Let
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2. Preliminaries

In this section, we will give some lemmas which are useful in proving our main results.
To obtain solutions of the BVP (1)-(2), we let
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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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3. Main Results

First we give the following assumptions:
(A1)
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4. Examples

Some examples are given to illustrate our main results.
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Citing 'EXISTENCE OF MULTIPLE POSITIVE SOLUTIONS FOR THE SYSTEMS OF HIGHER ORDER BOUNDARY VALUE PROBLEMS ON TIME SCALES
'

@article{ E1MCA9_2015_v33n1_2_1}
,title={EXISTENCE OF MULTIPLE POSITIVE SOLUTIONS FOR THE SYSTEMS OF HIGHER ORDER BOUNDARY VALUE PROBLEMS ON TIME SCALES}
,volume={1_2}
, url={http://dx.doi.org/10.14317/jami.2015.001}, DOI={10.14317/jami.2015.001}
, number= {1_2}
, journal={Journal of Applied Mathematics & Informatics}
, publisher={Korean Society of Computational and Applied Mathematics}
, author={Rao, A. KAMESWARA}
, year={2015}
, month={Jan}