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Multiple Positive Solutions for Nonlinear First-Order Impulsive Dynamic Equations on Time Scales with Parameter
Advances in Difference Equations volume 2009, Article number: 830247 (2009)
Abstract
By using the Leggett-Williams fixed point theorem, the existence of three positive solutions to a class of nonlinear first-order periodic boundary value problems of impulsive dynamic equations on time scales with parameter are obtained. An example is given to illustrate the main results in this paper.
1. Introduction
Let be a time scale, that is,
is a nonempty closed subset of
. Let
be fixed and
be points in
, an interval
denoting time scales interval, that is,
Other types of intervals are defined similarly. Some definitions concerning time scales can be found in [1–5].
In this paper, we are concerned with the existence of positive solutions for the following nonlinear first-order periodic boundary value problem on time scales:

where is a positive parameter,
,
is right-dense continuous,
,
and for each
and
represent the right and left limits of
at
.
The theory of impulsive differential equations is emerging as an important area of investigation, since it is a lot richer than the corresponding theory of differential equations without impulse effects. Moreover, such equations may exhibit several real world phenomena in physics, biology, engineering, and so forth, (see [6–8]). At the same time, the boundary value problems for impulsive differential equations and impulsive difference equations have received much attention [9–19]. On the other hand, recently, the theory of dynamic equations on time scales has become a new important branch (see, e.g., [1–5]). Naturally, some authors have focused their attention on the boundary value problems of impulsive dynamic equations on time scales [20–27]. In particular, for the first-order impulsive dynamic equations on time scales

where is a time scale which has at least finitely-many right-dense points,
is regressive and right-dense continuous,
is given function,
. The paper [21] obtained the existence of one solution to problem (1.2) by using the nonlinear alternative of Leray-Schauder type.
In [22], Benchohra et al. considered the following impulsive boundary value problem on time scales

They proved the existence of one solution to the problem (1.3) by applying Schaefer's fixed point theorem and the nonlinear alternative of Leray-Schauder type.
In [26], Li and Shen studied the problem (1.3). Some existence results to problem (1.3) are established by using a fixed point theorem, which is due to Krasnoselskii and Zabreiko, and the Leggett-Williams fixed point theorem.
In [27], the first author studied the problem (1.1) when . The existence of positive solutions to the problem (1.1) was obtained by means of the well-known Guo-Krasnoselskii fixed point theorem.
Recently, Sun and Li [28] considered the following periodic boundary value problem:

By using the fixed point index, some existence, multiplicity and nonexistence criteria of positive solutions to the problem (1.4) were obtained for suitable .
Motivated by the results mentioned above, in this paper, we shall show that the problem (1.1) has at least three positive solutions for suitable by using the Leggett-Williams fixed point theorem [29]. We note that for the case
and
problem (1.1) reduces to the problem studied by [30].
In the remainder of this section, we state the following theorem, which are crucial to our proof.
Let be a real Banach space and
be a cone. A function
is called a nonnegative continuous concave functional if
is continuous and

for all and
.
Let be constants,
Theorem 1.1 (see [29]).
Let be a completely continuous map and
be a nonnegative continuous concave functional on
such that
Suppose there exist
with
such that
-
(i)
and
-
(ii)
-
(iii)
with
Then has at least three fixed points
in
satisfying

2. Preliminaries
Throughout the rest of this paper, we always assume that the points of impulse are right-dense for each
We define

where is the restriction of
to
and
Let

with the norm Then X is a Banach space.
Definition 2.1.
A function is said to be a solution of the problem (1.1) if and only if
satisfies the dynamic equation

the impulsive conditions

and the periodic boundary condition
Lemma 2.2.
Suppose is
-continuous, then
is a solution of

where

if and only if is a solution of the boundary value problem

Proof.
Since the method is similar to that of in [27, Lemma  3.1], we omit it here.
Lemma 2.3.
Let be defined as Lemma 2.2, then

Proof.
It is obvious, so we omit it here.
Let

where It is not difficult to verify that
is a cone in
We define an operator by

By [27, Lemmas  3.3 and 3.4], it is easy to see that is completely continuous.
3. Main Result
Notation 1.
Let

and for we define
Theorem 3.1.
Assume that there exists a number such that the following conditions:
(H1) for
(H2) hold. Then the problem (1.1) has at least three positive solutions for

Proof.
Let it is easy to see that
is a nonnegative continuous concave functional on
such that
First, we assert that there exists such that
is completely continuous.
In fact, by the condition of (H2), there exist
and
such that

Let if
then
and we have

Take then the set
is a bounded set. According to that
is completely continuous, then
maps bounded sets into bounded sets and there exists a number
such that

If we deduce that
is completely continuous. If
then from (3.4), we know that for any
and
hold. Then we have
is completely continuous. Take
then
and
are completely continuous.
Second, we assert that and
for all
In fact, take so
Moreover, for
then
and we have

Third, we assert that there exist such that
if
Indeed, by the condition of (H2), there exist
and
such that

Then we get

Finally, we assert that if
and
To do this, if and
then

To sum up, all the hypotheses of Theorem 1.1 are satisfied by taking Hence
has at least three fixed points, that is, the problem (1.1) has at least three positive solutions
and
such that

Corollary 3.2.
Using (H3) instead of (H2) in Theorem 3.1, the conclusion of Theorem 3.1 remains true.
4. Example
Example 4.1.
Let We consider the following problem on

where is a positive parameter,
and

Taking then by
it is easy to see that
So, for all
we have
Obviously, we have
Therefore, together with Corollary 3.2, it follows that the problem (4.1) has at least three positive solutions for .
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The authors express their gratitude to the anonymous referee for his/her valuable suggestions.
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Wang, DB., Guan, W. Multiple Positive Solutions for Nonlinear First-Order Impulsive Dynamic Equations on Time Scales with Parameter. Adv Differ Equ 2009, 830247 (2009). https://doi.org/10.1155/2009/830247
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DOI: https://doi.org/10.1155/2009/830247
Keywords
- Dynamic Equation
- Fixed Point Theorem
- Real Banach Space
- Scale Interval
- Impulsive Differential Equation