- Research Article
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Existence of Periodic Solutions for
-Laplacian Equations on Time Scales
Advances in Difference Equations volumeĀ 2010, ArticleĀ number:Ā 584375 (2009)
Abstract
We systematically explore the periodicity of LiƩnard type -Laplacian equations on time scales. Sufficient criteria are established for the existence of periodic solutions for such equations, which generalize many known results for differential equations when the time scale is chosen as the set of the real numbers. The main method is based on the Mawhin's continuation theorem.
1. Introduction
In the past decades, periodic problems involving the scalar p-Laplacian were studied by many authors, especially for the second-order and three-order p-Laplacian differential equation, see [1ā8] and the references therein. Of the aforementioned works, Lu in [1] investigated the existence of periodic solutions for a p-Laplacian LiĆ©nard differential equation with a deviating argument
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584375/MediaObjects/13662_2009_Article_1309_Equ1_HTML.gif)
by Mawhin's continuation theorem of coincidence degree theory [3]. The author obtained a new result for the existence of periodic solutions and investigated the relation between the existence of periodic solutions and the deviating argument Cheung and Ren [4] studied the existence of
-periodic solutions for a p-Laplacian LiƩnard equation with a deviating argument
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584375/MediaObjects/13662_2009_Article_1309_Equ2_HTML.gif)
by Mawhin's continuation theorem. Two results for the existence of periodic solutions were obtained. Such equations are derived from many fields, such as fluid mechanics and elastic mechanics.
The theory of time scales has recently received a lot of attention since it has a tremendous potential for applications. For example, it can be used to describe the behavior of populations with hibernation periods. The theory of time scales was initiated by Hilger [9] in his Ph.D. thesis in 1990 in order to unify continuous and discrete analysis. By choosing the time scale to be the set of real numbers, the result on dynamic equations yields a result concerning a corresponding ordinary differential equation, while choosing the time scale as the set of integers, the same result leads to a result for a corresponding difference equation. Later, Bohner and Peterson systematically explore the theory of time scales and obtain many perfect results in [10] and [11]. Many examples are considered by the authors in these books.
But the research of periodic solutions on time scales has not got much attention, see [12ā16]. The methods usually used to explore the existence of periodic solutions on time scales are many fixed point theory, upper and lower solutions, Masseras theorem, and so on. For example, Kaufmann and Raffoul in [12] use a fixed point theorem due to Krasnosel'ski to show that the nonlinear neutral dynamic system with delay
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584375/MediaObjects/13662_2009_Article_1309_Equ3_HTML.gif)
has a periodic solution. Using the contraction mapping principle the authors show that the periodic solution is unique under a slightly more stringent inequality.
The Mawhin's continuation theorem has been extensively applied to explore the existence problem in ordinary differential (difference) equations but rarely applied to dynamic equations on general time scales. In [13], Bohner et al. introduce the Mawhin's continuation theorem to explore the existence of periodic solutions in predator-prey and competition dynamic systems, where the authors established some suitable sufficient criteria by defining some operators on time scales.
In [14], Li and Zhang have studied the periodic solutions for a periodic mutualism model
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584375/MediaObjects/13662_2009_Article_1309_Equ4_HTML.gif)
on a time scale by employing Mawhin's continuation theorem, and have obtained three sufficient criteria.
Combining Brouwer's fixed point theorem with Horn's fixed point theorem, two classes of one-order linear dynamic equations on time scales
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584375/MediaObjects/13662_2009_Article_1309_Equ5_HTML.gif)
are considered in [15] by Liu and Li. The authors presented some interesting properties of the exponential function on time scales and obtain a sufficient and necessary condition that guarantees the existence of the periodic solutions of the equation
In [16], Bohner et al. consider the system
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584375/MediaObjects/13662_2009_Article_1309_Equ6_HTML.gif)
easily verifiable sufficient criteria are established for the existence of periodic solutions of this class of nonautonomous scalar dynamic equations on time scales, the approach that authors used in this paper is based on Mawhin's continuation theorem.
In this paper, we consider the existence of periodic solutions for p-Laplacian equations on a time scales
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584375/MediaObjects/13662_2009_Article_1309_Equ7_HTML.gif)
where is a constant,
and
is a function with periodic
is a periodic time scale which has the subspace topology inherited from the standard topology on
Sufficient criteria are established for the existence of periodic solutions for such equations, which generalize many known results for differential equations when the time scales are chosen as the set of the real numbers. The main method is based on the Mawhin's continuation theorem.
If (1.7) reduces to the differential equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584375/MediaObjects/13662_2009_Article_1309_Equ8_HTML.gif)
We will use Mawhin's continuation theorem to study (1.7).
2. Preliminaries
In this section, we briefly give some basic definitions and lemmas on time scales which are used in what follows. Let be a time scale (a nonempty closed subset of
). The forward and backward jump operators
and the graininess
are defined, respectively, by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584375/MediaObjects/13662_2009_Article_1309_Equ9_HTML.gif)
We say that a point is left-dense if
and
If
and
then
is called right-dense. A point
is called left-scattered if
while right-scattered if
If
has a left-scattered maximum
then we set
otherwise set
If
has a right-scattered minimum
then set
otherwise set
A function is right-dense continuous (rd-continuous) provided that it is continuous at right-dense point in
and its left side limits exist at left-dense points in
If
is continuous at each right-dense point and each left-dense point, then
is said to be continuous function on
Definition 2.1 (see [10]).
Assume is a function and let
We define
to be the number (if it exists) with the property that for a given
there exists a neighborhood
of
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584375/MediaObjects/13662_2009_Article_1309_Equ10_HTML.gif)
We call the delta derivative of
at
If is continuous, then
is right-dense continuous, and if
is delta differentiable at
then
is continuous at
Let be right-dense continuous. If
for all
then we define the delta integral by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584375/MediaObjects/13662_2009_Article_1309_Equ11_HTML.gif)
Definition 2.2 (see [12]).
We say that a time scale is periodic if there is
such that if
then
For
the smallest positive
is called the period of the time scale.
Definition 2.3 (see [12]).
Let be a periodic time scale with period
We say that the function
is periodic with period
if there exists a natural number
such that
for all
and
is the smallest number such that
If
we say that
is periodic with period
if
is the smallest positive number such that
for all
Lemma 2.4 (see [10]).
If , and
then
-
(A1)
-
(A2)
if
for all
then
-
(A3)
if
on
then
Lemma 2.5 (Hlder's inequality [11]).
Let For rd-continuous functions
one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584375/MediaObjects/13662_2009_Article_1309_Equ12_HTML.gif)
where and
For convenience, we denote
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584375/MediaObjects/13662_2009_Article_1309_Equ13_HTML.gif)
where is an
-periodic real function, that is,
for all
Next, let us recall the continuation theorem in coincidence degree theory. To do so, we introduce the following notations.
Let be real Banach spaces,
a linear mapping,
a continuous mapping. The mapping
will be called a Fredholm mapping of index zero if
and
is closed in
If
is a Fredholm mapping of index zero and there exist continuous projections
such that
then it follows that
is invertible. We denote the inverse of that map by
If
is an open bounded subset of
the mapping
will be called
-compact on
if
is bounded and
is compact. Since
is isomorphic to
there exists an isomorphism
Lemma 2.6 (continuation theorem).
Suppose that and
are two Banach spaces, and
is a Fredholm operator of index 0. Furthermore, let
be an open bounded set and
L-compact on
If
-
(B1)
-
(B2)
-
(B3)
where
is an isomorphism,
then the equation has at least one solution in
Lemma 2.7 (see [13]).
Let and
If
is
-periodic, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584375/MediaObjects/13662_2009_Article_1309_Equ14_HTML.gif)
In order to use Mawhin's continuation theorem to study the existence of -periodic solutions for (1.7), we consider the following system:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584375/MediaObjects/13662_2009_Article_1309_Equ15_HTML.gif)
where is a constant with
Clearly, if
is an
-periodic solution to (2.7), then
must be an
-periodic solution to (1.7). Thus, in order to prove that (1.7) has an
-periodic solution, it suffices to show that (2.7) has an
-periodic solution.
Now, we set with the norm
It is easy to show that
is a Banach space when it is endowed with the above norm
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584375/MediaObjects/13662_2009_Article_1309_Equ16_HTML.gif)
Then it is easy to show that and
are both closed linear subspaces of
We claim that
and
Since for any
we have
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584375/MediaObjects/13662_2009_Article_1309_Equ17_HTML.gif)
so we obtain
Take Define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584375/MediaObjects/13662_2009_Article_1309_Equ18_HTML.gif)
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584375/MediaObjects/13662_2009_Article_1309_Equ19_HTML.gif)
and by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584375/MediaObjects/13662_2009_Article_1309_Equ20_HTML.gif)
Define the operator and
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584375/MediaObjects/13662_2009_Article_1309_Equ21_HTML.gif)
It is easy to see that (2.7) can be converted to the abstract equation
Then and
Since
is closed in
it follows that
is a Fredholm mapping of index zero. It is not difficult to show that
and
are continuous projections such that
and
Furthermore, the generalized inverse (to
)
exists and is given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584375/MediaObjects/13662_2009_Article_1309_Equ22_HTML.gif)
Since for every we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584375/MediaObjects/13662_2009_Article_1309_Equ23_HTML.gif)
from the definition of and the condition that
then
Thus, we get
Similarly, we can prove that
for every
So the operator
is well defined. Thus,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584375/MediaObjects/13662_2009_Article_1309_Equ24_HTML.gif)
Denote We have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584375/MediaObjects/13662_2009_Article_1309_Equ25_HTML.gif)
Clearly, and
are continuous. Since
is a Banach space, it is easy to show that
is a compact for any open bounded set
Moreover,
is bounded. Thus,
is
-compact on
3. Main Results
In this section, we present our main results.
Theorem 3.1.
Suppose that there exist positive constants and
such that the following conditions hold:
-
(i)
-
(ii)
then (1.7) has at least one -periodic solution.
Proof.
Consider the equation where
and
are defined by the second section. Let
If then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584375/MediaObjects/13662_2009_Article_1309_Equ26_HTML.gif)
From the first equation of (3.1), we obtain and then by substituting it into the second equation of (3.1), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584375/MediaObjects/13662_2009_Article_1309_Equ27_HTML.gif)
Integrating both sides of (3.2) from to
noting that
and applying Lemma 2.4, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584375/MediaObjects/13662_2009_Article_1309_Equ28_HTML.gif)
that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584375/MediaObjects/13662_2009_Article_1309_Equ29_HTML.gif)
There must exist such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584375/MediaObjects/13662_2009_Article_1309_Equ30_HTML.gif)
From conditions (i) and (ii), when we have
and
which contradicts to (3.5). Consequently
Similarly, there must exist
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584375/MediaObjects/13662_2009_Article_1309_Equ31_HTML.gif)
Then we have Applying Lemma 2.7, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584375/MediaObjects/13662_2009_Article_1309_Equ32_HTML.gif)
Let Then (3.7) equals to the following inequality:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584375/MediaObjects/13662_2009_Article_1309_Equ33_HTML.gif)
Let
Consider the second equation of (3.1) and (3.8), then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584375/MediaObjects/13662_2009_Article_1309_Equ34_HTML.gif)
Applying Lemma 2.5, we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584375/MediaObjects/13662_2009_Article_1309_Equ35_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584375/MediaObjects/13662_2009_Article_1309_Equ36_HTML.gif)
That is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584375/MediaObjects/13662_2009_Article_1309_Equ37_HTML.gif)
Thus,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584375/MediaObjects/13662_2009_Article_1309_Equ38_HTML.gif)
Since then we obtain that there exists a positive constant
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584375/MediaObjects/13662_2009_Article_1309_Equ39_HTML.gif)
Therefore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584375/MediaObjects/13662_2009_Article_1309_Equ40_HTML.gif)
Let If
then
is a constant vector with
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584375/MediaObjects/13662_2009_Article_1309_Equ41_HTML.gif)
From the second equation of (3.16) we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584375/MediaObjects/13662_2009_Article_1309_Equ42_HTML.gif)
that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584375/MediaObjects/13662_2009_Article_1309_Equ43_HTML.gif)
By assumptions (i) and (ii), we see that and
which implies
Now, we set Then
Thus from (3.8) and (3.14), we see that conditions (B1) and (B2) of Lemma 2.6 are satisfied. The remainder is verifying condition (B3) of Lemma 2.6. In order to do it, let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584375/MediaObjects/13662_2009_Article_1309_Equ44_HTML.gif)
Set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584375/MediaObjects/13662_2009_Article_1309_Equ45_HTML.gif)
It is easy to see that the equation that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584375/MediaObjects/13662_2009_Article_1309_Equ46_HTML.gif)
has no solution in So
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584375/MediaObjects/13662_2009_Article_1309_Equ47_HTML.gif)
If then we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584375/MediaObjects/13662_2009_Article_1309_Equ48_HTML.gif)
so we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584375/MediaObjects/13662_2009_Article_1309_Equ49_HTML.gif)
Then we see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F584375/MediaObjects/13662_2009_Article_1309_Equ50_HTML.gif)
so the condition (B3) of Lemma 2.6 is satisfied, the proof is complete.
When where
we have the following result.
Corollary 3.2.
Suppose that the following conditions hold:
-
(i)
-
(ii)
then (1.7) has at least one -periodic solution.
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Acknowledgments
The authors sincerely thank the reviewers for their valuable suggestions and useful comments that have led to the present improved version of the original manuscript. This research is supported by the Natural Science Foundation of China (60774004, 60904024), China Postdoctoral Science Foundation Funded Project (20080441126, 200902564), Shandong Postdoctoral Funded Project (200802018) and supported by Shandong Research Funds (Y2008A28), also supported by University of Jinan Research Funds for Doctors (B0621, XBS0843).
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Cao, F., Han, Z. & Sun, S. Existence of Periodic Solutions for -Laplacian Equations on Time Scales.
Adv Differ Equ 2010, 584375 (2009). https://doi.org/10.1155/2010/584375
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DOI: https://doi.org/10.1155/2010/584375