Existence of Periodic Solutions for -Laplacian Equations on Time Scales

Abstract

We systematically explore the periodicity of Liénard 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 [1–8] and the references therein. Of the aforementioned works, Lu in [1] investigated the existence of periodic solutions for a p-Laplacian Liénard differential equation with a deviating argument

(1.1)

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 Liénard equation with a deviating argument

(1.2)

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 [12–16]. 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

(1.3)

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

(1.4)

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

(1.5)

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

(1.6)

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

(1.7)

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

(1.8)

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

(2.1)

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

(2.2)

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

(2.3)

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

(2.4)

where and

For convenience, we denote

(2.5)

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

(2.6)

In order to use Mawhin's continuation theorem to study the existence of -periodic solutions for (1.7), we consider the following system:

(2.7)

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

(2.8)

Then it is easy to show that and are both closed linear subspaces of We claim that and Since for any we have and

(2.9)

so we obtain

Take Define

(2.10)

by

(2.11)

and by

(2.12)

Define the operator and by

(2.13)

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

(2.14)

Since for every we have

(2.15)

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,

(2.16)

Denote We have

(2.17)

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

(3.1)

From the first equation of (3.1), we obtain and then by substituting it into the second equation of (3.1), we get

(3.2)

Integrating both sides of (3.2) from to noting that and applying Lemma 2.4, we have

(3.3)

that is,

(3.4)

There must exist such that

(3.5)

From conditions (i) and (ii), when we have and which contradicts to (3.5). Consequently Similarly, there must exist such that

(3.6)

Then we have Applying Lemma 2.7, we get

(3.7)

Let Then (3.7) equals to the following inequality:

(3.8)

Let

Consider the second equation of (3.1) and (3.8), then we have

(3.9)

Applying Lemma 2.5, we obtain that

(3.10)

where

(3.11)

That is,

(3.12)

Thus,

(3.13)

Since then we obtain that there exists a positive constant such that

(3.14)

Therefore,

(3.15)

Let If then is a constant vector with

(3.16)

From the second equation of (3.16) we get

(3.17)

that is,

(3.18)

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

(3.19)

Set

(3.20)

It is easy to see that the equation that is,

(3.21)

has no solution in So

Let

(3.22)

If then we get

(3.23)

so we have

(3.24)

Then we see that

(3.25)

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.