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Existence of periodic solutions for a prescribed mean curvature Liénard p-Laplacian equation with two delays
Advances in Difference Equations volume 2014, Article number: 290 (2014)
This paper is concerned with the prescribed mean curvature Liénard type p-Laplacian equation with two arguments. By employing Mawhin’s coincidence degree theorem and the analysis techniques, some new existence results of periodic solutions are obtained. We also give an example to illustrate the application of our main results.
Recently, the study of the periodic solutions of prescribed mean curvature equations has become very active; see [1–5] and the references therein. Meanwhile, the Liénard equation, Liénard system, and p-Laplacian equations are also studied by many people; see for example [6–8]. These kinds of equations have wide applications in many fields, such as physics, mechanics, and engineering.
In  Wang studied the following prescribed mean curvature Rayleigh equation with one deviating argument:
under the assumptions:
By using the theory of coincidence degree, the author obtained some sufficient conditions for the existence and uniqueness of periodic solution in the case of .
Stimulated by , we study the periodic solutions for a kind of prescribed mean curvature Liénard p-Laplacian equation with two deviating arguments in the form
where , , τ, γ are continuous functions with period T, , , and , .
To the best of our knowledge, there are few results on this topic. The purpose of this paper is to establish a criterion to guarantee the existence of T-periodic solution. Our methods and results are different from the corresponding ones of . So our results are essentially new.
Let X and Y be real Banach spaces and be a Fredholm operator with index zero, that is, and . Furthermore, let and be the continuous projectors. Clearly, , thus the restriction is invertible. Denote by K the inverse of .
Let Ω be an open bounded subset of X with . A map is called L-compact in if is bounded and the operator is compact. The following, Mawhin’s continuation theorem, is well known.
Lemma 2.1 (Gaines and Mawhin )
Suppose that X and Y are two Banach spaces, and is a Fredholm operator with index zero. Furthermore, is an open bounded subset and is L-compact in . If all the following conditions hold:
, , ;
, where is an isomorphism.
Then the equation has a solution in .
To use Mawhin’s continuation theorem to study (1.3), we firstly transform (1.3) into the following form:
Denote by , , , then prescribed mean curvature p-Laplacian operator is marked by . Clearly, if is a T-periodic solution to (2.1), then must be a T-periodic solution to (1.3).
Set with the norm , with the norm , . Then X and Y are Banach spaces. Let
It is easy to see that , . So L is a Fredholm operator with index zero. Define projectors and by
and let K be the inverse of . Obviously ,
From (2.3) and (2.4), one can easily see that N is L-compact on Ω, where Ω is an open bounded subset of X. The following lemma is useful to estimate a priori bounds of periodic solutions of (2.1).
Lemma 2.2 (Liu et al. )
Suppose that , and that there is a point such that , then
Lemma 2.3 ()
Let , . If then
where , satisfies the above inequality and .
At the end of this section, we list the basic assumptions which will be used in Section 3.
(H1) , and , , and .
(H2) There is a constant such that if and only if .
(H3) There exist nonnegative constants , , , such that
(H4) There is a constant such that for all .
3 Main results and the proof
In this section we state the main results and give its proof.
Theorem 3.1 Suppose the assumptions (H1)-(H4) hold. Then (1.3) has at least one T-periodic solution provided .
Proof Let , suppose , then
Integrating both sides of the second equation of (3.1) from 0 to T, we have
which implies that there exists a point such that
Now we claim that there must exist a point such that
In fact, let , , then from (3.2), . If holds, (3.3) is clearly true. Now assume .
Set , according to (H1), one has
namely , then there must exist a such that . That is, . According to (H2), (3.3) holds. If holds, by a similar method, we can show that (3.3) is also true.
Let , where and n is an integer. Then
Thus we get
On the other hand, multiplying both sides of the second equation of (3.1) by and integrating over , we have
where . Furthermore, in view of (H4), we have
Denote , then
Together with (3.6) and the fact yields
In view of Lemma 2.3, we obtain
where , are defined by Lemma 2.3, and
From (3.3), there exists a point such that . Let , we have , , and ; then, by Lemma 2.2,
By the Minkowski inequality, we get
Combining (3.7) and (3.8), we obtain
Since , there is a constant independent of λ such that , i.e., .
By the first equation of (3.1), we have
together with , which implies that there is a constant such that . Hence
By the two equations of (3.1) and Hölder’s inequality, we obtain
where , , . By (3.10), (3.11), .
Let . If , then . In view of , we have
So . Together with (H2) yields
Let . Then and Ω is a bounded open set of X. So (1) and (2) of Lemma 2.1 are satisfied.
In the next step we show that condition (3) of Lemma 2.1 holds. Define a linear isomorphism by , and let
The direct computation and (H2) show that for ,
Thus, for , which implies
Condition (3) of Lemma 2.1 holds. By Lemma 2.1, the equation has a solution. This completes the proof of Theorem 3.1. □
We can use a similar method to conclude the following result, the details are omitted.
Theorem 3.2 Suppose (H3), (H4), and the following assumptions hold:
() , , and , , , and .
() There is a constant such that if and only if .
Then (1.3) has at least one T-periodic solution if .
4 Uniqueness results
It is difficult to establish the uniqueness of the T-periodic solution of (1.3), but in the special case we obtain the uniqueness of (1.3).
Theorem 4.1 Assume () holds, and , (, are sufficiently small constants). Then (1.3) has at most one T-periodic solution.
Then (1.3) can be converted into the following form:
Suppose that () are two T-periodic solutions of (1.3). Then from (4.1), we have
Let , , which together with (4.2) yields
Now, we claim that
We argue by contradiction, in view of and , , we obtain .
Then there must exist such that
Noticing (), that is, , , , sufficiently small, it follows from the second equation of (4.4) that
Then from the third equation of (4.4), we get
In view of , , , , are increasing on x, and , are sufficiently small, thus we have
This contradicts the third equation of (4.4), which implies that
By using a similar argument, we can also show that .
Hence, we obtain
Then from (4.3) and , we have , , . That is
Therefore, (1.3) has at most one T-periodic solution. The proof of Theorem 4.1 is now complete. □
From Theorem 3.2 and Theorem 4.1, we see that (1.3) has an unique T-periodic solution when , .
5 An example
In this section we give an example to illustrate the application of Theorem 3.1.
Example 5.1 Consider the prescribed mean curvature Liénard equation with two deviating arguments:
where , , , , , , . So . Choosing , , , , then . It is obvious that (H1), (H2), (H3), and (H4) hold. Then it follows from Theorem 3.1 that (5.1) has at least one -periodic solution.
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This work was sponsored by the NSFC (11471073) and the Fundamental Research Funds for the Central Universities (2014B11414). The authors are very grateful to the referee for his/her careful reading of the manuscript and for his/her valuable suggestions on improving this article.
The authors declare that they have no competing interests.
All authors have equally contributed in obtaining new results in this article and also read and approved the final manuscript.
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Li, Z., An, T. & Ge, W. Existence of periodic solutions for a prescribed mean curvature Liénard p-Laplacian equation with two delays. Adv Differ Equ 2014, 290 (2014). https://doi.org/10.1186/1687-1847-2014-290
- periodic solution
- Mawhin’s continuation theorem
- Liénard type p-Laplacian equation