- Open Access
Existence of periodic solutions for a prescribed mean curvature Liénard p-Laplacian equation with two delays
© Li et al.; licensee Springer. 2014
Received: 2 September 2014
Accepted: 13 November 2014
Published: 25 November 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.
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 .
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 )
, , ;
, where is an isomorphism.
Then the equation has a solution in .
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).
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. )
Lemma 2.3 ()
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 .
(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 .
In fact, let , , then from (3.2), . If holds, (3.3) is clearly true. Now assume .
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.
Since , there is a constant independent of λ such that , i.e., .
where , , . By (3.10), (3.11), .
Let . Then and Ω is a bounded open set of X. So (1) and (2) of Lemma 2.1 are satisfied.
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.
We argue by contradiction, in view of and , , we obtain .
By using a similar argument, we can also show that .
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.
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.
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.
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