Abstract
By applying the Manásevich-Mawhin continuation theorem, we establish some sufficient conditions for the existence and uniqueness of positive periodic solutions for the Liénard type ϕ-Laplacian operator equation.
MSC: 34K13, 34C25.
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1 Introduction
Consider the following Liénard equation:
are the -Carathéodory functions, which means that they are measurable in the first variable and continuous in the second variable, and for every there exists such that for all and a.e. ; and f, g are the T-periodic functions about t. is T-periodic.
Here let be a continuous function and which satisfies:
(A1):, , .
(A2): There exists a function , as , such that , .
It is easy to see that ϕ represents a large class of nonlinear operators, including which is a p-Laplacian, i.e., for .
As is well known, the existence of periodic solutions for a p-Laplacian differential equation was extensively studied (see [1]–[7] and the references therein). In recent years, there also appeared some results on the ϕ-Laplacian differential equation; see [8]–[10]. In [8], Ding et al. investigate the existence of periodic solutions for the Liénard type ϕ-Laplacian differential equation (1.1) with the following assumption:
(H0):, and , for all .
However, for the existence of periodic solutions to (1.1) without (H0), the results are scarce. Thus, it is worthwhile to study (1.1) in this case.
In this paper, by using some analysis techniques, we establish some sufficient condition for the existence and uniqueness of positive T-periodic solutions of (1.1). The results of this paper complement the results previously found in [8].
2 Main results
For convenience, define
which is a Banach space endowed with the norm defined by , for all x, and
For the T-periodic boundary value problem
ϕ is defined as above and is assumed to be Carathéodory.
Lemma 2.1
(Manásevich-Mawhin [11])
Let Ω be an open bounded set in. If
-
(i)
for each the problem
has no solution on ∂ Ω;
-
(ii)
the equation
has no solution on;
-
(iii)
the Brouwer degree of F is
then the periodic boundary value problem (2.1) has at least one periodic solution on.
Lemma 2.2
Ifis bounded, then x is also bounded.
Proof
Since is bounded, there exists a positive constant N such that . From (A2), we have . Hence, we can get for all . If x is not bounded, then from the definition of α, we get for some , which is a contradiction. So x is also bounded. □
For the sake of convenience, we list the following assumptions which will be used repeatedly in the sequel:
(H1): There exists a positive constant D such that for and , for and .
(H2): There exists a constant such that .
(H3): There exist positive constants a, b, B such that
(H4):g is a continuous differentiable function defined on ℝ, and , where .
Applying Lemmas 2.1-2.2, we obtain the following.
Theorem 2.1
Assume that (H1)-(H3) hold. Then (1.1) has positive T-periodic solution if.
Proof
Consider the homotopic equation of (1.1) as follows:
Firstly, we will claim that the set of all possible T-periodic solutions of (2.2) is bounded. Let be an arbitrary solution of (2.2) with period T. As , there exists such that , while , and we see
where .
Consider the equivalent system of (2.2)
We first claim that there is a constant such that
In view of , we know that there exist two constants such that
From the first equation of (2.4) and (A2), we know
Let be, respectively, a global maximum and minimum point of ; clearly, we have
From (H2) we know f will not change sign for . Without loss of generality, suppose for and upon substitution of (2.6) into the second equation of (2.4), we have
Since , from (A2), we know that . So, we have
i.e.
From (H1), we know that
Similarly, from (2.7) we have
and again by (H1)
Case (1): If , define , obviously, .
Case (2): If , from (2.8) and the fact that is a continuous function in ℝ, there exists a constant ξ between and such that . This proves (2.5).
Then we have
and
Combining the above two inequalities, we obtain
Since is T-periodic, multiplying and (2.2) and then integrating it from 0 to T, we have
In view of (2.10), we have
From (H2), we know
Set
From (H3), we have
where , .
Since , it is easy to see that there is a constant (independent of λ) such that
By applying Hölder’s inequality and (2.9), we have
In view of (2.3), we have
where .
Thus, from Lemma 2.2, we know that there exists some positive constant such that, for all ,
Set ; we have
and we know that (2.2) has no solution on ∂ Ω as and when , or , from (2.9) we know that . So, from (H1) we see that
So condition (ii) is also satisfied. Set
where , , and we have
and thus is a homotopic transformation and
So condition (iii) is satisfied. In view of Lemma (2.1), there exists a solution with period T.
Suppose that is the T-periodic solution of (1.1). Let be the global minimum point of on . Then and we claim that
If not, i.e., , then there exists such that for . Therefore, is strictly decreasing for . From (A1), we know that is strictly decreasing for . This contradicts the definition of . Thus, (2.12) is true. From (1.1) and (2.12), we have
In view of (H1), (2.13) implies . Thus,
which implies that (1.1) has at least one positive solution with period T. This completes the proof. □
Next, we consider , then (1.1) is transformed into
Set
and we can rewrite (2.14) in the following form:
Lemma 2.3
If (H4) holds, then (2.14) has at most one T-periodic solution in.
Proof
Assume that and are two T-periodic solutions of (2.14). Then we obtain
Set
It follows from (2.16) that
Now, we prove that
In contrast, in view of , for , we obtain
Then there must exist (for convenience, we can choose ) such that
We claim that . In contrast, we obtain and there exists a constant such that for . Therefore, is strictly increasing for , which implies that
This contradicts the definition of . Thus, we get .
This implies that
Since , from (2.17) and the first equation of (2.18), we get . Then, from the second equation of (2.18), we have
In view of , , it follows from (2.19) that
From (A1), we can see that
From the first equation of (2.17), we know
which contradicts (2.20). This contradiction implies that
By using a similar argument, we can also show that
Therefore, we obtain
Then, from (2.17), we get
Again, from , we find
Hence, (2.14) has at most one T-periodic solution in . The proof of Lemma 2.3 is now complete. □
By Lemma 2.3 and Theorem 2.1, we get the following.
Theorem 2.2
Assume (H1)-(H4) hold. Then (2.14) has a unique positive T-periodic solution if.
We illustrate our results with some examples.
Example 2.1
Consider the following second-order p-Laplacian Liénard equation:
where .
Comparing (2.21) to (1.1), we see that , , , . Obviously, we know is an homeomorphism for ℝ to ℝ, satisfying (A1) and (A2). Moreover, it is easily seen that there exists a constant such that (H1) holds. We have , then (H2) holds. Choose ; we have , here , , then (H3) holds and . So, by Theorem 2.1, we find that (2.21) has a positive periodic solution.
Example 2.2
Consider the following second-order ϕ-Laplacian Liénard equation:
where .
Comparing (2.22) to (2.14), we see that , , , . Obviously, we get
and
So, we know (A1) and (A2) hold. We know that ; then (H4) holds. Moreover, it is easily seen that there exists a such that (H1) holds. We have , then (H2) holds. Choose ; we have , here , , then (H3) holds and . Therefore, by Theorem 2.2, we know that (2.22) has a unique positive periodic solution.
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Acknowledgements
YX, XH, and ZC would like to thank the referee for invaluable comments and insightful suggestions. This work was supported by NSFC project (Nos. 11326124, 11271339) and Education Department of Henan Province project (No. 14A110002).
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YX, XH, and ZC worked together in the derivation of the mathematical results. All authors read and approved the final manuscript.
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Xin, Y., Han, X. & Cheng, Z. Existence and uniqueness of positive periodic solution for ϕ-Laplacian Liénard equation. Bound Value Probl 2014, 244 (2014). https://doi.org/10.1186/s13661-014-0244-x
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DOI: https://doi.org/10.1186/s13661-014-0244-x