Abstract
By using the Manásevich-Mawhin continuation theorem and some analysis skills, we establish some sufficient condition for the existence and uniqueness of positive T-periodic solutions for a generalized Rayleigh type ϕ-Laplacian operator equation. The results of this paper are new and they complement previous known results.
MSC:34K13, 34C25.
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1 Introduction
During the past few years, many researchers have discussed the periodic solutions of a Rayleigh type differential equation (see [1–10]). For example, in 2009, Xiao and Liu [7] studied the Rayleigh type p-Laplacian equation with a deviating argument of the form
By using the coincidence degree theory, we establish new results on the existence of periodic solutions for the above equation. Afterward, Xiong and Shao [9] used the coincidence degree theory to establish new results on the existence and uniqueness of positive T-periodic solutions for the Rayleigh type p-Laplacian equation of the form
In this paper, we consider the following Rayleigh type ϕ-Laplacian operator equation:
where the function is continuous and . is an -Carathéodory function and , , which means it is measurable in the first variable and continuous in the second variable. For every , there exists such that for all and a.e. ; and f, g is a T-periodic function about t and . and is T-periodic.
Here is a continuous function and , which satisfies
(A1) for , ;
(A2) there exists a function , as , such that for .
It is easy to see that ϕ represents a large class of nonlinear operators, including is a p-Laplacian, i.e., for .
We know that the study on ϕ-Laplacian is relatively infrequent, the main difficulty lies in the fact that the ϕ-Laplacian operator typically possesses more uncertainty than the p-Laplacian operator. For example, the key step for to get a priori solutions, , is no longer available for general ϕ-Laplacian. So, we need to find a new method to get over it.
By using the Manásevich-Mawhin continuation theorem and some analysis skills, we establish some sufficient condition for the existence of positive T-periodic solutions of (1.1). The results of this paper are new and they complement previous known results.
2 Main results
For convenience, define
which is a Banach space endowed with the norm ; define for all x, and
For the T-periodic boundary value problem
here 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
Then the periodic boundary value problem (2.1) has at least one periodic solution on .
Lemma 2.2 If is bounded, then x is also bounded.
Proof Since is bounded, then 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. □
Lemma 2.3 Suppose that the following condition holds:
(A3) for all t, , .
Then (1.1) has at most one T-periodic solution in .
Proof Assume that and are two T-periodic solutions of (1.1). Then we obtain
Set . Now, we claim that
In contrast, in view of , for , we obtain
Then there must exist (for convenience, we can choose ) such that
which implies that
and
By hypothesis (A3) and (2.2), we have
and there exists such that for all . Therefore, is strictly increasing for , which implies that
From (A1) we get
This contradicts the definition of . Thus,
By using a similar argument, we can also show that
Therefore, we obtain
Hence, (1.1) has at most one T-periodic solution in . The proof of Lemma 2.3 is now complete. □
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 exist constants and such that for ;
(H3) there exist positive constants ρ and γ such that for ;
(H4) there exist positive constants α, β, B such that
By using Lemmas 2.1-2.3, we obtain our main results.
Theorem 2.1 Assume that conditions (H1)-(H4) and (A3) hold. Then (1.1) has a unique positive T-periodic solution if .
Proof Consider the homotopic equation of (1.1) as follows:
By Lemma 2.3, it is easy to see that (1.1) has at most one T-periodic solution in . Thus, to prove Theorem 2.1, it suffices to show that (1.1) has at least one T-periodic solution in . To do this, we are going to apply Lemmas 2.1 and 2.2. Firstly, we will claim that the set of all possible T-periodic solutions of (2.3) is bounded. Let be an arbitrary solution of (2.3) with period T. As , there exists such that , while , we see
where .
We claim that there is a constant such that
Let , be, respectively, the global maximum point and the global minimum point of on ; then , and we claim that
Assume, by way of contradiction, that (2.6) does not hold. Then and there exists such that for . Therefore is strictly increasing for . From (A1) we know that is strictly increasing for . This contradicts the definition of . Thus, (2.6) is true. From , (2.3) and (2.6), we have
Similarly, we get
In view of (H1), (2.7) and (2.8) imply that
Case (1): If , define , obviously, .
Case (2): If , from , we know . Define , we have . This proves (2.5).
Then we have
and
Combining the above two inequalities, we obtain
Since is T-periodic, multiplying and (2.3) and then integrating it from 0 to T, we have
In view of (2.10), we have
From (H2), we know
Set
From (H4), we have
where , .
For the constant , which is only dependent on , we have
So, from (2.11), we have
Since , so 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 (2.4) and (H3), we have
where .
Thus, from Lemma 2.2, we know that there exists some positive constant such that, for all ,
Set , we have
we know that (2.4) has no solution on ∂ Ω as and when , or , from (2.11) we know that . So, from (H1) we see that
So condition (ii) is also satisfied. Set
where , , we have
and thus is a homotopic transformation and
So condition (iii) is satisfied. In view of Lemma 2.1, there exists at least one solution with period T.
Suppose that is the T-periodic solution of (1.1). We can easily show that (2.8) also holds. Thus,
which implies that (1.1) has a unique positive solution with period T. This completes the proof. □
We illustrate our results with some examples.
Example 2.1 Consider the following second-order p-Laplacian-like Rayleigh equation:
where .
Comparing (2.12) to (1.1), we see that , , , . Obviously, we know that is a homeomorphism from ℝ to ℝ satisfying (A1) and (A2). Consider for , then (A3) holds. Moreover, it is easily seen that there exists a constant such that (H1) holds. Consider , here , , and , here , . So, we can get that conditions (H2) and (H3) hold. Choose , we have , here , , then (H4) holds and . So, by Theorem 2.1, we can get that (2.12) has a unique positive periodic solution.
Example 2.2 Consider the following second-order p-Laplacian-like Rayleigh equation:
where .
Comparing (2.13) to (1.1), we see that , , , . Obviously, we get
and
So, we know that (A1) and (A2) hold. Consider for , then (A3) holds. Moreover, it is easily seen that there exists a constant such that (H1) holds. Consider , here , , and , here , . So, we can get that conditions (H2) and (H3) hold. Choose , we have , here , , then (H4) holds and . Therefore, by Theorem 2.1, we know that (2.13) has a unique positive periodic solution.
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Acknowledgements
Research is supported by the National Natural Science Foundation of China (Nos. 11326124, 11271339).
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YX and ZBC worked together in the derivation of the mathematical results. All authors read and approved the final manuscript.
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Xin, Y., Cheng, Z. Existence and uniqueness of a positive periodic solution for Rayleigh type ϕ-Laplacian equation. Adv Differ Equ 2014, 225 (2014). https://doi.org/10.1186/1687-1847-2014-225
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DOI: https://doi.org/10.1186/1687-1847-2014-225