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Existence and uniqueness of a positive periodic solution for Rayleigh type ϕ-Laplacian equation
Advances in Difference Equations volume 2014, Article number: 225 (2014)
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.
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  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  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 )
Let Ω be an open bounded set in . If
for each , the problem
has no solution on ∂ Ω;
has no solution on ;
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
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
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
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
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
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:
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:
Comparing (2.13) to (1.1), we see that , , , . Obviously, we get
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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Research is supported by the National Natural Science Foundation of China (Nos. 11326124, 11271339).
The authors declare that they have no competing interests.
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
- positive periodic solution
- Rayleigh equation