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Existence and Uniqueness of Periodic Solutions for a Class of Nonlinear Equations with -Laplacian-Like Operators
Advances in Difference Equations volume 2010, Article number: 197263 (2010)
We investigate the following nonlinear equations with -Laplacian-like operators : some criteria to guarantee the existence and uniqueness of periodic solutions of the above equation are given by using Mawhin's continuation theorem. Our results are new and extend some recent results due to Liu (B. Liu, Existence and uniqueness of periodic solutions for a kind of Lienard type -Laplacian equation, Nonlinear Analysis TMA, 69, 724–729, 2008).
In this paper, we deal with the existence and uniqueness of periodic solutions for the following nonlinear equations with -Laplacian-like operators:
where , are continuous functions on , and is a continuous function on with period ; moreover, is a continuous function satisfying the following:
(H1) for any and ;
(H2) there exists a function such that and
It is obvious that under these two conditions, is an homeomorphism from onto and is increasing on
Recall that -Laplacian equations have been of great interest for many mathematicians. Especially, there is a large literature (see, e.g., [1–7] and references therein) about the existence of periodic solutions to the following -Laplacian equation:
and its variants, where for and . Obviously, (1.3) is a special case of (1.1).
However, there are seldom results about the existence of periodic solutions to (1.1). The main difficulty lies in the -Laplacian-like operator of (1.1), which is more complicated than in (1.3). Since there is no concrete form for the -Laplacian-like operator of (1.1), it is more difficult to prove the existence of periodic solutions to (1.1).
Therefore, in this paper, we will devote ourselves to investigate the existence of periodic solutions to (1.1). As one will see, our theorem generalizes some recent results even for the case of (see Remark 2.2).
Next, let us recall some notations and basic results. For convenience, we denote
which is a Banach space endowed with the norm , where
In the proof of our main results, we will need the following classical Mawhin's continuation theorem.
Lemma 1.1 ().
Let (H1), (H2) hold and is Carathéodory. Assume that is an open bounded set in such that the following conditions hold.
(S1) For each , the problem
has no solution on .
(S2) The equation
has no solution on
(S3) The Brouwer degree
Then the periodic boundary value problem
has at least one -periodic solution on .
2. Main Results
In this section, we prove an existence and uniqueness theorem for (1.1).
Suppose the following assumptions hold:
(A1) and for all ;
(A2) there exist a constant and a function such that for all and ,
Then (1.1) has a unique -periodic solution.
Existence. For the proof of existence, we use Lemma 1.1. First, let us consider the homotopic equation of (1.1):
Let be an arbitrary solution of (2.2). By integrating the two sides of (2.2) over , and noticing that and , we have
Since is continuous, there exists such that
In view of (A1), we obtain
where . Then, for each , we have
which gives that
Since , there is a constant such that
In view of (A2) and
By (2.9), we have
there exists a constant such that
On the other hand, it follows from
that for . In addition, since , there exists such that . Thus .
Then, for all , we have
For the above , it follows from that there exists such that
Combining this with , we get
which yields that .
Now, we have proved that any solution of (2.2) satisfies
Since , we have
In view of being strictly decreasing, we get
Then, we know that (2.2) has no solution on for each , that is, the assumption (S1) of Lemma 1.1 holds. In addition, it follows from (2.24) that
So the assumption (S2) of Lemma 1.1 holds. Let
For and , by (2.24), we have
Thus, is a homotopic transformation. So
that is, the assumption (S3) of Lemma 1.1 holds. By applying Lemma 1.1, there exists at least one solution with period to (1.1).
Then (1.1) is transformed into
Let and being two -periodic solutions of (1.1); and
Then we obtain
it follows from (2.33) that
Now, we claim that
If this is not true, we consider the following two cases.
There exists such that
which implies that
By (A1), . So it follows from that . Thus, in view of
and (H1), we obtain
which contradicts with .
Also, we have and . Then, similar to the proof of Case 1, one can get a contradiction.
Now, we have proved that
Analogously, one can show that
So we have . Then, it follows from (2.35) that
which implies that
Hence, (1.1) has a unique -periodic solution. The proof of Theorem 2.1 is now completed.
In Theorem 2.1, setting , then (A2) becomes as follows:
(A2′) there exists a constant such that for all and ,
In addition, we have the following interesting corollary.
Suppose (A1) and
(A2″) there exist a constant such that
hold. Then (1.1) has a unique -periodic solution.
Let . Noticing that
we know that (A2) holds with . This completes the proof.
At last, we give two examples to illustrate our results.
Consider the following nonlinear equation:
where , , , and . One can easily check that satisfy (H1) and (H2). Obviously, (A1) holds. Moreover, since
it is easy to verify that (A2) holds. By Theorem 2.1, (2.49) has a unique -periodic solution.
Consider the following -Laplacian equation:
where , and . Obviously, (A1) holds. Moreover, we have
So () holds. Then, by Corollary 2.3, (2.51) has a unique -periodic solution.
In Example 2.5, , we have
Lu S: New results on the existence of periodic solutions to a -Laplacian differential equation with a deviating argument. Journal of Mathematical Analysis and Applications 2007,336(2):1107-1123. 10.1016/j.jmaa.2007.03.032
Lu S:Existence of periodic solutions to a -Laplacian Liénard differential equation with a deviating argument. Nonlinear Analysis: Theory, Methods & Applications 2008,68(6):1453-1461. 10.1016/j.na.2006.12.041
Liu B:Periodic solutions for Liénard type -Laplacian equation with a deviating argument. Journal of Computational and Applied Mathematics 2008,214(1):13-18. 10.1016/j.cam.2007.02.004
Cheung W-S, Ren J:Periodic solutions for -Laplacian Liénard equation with a deviating argument. Nonlinear Analysis: Theory, Methods & Applications 2004,59(1-2):107-120.
Cheung W-S, Ren J:Periodic solutions for -Laplacian differential equation with multiple deviating arguments. Nonlinear Analysis: Theory, Methods & Applications 2005,62(4):727-742. 10.1016/j.na.2005.03.096
Cheung W-S, Ren J:Periodic solutions for -Laplacian Rayleigh equations. Nonlinear Analysis: Theory, Methods & Applications 2006,65(10):2003-2012. 10.1016/j.na.2005.11.002
Liu B:Existence and uniqueness of periodic solutions for a kind of Liénard type -Laplacian equation. Nonlinear Analysis: Theory, Methods & Applications 2008,69(2):724-729. 10.1016/j.na.2007.06.007
Manásevich R, Mawhin J:Periodic solutions for nonlinear systems with -Laplacian-like operators. Journal of Differential Equations 1998,145(2):367-393. 10.1006/jdeq.1998.3425
The authors are grateful to the referee for valuable suggestions and comments, which improved the quality of this paper. The work was supported by the NSF of China, the NSF of Jiangxi Province of China (2008GQS0057), the Youth Foundation of Jiangxi Provincial Education Department (GJJ09456), and the Youth Foundation of Jiangxi Normal University.
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Ding, HS., Ye, GR. & Long, W. Existence and Uniqueness of Periodic Solutions for a Class of Nonlinear Equations with -Laplacian-Like Operators. Adv Differ Equ 2010, 197263 (2010). https://doi.org/10.1155/2010/197263
- Differential Equation
- Partial Differential Equation
- Ordinary Differential Equation
- Functional Analysis
- Periodic Solution