Abstract
In this paper, we consider the following high-order p-Laplacian generalized neutral differential equation
where , for and ; is a continuous periodic function with , and for all . is a continuous periodic function with and , c is a constant and , and δ is a T-periodic function, T is a positive constant; n is a positive integer. By applications of coincidence degree theory and some analysis skills, sufficient conditions for the existence of periodic solutions are established.
MSC:34K13, 34K40, 34C25.
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1 Introduction
In recent years, there has been a good amount of work on periodic solutions for neutral differential equations (see [1–9] and the references cited therein). For example, in [1], Cao and He investigated a class of high-order neutral differential equations
By using the Fourier series method and inequality technique, they obtained the existence of a periodic solution for (1.1). In [8], applying Mawhin’s continuation theorem, Wang and Lu studied the existence of a periodic solution for a high-order neutral functional differential equation with distributed delay as follows:
here . Recently, in [5] and [6], Ren et al. observed the high-order p-Laplacian neutral differential equation
and presented sufficient conditions for the existence of periodic solutions for (1.3) in the critical case (i.e., ) and in the general case (i.e., ), respectively.
In this paper, we consider the following high-order p-Laplacian generalized neutral differential equation
where , for and ; is a continuous periodic function with , and for all . is a continuous periodic function with and , c is a constant and , and δ is a T-periodic function, T is a positive constant; n is a positive integer.
In (1.4), the neutral operator is a natural generalization of the operator , which typically possesses a more complicated nonlinearity than . For example, is homogeneous in the following sense , whereas A in general is inhomogeneous. As a consequence, many of the new results for differential equations with the neutral operator A will not be a direct extension of known theorems for neutral differential equations.
The paper is organized as follows. In Section 2, we first give qualitative properties of the neutral operator A which will be helpful for further studies of differential equations with this neutral operator; in Section 3, by applying Mawhin’s continuation theory and some new inequalities, we obtain sufficient conditions for the existence of periodic solutions for (1.4), an example is also given to illustrate our results.
2 Lemmas
Let with the norm . Define difference operators A and B as follows:
Lemma 2.1 (see [10])
If , then the operator A has a continuous inverse on , satisfying
Let X and Y be real Banach spaces and let be a Fredholm operator with index zero, here denotes the domain of L. This means that ImL is closed in Y and . Consider supplementary subspaces , of X, Y respectively such that , . Let and denote the natural projections. Clearly, and so the restriction is invertible. Let K denote the inverse of .
Let Ω be an open bounded subset of X with . A map is said to be L-compact in if is bounded and the operator is compact.
Lemma 2.2 (Gaines and Mawhin [11])
Suppose that X and Y are two Banach spaces, and is a Fredholm operator with index zero. Let be an open bounded set and be L-compact on . Assume that the following conditions hold:
-
(1)
, , ;
-
(2)
, ;
-
(3)
, where is an isomorphism.
Then the equation has a solution in .
Lemma 2.3 (see [12])
If and , then
where , and p is a fixed real number with .
Remark 2.1 When , , then (2.1) is transformed into .
In order to apply Mawhin’s continuation degree theorem, we rewrite (1.4) in the form
where . Clearly, if is a T-periodic solution to (2.2), then must be a T-periodic solution to (1.4). Thus, the problem of finding a T-periodic solution for (1.4) reduces to finding one for (2.2).
Now, set with the norm ; with the norm . Clearly, X and Y are both Banach spaces. Meanwhile, define
by
and by
Then (2.2) can be converted into the abstract equation . From the definition of L, one can easily see that
So, L is a Fredholm operator with index zero. Let and be defined by
then , . Setting and denotes the inverse of , then
where () are defined by the following
From (2.3) and (2.4), it is clear that QN and are continuous, is bounded and then is compact for any open bounded which means N is L-compact on .
3 Existence of periodic solutions for (1.4)
For the sake of convenience, we list the following assumptions which will be used repeatedly in the sequel:
(H1) There exists a constant such that
(H2) There exists a constant such that
(H3) There exist non-negative constants , , , m such that
(H4) There exist non-negative constants , , such that
for all .
Theorem 3.1 Assume that (H1) and (H3) hold, then (1.4) has at least one non-constant T-periodic solution if and , here .
Proof Consider the equation
Set . If , then
Substituting into the second equation of (3.1), we get
Integrating both sides of (3.2) from 0 to T, we have
From (3.3), there exists a point such that
In view of (H1), we obtain
Then we have
and
Combining the above two inequalities, we obtain
Since , we have
and
By applying Lemma 2.1, we have
where . Since , then
On the other hand, from , there exists a point such that , which together with the integration of the second equation of (3.1) on interval yields
For a given constant , which is only dependent on , we have
From (3.5) and (3.6), we have
Since , there exists a point such that . From (3.4) and Remark 2.1, we can easily get
Combination of (3.7) and (3.8) implies
Since and , there exists a positive constant (independent of λ) such that
From (3.5) and (3.9), we obtain that
Hence
From (3.6), we know
From (3.8), we can get
Let , and , then
If , then , or −M. But if , we know
From assumption (H1), we have , which yields a contradiction. Similarly, in the case , we also have , that is, , . So, conditions (1) and (2) of Lemma 2.2 are both satisfied. Define the isomorphism as follows:
Let , , then ,
We have and then
From (H1), it is obvious that , . Hence,
So, condition (3) of Lemma 2.2 is satisfied. By applying Lemma 2.2, we conclude that equation has a solution on , i.e., (2.2) has a T-periodic solution .
Finally, observe that is not a constant. For if (constant), then from (1.4) we have , which contradicts the assumption that . The proof is complete. □
Similarly, we can get the following result.
Theorem 3.2 Assume that (H2) and (H3) hold, then (1.4) has at least one non-constant T-periodic solution if and .
We illustrate our results with an example.
Example 3.1 Consider the following neutral functional differential equation
Here . It is clear that , , , , , , then we can get , , and . Choose such that (H1) holds. Now we consider the assumption (H3), it is easy to see
which means that (H3) holds with , , , . Obviously,
By Theorem 3.1, (3.10) has at least one nonconstant -periodic solution.
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Acknowledgements
CZB and RJL would like to thank the referee for invaluable comments and insightful suggestions. This work was supported by NSFC project (No. 11271339) and NCET Program (No. 10-0141).
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CZB and RJL worked together in the derivation of the mathematical results. Both authors read and approved the final manuscript.
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Cheng, Z., Ren, J. Some results for high-order generalized neutral differential equation. Adv Differ Equ 2013, 202 (2013). https://doi.org/10.1186/1687-1847-2013-202
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DOI: https://doi.org/10.1186/1687-1847-2013-202