Some results for high-order generalized neutral differential equation
© Cheng and Ren; licensee Springer 2013
Received: 3 April 2013
Accepted: 20 June 2013
Published: 8 July 2013
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.
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.
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.
Lemma 2.1 (see )
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 )
, , ;
, where is an isomorphism.
Then the equation has a solution in .
Lemma 2.3 (see )
where , and p is a fixed real number with .
Remark 2.1 When , , then (2.1) is transformed into .
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).
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:
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 .
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.
By Theorem 3.1, (3.10) has at least one nonconstant -periodic solution.
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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