Periodic solutions for a kind of neutral functional differential systems
© He and Du; licensee Springer 2014
Received: 23 March 2014
Accepted: 3 June 2014
Published: 23 September 2014
In this paper, we analyze some properties of the linear difference operator , , and then, by using the coincidence degree theory of Mawhin, a kind of neutral differential systems with non-constant matrix is studied. Some new results on the existence of periodicity are obtained. It is worth noting that is no longer a constant matrix, which is different from the corresponding ones of past work.
The field of neutral functional equations (in short NFDEs) is making significant breakthroughs in its practice; it is no longer only a specialist’s field. In many practical systems, models of systems are described by NFDEs in which the models depend on the delays of state and state derivatives. Practical examples for neutral systems include population ecology, heat exchanges, mechanics, and economics; see –. In particular, qualitative analysis such as periodicity and stability of solutions of NFDEs has been studied extensively by many authors. We refer to – for some recent work on the subject of periodicity and stability of neutral equations.
where , , ; , ; , ; , ; T, and τ are given constants with .
; , ;
- (2)with the norm
- (3)with the norm
Clearly, and are Banach spaces.
2 Main lemmas
where, is a real symmetric matrix.
We will give some properties of .
- (1)Since is a real symmetric matrix, there exists an orthogonal matrix such that
- (2)Similar to the above proof, when , we get
Let X and Y be two Banach spaces and let be a linear operator, a Fredholm operator with index zero (meaning that ImL is closed in Y and ). If L is a Fredholm operator with index zero, then there exist continuous projectors , such that , , and is invertible. Denote by the inverse of .
Let Ω be an open bounded subset of X, a map is said to be L-compact in if is bounded and the operator is relatively compact. We first give the famous Mawhin continuation theorem.
, , ,
then the equationhas a solution on.
3 Main results
Suppose that, is a nonzero periodic solution of (3.1) and there exists a constantsuch that
(H2): (or <0), whenever, .
Since and , is an embedding operator. Hence is a completely continuous operator in ImL. By the definitions of Q and N, one knows that is bounded on . Hence the nonlinear operator N is L-compact on . We complete the proof by three steps.
- (1)If , by (3.7) we have
- (2)If , from (3.8) and the fact that is continuous on ℝ, there is a point between and such that . This also proves (3.6). Let , , . Then . Hence we get
which is a contradiction; see (3.14). Hence is a bounded set.
Applying Lemma 2.3, we reach the conclusion. □
When or , there are no existence results for periodic solutions for system (1.1). We hope that there is interest in doing further research on this issue.
By using Theorem 3.1, when , we know that system (3.15) has at least one 2π-periodic solution.
The authors wish to thank the anonymous referee for his/her valuable suggestions to this paper. This work is supported by the NSFC of China (11171085).
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