Periodic solutions of delay differential equations with feedback control for enterprise clusters based on ecology theory
© Soltan Mohamadi et al.; licensee Springer. 2014
Received: 5 October 2013
Accepted: 25 June 2014
Published: 20 August 2014
By utilizing a fixed point theorem of strict-set-contraction, some criteria are established for the existence of positive periodic solutions of enterprise clusters based on ecology theory with time-varying delays and feedback controls.
where , represent the output of the satellite enterprises, , and the core enterprise, Ay, respectively, , r are the intrinsic growth rates, , a account for their respective self-regulations, accounts for the rates of inter-enterprises competition, c represents the rate of intra-enterprise competition from Ay, represents the rate of conversion of a commodity into the reproduction of enterprise , and d represent the initial production of the enterprises, respectively, , v are the control variables, .
where h is a continuous w-periodic function.
In this paper we shall use the following hypotheses.
In order to obtain our main results, we first make the following preparations.
, for all and all ,
where denotes the diameter of the set .
T is called strict-set-contractive if it is k-set-contractive for some .
Then T has at least one fixed point in .
2 Main results
In this section, at first based on a fixed point theorem for the strict-set-contraction, we shall study the existence of at least one positive periodic solution of the system, then we state and prove our main results.
If , , then is well defined.
If (H4) holds and , , then is well defined.
- (ii)In view of the proof of (i) we need to prove that
If , , then is strict-set-contractive.
If (H4) holds and , , then is strict-set-contractive.
where and .
Therefore, T is strict-set-contractive. □
If , , then the system has at least one positive w-periodic solution.
If (H4) holds and , , then system has at least one positive w-periodic solution.
Proof We only need to prove (i), since the proof of (ii) is similar. Let and . Then we find . It follows from Lemmas 2.1 and 2.2 that is strict-set-contractive.
If there exists or such that then the system has at least one positive w-periodic solution.
Now, we shall show that condition (ii) of Lemma 1.1 holds.
This is a contradiction.
which is a contradiction. Therefore, conditions (i) and (ii) hold. By Lemma 2.1, we see that T has at least one nonzero fixed point in . Thus the system has at least one positive w-periodic solution. The proof of Theorem 2.1 is complete. □
The authors are grateful to the referee for useful comments, which improved the manuscript, and for pointing out a number of misprints.
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