Multiple Positive Periodic Solutions for Delay Differential System
We obtain some existence results for multiple positive periodic solutions of some delay differential systems. Examples are presented as applications. For a general positive integer , main results of this paper do not appear in former literatures as we know. Comparing with the existing results, our results are new also when .
It is known that multiple delay Logistic equations
To the best of our knowledge, few papers concerning the existence of multiple positive solutions of (1.1) can be found in literature. Furthermore, no papers have yet deal with the more general nonautonomous delay differential systems
where are all positive integer and
are given functions and signs , are given as follows:
The extension to systems is a natural one; for example, many occurrences in nature involve two or more populations coexisting in an environment, with the model being best described by a system of differential equations (see ).
The aim of this paper is to investigate systems (1.4) and (1.5). In what follows we only discuss the existence of positive periodic solutions of system (1.4); similar results can be obtained for system (1.5). By using multiple fixed-point theorems (see Lemmas 2.1 and 2.2), which are different from those used in [8–10], we obtain the existence of multiple positive periodic solutions of system (1.4) (see Theorems 3.1, 4.1, and 4.3). Some examples are given also to illustrate our main theorems. Main results of this paper are new also even if (see Remark 4.5).
This paper is organized as follows. In Section 2, we make some preliminaries. In Section 3, we derive existence result (see Theorem 3.1) for two positive periodic solutions of system (1.4). Example 3.2 is given below Theorem 3.1. The existence of three positive periodic solutions of system (1.4) is presented in Section 4 (see Theorems 4.1 and 4.3). Applications of Theorems 4.1 and 4.3 may be seen from Examples 4.2 and 4.4.
We make the basic assumption throughout this paper that
Let us now provide some preparations. Let be a real Banach space and let be a cone in . A map is said to be a nonnegative continuous concave functional on cone if is continuous and
For numbers , such that and a nonnegative continuous concave functional on cone , we define
Setting we define
Then and are all Banach spaces and is a cone in . Set
It is easy to see that for any , functions , have properties
Now we define an operator as follows:
signs , are given in (1.7) and we often use them in the remainder of this paper. It is easy to say that a -periodic solution of operator equation
on , that is, a fixed point of operator , is a -positive periodic solution of system (1.4). So, our main results concerning multiple positive solutions of system (1.4) will arise as application of the following fixed-point theorem.
Lemma 2.1 (see ). Let be a cone in a real Banach space . Let and be increasing, nonnegative, continuous functionals on , and let be a nonnegative continuous functional on with such that, for some and ,
Suppose there exists a completely continuous operator and such that
and(i), for all (ii) for all (iii) and for all
Then has at least two fixed points and belonging to such that
Lemma 2.2 (see ). Let be a cone in a real Banach space , let be completely continuous, and let be a nonnegative continuous concave functional on with for all Suppose that there exists such that(i) and for ;(ii) for all ;(iii) for with
Then has at least three fixed points , , satisfying
3. Existence of Two Positive Solutions of System (1.4)
Theorem 3.1. Assume that there exist numbers such that the following three assumptions are satisfied.One has
where is fixed and
, , , and imply
, , , and imply
Then system (1.4) has at least two -positive periodic solutions.
Proof. We begin by defining
Clearly, and are increasing, nonnegative, continuous functionals on , and is nonnegative a continuous functional on with . Moreover, we observe that
Now, we proceed to show that other conditions of Lemma 2.1 are also satisfied.
Firstly, we will show that In fact, we have from (2.8), for any , which yields Hence for all Furthermore, we know from the continuity of functions , , , that the operator is completely continuous. Hence, we conclude that (3.10) holds.
Secondly, let us prove For any so that we get, in view of (1.7), (2.4) and (3.8), Consequently, for any condition and (3.14) imply that which gives (3.13).
Thirdly, we verify As before, and (1.7), (2.4), and (3.8) also tell us that Then condition , (3.17), and the fact that the function is concave imply Thus (3.16) holds.
Finally, let us prove Obviously, In addition, for any we get since and . So we have from condition that Hence (3.19) holds.
To sum up, (3.6)–(3.10), (3.13), (3.16), and (3.19) tell us that conditions of Lemma 2.1 all hold here. Consequently, system (1.4) has at least two -positive periodic solutions and belonging to such that
As an application of Theorem 3.1, we provide the following example. For convenience, all examples in this paper are given when
Example 3.2. Assume that is a fixed constant. Consider the following system: where We set Then We may verify that conditions , , and are all satisfied. Hence, Theorem 3.1 tells us that system (3.24) has at least two -positive periodic solutions and such that
4. Existence of Three Positive Solutions of System (1.4)
For the sake of convenience we list the assumptions to be used in this section as follows.There exists a number such that
where and , are given in .There exist numbers and such that
There exists a number such that,
Let us now state the first existence result of three positive solutions of system (1.4).
Theorem 4.1. Assume that conditions , , and hold. Then system (1.4) has at least three -positive periodic solutions.
Proof. Firstly, we set
Obviously, is a nonnegative continuous concave functional on and
Secondly, condition implies that there exists a number such that In fact, we know from condition that there exist numbers satisfying So Set Then Let us choose Then for any , we have which implies for all . Moreover, we know from the proof of (3.10) that is completely continuous.
Thirdly, let us show that numbers satisfy conditions (i), (ii), and (iii) of Lemma 2.2.
Step 1. We prove that Clearly, Moreover, for any we have Then condition , (4.6), and (4.18) imply that which gives for And then we arrive at (4.17).Step 2. Condition implies In fact, for any , that is, , from and condition we have which yields (4.20).Step 3. for with This is the case because implies
At present, we may say that hypotheses of Lemma 2.2 (the Leggett-Willaims theorem) are satisfied. Hence system (1.4) has at least three -positive periodic solutions: such that
We give the following example to illustrate Theorem 4.1.
Example 4.2. Consider the following system:
where is a fixed constant and
We set Then We may verify also that conditions , , and hold. Hence, Theorem 4.1 tells us that system (4.26) has at least three -positive periodic solutions: such that
The second existence result of three positive solutions of system (1.4) is as follows.
Theorem 4.3. Assume that conditions , , and hold. Then system (1.4) has at least three -positive periodic solutions.
Proof. If we can get (4.8) with replaced by in this case, then the proof is complete. In fact, for any , condition implies Then as desired. This ends the proof.
The following example is an application of Theorem 4.3.
Example 4.4. Consider system
where is a fixed constant and
If we set then We choose Then assumptions , , and hold. Hence we know from Theorem 4.3 that system (4.34) has at least three -positive periodic solutions: such that
We end this paper by the following remark.
Remark 4.5. For a general positive integer , main results of this paper do not appear in former literatures as we know. Comparing with the existing results, our Theorems 3.1, 4.1, and 4.3 are new also when .
Z.-C. Hao acknowledges support from NSFC (10771117), and PH.D. Programs Foundation of Ministry of Education of China(20093705120002), and NSF of Shandong Province of China (Y2008A24), China Postdoctoral Science Foundation (20090451290), Shandong Province Postdoctoral Foundation (200801001), Foundation of Qufu Normal University (BSQD07026).
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