In this section, we consider the nonhomogeneous equation
We say that (1.2)-(1.3) is nonresonant if its unique T-periodic solution is the trivial one. When (1.2)-(1.3) is nonresonant, as a consequence of Fredholm’s alternative, equation (2.1) admits a unique T-periodic solution which can be written as
where , is the Green function of (2.1), associated with (1.3), and we will prove its positiveness. Throughout this paper, we assume that the following condition is satisfied:
, are continuous functions and for all .
Lemma 2.1 Let be a continuous function. Then, for any nonnegative continuous function defined on , the integral equation
has a unique solution , which is continuous on and satisfies the following inequality:
Proof We solve equation (2.2) by the method of successive approximations. Let
One can easily verify that
which implies that the series converges uniformly with respect to . Obviously, is a continuous solution of (2.2). Moreover, inequality (2.3) holds because and are nonnegative functions.
Next we prove the uniqueness. To do so, we first show that the solution of (2.2) is unique on with . Then the uniqueness of the solution on is direct using the continuation property.
On the contrary, suppose that (2.2) has two solutions and on . Then, for , we have
Hence it follows that for all . □
Let us denote by and the solutions of (1.2) satisfying the initial conditions
Lemma 2.2 and satisfy the following integral equations:
Proof Since is a solution of (1.2), we have
Integrating (2.6) from 0 to t and noticing , we obtain
To obtain (2.4), we only need to integrate the above equality from 0 to t and notice . In a similar way, we can prove (2.5). □
Lemma 2.3 For the solution of boundary value problem (2.1)-(1.3), the formula
is the Green function, the number D is defined by .
Proof It is easy to see that the general solution of equation (2.1) has the form
where and are arbitrary constants. Substituting this expression for in boundary condition (1.3), we can obtain
After not very complicated calculations, we can get (2.7) and (2.8). □
Remark 2.4 As a direct application of Lemma 2.3, if , , then the Green function of boundary value problem (2.1)-(1.3) has the form
Lemma 2.5 Assume that (H) holds. Then the Green function associated with (2.1)-(1.3) is positive for all .
Proof Since , it is enough to prove that for . Recall that and satisfy integral equations (2.4) and (2.5). By condition (H) and Lemma 2.1, it follows that
Now from (2.9) we get . Setting
for , we have
Evidently, , for holds. Let us now show that
To prove (2.10), we note that for fixed ,
Hence it follows that for all , we have
Using Lemma 2.1, we get from (2.12) that for all .
Next, we prove (2.11), note that for fixed ,
Hence it follows that, for all , we have
Again using Lemma 2.1, we get from (2.13) that if , and the proof is completed. □
Under hypothesis (H), we always define
Thus and . When , , and a direct calculation shows that