We make assumptions involving , and κ as follows.
(H1) , , .
Definition 2.1 is called a lower system of solutions of differential system (1.1) if
Analogously, is called an upper system of solutions of differential system (1.1) if
In what follows, we assume that
and define that sector
where the vectorial inequalities mean that the same inequalities hold between their corresponding components.
We present new comparison results and lemmas which are crucial for our discussion.
Lemma 2.1 [11, 14]
Suppose that . If there exist and satisfying
then , .
Lemma 2.2 Suppose that (H1) holds. Let satisfy
where . Then , , .
Proof Suppose the contrary. By Lemma 2.1, one easily sees that there are only three cases to consider:
Case 1. and . By Lemma 2.1, for all . Then , which contradicts .
Case 2. and . By Lemma 2.1, for all . So, we have , which contradicts .
Case 3. and . There are such that
It follows from (2.2) that
Hence, we have
The last two inequalities give
which implies that , a contradiction. Hence, , , . □
Lemma 2.3 Suppose that (H1) holds. Let satisfy
where . Then , .
The proof of Lemma 2.3 is easy, so we omit it.
Consider the differential system of BVPs
Lemma 2.4 Assume that (H1) holds. Then is a system of solutions of BVPs (2.3) if and only if is a system of solutions of the integral equation
Proof First, suppose that is a system of solutions of BVPs (2.3). It is easy to see that (2.3) is equivalent to the system of integral equations
Integrating (2.5) and (2.6) with respect to and respectively on gives
Substituting (2.7) into (2.5) and (2.6), we have
which is equivalent to system (2.4).
Conversely, assume that is a system of solutions of an integral equation. Direct differentiation on (2.4) implies
Making use of the fact for , we obtain
Simple computations yield
Hence, is a system of solutions of BVPs (2.3). □
Consider the linear differential system of BVPs
where and .
Lemma 2.5 Assume that (H1) holds. Then there exists a unique system of solutions to BVPs (2.8).
Proof It follows from Lemma 2.4 that (2.8) is equivalent to the operator equation
By using standard arguments, we can easily show that is linear completely continuous. By Lemma 2.3, the operator equation has only the zero solution. Then, by the Fredholm theorem, for given , the operator equation has only one solution in . Hence, (2.8) has exactly one system of solutions . □