Recalling that δ was defined after Lemma 2.1, for convenience, we introduce the following notations. Assume that the constant and γ is some positive function on J,
Theorem 3.1 Assume that there exist positive constants α, β such that , , , and
(3.1)
Then (1.1) has at least one positive solution u such that .
Proof Clearly, , let , . Define the open sets
Then is completely continuous. By (3.1) and the definition of , , , , there exists such that
and
(3.4)
Let . We show that
(3.5)
If not, there exist and such that . Let . Noting that for any , we obtain that for ,
which implies that , a contradiction.
On the other hand, for , , we have
From Lemma 2.4 it follows that Φ has a fixed point . Furthermore, and , which means that is a positive solution of Eq. (1.1). The proof is complete. □
In the next theorem, we make use of the eigenvalue and the corresponding eigenfunction ϕ introduced in Lemma 2.1.
Theorem 3.2 Assume that there exist positive constants α, β such that , , , and
(3.6)
here on J. Then (1.1) has at least one positive solution u such that .
Proof Obviously, , put , . Define the open sets
At first, we show that . For any , from (), we have
On the other hand,
It is easy to check that is completely continuous.
Next, we show that
(3.7)
If not, there exist and such that . Hence,
(3.8)
Multiplying the first equation of (3.8) by ϕ and integrating from 0 to ω, we obtain that
(3.9)
One can find that
(3.10)
Substituting (3.10) into (3.9), we get
Noting that , therefore,
which implies that
a contradiction.
Finally, we show that
Since and are negative for and , the condition (3.6) implies that . Hence, for and for any ,
Suppose that there exist and such that , that is,
(3.11)
Multiplying the first equation of (3.11) by ϕ and integrating from 0 to ω, we obtain that
(3.12)
One can get that
(3.13)
Substituting (3.13) into (3.12), we get
Noting that , therefore,
It is impossible for . When ,
a contradiction.
From Lemma 2.3 it follows that Φ has a fixed point . Furthermore, and , which means that is a positive solution of Eq. (1.1). The proof is complete. □
Corollary 3.1 Assume that , , , and
or
here on J. Then (1.1) has at least one positive solution.
Corollary 3.2 Assume that there exists a constant α such that , (, α and ∞) and
here on J. Then there exists one open interval such that (1.1) has at least two positive solutions for .
Example 1 Consider the equation
(3.14)
where , and
here and . Since , and , by Theorem 3.1, (3.14) has at least one positive solution for any .
Example 2 Consider the equation
(3.15)
where , .
It is well known that, for the problem consisting of the equation , , and the boundary condition
(3.16)
the first eigenvalue is 0 (see, for example, [[19], p.428]). It follows that the first eigenvalue is for the problem consisting of the equation
and the boundary condition (3.16). Meanwhile, we can obtain the positive eigenfunction corresponding to . It is also easy to check that , , and (here ). So, the right-hand side of the inequality in Corollary 3.2 is obviously satisfied. Considering the monotonicity of and , we can choose a sufficiently small positive constant α such that the left-hand side of the inequality is true. Therefore, by a direct application of Corollary 3.2, there exists one open interval such that (3.15) has at least two positive solutions for .