An Existence Result of Positive Solutions for Fully Second-Order Boundary Value Problems

the existence of positive solutions has been discussed by many authors; see [1–7]. In these works, the positivity of the corresponding Green function G(t, s) plays an important role. The positivity guarantees that BVP (2) can be converted to a fixed point problem of a cone mapping in C(I), where I = [0, 1]. Hence, these authors can apply the fixed point theorems of cone mapping to obtain the existence of positive solutions for BVP (2). But their argument methods are not applicable to BVP (1), since these methods cannot deal with the derivative term u. For the more general BVP (1), the existence of solutions and multiple solutions has been discussed by some authors; see [8–11]. In [8–11], the authors have obtained some of the existence results of one solution or multiple solutions by using lower and upper solutions method. But, there are only a few results [12, 13] on the existence of positive solutions to the general Dirichlet BVP (1). In [12], Zhang discussed the existence of positive solutions by using Leray-Schauder degree theory andproved that BVP (1) has at least one positive solution iff(t, x, y) is nonnegative and sublinear growth onx andy; see [12,Theorem 2.1]. Usually the superlinear problems aremore difficult to treat than the sublinear problems. In [13], Agarwal et al. obtained existence results of positive solutions of BVP (1) by using the fixed point index in cones when f(t, x, y) is nonnegative. They allow that f(t, x, y) may be superlinear growth on x rather than y. The purpose of this paper is to obtain existence result of positive solution for BVP (1) under themore general case that f(t, x, y)may be sign-changing and superlinear growth on x and y. Our main result is as follows.

For the more general BVP (1), the existence of solutions and multiple solutions has been discussed by some authors; see [8][9][10][11].In [8][9][10][11], the authors have obtained some of the existence results of one solution or multiple solutions by using lower and upper solutions method.But, there are only a few results [12,13] on the existence of positive solutions to the general Dirichlet BVP (1).In [12], Zhang discussed the existence of positive solutions by using Leray-Schauder degree theory and proved that BVP (1) has at least one positive solution if (, , ) is nonnegative and sublinear growth on  and ; see [12,Theorem 2.1].Usually the superlinear problems are more difficult to treat than the sublinear problems.In [13], Agarwal et al. obtained existence results of positive solutions of BVP (1) by using the fixed point index in cones when (, , ) is nonnegative.They allow that (, , ) may be superlinear growth on  rather than .
The purpose of this paper is to obtain existence result of positive solution for BVP (1) under the more general case that (, , ) may be sign-changing and superlinear growth on  and .Our main result is as follows.
(F3) given any  > 0, there is a positive continuous function such that then BVP (1) has at least one positive solution.
In Theorem 1, besides that nonlinearity  may be signchanging, condition (F3), a Nagumo-type condition, allows that (, , ) may be superlinear growth on  and  but restricts  on  to quadric growth.See Example 4. This case has not been discussed in [12,13].
The proofs of Theorem 1 are based on the method of lower and upper solution.If a function V ∈  2 () satisfies we call it a lower solution of BVP (1).If all of the inequality in ( 7) is inverse, we call it an upper solution of BVP (1).We will use the following well-known lower and upper solution theorem to prove Theorem 1 in the next section.and BVP (1) has a lower solution V 0 and an upper solution  0 with V 0 ≤  0 .If the nonlinearity  satisfies the Nagumo-type condition (F3), then BVP ( 1) has at least one solution between V 0 and  0 .
Proof of Theorem 1.We use Theorem A to prove Theorem 1.
and hence  0 is a positive upper solution of BVP (1).