Positive solutions for the (n, p) boundary value problem

We consider the (n, p) boundary value problem in this paper. Some new upper estimates to positive solutions for the problem are obtained. Existence and nonexistence results for positive solutions of the problem are obtained by using the Krasnosel’skii fixed point theorem. An example is included to illustrate the results.

The (n, p) problem has been considered by many authors.For example, in 1995, Eloe and Henderson [5] studied a special case of the (n, p) problem in which p = n − 2. In 2000, Agarwal, O'Regan, and Lakshmikantham [1] considered the existence of positive solutions for the singular (n, p) problem.In 2003, Baxley and Houmand [3] considered the existence of multiple positive solutions for the (n, p) problem.
Motivated by these works, we in this paper consider the existence and nonexistence of positive solutions to the problem (1)- (2).By a positive solution, we mean a solution u(t) such that u(t) > 0 on (0, 1).The purpose of this paper is twofold.First we shall prove some new upper estimates for positive solutions of the problem (1)- (2).Then, using these new upper estimates, we obtain some new existence and nonexistence results for positive solutions of the problem (1)- (2).
The Green's function 2) is given by (see [1]) And the problem (1)-( 2) is equivalent to the integral equation To prove some of our results, we will need the following fixed point theorem, which is due to Krasnosel'skii [9].Theorem 1.1 Let X be a Banach space over the reals, and let P ⊂ X be a cone in X.Let H 1 and H 2 be real numbers such that H 2 > H 1 > 0, and let Suppose L : P ∩ (Ω 2 − Ω 1 ) → P is a completely continuous operator such that, either Then L has a fixed point in P ∩ ( Ω 2 − Ω 1 ).
Throughout the paper, we let X = C[0, 1] be equipped with norm Obviously X is a Banach space.Also we define These constants will be used later in the statements of the existence and nonexistence theorems.This paper is organized as follows.In Section 2, we obtain some new upper estimates to positive solutions to the (n, p) problem.In Sections 3 and 4, we establish some new existence and nonexistence results for positive solutions of the problem.An example is given at the end of the paper to illustrate the main results of the paper.

Estimates for Positive Solutions
We begin with some definitions.Throughout the paper, we define the functions a : The functions a(t), b(t), and c(t) will be used to estimate positive solutions of the problem (1)- (2).It is easy to verify the following facts (3) a(t), b(t), and c(t) are increasing nonnegative functions; For example, we have and We leave the other details to the reader.The next lemma was proved by Agarwal, O'Regan, and Lakshmikantham in [1].For details of the proof, see Theorem 1.3 of [1].2), and As a direct consequence of Lemma 2.1, we have 2) and ( 4), then
One implication of the above lemmas is that if u(t) is a positive solution to the problem ( 1)-( 2), then u(t) ≥ a(t) u .This provides a nice lower estimate to positive solutions for the (n, p) problem.To our knowledge, no satisfactory upper estimates for positive solutions for the (n, p) problem have been given in the literature.2) and ( 4).If we define then To prove the lemma, it suffices to show that h(t) ≥ 0 for 0 ≤ t ≤ 1. Assume the contrary that h(t 0 ) < 0 for some t 0 ∈ (0, 1).If we can show that this leads to a contradiction, then we are done.
If we define Therefore, h (n) (t) is nondecreasing.
Theorem 2.1 Suppose that, in addition to (H1) and (H2), the following condition holds.
(H3) Both f and g are non-decreasing functions.

Throughout we define
The next theorem is our first existence result.Theorem 3.1 Suppose that (H1) and (H2) hold.If BF 0 < 1 < Af ∞ , then the problem (1)-( 2) has at least one positive solution.
For each u ∈ P with u = H 1 , we have To construct Ω 2 , we choose β ∈ (0, 1/4) and δ > 0 such that There exists Therefore, if u ∈ P with u = H 2 , then Then the condition (K1) of Theorem 1.1 is satisfied.So there exists a fixed point of T in P .The proof is complete.
Proof.Choose ε > 0 such that (f 0 − ε)A ≥ 1.There exists H 1 > 0 such that So, for u ∈ P with u = H 1 we have To construct Ω 2 , we choose δ ∈ (0, 1) such that ((F ∞ + δ)B + δ) ≤ 1.There exists an and let If u ∈ P with u = H 2 , then we have Now from Theorem 1.1 we see that problem (1)-( 2) has at least one positive solution.
The proof is complete.
The proofs of Theorems 3.3 and 3.4 are very similar to those of Theorems 3.1 and 3.2.The only difference is that we use the positive cone Q, instead of P , in the proofs of Theorems 3.3 and 3.4.

( 1 )
u(1) = u ; EJQTDE Spec.Ed.I, 2009 No. 31 EJQTDE Spec.Ed.I, 2009 No. 31 EJQTDE Spec.Ed.I, 2009 No. 31 Theorem 2.2 follows directly from Lemma 2.3.Note that Theorems 2.1 and 2.2 provide some upper estimates for positive solutions for the (n, p) boundary value problem.These upper estimates are new and have not been obtained before.
follows immediately from Lemma 2.4 that u(t) ≤ c(t)u(1) for 0 ≤ t ≤ 1.The proof is now complete.EJQTDE Spec.Ed.I, 2009 No. 31 3− t 4 , It is easy to see that λx < f (x) < 3λx for x > 0. Using Maple or Mathematica, we can easily compute the constants: Note that the function g(t) is increasing in t, and f (x) is increasing in x for each fixed λ > 0, therefore Theorems 3.3 and 4.3 apply.From Theorem 3.3 we see that if