METHODS OF EXTENDING LOWER ORDER PROBLEMS TO HIGHER ORDER PROBLEMS IN THE CONTEXT OF SMALLEST EIGENVALUE COMPARISONS

The theory of u 0-positive operators with respect to a cone in a Ba-nach space is applied to the linear differential equations u (4) + λ 1 p(x)u = 0 and u (4) + λ 2 q(x)u = 0, 0 ≤ x ≤ 1, with each satisfying the boundary conditions u(0) = u ′ (r) = u ′′ (r) = u ′′′ (1) = 0, 0 < r < 1. The existence of smallest positive eigenvalues is established, and a comparison theorem for smallest positive eigenval-ues is obtained. These results are then extended to the nth order problem using two different methods. One method involves finding sign conditions for the Green's function for −u (n) = 0 satisfying the higher order boundary conditions, and the other involves making a substitution that allows us to work with a variation of the fourth order problem.


Introduction
In this paper, we will consider the eigenvalue problems The focus of this paper will be on comparing the smallest eigenvalues for these eigenvalue problems. First, using the theory of u 0 -positive operators with respect to a cone in a Banach space, we establish the existence of smallest eigenvalues for (1.1), (1.3), and (1.2), (1.3), and then compare these smallest eigenvalues after assuming a relationship between p(x) and q(x). We then extend these results to the nth order case using two different methods. First, we establish the sign properties of the Green's function for the nth order problem, and by using these properties, we are EJQTDE, 2011 No. 99, p. 1 Theorem 2.1 can be found in Krasnosel'skii's book [19], and the proof of Theorem 2.2 is provided by Keener and Travis [17] as an extension of Krasonel'skii's results.
Lemma 2.1. Let B be a Banach space over the reals, and let P ⊂ B be a solid cone. If M : B → B is a linear operator such that M : P\{0} → P • , then M is u 0 -positive with respect to P.
Proof. Choose any u 0 ∈ P\{0}, and let u ∈ P\{0}. So Mu ∈ Ω ⊂ P • . Choose k 1 > 0 sufficiently small and k 2 sufficiently large so that Mu − k 1 u 0 ∈ P • and u 0 − 1 k 2 Mu ∈ P • . So k 1 u 0 ≤ Mu with respect to P and Mu ≤ k 2 u 0 with respect to P. Thus k 1 u 0 ≤ Mu ≤ k 2 u 0 with respect to P and so M is u 0 -positive with respect to P . Theorem 2.1. Let B be a real Banach space and let P ⊂ B be a reproducing cone. Let L : B → B be a compact, u 0 -positive, linear operator. Then L has an essentially unique eigenvector in P, and the corresponding eigenvalue is simple, positive, and larger than the absolute value of any other eigenvalue.
Theorem 2.2. Let B be a real Banach space and P ⊂ B be a cone. Let both M, N : B → B be bounded, linear operators and assume that at least one of the operators is u 0 -positive. If M ≤ N, Mu 1 ≥ λ 1 u 1 for some u 1 ∈ P and some λ 1 > 0, and Nu 2 ≤ λ 2 u 2 for some u 2 ∈ P and some λ 2 > 0, then λ 1 ≤ λ 2 . Futhermore, λ 1 = λ 2 implies u 1 is a scalar multiple of u 2 .

The Fourth Order Problem
In this section, we consider the fourth order eigenvalue problems satisfying the boundary conditions where 0 < r < 1, and p(x) and q(x) are continuous nonnegative functions on [0, 1], where neither p(x) nor q(x) vanishes identically on any compact subinterval of [0, 1]. We derive comparison results for these fourth order eigenvalue problems by applying the theorems previously mentioned. To do this, we will define integral operators whose kernel is the Green's function for −u (4) = 0 satisfying (3.3). EJQTDE, 2011 No. 99, p. 3 This Green's function is given by , s ≤ r, s > x, To apply Theorems 2.1 and 2.2, we need to define a Banach space B and a cone P ⊂ B. Define the Banach space B by Define the cone P to be Notice that for u ∈ B, 0 ≤ x ≤ 1, and so sup 0≤x≤1 |u(x)| ≤ ||u||.
Lemma 3.1. The cone P is solid in B and hence reproducing.
EJQTDE, 2011 No. 99, p. 4 Next, we define our linear operators M, N : B → B by and A standard application of the Arzelá-Ascoli theorem shows that M and N are compact.
and so Mu ∈ Ω ⊂ P • . So M : P\{0} → Ω ⊂ P • . Therefore by Lemma 2.1, M is u 0 -positive with respect to P. A similar argument for N completes the proof.   Proof. Since M is a compact linear operator that is u 0 -positive with respect to P, by Theorem 2.1, M has an essentially unique eigenvector, say u ∈ P, and eigenvalue Λ with the above properties. Since Proof. Let p(x) ≤ q(x) on [0, 1]. So for any u ∈ P and x ∈ [0, 1], So Nu − Mu ∈ P for all u ∈ P, or M ≤ N with respect to P. Then by Theorem 2.2, By Remark 3.1, the following theorem is an immediate consequence of Theorems 3.1 and 3. , respectively, each of which is simple, positive, and less than the absolute value of any other eigenvalue of the corresponding problems. Also, eigenfunctions corresponding to λ 1 and λ 2 may be chosen to belong to P • . Finally, λ 1 ≥ λ 2 , and

Extending the Fourth Order Problem Using Sign Properties of the Green's Function
Let n ∈ N, n ≥ 5. In this section, we will consider the eigenvalue problems satisfying the boundary conditions where 0 < r < 1, and p(x) and q(x) are continuous nonnegative functions on [0, 1], where neither p(x) nor q(x) vanish identically on any compact subinterval of [0, 1]. Here we will use methods similar to the methods used previously to derive comparison theorems for these nth order eigenvalue problems. We will do this by finding the the sign properties of the Green's function, which we will call G n (x, s), for −u (n) = 0 satisfying Notice that for u ∈ B, 0 ≤ x ≤ 1, and so sup 0≤x≤1 |u (n−4) (x)| ≤ ||u||. Note Ω ⊂ P. Choose u ∈ Ω and define B ǫ (u) = {v ∈ B | ||u − v|| < ǫ} for ǫ > 0.
Next, we define our linear operators M and N by    Proof. Since M is a compact linear operator that is u 0 -positive with respect to P, by Theorem 2.1, M has an essentially unique eigenvector, say u ∈ P, and eigenvalue Λ with the above properties. Since u = 0, Mu ∈ Ω ⊂ P • and u = M 1 Λ u ∈ P • .
So Nu − Mu ∈ P for all u ∈ P, or M ≤ N with respect to P. Then by Theorem 2.2, By Remark 4.1, the following theorem is an immediate consequence of Theorems 4.1 and 4.2. , respectively, each of which is simple, positive, and less than the absolute value of any other eigenvalue of the corresponding problems. Also, eigenfunctions corresponding to λ 1 and λ 2 may be chosen to belong to P • . Finally, λ 1 ≥ λ 2 and λ 1 = λ 2 if and only if p(x) = q(x) for 0 ≤ x ≤ 1.

Extending the Fourth Order Problem Using Substitution
Instead of using the sign properties of the Green's function for the nth order equation to derive the comparison theorems, we will instead make a substitution and work with a variation of the fourth order problem. This method has its benefits, since we do not need to find the sign properties of the Green's function of the nth order problem, and can instead work with the fourth order problem. The techniques used in this section have been used previously by Henderson and Parmjet [11] and by Maroun [21] to reduce the order of singular problems. However, they have not been used in the context of smallest eigenvalue comparisons.
EJQTDE, 2011 No. 99, p. 10 Let n ∈ N, n ≥ 5. We consider the eigenvalue problems satisfying the boundary conditions and the eigenvalue problems satisfying the boundary condtions For these reasons, we will derive comparison theorems for eigenvalue problems To apply Theorems 2.1 and 2.2, we need to define a Banach space B and a cone P ⊂ B. Define the Banach space B by Define the cone P to be Notice that for v ∈ B, 0 ≤ x ≤ 1, and so sup 0≤x≤1 |v(x)| ≤ ||v||.
Lemma 5.1. The cone P is solid in B and hence reproducing.
It was shown earlier that Ω ⊂ P • . Therefore P is solid in B.
Next, we define our linear operators M, N : B → B by A standard application of the Arzelá-Ascoli theorem shows that M and N are compact.
and so Mv ∈ Ω ⊂ P • . So M : P\{0} → Ω ⊂ P • . Therefore by Lemma 2.1, M is u 0 -positive with respect to P. A similar argument for N completes the proof.
Theorem 5.1. Let B, P, M, and N be defined as earlier. Then M (and N) has an eigenvalue that is simple, positive, and larger than the absolute value of any other eigenvalue, with an essentially unique eigenvector that can be chosen to be in P • .
Proof. Since M is a compact linear operator that is u 0 -positive with respect to P, by Theorem 2.1, M has an essentially unique eigenvector, say v ∈ P, and eigenvalue Λ with the above properties. Since v = 0, Mv ∈ Ω ⊂ P • and v = M 1 Λ v ∈ P • . So Nv − Mv ∈ P for all v ∈ P, or M ≤ N with respect to P. Then by Theorem 2.2, Λ 1 ≤ Λ 2 .
By Remark 5.1, the following theorem is an immediate consequence of Theorems 5.1 and 5.2.